Special Relativity -- Applications to astronomy and the accelerator physics
SSPECIAL RELATIVITYApplications to astronomy and the acceleratorphysics
Evgeny Saldin
Preprint submitted to 16 July 2020 a r X i v : . [ phy s i c s . c l a ss - ph ] J u l Working draft from August 2020 (still a work-in-progress)
This is a working draft, which I will continue to update. Pleasesend an email to [email protected] if you notice errors. ff erence in relativistic kinematics be-tween rotating and non-rotating frames of reference is of great practical aswell as theoretical significance. A correct solution of this problem requiresthe use of relativistic principles even at low velocities since the first-orderterm in ( v / c ) play a fundamental role in the non-inertial relativistic kine-matics of light propagation.All the results presented here are derived from the ”first principles”, and allsteps involving physical principles are given. To preserve a self-consistentstyle, I place the derivation of auxiliary results in appendices. To help readersform their own opinion on the topics discussed, the end of each chapterhas a suggested bibliography together with relevant remarks. The list ofreferences includes only the papers I have consulted directly. A lot of papersremain unmentioned, and for this I apologize.I am grateful to my longtime friends Gianluca Geloni and Vitaly Kocharyanfor discussions over many years about much of the material in this book.I should also like to express my thanks to DESY (Deutsches Electronen-Synchrotron) for enabling me to work in this interesting field.3 ontents ff erent Approaches to Special Relativity 132.3 Myth About the Incorrectness of Galilean Transformations 162.4 Myth About the Non-Relativistic Limit of the LorentzTransformations 172.5 Myth about the Constancy of the Speed of Light 182.6 Convention-Dependent and Convention-Invariant Parts of theTheory 182.7 Myth about the Reality of Relativistic Time Dilation and LengthContraction 202.8 Relativistic Particle Dynamics 202.9 Mistake in Commonly Used Method of Coupling Fields andParticles 222.10 Clarification of the True Content of the ”Single Frame” Theory 232.11 Bibliography and Notes 253 Space-Time and Its Coordinatization 283.1 Introductory Remarks 283.2 Choice of Coordinates System in an Inertial Frame 283.3 Bibliography and Notes 344 Aberration of Light Phenomenon 364.1 Inertial Frame of Reference 364.2 Noninertial Systems of Reference 594.3 Explanation of the Aberration on the Basis of the Ether Theory 724.4 The Physics of Coordinate Transformations 74 .5 Bibliography and Notes 905 Stellar Aberration 955.1 A New Approach to the Stellar Aberration 955.2 Heliocentric Inertial Frame of Reference 985.3 Earth-Based Non-Inertial Frame of Reference 995.4 Region of Applicability 1015.5 Bibliography and Notes 1016 Relativistic Dynamics and Electrodynamics 1046.1 Relativistic Particle Dynamics 1046.2 Relativity and Electrodynamics 1156.3 An Illustrative Example 1196.4 Bibliography and Notes 1287 Synchrotron Radiation 1317.1 Introductory Remarks 1317.2 Paraxial Approximation for the Radiation Field 1327.3 Undulator Radiation 1357.4 Synchrotron Radiation from Bending Magnets 1447.5 How to Solve Problems Involving Many Trajectory Kicks 1527.6 Synchrotron Radiation in the Case of Particle Motion on a Helix 1577.7 Bibliography and Notes 1638 Relativity and X-Ray Free Electron Lasers 1648.1 Introductory Remarks 1648.2 Modulation Wavefront Orientation 1678.3 XFEL Radiation Setup 1688.4 Modulation Wavefront Tilt in Maxwell’s Theory. LogicalInconsistency 176 .5 Bibliography and Notes 1779 Relativistic Spin Kinematics 1809.1 Introductory Remarks 1809.2 The Commutativity of Collinear Lorentz Boosts 1809.3 The Noncommutativity of Two Lorentz Boosts in NonparallelDirections 1819.4 Wigner Rotation 1829.5 Bibliography and Notes 18610 Relativistic Spin Dynamics 18810.1 Introductory Remarks 18810.2 Magnetic Dipole at Rest in an Electromagnetic Field 18810.3 Derivation of the Covariant (BMT) Equation of Motion of Spin 18910.4 Change Spin Variables 19110.5 An Alternative Approach to the BMT Theory 19210.6 Spin Tracking 19410.7 Spin Rotation in the Limit g → ff ect 19710.8 Incorrect Expression for Wigner Rotation. Myth About ExperimentalTest 19910.9 Bibliography and Notes 202References 204 Introduction
The standard books on special relativity do not usually address the questionsof the physical meaning of relativistic e ff ects and the nature of space-time.The presentation of the subject in the present book di ff ers somewhat fromthe usual one in that the four-dimensional geometric formulation of thetheory plays a dominant role than in most of the current textbooks.The book begins with a critical survey of the present approaches to spe-cial relativity. The established way of looking at special relativity is basedon Einstein postulates: the principle of relativity and the constancy of thevelocity of light. In the most general geometric approach to the theory ofspecial relativity, the principle of relativity, in contrast to Einstein formu-lation, is only a consequence of the geometry of space-time. We point thatthe essence of the special theory of relativity consists in the following pos-tulate: all physical processes proceed in four-dimensional space-time, thegeometry of which is pseudo-Euclidean.The space-time geometric approach to special relativity deals with all pos-sible choices of coordinates of the chosen reference frames, and thereforethe second Einstein postulate, referred to as the constancy of the coordinatespeed of light, does not have a place in this more general formulation. Onlyin Lorentz coordinates, when Einstein’s synchronization of distant clocksand Cartesian space coordinates are used, the coordinate speed of light isisotropic and constant. Thus, the basic elements of the space-time geometricformulation of the special relativity and the usual Einstein’s formulation,are quite di ff erent.It should be emphasized that in practical applications there are two choicesof clock synchronization convention useful to consider:(a) Einstein’s convention, leading to the Lorentz transformations betweenframes.(b) Absolute time convention, leading to the Galilean transformations be-tween frames.Absolute time (or simultaneity) can be introduced in special relativity with-out a ff ecting neither the logical structure, no the (convention-independent)predictions of the theory. In the theory of relativity, this choice may seemquite unusual, but it is usually most convenient when one wants to connectto laboratory reality.There is a widespread view that only philosophers of physics discuss theissue of distant clock synchronization. Indeed, a typical physical laboratory7ontains no space-time grid. It should be clear that a rule-clock structureexist only in our mind and manipulations with non existing clocks in thespecial relativity are an indispensable prerequisite for the application of dy-namics and electrodynamics theory in the coordinate representation. Suchsituation usually forces physicist to believe that the application of the theoryof relativity to the study of physical processes is possible without detailedknowledge of the clocks synchronization procedure.However, many problems in special relativity can be adequately treated onlyby an approach which uses the non-standard absolute time synchronization.One of the features that is unique to this book is its treatment of the absolutetime coordinatization. No other books deals with Galilean transformationsin the framework of special relativity.Third chapter presents an ”operational interpretation” of the Lorentz andabsolute time coordinatizations. This is probably the most important andcomplicated chapter of this book. Today the statement about correctnessof Galilean transformations is a ”shocking heresy”, which o ff ends the ”rel-ativistic” intuition and the generally accepted way of looking at specialrelativity of most physicists. The di ff erence between absolute time synchro-nization and Einstein’s time synchronization from the operational pointof view will be an important discovery for every special relativity expert.To our knowledge, neither operational interpretation of the absolute timecoordinatization nor the di ff erence between absolute time synchronizationand Einstein’s time synchronization from the operational point of view, aregiven elsewhere in the literature.We use synthetic approach to present the material: some simple modelsare studied first, and more complicated ones are introduced gradually. Thederivation of relativistic kinematics in the first part of the book (Chapter 3-5)is fairly elementary from a mathematical point of view, but it is conceptuallysubtle.We start with aberration of light phenomena. The e ff ect of light aberrationis a change in the direction of light propagation ascribed to boosted lightsources. Light, being a special case of electromagnetic waves, is describedby the electrodynamics theory. It is well known that the electrodynamicstheory meets all requirements of the theory of relativity and therefore mustaccurately describe the properties of such a typical relativistic object as light.In the fourth chapter, we present a critical reexamination of the existingaberration of light theory. The phenomenon of aberration of light is by nomeans simple to describe, even in the first order in v / c : a large number ofincorrect results can be found in the literature. The utilization of the elec-trodynamics in the absolute time coordinatization becomes indispensable8hen we consider optical phenomena associated with a relative motion oftwo or more bodies. Questions related to reflection of light from transverselymoving mirror (or transmission through the transversely moving end of atelescope barrel) lead to serious misunderstanding, which is actually due toan inadequate understanding of several complicated aspects of the theoryof relativity, among which is the aberration of light.The Chapter 5 deals with astronomical applications. The e ff ect of stellaraberration seems to be one of the simplest phenomena in astronomicalobservations. In spite of its apparent simplicity, aberration seems to beone of the most intricate e ff ects in special relativity. It is widely believedthat stellar aberration depends on the relative velocity of the source (star)and observer. Observations show clearly that stellar aberration does notdepend on the relative motion between star and telescope on the earth. Thelack of symmetry, between the cases when either the source or telescope ismoving is shown clearly on the basis of the separation of binary stars. Therelative motion of these stars with respect to each other (and hence, withrespect to the earth) is never followed by any aberration, although motion ofthese stars is, sometimes, much faster than that of the earth around the sun.Contradiction is so obvious that some astronomers use this fact to argue thatstellar aberration contradicts the special theory of relativity. We demonstratethat the fact that we do not see myriads of widely separated binaries in wildgyration does not require any fundamental change of outlook, but it doesrequire that aberration of ”distant” stars is treated in the framework ofspace-time geometric approach.The second part of the book (Chapter 6 - 10) deals with accelerator physics.This part is technically more challenging, but perhaps also more practical.The study of relativistic particle motion in a constant magnetic field ac-cording to usual accelerator engineering, is intimately connected with theold (Newtonian) kinematics: the Galilean vectorial law of addition of ve-locities is actually used. A non-covariant approach to relativistic particledynamics is based on the absolute time coordinatization, but this is actuallya hidden coordinatization. The absolute time synchronization conventionis self-evident and this is the reason why this subject is not discussed in ac-celerator physics. There is a reason to prefer the non-covariant way withinthe framework of dynamics only. In this approach we have no mixture ofpositions and time. This (3 +
1) dimensional non-covariant particle trackingmethod is simple, self-evident, and adequate to the laboratory reality. It canbe demonstrated that there is no principle di ffi culty with the non-covariantapproach in mechanics and electrodynamics. It is perfectly satisfactory. Itdoes not matter which transformation is used to describe the same reality.What matter is that, once fixed, such convention should be applied and keptin a consistent way for both dynamics and electrodynamics.9he common mistake made in accelerator physics is connected with theincorrect algorithm for solving the electromagnetic field equations. If onewants to use the usual Maxwell’s equations, only the solution of the dy-namics equations in covariant form (i.e. in Lorentz coordinates) gives thecorrect coupling between the Maxwell’s equations and particle trajectoriesin the lab frame.Accelerator physicists, who try to understand the situation related to theuse of the theory of relativity in the synchrotron radiation phenomena,are often troubled by the fact that the di ff erence between covariant andnon-covariant particle trajectories was never understood, and that nobodyrealized that there was a contribution to the synchrotron radiation from rela-tivistic kinematics e ff ects. Accelerator physics was always thought in termsof the old (Newtonian) kinematics that is not compatible with Maxwell’sequations. At this point, a reasonable question arises: since storage rings aredesigned without accounting for the relativistic kinematics e ff ects, how canthey actually operate? In fact, electron dynamics in storage ring is greatlyinfluenced by the emission of radiation. I answer this question in great de-tail in the Chapter 7. The main emphasis of this chapter is on spontaneoussynchrotron radiation from bending magnets and undulators.The theory of relativity shows that the relativity of simultaneity, which isa most fundamental relativistic kinematics e ff ect, is related with extendedrelativistic objects. But up to 21 st century there were no macroscopic objectspossessing relativistic velocities, and there was a general belief that onlymicroscopic particles in experiments can travel at velocities close to that oflight. The 2010s saw a rapid development of new laser light sources in theX-ray wavelength range. An X-ray free electron laser (XFEL) is an examplewhere improvements in accelerator technology makes it possible to developultrarelativistic macroscopic objects with an internal structure (modulatedelectron bunches), and the relativistic kinematics plays an essential role intheir description.In Chapter 8 we present a critical reexamination of existing XFEL theory.It is mainly addressed to readers with limiting knowledge of acceleratorphysics. Fortunately, the principle of XFEL operation does not require spe-cific knowledge of undulator radiation theory presented in the Chapter 7and can be explained in a very simple way.Relativistic kinematics enters XFEL physics in a most fundamental waythrough the rotation of the modulation wavefront, which, in ultrarelativisticapproximation, is closely associated to the relativity of simultaneity. Whenthe trajectories of particles calculated in the Lorentz reference frame (i.e. aninertial frame where Einstein synchronization procedure is used to assignvalues to the time coordinate) they must include relativistic kinematics ef-10ects such as relativity of simultaneity. In the ultrarelativistic asymptote, theorientation of the modulation wavefront , i.e the orientation of the plane ofsimultaneity, is always perpendicular to the electron beam velocity whenthe evolution of the modulated electron beam is treated using Lorentz co-ordinates.We should remark that Maxwell’s equations are valid only in Lorentz refer-ence frames. Einstein’s time order should obviously be applied and kept inconsistent way both in dynamics and electrodynamics. It is important at thispoint to emphasize that the theory of relativity dictates that a modulatedelectron beam in the ultrarelativistic asymptote has the same kinematics, inLorentz coordinates, as a laser beam. According to Maxwell’s equations, thewavefront of a laser beam is always orthogonal to the propagation direction.Experiments show that this prediction is, in fact, true.The theory of relativity as a theory of space-time with pseudo-Euclideangeometry has had more than hundred years of development, and rathersuddenly it has begun to be fully exploited in practical ways in XFEL physics.The kinematics tools needed to study the motion of charged elementaryspinning particles in the storage ring are the main topic of final two chap-ters of the book. As known, a composition of noncollinear Lorentz boostsdoes not result in a di ff erent boost but in a Lorentz transformation involvinga boost and a spatial rotation, the Wigner rotation. The results for the Wignerrotation in the Lorentz lab frame obtained by many experts on special rela-tivity are incorrect. They overestimate the angle of the Wigner rotation by afactor γ compared to its real value, and the direction of the rotation is alsodetermined incorrectly.In 1959, Bargman, Michel, and Telegdi (BMT) proposed a consistent rela-tivistic theory for the dynamics of the spin as observed in the lab frame,which was successfully tested in experiments. It is commonly believed thatthe BMT equation contains the standard (and incorrect) result for the Wignerrotation in the Lorentz lab frame. The existing textbooks then suggest thatthe experimental test of the BMT equation is a direct proof of validity for thestandard expression for Wigner rotation. We demonstrate that the notionthat the standard (incorrect) result for the Wigner rotation as an integral partof the BMT equation in most texts is based, in turn, on an incorrect phys-ical argument. The aim of the final chapter is to analyze the complicatedsituation relating to the use of the Wigner rotation theory in acceleratorphysics. 11 A Critical Survey of the Present Approaches to Special Relativity
The laws of physics are invariant with respect to Lorentz transformations.This is a restrictive principle and does not determine the exact form ofthe dynamics in question. Understanding the postulates of the theory ofrelativity is similar to understanding energy conservation: at first we learnthis as a principle and later on we study microscopic interpretations thatmust be consistent with this principle. For any system to which the energyconservation principle can be applied, a deeper theory should exist whichyields insight into the detailed physical processes involved. Of course, thisdeeper theory must lead to energy conservation.The principle of conservation of energy is very useful in making analyseswithout knowing all the formulas of the fundamental theory. A method-ological analogy with the postulates of the special relativity emerges byitself. Suppose we do not know why a muon disintegrates, but we knowthe law of decay in the Lorentz rest frame. This law would then be a phe-nomenological law. The relativistic generalization of this law to any Lorentzframe allows us to make a prediction on the average distance traveled bya muon. In particular, when a Lorentz transformation of the decay law istried, one obtains the prediction that after the travel distance γ v τ , the popu-lation in the lab frame would be reduced to 1 / τ to γτ .However, the theory of relativity is necessary incomplete. Constructive (mi-croscopic) theories like electrodynamics or quantum field theory providemore insight into the nature of things than restrictive theories like specialrelativity. Relativistic kinematics is only an interpretation of the behaviorof the dynamical matter fields in the view of di ff erent observers. The pointis that one can, in principle compute any relativistic quantity directly fromthe underlying theories of matter without involving relativity at all. Forexample, muons in motion behave relativistically because the field forcesthat are responsible for the muon disintegration satisfy quantum field equa-tions that are Lorentz covariant. Of course, in the ”microscopic” approachto relativistic phenomena, Lorentz covariance of all the fundamental lawsof physics remains, similarly to energy conservation, an unexplained fact,but all explanation must stop somewhere.12 .2 Di ff erent Approaches to Special Relativity In literature, three approaches to special relativity are discussed: Einstein’sapproach, the usual covariant approach, and the space-time geometric ap-proach.Einstein formulation is based on postulates: the principle of relativity andthe constancy of the velocity of light. The usual covariant formulation of thetheory of relativity deals with the pseudo-Eucledian space-time geometryand with the invariance of interval ds , but it is understood only in a limitedsense when the metric is strictly diagonal. Assuming diagonality of themetric we also automatically assume Lorentz coordinates, and that di ff erentinertial frames are related by Lorentz transformations.In space-time geometric approach, primary importance is attributed to thegeometry of space-time; it is supposed that the geometry of space-time isa pseudo-Euclidean geometry in which only 4-tensors quantities do havereal physical meaning. In this most general approach the principle of rela-tivity in contrast to Einstein formulation of the special relativity is a simpleconsequence of the space-time geometry. Since the space-time geometricapproach deals with all possible choices of coordinates of the chosen refer-ence frames, the second Einstein postulate referred to the constancy of thecoordinate velocity of light does not hold in this formulation of the theoryof relativity. Only in Lorentz coordinates, when Einstein’s synchronizationof distant clocks and Cartesian space coordinates are used, the coordinatespeed of light is isotropic and constant. Traditionally, the special theory of relativity is built on the principle ofrelativity and on a second additional postulate concerning the velocity oflight:1. Principle of relativity. The laws of nature are the same (or take the sameform) in all inertial frames2. Constancy of the speed of light. Light propagates with constant velocity c independently of the direction of propagation, and of the velocity of itssource.The constancy of the light velocity in all inertial systems of reference is nota fundamental statement of the theory of relativity. The central principleof special relativity is the Lorentz covariance of all the fundamental lawsof physics. It it important to stress at this point that the second postulate,13ontrary to the view presented in textbooks, is not a separate physical as-sumption, but a convention that cannot be the subject of experimental tests.Assuming postulate 2 on the constancy of the speed of light in all inertialframes we also automatically assume Lorentz coordinates, and that di ff er-ent inertial frames are related by Lorentz transformations. In other words,according to such limiting understanding of the theory of relativity it isassumed that only Lorentz transformations must be used to map the coor-dinates of events between inertial observers. In the usual covariant approach the special of relativity is understood asthe theory of space-time with pseudo-Euclidean geometry. Quantities ofphysical interest are represented by tensors in a four-dimensional space-time, i.e. by covariant quantities, and the laws of physics are written inmanifestly covariant way as four-tensor equations.In order to develop space-time geometry, it is necessary to introduce a metricor a measure ds of space-time intervals. The type of measure determinesthe nature of the geometry. Any event in the usual covariant approach ismathematically represented by a point in space-time, called world-point.The evolution of a particle is, instead, represented by a curve in space-time,called world-line. If ds is the infinitesimal displacement along a particleworld-line, then ds = c dT − dX − dY − dZ , (1)where we have selected a special type of coordinate system (a Lorentzcoordinate system), defined by the requirement that Eq. (1) holds.To simplify our writing we will use, instead of variables T , X , Y , Z , variables X = cT , X = X , X = Y , X = Z . Then, by adopting the tensor notation, Eq.(1) becomes ds = η ij dX i dX j , where Einstein summation is understood. Here η ij are the Cartesian components of the metric tensor and by definition, inany Lorentz system, they are given by η ij = diag[1 , − , − , − F µν is actu-ally not a tensor since F µν are only components implicitly taken in standard(orthogonal) basis. The components are coordinate quantities and they donot contain the whole information about the physical quantity, since a basisof the space-time is not included. This is no problem only in the limitingcase when transformations from one orthogonal basis to another orthogo-nal basis are selected i.e. only assuming that Lorentz transformations mustbe used to map the coordinates of events. According to the usual covari-ant approach, another transformations from standard to non standard (notorthogonal) basis, like Galilean transformations, are ”incorrect”. We emphasize the great freedom one has in the choice of a Minkovskispace-time coordinatization. The space-time continuum, determined by theinterval Eq. (1) can be described in arbitrary coordinates and not only inLorentz coordinates. In the transition to arbitrary coordinates, the geome-try of four-dimensional space-time obviously does not change, and in thespecial theory of relativity we are not limited in any way in the choice ofa coordinates system. The space coordinates x , x , x can be any quantitiesdefining the position of particles in space, and the time coordinate x can bedefined by an arbitrary running clock. The components of the metric tensorin the coordinate system x i can be determined by performing the transfor-mation from the Lorentz coordinates X i to the arbitrary variables x j , whichare fixed as X i = f i ( x j ). One then obtains ds = η ij dX i dX j = η ij ∂ X i ∂ x k ∂ X j ∂ x m dx k dx m = g km dx k dx m , (2)This expression represents the general form of the pseudo-Euclidean metric.In textbooks and monographs, the special theory of relativity is generallypresented in relation to an interval ds in the Minkowski form Eq.(1), whileEq.(2) is ascribed to the theory of general relativity.However, in the space-time geometric approach, special relativity is under-15tood as a theory of four-dimensional space-time with pseudo-Euclideangeometry. In this formulation of the theory of relativity the space-time con-tinuum can be described equally well from the point of view of any co-ordinate system, which cannot possibly change ds . At variance, the usualformulation of the theory of relativity also deals with the invariance of ds , but it is understood only in a limited sense when the metric is strictlydiagonal.Common textbook presentations of the special theory of relativity use theEinstein approach or, as generalization, the usual covariant approach whichdeals, as discussed above, only with components of the 4-tensors in spe-cific (orthogonal) Lorentz basis. The fact that in the process of transition toarbitrary coordinates the geometry of the space-time does not change, isnot considered in textbooks. As a consequence there is a widespread beliefamong experts that a transformation from an orthogonal Lorentz basis toa non orthogonal basis is incorrect, while a Lorentz transformation, whichis a transformation from an orthogonal Lorentz basis to another orthogonalLorentz basis, is correct. This is not true. We can describe physics in anyarbitrary coordinates system. The di ff erent transformations of coordinatesonly correspond to a change in the way of components of 4-tensors arewritten, but not influence of 4-tensors themselves. Although the Einsteinsynchronization i.e. Lorentz coordinates choice, is preferred by physicistsdue to its simplicity and symmetry, it is nothing more ”physical” than anyother. A particularly very unusual choice of coordinates, the absolute timecoordinate choice, will be considered and exploited in this book. The use of Galilean transformations within the theory of relativity requiressome special discussion. Many physicists still tend to think of Galileantransformations (which is actually a transformations from an orthogonalLorentz basis to a non-orthogonal basis) as old, incorrect transformationsbetween spatial coordinates and time. A widespread argument used tosupport the incorrectness of Galilean transformations is that they do notpreserve the form-invariance of Maxwell’s equations under a change ofinertial frame. This idea is a part of the material in well-known books andmonographs (1) .Authors of textbooks are mistaken in their belief about the incorrectness ofGalilean transformations. The special theory of relativity is the theory offour-dimensional space-time with pseudo-Euclidean geometry. From thisviewpoint, the principle of relativity is a simple consequence of the space-time geometry, and the space-time continuum can be described in arbitrary16oordinates (2) . Therefore, contrary to the view presented in most textbooks,Galilean transformations are actually compatible with the principle of rela-tivity although, of course, they alter the form of Maxwell’s equations.
It is generally believed that a Lorentz transformation reduces to a Galileantransformation in the non-relativistic limit. We state that this typical text-book statement is incorrect and misleading. Kinematics is a comparativestudy which requires two coordinate systems, and one needs to assign timecoordinates to the two systems. Di ff erent types of clock synchronizationprovide di ff erent time coordinates. The convention on the clock synchro-nization amounts to nothing more than a definite choice of the coordinatesystem in an inertial frame of reference in Minkowski space-time. Pragmaticarguments for choosing one coordinate system over another may thereforelead to di ff erent choices in di ff erent situations. Usually, in relativistic engi-neering, we have a choice between absolute time coordinate and Lorentztime coordinate. The space-time continuum can be described equally wellin both coordinate systems. This means that for arbitrary particle speed, theGalilean coordinate transformations well characterize a change in the ref-erence frame from the lab inertial observer to a comoving inertial observerin the context of the theory of relativity. Let us consider the non relativisticlimit. The Lorentz transformation, for v / c so small that v / c is neglectedcan be written as x (cid:48) = x − vt , t (cid:48) = t − xv / c . This infinitesimal Lorentz trans-formation di ff ers from the infinitesimal Galilean transformation x (cid:48) = x − vt , t (cid:48) = t . The di ff erence is in the term xv / c in the Lorentz transformation fortime, which is a first order term.We only wish to emphasize here the following point. An infinitesimalLorentz transformation di ff ers from Galilean transformation only by theinclusion of the relativity of simultaneity, which is the only relativistic e ff ectthat appears in the first order in v / c . All other higher order e ff ects, thatare Lorentz-Fitzgerald contraction, time dilation, and relativistic correctionin the law of composition of velocities, can be derived mathematically, byiterating this infinitesimal transformation (3) .The main di ff erence between the Lorentz coordinatization and the absolutetime coordinatization is that the transformation laws connecting coordinatesand times between relatively moving systems are di ff erent. It is impossible toagree with the textbook statement that there is reduction of t (cid:48) = γ ( t − vx / c ) toGalilean relation t (cid:48) = t in the non-relativistic limit. This would mean that inthe non-relativistic limit infinitesimal Lorentz transformations are identicalto infinitesimal Galilean transformations. This statement is absurd conclu-17ion from a mathematical standpoint. The essence of Lorentz (or Galilean)transformations consists in their infinitesimal form: relativistic kinematicse ff ects cannot be found by the mathematical procedure of iterating the in-finitesimal Galilean transformations. It is generally believed that experiments show that the speed of light in vac-uum is independent of the source or observer (4) . This statement presentedin most textbooks and is obviously incorrect. The constancy of the speed oflight is related to the choice of synchronization convention, and cannot besubject to experimental tests (5) .In fact, in order to measure the one-way speed of light one has first to syn-chronize the infinity of clocks assumed attached to every position in space,which allows us to perform time measurements. Obviously, an unavoidabledeadlock appears if one synchronizes the clocks by assuming a-priori thatthe one-way speed of light is c . In fact, in that case, the one-way speed oflight measured with these clocks (that is the Einstein speed of light) cannotbe anything else but c : this is because the clocks have been set assuming thatparticular one-way speed in advance.Therefore, it can be said that the value of the one-way speed of light is justa matter of convention without physical meaning. In contrast to this, thetwo-way speed of light, directly measurable along a round-trip, has physi-cal meaning, because round-trip experiments rely upon the observation ofsimultaneity or non-simultaneity of events at a single point in space andnot depends on clock synchronization convention. All well known methodsto measure the speed of light are, indeed, round-trip measurements. Thecardinal example is given by the Michelson-Morley experiment: this exper-iment uses, indeed, an interferometer where light beams are compared in atwo-way fashion. Consider the motion of charged particle in a given magnetic field. The theoryof relativity says that the particle trajectory (cid:126) x ( t ) in the lab frame dependson the choice of a convention, namely the synchronization convention ofclocks in the lab frame. Whenever we have a theory containing an arbitraryconvention, we should examine what parts of the theory depend on thechoice of that convention and what parts do not. We may call the former18onvention-dependent, and the latter convention-invariant parts. Clearly,physically meaningful measurement results must be convention-invariant.Consider the motion of two charged particles in a given magnetic field,which is used to control the particle trajectories. Suppose there are twoapertures at point A and at point A (cid:48) . From the solution of the dynamicsequation of motion we may conclude that the first particle gets throughthe aperture at A and the second particle gets through the aperture at A (cid:48) simultaneity. The two events, i.e. the passage of particles at point A andpoint A (cid:48) have exact objective meaning i.e. convention-invariant. However,the simultaneity of these two events is convention-dependent and has noexact objective meaning. It is important at this point to emphasize that,consistently with the conventionality of simultaneity, also the value of thespeed of particle is a matter of convention and has no definite objectivemeaning.In order to examine what parts of the dynamics theory depend on thechoice of that convention and what parts do not, we want to show thedi ff erence between the notions of path and trajectory. Let us consider themotion of a particle in three-dimensional space using the vector-valuedfunction (cid:126) x ( t ). We have a prescribed curve (path) along which the particlemoves. The motion along the path is described by l ( t ), where l is a certainparameter (in our case of interest the length of the arc). The trajectory of aparticle conveys more information about its motion because every positionis described additionally by the corresponding time instant. The path israther a purely geometrical notion. If we take the origin of the (Cartesian)coordinate system and we connect the point to the point laying on the pathand describing the motion of the particle, then the creating vector will be aposition vector (cid:126) x ( l ) (6) .The di ff erence between trajectory (cid:126) x ( t ) and path (cid:126) x ( l ) is very interesting. Thepath has exact objective meaning i.e. it is convention-invariant. In contrastto this, and consistently with the conventionality intrinsic in the velocity,the trajectory (cid:126) x ( t ) of the particle is convention dependent and has no exactobjective meaning.In order to avoid being to abstract for to long we have given some examples:just think of the experiments related with accelerator physics. Suppose wewant to perform a particle momentum measurement. A uniform magneticfield can be used in making a ”momentum analyzer” for high-energy chargeparticles, and it must be recognized that this method for determining the par-ticle’s momentum is convention-independent. In fact, the curvature radiusof the path in the magnetic field (and consequently the three-momentum)has obviously an objective meaning, i.e. is convention-invariant. Dynamicstheory contains a particle trajectory that we do not need to check directly,19ut which is used in the analysis of electrodynamics problem. Generally, experts on the theory of relativity erroneously identify the proper-ties of Minkovski space-time with the familiar form that certain convention-dependent quantities assume under the standard Lorentz coordinatization.These quantities usually are called ”relativistic kinematics e ff ects”. There isa widespread belief that the convention-dependent quantities like the timedilation, length contraction, and Einstein’s addition of velocities have di-rect physical meaning. We found that statement like ” moving clocks runslow” is not true under the adopted absolute time clock synchronization,and, hence, are by no means intrinsic features of Minkowski space-time.Relativistic kinematic e ff ects are coordinate (i.e. convention-dependent) ef-fects and have no exact objective meaning (7) . In the case of Lorentz co-ordinatization, one will experience e.g. the time dilation phenomenon. Incontrast to this, in the case of absolute time coordinatization there are norelativistic kinematics e ff ects and no time dilation will be found. However,all coordinate-independent quantities like the particle path (cid:126) x ( l ) and momen-tum | (cid:126) p | remain independent of such a change in clock synchronization. The accelerated motion is described by a covariant equation of motion for arelativistic charged particle under the action of the four-force in the Lorentzlab frame. The trajectory of a particle (cid:126) x cov ( t ) is viewed from the Lorentz labframe as a result of successive infinitesimal Lorentz transformations. Thelab frame time t in the equation of motion cannot be independent from thespace variables. This is because Lorentz transformations lead to a mixtureof positions and time, and the relativistic kinematics e ff ects are consideredto be a manifestation of the relativity of simultaneity. Let us consider the conventional particle tracking approach. It is generallyaccepted that in order to describe dynamics of relativistic particles in the labreference frame, which we assume inertial, can be described by taking intoaccount the relativistic dependence of the particle momentum on the veloc-ity. The treatment of relativistic particle dynamics involves only corrected20ewton’s second law. In a given lab frame, there is an electric field (cid:126) E andmagnetic field (cid:126) B . They push on a particle in accordance with d (cid:126) pdt = e (cid:32) (cid:126) E + (cid:126) vc × (cid:126) B (cid:33) ,(cid:126) p = m (cid:126) v (cid:32) − v c (cid:33) − / , (3)where here the particle’s mass, charge, and velocity are denoted by m , e , and (cid:126) v respectively. The Lorentz force law, plus measurements on the componentsof acceleration of test particles, can be viewed as defining the components ofthe electric and magnetic fields. Once field components are known from theacceleration of test particles, they can be used to predict the accelerations ofother particles.This solution of the dynamics problem in the lab frame makes no referenceto Lorentz transformations. Conventional particle tracking treats the space-time continuum in a non-relativistic format, as a (3 +
1) manifold. In otherwords, in this approach, introducing as only modification to the classicalcase the relativistic mass, time di ff er from space. In fact, we have no mixtureof positions and time (8) .Most of the interesting phenomena in which charges move under the actionof electromagnetic fields occur in very complicated situations. But here wejust want to discuss the simple problem of the accelerated motion of parti-cles in a constant magnetic field. According to the non-covariant treatment,the magnetic field is only capable of altering the direction of motion, but notthe speed (i.e. mass) of a particle. This study of relativistic particle motion ina constant magnetic field, usual for accelerator engineering, looks preciselythe same as in nonrelativistic Newtonian dynamics and kinematics. Thetrajectory of a particle (cid:126) x ( t ), which follows from the solution of the correctedNewton’s second law, does not include relativistic kinematics e ff ects as rela-tivity of simultaneity and the Galilean vectorial law of addition of velocitiesis actually used.Let us discuss the important problem of the addition of velocities in relativ-ity. Suppose that in the case of accelerated motion one introduces an infinitesequence of co-moving frames. At each instant, the rest frame is a Lorentzframe centered on the particle and moving with it. Suppose that in inertialframe where particle is at rest at a given time, the traveler was observinglight itself. In other words measured speed of light v = c , and yet the frameis moving relative the lab frame. How will it look to the observer in thelab frame? According to Einstein’s law of addition of velocity the answer21ill be c . Maxwell’s equations remain in the same form when Lorentz trans-formations are applied to them, but Lorentz transformations give rise tonon-Galilean transformation rules for velocities, and therefore the theory ofrelativity shows that, if Maxwell’s equations is to be valid in the lab frame,the trajectories of the particles must include relativistic kinematics e ff ects.In other words, Maxwell,s equations can be applied in the lab frame onlyin the case when particle trajectories are viewed, from the lab frame as theresult of successive infinitesimal Lorentz transformations.The absence of relativistic kinematics e ff ects is the prediction of conven-tional non-covariant theory and is obviously absurd from the viewpoint ofMaxwell’s electrodynamics. Therefore, something is fundamentally, power-fully, and absolutely wrong in coupling fields and particles within a ”singleinertial frame”. It is generally believed that the electrodynamics problem can be treatedwithin the same ”single inertial frame” description without reference toLorentz transformations. In all standard derivations it is assumed that usualMaxwell’s equations and corrected Newton’s second law can explain allexperiments that are performed in a single inertial frame, for instance thelab reference frame.Going to electrodynamics problem, the di ff erential form of Maxwell’s equa-tions describing electromagnetic phenomena in the same inertial lab frame(in cgs units) is given by the following expressions: (cid:126) ∇ · (cid:126) E = πρ ,(cid:126) ∇ · (cid:126) B = ,(cid:126) ∇ × (cid:126) E = − c ∂(cid:126) B ∂ t ,(cid:126) ∇ × (cid:126) B = π c (cid:126) j + c ∂(cid:126) E ∂ t . (4)Here the charge density ρ and current density (cid:126) j are written as ρ ( (cid:126) x , t ) = (cid:88) n e n δ ( (cid:126) x − (cid:126) x n ( t )) ,(cid:126) j ( (cid:126) x , t ) = (cid:88) n e n (cid:126) v n ( t ) δ ( (cid:126) x − (cid:126) x n ( t )) , (5)22here δ ( (cid:126) x − (cid:126) x n ( t )) is three-dimensional delta function, m n , e n , (cid:126) x n ( t ), and (cid:126) v n = d (cid:126) x n ( t ) / dt denote the mass, charge, position, and the velocity of the n th parti-cle, respectively. To evaluate radiation fields arising from an external sourcesEq. (4) we need the velocity (cid:126) v n and the position (cid:126) x n as a function of lab frametime t . It is generally accepted by physics community that equation of mo-tion, which describes how coordinates of the particle carrying the chargechange with time t , is described by corrected Newton’s second law Eq. (3).This coupling of Maxwell’s equations and corrected Newton’s equation iscommonly accepted as useful method in accelerator physics and, in partic-ular, in analytical and numerical calculations of radiation properties. Suchapproach to relativistic dynamics and electrodynamics usually forces the ac-celerator physicist to believe that the design of particle accelerators possiblewithout detailed knowledge of the theory of relativity.However, there is a common mistake made in accelerator physics connectedwith the di ff erence between (cid:126) x ( t ) and (cid:126) x cov ( t ) trajectories. Let us look at this dif-ference from the point of view of electrodynamics of relativistically movingcharges. To evaluate fields arising from external sources we need to knowtheir velocity and positions as a function of the lab frame time t . Suppose onewants to calculate properties of radiation. Given our previous discussion thequestion arises, whether one should solve the usual Maxwell’s equations inthe lab frame with current and charge density created by particle movingalong non-covariant trajectories like (cid:126) x ( t ). I claim that the answer to this ques-tion is negative. This algorithm for solving usual Maxwell’s equations in thelab frame, which is considered in all standard treatments as relativisticallycorrect, is at odds with the principle of relativity. This essential point hasnever received attention in the physical community. Only the solution of thedynamics equations in covariant form gives the correct coupling betweenthe usual Maxwell’s equations and particle trajectories in the lab frame. Let us now examine the logical content of the concept of a ”single inertialframe”. If a traveler in a co moving frame, similar to an observer in the labframe, introduces a definite coordinate-time grid, there is always a definitetransformation between these two four-dimensional coordinate systems.Thus, particle trajectories are always viewed from the lab frame as a result ofsuccessive transformations, and the form of these transformations dependson the choice of coordinate systems in the comoving and the lab frame.One might well wonder why it is necessary to discuss how di ff erent inertialframes are related to one another. The point is that all natural phenomena23ollow the principle of relativity, which is a restrictive principle: it says thatthe laws of nature are the same (or take the same form) in all inertial frames.In agreement with this principle, usual Maxwell’s equations can always beexploited in any inertial frame where electromagnetic sources are at restusing Einstein synchronization procedure in the rest frame of the source.The fact that one can deduce electromagnetic field equations for arbitrarymoving sources by studying the form taken by Maxwell’s equations underthe transformation between rest frame of the source and the frame wherethe source is moving is a practical application of the principle of relativity.The question now arises how to assign a time coordinate to the lab frame.Coordinates serve the purpose of labeling events in an unambiguous way,and this can be done in infinitely many di ff erent ways. The principle of rela-tivity dictates that Maxwell’s equations can be applied in the lab frame onlyin the case when Lorentz coordinates are assigned and particle trajectoriesare viewing from the lab frame as a result of successive infinitesimal Lorentztransformations between the lab and comoving inertial frames.A ”single frame” (non-covariant 3 +
1) approach to relativistic particle dy-namics has been used in particle tracking calculations for about seventyyears. However, the type of clock synchronization which provides the timecoordinate t in the corrected Newton’s equation has never been discussedin literature. It is clear that without an answer to the question about themethod of synchronization used, not only the concept of velocity, but alsothe dynamics law has no physical meaning. A ”single frame” approachto relativistic particle dynamics is forcefully based on a definite synchro-nization assumption but this is actually a hidden assumption. Accordingto conventional particle tracking, the dynamical evolution in the lab frameis based on the use of the lab frame time t as an independent variable, in-dependent in the sense that t is not related to the spatial variables. Suchapproach to relativistic particle dynamics is actually based on the use ofa not standard (not Einstein) clock synchronization assumption in the labframe.In fact, the usual for accelerator engineering study of relativistic particlemotion in a constant magnetic field looks precisely the same as in non-relativistic Newtonian mechanics and the trajectories of the electrons doesnot include relativistic kinematics e ff ects. According to textbooks, this is noproblem. If no more than one frame is involved, one does not need to use(and does not need to know) the theory of relativity. Only when one passesfrom one reference frame to another the relativistic context is important.Conventional particle tracking in a constant magnetic field is actually basedon classical Newton mechanics. It is generally believed that the electrody-namics problem, similar to conventional particle tracking, can be treatedwithin a description involving a single inertial frame and one should solve24he usual Maxwell’s equations in the lab frame with current and chargedensity created by particles moving along the non-covariant (single frame)trajectories.This is misconception. The situation when only one frame is involved andthe relativistic context is unimportant cannot be realized. The lab observermay argue, ”I don’t care about other frames.” Perhaps the lab observerdoesn’t, but nature knows that, according to the principle of relativity,Maxwell’s equations are always valid in the Lorentz comoving frame. Elec-trodynamics equations can be written down in the lab frame only whena space-time coordinate system has been specified. An observer in the labframe has only one freedom. This is the choice of a coordinate system (i.e.the choice of clock synchronization convention) in the lab frame. After this,the theory of relativity states that the electrodynamics equations in the labframe are the result of transformation of Maxwell’s equations from theLorentz comoving frame to the lab frame.
1. Many physicists tend to think of Galilean transformations as pre-relativistictransformations between spatial coordinates and time that are not compat-ible with the special theory of relativity. To quote e.g. Bohm [1] ”... theGalilean law of addition of velocities implies that the speed of light shouldvary with the speed of the observing equipment. Since this predicted varia-tion is contrary to the fact, the Galilean transformations evidently cannot bethe correct one.”. Similar statements can also be found in recently publishedpedagogical papers. To quote e.g. Drake and Purvis [2] ”One of the greatinsights to come relativity theory was the realization that Galilean transfor-mations are wrong. The correct way to translate the space-time measure ofevents between inertial frames is with the Lorentz transformations” How-ever, this is not true. Galilean transformations are simply transformationsrelating a given coordinate set to another coordinate set. The space-timecontinuum can be described in arbitrary coordinates, and choice of this setof coordinates cannot change the geometry of space-time.2. The mathematical argument that in the process of transition to arbitrarycoordinates the geometry of the space-time does not change, is consideredin textbooks as erroneous. To quote L. Landau and E. Lifshitz [3]: ”Thisformula is called the Galileo transformation. It is easily to verify that thistransformation, as was to be expected, does not satisfy the requirementsof the theory of relativity; it does not leave the interval between eventsinvariant.”. This fact is ascribed to a lack of understanding of the di ff er-ence between convention-dependent and convention-invariant parts of the25heory. In pseudo-Euclidean geometry the interval between events is an in-variant in arbitrary coordinates. A comparison with three-dimensional Eu-clidean space might help here. In the usual 3D Euclidean space, one can con-sider a Cartesian coordinate system ( x , y , z ), a cylindrical coordinate system( r , φ, z ), a spherical coordinate system ( ρ, θ, φ ), or any other. Depending onthe choice of the coordinate system one respectively has ds = dx + dy + dz , ds = dr + r d φ + dz , ds = d ρ + ρ d θ + ρ sin θ d φ . The metric actuallydoes not change, but the components of the metric do, depending only onthe choice of coordinates. In general, in fact, we write ds = g ik dx i dx k . Consid-ering Cartesian coordinates we will always have g ij = diag(1 , , t (cid:48) = γ ( t − vx / c ) to Galilean relation t (cid:48) = t requires x (cid:28) ct as well as v / c (cid:28) v / c .” This state-ment is absurd conclusion even from a mathematical standpoint. This hasbeen recognized by some expects, perhaps most explicitly by Baierlein whostates [6] that ”If the Lorentz transformation for infinitesimal v / c were toreduce to the Galilean transformation, then iterative process could nevergenerate a finite Lorentz transformation that is radically di ff erent from theGalilean transformation. But the finite transformations are radically di ff er-ent, and so -however subtly-the infinitesimal Lorentz transformation mustdi ff er significantly from the Galilean transformation.”4. Many authors of textbooks still attribute a measurable status to the con-ventional quantities. To quote e.g. Cristodoulides [7] ”The fact that Galileantransformation does not leave Maxwell’s equations has already been men-tioned [...] On the other hand, experiments show that the speed of light invacuum is independent of the source or observer.”.5. Because we have empirical access only to the round-trip average speedof light, statements about the magnitude and isotrophy of the one-wayspeed of light must reflect the assumptions made in the choice of timecoordinatization, and such entities change as the theory is re-synchronized.To quote C. Anderson, I. Vetharaniam, and G. Stedman [8]: ”No experiment,then, is a ”one-way” experiment. An empirical test of any property of theone-way speed of light is not possible. Such quantities as the one-way speedof light are irreducibly conventional in nature, and recognizing this aspectis to recognize a profound feature of nature”.26. For a general discussion of the di ff erence between path and trajectory wesuggest reading the book [9].7. The standard textbooks erroneously identify the properties of space-timewith the familiar form that certain synchronization-dependent quantitiesassume under the Einstein’s clock synchronization. These quantities are, forexample, the formulae for ”time dilation” and ”length contraction”. Fortu-nately, it has also been occasionally stressed in the literature that the formsof the above ”relativistic e ff ects” are coordinate-dependent, their forms de-pend in turn on the kind of synchronization procedure adopted. To quoteLeubner, K. Auflinger, and P. Krumm [10]: ”... there is a widespread beliefamong students that the familiar form of coordinate-dependent quantitieslike the measured velocity of light, the Lorentz transformation between twoobservers, ”addition of velocities”, ”time dilation”, ”length contraction”which they assume under the standard clock synchronization, is relativityproper. In order to demonstrate that this is by no means so, this paper studiesthe consequences of a non-standard synchronization, and it is shown thatdrastic changes in the appearance of all these quantities are thus induced.”8. Clarification of the true content of the non-covariant theory can be foundin various advanced textbooks. To quote e.g. Ferrarese and Bini [11]: ”...within a single inertial frame, the time is an absolute quantity in specialrelativity also. As a consequence, if no more than one frame is involved , onewould not expect di ff erences between classical and relativistic kinematics.But in the relativistic context there are di ff erences in the transformationlaws of the various relative quantities (of kinematics or dynamics), whenpassing from one reference frame to another.” We see that authors give aspecial role to concept of a ”single inertial frame”. The name ”single inertialframe” tends to suggest that a distinctive trait of non-covariant theory isthe absence of relativistic kinematics in the description of particle motion.This point was expressed by Friedman [14]: ”Within any single inertialframe, things looks precisely the same as in Newtonian kinematics: there isan enduring Euclidean three-space, a global (i.e. absolute) time t , and lawof motion. But di ff erent inertial frames are related to one another in a non-Newtonian fashion.” According to conventional particle tracking, within the”single” frame there is no relativistic kinematics e ff ects. This, as we alreadymentioned, contradicts the Maxwell’s electrodynamics. We cannot take oldkinematics for mechanics and Einstein’s kinematics for electrodynamics.If one wants to use the usual Maxwell’s equations, only solution of thedynamics equations in covariant form gives the correct coupling betweenMaxwell’s equations and particle trajectories in the ”single frame”.27 Space-Time and Its Coordinatization
Let us discuss an ”operational interpretation” of the Lorentz and absolutetime coordinatizations. We should underline that we claim the non covari-ant approach to relativistic particle dynamics is actually based on the use ofa not standard and unusual clock synchronization assumption within thetheory of relativity. It is important to know how to operationally interpretthe absolute time convention i.e. how one should perform the clock syn-chronization in the lab frame. The result is very interesting, since it tell usabout di ff erence between absolute time synchronization and Einstein’s timesynchronization from the operational point of view. Each physical phenomenon occurs in space and time. A concrete methodfor representing space and time is a frame of reference. One-and-the samespace and time can be represented by various coordinate-time grids, i.e.,by various frames of references. Even the simplest space-time coordinatesystems require carefully description.Clocks reveal the motion of a particle through the coordinate-time grid.The general approach to the determination of the motion of a particle isthe following: at any instant a particle has a well-defined velocity (cid:126) v asmeasured in a laboratory frame of reference. How is a velocity of a particlefound? The velocity is determined once the coordinates in the lab frameare chosen, and is then measured at appropriate time intervals along theparticle’s trajectory. But how to measure a time interval between eventsoccurring at di ff erent points in space? In order to do so, and hence measurethe velocity of a particle within a single inertial lab frame, one first has tosynchronize distant clocks. The concept of synchronization is a key conceptin the understanding of special relativity. It is possible to think of variousmethods to synchronize distant clocks (1) . The choice of a convention onclock synchronization is nothing more than a definite choice of coordinatessystem in an inertial frame of reference of the Minkowski space-time.The space-time continuum can be described in arbitrary coordinates. Bychanging these arbitrary coordinates, the geometry of the four-dimensionalspace-time obviously does not change, and in the special theory of relativitywe are not limited in any way in the choice of a coordinates system. Relyingon the geometric structure of Minkowski space-time, one defines the class of28nertial frames and adopts a Lorentz frame with orthonormal basis vectors.Within the chosen Lorentz frame, Einstein’s synchronization procedure ofdistant clocks (which based on the constancy of the speed of light in allinertial framers) and Cartesian space coordinates are enforced: covariantparticle tracking is based on the use Lorentz coordinates. Let us give an ”operational interpretation” of the Lorentz coordinatizations.The fundamental laws of electrodynamics are expressed by Maxwell’s equa-tions, according to which, as well-known, light propagates with the samevelocity c in all directions. This is because Maxwell’s theory has no intrinsicanisotropy. It has been stated that in their original form Maxwell’s equa-tions are only valid in inertial frames. However, Maxwell’s equations can bewritten down in coordinate representation only if the space-time coordinatesystem has already been specified.The problem of assigning Lorentz coordinates to the lab frame in the case ofaccelerated motion is complicated. We would like to start with the simplerquestion of how to assign space-time coordinates to an inertial frame, wherea source of light is at rest. We need to give a ”practical”, ”operational” answerto this question. The most natural method of synchronization consists inputting all the ideal clocks together at the same point in space, where theycan be synchronized. Then, they can be transported slowly to their originalplaces (slow clock transport) (2) .The usual Maxwell’s equations are valid in any inertial frame where sourcesare at rest and the procedure of slow clock transport is used to assign val-ues to the time coordinate. The same considerations apply when chargedparticles are moving in non-relativistic manner. In particular, when oscillat-ing, charged particles emit radiation, and in the non-relativistic case, whencharges oscillate with velocities much smaller than c , dipole radiation isgenerated and described with the help of the Maxwell’s equations in theirusual form.Let’s examine in a more detail how the dipole radiation term comes about.The retardation time in the integrands of the expression for the radiationfield amplitude, can be neglected in the cases where the trajectory of thecharge changes little during this time. It is easy to find the conditions forsatisfying this requirement. Let us denote by a the order of magnitude ofthe dimensions of the system. Then the retardation time ∼ a / c . In order toensure that the distribution of the charges in the system does not undergo asignificant change during this time, it is necessary that a (cid:28) λ , where λ is theradiation wavelength. Thus, the dimensions of the system must be small29ompared to radiation wavelength. This condition can be written in stillanother form v (cid:28) c , where v is of the order of magnitude of the velocitiesof the charges. In accounting only for the dipole part of the radiation weneglect all information about the electron trajectory. Therefore, one shouldnot be surprised to find that dipole radiation theory gives fields very muchlike the instantaneous theory.The theory of relativity o ff ers an alternative procedure of clocks synchro-nization based on the constancy of the speed of light in all inertial frames.This is usually considered a postulate but, as we have seen, it is just a con-vention. The synchronization procedure that follows is the usual Einsteinsynchronization procedure. Suppose we have a dipole radiation source.When the dipole light source is at rest, the field equations are constituted bythe usual Maxwell’s equations. Indeed, in dipole radiation theory we con-sider the small expansion parameter v / c (cid:28) v / c .In other words, in dipole radiation theory we use zero order non relativisticapproximation. Einstein synchronization is defined in terms of light signalsemitted by the dipole source at rest, assuming that light propagate withthe same velocity c in all directions. Using Einstein synchronization proce-dure in the rest frame of the dipole source, we actually select the Lorentzcoordinate system.Slow transport synchronization is equivalent to Einstein synchronization ininertial system where the dipole light source is at rest (3) . In other words,suppose we have two sets of synchronized clocks spaced along the x axis.Suppose that one set of clocks is synchronized by using the slow clocktransport procedure and the other by light signals. If we would ride togetherwith any clock in either set, we could see that it has the same time as theadjacent clocks, with which its reading is compared. This is because in ourcase of interest, when light source is at rest, field equations are the usualMaxwell’s equations and Einstein synchronization is defined in terms oflight signals emitted by a source at rest assuming that light propagates withthe same velocity c in all directions. Using any of these synchronizationprocedures in the rest frame we actually select a Lorentz coordinate system.In this coordinate system the metric has Minkowski form ds = c dt (cid:48) − dx (cid:48) − dy (cid:48) − dz (cid:48) . In the rest frame, fields are expressed as a function of theindependent variables x (cid:48) , y (cid:48) , z (cid:48) , and t (cid:48) . Let us consider Maxwell’s equationsin free space. The electric field (cid:126) E (cid:48) of an electromagnetic wave satisfies theequation (cid:3) (cid:48) (cid:126) E (cid:48) = ∇ (cid:48) (cid:126) E (cid:48) − ∂ (cid:126) E (cid:48) /∂ ( ct (cid:48) ) = We now consider the case when the light source in the lab frame is accel-erated from rest up to velocity v along the x -axis. A fundamental question30o ask is whether our lab clock synchronization method depends on thestate of motion of the light source or not. The answer simply fixes a conven-tion. The simplest method of synchronization consists in keeping, withoutchanges, the same set of uniformly synchronized clocks used in the casewhen the light source was at rest, i.e. we still enforce the clock transportsynchronization ( or Einstein synchronization which is defined in terms oflight signals emitted by the dipole source at rest). This choice is usually themost convenient one from the viewpoint of connection to laboratory reality.This synchronization convention preserves simultaneity and is actuallybased on the absolute time (or absolute simultaneity) convention. Afterthe boost along the x axis, the Cartesian coordinates of the emitter trans-form as x (cid:48) = x − vt , y (cid:48) = y , z (cid:48) = z . This transformation completes with theinvariance of simultaneity, ∆ t (cid:48) = ∆ t . The absolute character of the temporalcoincidence of two events is a consequence of the absolute concept of time,enforced by t (cid:48) = t . As a result of the boost, the transformation of time andspatial coordinates of any event has the form of a Galilean transformation.In the comoving frame, fields are expressed as a function of the indepen-dent variables x (cid:48) , y (cid:48) , z (cid:48) , and t (cid:48) . According to the principle of relativity, theMaxwell’s equations always valid in the Lorentz comoving frame. Theelectric field (cid:126) E (cid:48) of an electromagnetic wave satisfies the equation (cid:3) (cid:48) (cid:126) E (cid:48) = ∇ (cid:48) (cid:126) E (cid:48) − ∂ (cid:126) E (cid:48) /∂ ( ct (cid:48) ) =
0. However, the variables x (cid:48) , y (cid:48) , z (cid:48) , t (cid:48) can be expressedin terms of the independent variables x , y , z , t by means of a Galilean trans-formation, so that fields can be written in terms of x , y , z , t . From the Galileantransformation x (cid:48) = x − vt , y (cid:48) = y , z (cid:48) = z , t (cid:48) = t , after partial di ff erentiation,one obtains ∂/∂ t = ∂/∂ t (cid:48) − v ∂/∂ x (cid:48) , ∂/∂ x = ∂/∂ x (cid:48) . Hence the wave equationtransforms into (cid:3) (cid:126) E = (cid:32) − v c (cid:33) ∂ (cid:126) E ∂ x − (cid:18) vc (cid:19) ∂ (cid:126) E ∂ t ∂ x + ∂ (cid:126) E ∂ y + ∂ (cid:126) E ∂ z − c ∂ (cid:126) E ∂ t = , (6)where coordinates and time are transformed according to a Galilean trans-formation. The solution of this equation F [ x − ( c + v ) t ] + G [ x + ( − c + v ) t ] is thesum of two arbitrary functions, one of argument x − ( c + v ) t and the otherof argument x + ( − c + v ) t . Here we obtained the solution for waves whichmove in the x direction by supposing that the field does not depend on y and z . The first term represents a wave traveling forward in the positive x direction, and the second term a wave traveling backwards in the negative x direction.We conclude that the speed of light emitted by a moving source measured inthe lab frame ( t , x ) depends on the relative velocity of source and observer,in our example v . In other words, the speed of light is compatible with the31alilean law of addition of velocities. In fact, the coordinate velocity of lightparallel to the x -axis is given by dx / dt = c + v in the positive direction, and dx / dt = − c + v in the negative direction. The reason why it is di ff erent fromthe electrodynamics constant c is due to the fact that the clocks are synchro-nized following the absolute time convention, which is fixed because ( t , x )is related to ( t (cid:48) , x (cid:48) ) via a Galilean transformation (4) .After properly transforming the d’Alembertian, which changes the initialcoordinates ( x (cid:48) , y (cid:48) , z (cid:48) , t (cid:48) ) into ( x , y , z , t ), we can see that the homogeneouswave equation for the field in the lab frame has nearly but not quite theusual, standard form that takes when there is no uniform translation inthe transverse direction with velocity v . The main di ff erence consists in thecrossed term ∂ /∂ t ∂ x , which complicates the solution of the equation. Toget around this di ffi culty, we observe that simplification is always possible.The trick needed here is to further make a change of the time variableaccording to the transformation t (cid:48) = t − xv x / c . In the new variables in i.e.after the Galilean coordinate transformation and the time shift we obtainthe d’Alembertian in the following form (cid:3) = (cid:32) − v x c (cid:33) ∂ ∂ x + ∂ ∂ y + ∂ ∂ z − (cid:32) − v x c (cid:33) c ∂ ∂ t . (7)A further change of a factor γ in the scale of time and of the coordinate alongthe direction of uniform motion leads to the usual wave equation.We have, then, a general method for finding solution of electrodynamicsproblem in the case of the absolute time coordinatization. Since the Galileantransformation x = x (cid:48) + vt (cid:48) , t = t (cid:48) , completed by the introduction of the newvariables ct L = (cid:104) √ − v / c ct + ( v / c ) x / √ − v / c (cid:105) , and x L = x / √ − v / c , ismathematically equivalent to a Lorentz transformation x L = γ ( x (cid:48) + vt (cid:48) ) , t L = γ ( t (cid:48) + vx (cid:48) / c ), it obviously follows that transforming to new variables x L , t L leads to the usual Maxwell’s equations. In particular, when coordinates andtime are transformed according to a Galilean transformation followed by thevariable changes specified above, the d’Alembertian (cid:3) (cid:48) = ∇ (cid:48) − ∂ /∂ ( ct (cid:48) ) transforms into (cid:3) L = ∇ L − ∂ /∂ ( ct L ) . As expected, in the new variables thevelocity of light is constant in all directions, and equal to the electrodynamicsconstant c .The overall combination of Galileo transformation and variable changesactually yields the Lorentz transformation in the case of absolute time coor-dinatization in the lab frame, but in this context this transformation are onlyto be understood as useful mathematical device, which allow one to solvethe electrodynamics problem in the choice of absolute time synchronizationwith minimal e ff ort. 32e can now rise an interesting question: do we need to transform the resultsof the electrodynamics problem solution into the original variables? Westate that the variable changes performed above have no intrinsic meaning- their meaning only being assigned by a convention. In particular, one cansee the connection between the time shift t = t (cid:48) + x (cid:48) v / c and the issue ofclock synchrony. Note that the final change in the scale of time and spatialcoordinates is unrecognizable also from a physical viewpoint. It is clear thatthe convention-independent results of calculations are precisely the same inthe new variables. As a consequence, we should not care to transform theresults of the electrodynamics problem solution into the original variables.The question now arises how to operationally interpret these variable changesi.e. how one should change the rule-clock structure of the the lab referenceframe. In order to assign a Lorentz coordinate system in the lab frame afterthe Galilean boost x = x (cid:48) + vt (cid:48) , t = t (cid:48) , one needs to perform additionally achange scale of reference rules x → γ x , accounting for length contraction.After this, one needs to change the rhythm of all clocks t → t /γ , thus ac-counting for time dilation. The transformation of the rule-clock structurecompletes with the distant clock resynchronization t → t + xv / c . This newspace-time coordinates in the lab frame are interpreted, mathematically, bysaying that the wave equation is now diagonal and the speed of light fromthe moving source is isotropic and equal to c .So, from an operational point of view, the new coordinates in the lab frameafter the clocks resynchronization are impeccable. However, from the theoryof relativity we know that if we wish to assign Lorentz coordinates to aninertial lab frame, the synchronization must be defined in terms of lightsignals. The following important detail of such synchronization can hardlybe emphasized enough. If the source of light is in motion, we see that theprocedure for distant clocks synchronizing must be performed by using amoving light source. The constant value of c for the speed of light emittedby the moving source destroys the simultaneity introduced by light signalsemitted by the (dipole) source at rest. The coordinates reflecting the constantspeed of light c from a moving source are Lorentz coordinates for thatparticular source.Consider now two light sources say ”1” and ”2”. Suppose that in the labframe the velocities of ”1” and ”2” are (cid:126) v , (cid:126) v and (cid:126) v (cid:44) (cid:126) v . The question nowarises how to assign a time coordinate to the lab reference frame. We havea choice between an absolute time coordinate and a Lorentz time coordi-nate. The most natural choice, from the point of view of connecting to thelaboratory reality, is the absolute time synchronization. In this case simul-taneity is absolute, and for this we should prepare, for two sources, only oneset of synchronized clocks in the lab frame. On the other hand, Maxwell’sequations are not form-invariant under Galilean transformations, that is,33heir form is di ff erent on the lab frame. In fact, the use of the absolute timeconvention, implies the use of much more complicated field equations, andthese equations are di ff erent for each source. Now we are in the position toassign Lorentz coordinates. The only possibility to introduce Lorentz coordi-nates in this situation consists in introducing individual coordinate systems(i.e. individual set of clocks) for each source. It is clear that if operationalmethods are at hand to fix the coordinates (clock synchronization in the labframe) for the first source, the same methods can be used to assign values tothe coordinates for the second source and these will be two di ff erent Lorentzcoordinate systems.
1. According to the thesis of conventionality of simultaneity [12–16], simul-taneity of distant events is a conventional matter, as it can be legitimatelyfixed in di ff erent manners in any given inertial reference frame. To quote e.g.Moeller [12]: ”All methods for the regulation of clocks meet with the samefundamental di ffi culty. The concept of simultaneity between two events indi ff erent places obviously has no exact objective meaning at all, since wecannot specify any experimental method by which this simultaneity couldbe ascertained. The same is therefore true also for concept of velocity.”2. In the text first published in 1923, Eddington discussed, apparently for thefirst time, a procedure for synchronization using slow transport of clocks[17]. The details can be found in review [8] ( see also [18]).3. We already pointed that we have empirical access only to the round-tripaverage speed of light. An empirical test of any property of the one-wayspeed of light is not possible. Many authors of textbooks still attribute areality status of the one-way speed of light. To quote Hrasko [19]: ”It issometimes claimed that Einstein synchronization of distant clocks A ans B is circular. The argument is very simple: Einstein synchronization is basedon the equality of light velocity on the path from A to B and back from B to A , but measurement of light velocity in one direction between to distantpoints is impossible unless the clocks at these points have already beensynchronized. This argument is, however, fallacious. It is true that one-waymeasurement of light velocity can be performed only if clocks at the end-points are synchronized correctly. But since they need not show the correctcoordinate time, they can be synchronized without light signals by trans-porting them from common site in a symmetrical manner. The procedureconsists of following steps: ... . As we see, the thought experiment describedis capable to prove constancy of light speed if it is true, or to disprove it ifit is false. It provided, therefore, solid logical foundation for Einstein’s syn-34hronization prescription.” This logical argument is incorrect. Slow clocktransport synchronization is equivalent to Einstein’s synchronization in in-ertial system where the light source is at rest.4. Let us illustrate a particular special example of the simultaneity conven-tion. Let the synchronization of clocks in di ff erent spatial points be providedby light signals having, respectively, velocity c in the direction parallel tothe positive axis x , and velocity c in the opposite direction. Then, a light sig-nal sent from point A at time t A will arrive at point B at time t B = t A + x AB / c .The reflected signal will arrive back at point A at time t (cid:48) A = t B + x AB / c .From these two expressions we get t (cid:48) A − t A = x AB ( c + c ) / ( c c ). Summingup we have t B = t A + c ( t (cid:48) A − t A ) / ( c + c ). So we come to the synchronizationfirst proposed in [13]: t B = (cid:15) ( t (cid:48) A − t A ), where 0 < (cid:15) <
1. Einstein’s choicefor (cid:15) is (cid:15) = /
2. If (cid:15) (cid:44) /
2, then speed of light from A to B di ff ers fromthe speed of light from B to A . A nice pedagogical (but artificial) exampleof a non-standard synchronization is clock synchronization [10] in which (cid:15) =
0. In this book we call the choice (cid:15) = / Aberration of Light Phenomenon
The e ff ect of light aberration is usually understood as a change in the direc-tion of light propagation ascribed to boosted light sources. We will demon-strate that the aberration of light is a complex phenomenon which mustbe branched out into a number of varieties according to their origin. Thesebranching out takes place depending on what is the cause of aberration -whether it is a jump in the velocity (relative to the fixed stars) of the observeror of the light source. Aberration could undergo further splitting - depend-ing on the physical influence of the optical instrument on the measurement.We will describe the e ff ect of aberration by working only up to the firstorder v / c . The appearance of relativistic e ff ects in optical phenomena doesnot depend on a large speed of the radiation sources. Light is always arelativistic object, no matter how small the ratio v / c may be.The explanation of the e ff ect of aberration is actually based on the use ofthe model of single plane-wave emitter. As a simple model of a plane-waveemitter, we use a two-dimensional array of identical coherent elementarysources (dipoles), uniformly distributed on a given ( x − y ) plane P . We takethe elementary sources to start radiating waves simultaneously with respectto the lab reference frame where the plane P is at rest. Therefore we havea plane full of sources, oscillating together, with their motion in the planeand all having the same amplitude and phase.Let us suppose that the elementary sources are oscillating at frequency ω . Byletting the distance between each two adjacent elementary sources approachzero (i.e. much smaller with respect to the radiation wavelength λ = π c /ω ),we may consider this two-dimensional arrangement as an ideal plane-waveemitter.The concept of an (infinite) plane wave is widely used in physics. It is ananalytically well-behaved solution of Maxwell’s equations. However, it isnot a physically realizable solution, because the total energy content of sucha wave is infinite. In any physically realizable situation, one will have toconsider a finite source aperture. It is always hiddenly assumed that thedetector for the direction of the radiation is an energy propagation detectorand the size of the detector aperture is su ffi ciently large compared with theradiation beam size. Indeed, what is usually considered as an aberration is,in fact, an apparent deviation of the energy transport direction.36 .1.2 A Moving Emitter. The ”Single Frame” Description Let us consider the case when the ”plane-wave” emitter in the lab inertialframe is accelerated from rest up to velocity v along the x axis. Suppose thatan observer, which is at rest with respect to the inertial frame of referenceperforms the direction of the energy transport measurement.There could be two approaches to the analysis of the aberration o light ra-diated by a single moving emitter. The first one is the covariant approach.The explanation of the e ff ect of aberration of light in the case of a singlemoving emitter presented in the literature is actually based on the use of aLorentz boost to describe how the direction of a beam of light depends onthe velocity of the light source relative to the inertial frame of reference. An-other non-covariant approach consists in using a ”single frame” descriptionwithout reference to Lorentz transformations.The two approaches, treated according to Einstein’s or absolute time syn-chronization conventions give the same result in the case of a single movingemitter. The choice between these two di ff erent approaches in this particularcase is a matter of pragmatics.In this section we demonstrate both approaches. Let us start with non-covariant approach. We must emphasize that there is no principle di ffi cultywith the a non-covariant (3 +
1) approach in relativistic electrodynamics. It isperfectly satisfactory. The aberration of light problem can be treated withinthe same ”single inertial frame” description without reference to Lorentztransformations. Within a single inertial frame, the time is an absolute quan-tity in special relativity also. What does ”absolute” time mean? It means thatsimultaneity is absolute and there is no mixture of positions and time whensources change their velocities in the inertial frame. A distinctive trait ofour non-covariant theory is the absence of relativistic kinematics in the de-scription of aberration of light phenomena. When one has a transverselymoving emitter there is the deviation of the energy transport for radiatedlight. According to the ”single frame” approach, this e ff ect is a consequenceof the fact that the Doppler e ff ect is responsible for the angular frequencydispersion of the radiated light waves (Fig. 1 left).Let us consider the electrodynamics of the moving source. The explanationof the phenomenon of radiation in our case of interest consists in using aGalileo boost to describe the uniform translational motion of the source inthe inertial lab frame. Maxwell’s equations are not preserved in form by theGalilean transformation, i.e. Maxwell’s equations are not invariant underGalilean transformation. The new terms that have to be put into the fieldequations due to use of Galilean transformation lead to the prediction of theDoppler e ff ect. 37 ig. 1. The case when the ”plane wave” emitter in an inertial lab frame is acceleratedfrom rest up to velocity v along the x axis. It is assumed that there is no physicalinfluence of the detector for the direction of the radiation on the measurement. Theaberration increment is connected with the physical parameters by the relation: θ a = v / c . The explanation of the e ff ect of aberration is based on the use of a Galilean(left) and Lorentz (right) boost to describe how the direction of a light beam dependson the velocity of light source relative to the lab frame. If make a Lorentz boost, weautomatically introduce a time transformation t (cid:48) = t − xv / c and the e ff ect of thistransformation is just a rotation of the radiation phase front in the lab frame. One should not to be surprised to find that electrodynamics problem ofmoving emitter has intrinsic anisotropy. If fact, anisontropy results directlyfrom the time-dependence of the transverse position of the moving emitterwith finite aperture. What must be recognized is that in the time-dependentemitter problem, the results will depend on the direction of the velocityvector.The present approach to moving emitter problem uses the Fourier transformmethods. When we are dealing with linear systems it is useful to decomposea complicated input into a number of more simple inputs, to calculate theresponse of the system to each of these elementary functions, and to super-impose the individual responses to find the total response. Fourier analysisprovides the basic means for performing such a decomposition (1) .Consider the inverse transform relationship g ( x ) = (cid:82) ∞−∞ G ( K ) exp( iKx ) dK ex-pressing the profile function in terms of its wavenumber spectrum. We mayregard this expression as a decomposition of the function g ( x ) into a linearcombination (in our case into an integral) of elementary functions, eachwith specific form exp( iKx ). From this it is clear that the number G ( K ) is38imply a weighting factor that must be applied to elementary function ofwavenumber K to synthesize the desired g ( x ).The emitter with finite aperture is a kind of active medium which breaks upthe radiated beam into a number of di ff racted beams of plane waves. Eachof these beams corresponds to one of the Fourier components into whichan active medium can be resolved. Let us assume that the dipole densityof the elementary source varies according to the law ρ dip = g ( K ⊥ ) cos ( K ⊥ x ).We conclude, then, that the active medium of the elementary source issinusoidally space-modulated.Let us demonstrate that the new terms that have to be put into the fieldequations due to use of Galilean transformation lead to the prediction ofthe Doppler e ff ect. Consider as a possible solution a radiated plane waveexp( i (cid:126) k · (cid:126) r − i ω t ). With a plane wave exp( i (cid:126) k · (cid:126) r − i ω t ) with the wavenumber vector (cid:126) k and the frequency ω equation Eq.(6) becomes: (1 − v / c ) k x + vk x ω/ c + k z − ω / c =
0. From initial (time dependent) conditions we will find it necessaryto use (cid:126) k as independent variable and we will consider ω as a function of k x : ω = ω i + ∆ ω ( k x ), where ω i is the frequency of the emitter radiation beforethe acceleration. The wavenumber vector of the radiated plane wave isfixed by initial conditions. In fact, k z = (cid:113) ω i / c − k x , k x = K ⊥ , where K ⊥ isthe wavenumber of sinusoidally space-modulated dipole density. From thisdispersion equation, we find the requirement that the wavenumber K ⊥ andthe frequency change ∆ ω are related by ∆ ω = K ⊥ v .There is a di ff erent physical viewpoint on the Doppler e ff ect of the lightbeam radiated from a moving emitter that is equivalent to the presentedabove. The phase of the wave at the world point ( (cid:126) r , t ) cannot depend of onthe choice of a coordinate system. Therefore, the phase (cid:126) k · (cid:126) r − ω t must beinvariant of the Galilean transformation. Consequently, (cid:126) k (cid:48) · (cid:126) r (cid:48) − ω (cid:48) t (cid:48) = (cid:126) k · (cid:126) r − ω t .Substituting the Galilean transformation formulae x (cid:48) = x − vt , t (cid:48) = t intothe phase equality formula we obtain ∆ ω = K ⊥ v . This frequency changecoincides with the result derived directly from the dispersion equation, asmust be.Let us first remind the reader of the fact that the usual velocity of waves isdefined as given the phase di ff erence between the oscillations observed attwo di ff erent points in a free plane wave. It is primary used for computinginterference fringes that makes phase di ff erences visible. In a plane wavewe observe the phase velocity ω/ k . Another (group, or energy propagation)velocity can be defined, if we consider the propagation of a peculiarity, thatis change in amplitude impressed on a train of waves. A simple combinationof groups obtains when two waves ω = ω + ∆ ω, k = k + ∆ k and ω = ω − ∆ ω, k = k − ∆ k are superimposed. This represents a carrier with frequency39 and a modulation with frequency ∆ ω . The wave may be described as asuccession of moving beats (or groups). The carrier’s velocity is ω/ k , whilethe group velocity is given by v g = ∆ ω/ ∆ k → d ω/ dk .Many textbooks on electromagnetic theory discuss the aberration of lightphenomena in the context of plane wave. However, in dealing with planewave one will have an incorrect model of the aberration of light. When aninfinite sinusoidal wave travels, there is a uniform average energy densitythroughout the space. Does this energy remain where it is, or does it prop-agate through the space? It is impossible to know this. All experimentalmethods for measuring the aberration of light operate with light signals,and hence do not measure the phase velocity but the signal velocity andthis velocity coincides with group velocity.In our example, the plane waves with di ff erent wavenumber vectors prop-agate out from the moving emitter with the di ff erent frequencies. Thenequation ∆ ω/ ∆ k x = v holds for each scattered waves independently on thesign and the magnitude of the radiated angle. In fact, the ∆ ω in our case ofinterest is the Doppler shift ∆ ω = (cid:126) K ⊥ · (cid:126) v and the ∆ k x is simply the transversecomponent of the radiated wavenumber vector ∆ k x = K ⊥ . The last equationsstate that radiated light beam with finite transverse size moves along the x direction with group velocity d ω/ dk x = v . It should be note, however, that there is another satisfactory way of explain-ing the e ff ect of aberration of light from the moving source. The explanationconsists in using a Lorentz boost to describe the uniform translation motionof the light source in the lab frame. On the one hand, the Maxwell’s equa-tions remain invariant with respect to Lorentz transformations. On the otherhand, if make a Lorentz boost, we automatically introduce a time transfor-mation t (cid:48) = t − xv / c and the e ff ect of this transformation is just a rotation ofthe radiation phase front in the lab frame. This is because the e ff ect of thistime transformation is just a dislocation in the timing of processes, whichhas the e ff ect of rotating the plane of simultaneity on the angle v / c in thefirst order approximation.According to Maxwell’s electrodynamics, coherent radiation is always emit-ted in the direction normal to the radiation phase front. This is becauseMaxwell’s equations have no intrinsic anisotropy. In other words, when auniform translational motion of the source is treated according to Lorentztransformations, the aberration of light e ff ect is described in the languageof relativistic kinematics. According to the relativistic kinematics, the extraphase chirp d φ/ dx = k x = v ω i / c is introduced and the array of identical40lementary sources of the moving emitter now have di ff erent phases. Asa consequence of this, the plane wave wavefront rotates after the Lorentztransformation. Then, the radiated light beam is propagated at the angle v / c , yielding the phenomenon of the aberration of light (Fig. 1 right).We now ask about the group velocity of the radiated beam. With a planewave exp( i (cid:126) k · (cid:126) r − i ω t ) dispersion equation in the case of Maxwell’s electro-dynamics is reduced to k z + k x − ω / c =
0. From the initial conditions andthe Lorentz transformation we find that ω = γ ( ω i + vK ⊥ ), k z = (cid:113) ω i / c − K ⊥ , k x = γ ( v ω i / c + K ⊥ ), where K ⊥ is the wavenumber of sinusoidally space-modulated dipole density, ω i is the frequency of the emitter radiation beforethe acceleration. Substituting these expressions in dispersion equation wefind that the latter is satisfied, as must be.As one of the consequences of the Doppler e ff ect in the Lorentz coordinati-zation, we find an angular frequency dispersion of the light waves radiatedfrom the moving emitter with finite aperture. The Doppler shift, ∆ ω , of ra-diated light wave (in the first order approximation) is given by ∆ ω = (cid:126) K ⊥ · (cid:126) v ,where K ⊥ is the transverse component of the radiated wavenumber vector.The last equation state that radiated light beam with finite transverse sizemoves along the x direction with group velocity d ω/ dk x = v .It is interesting to discuss what it means that there are two di ff erent (covari-ant and non-covariant) approaches that produce the same group velocity.The point is that both approaches describe correctly the same physical re-ality and since the group velocity has obviously an objective meaning (i.e.convention-invariant), both approaches yield the same physical results.The di ff erence between the absolute time coordinatization and the Lorentzcoordinatization is very interesting. In the Chapter 3 we already discussedhow one can transform the absolute time coordinatization to Lorentz co-ordinatization. We can interpret manipulations with rule-clock structure inthe lab frame simply as a change of the time variable according to the trans-formation t → t + xv x / c . The overall combination of Galileo transformationand variable changes actually yields the Lorentz transformation in the caseof absolute time coordinatization in the lab frame. This variable change hasno intrinsic meaning. One can see the connection between the time shiftand the issue of clock synchrony. The convention-independent results ofcalculations are precisely the same in the new variables. As a consequence,we should not care to transform the results of the electrodynamics problemsolution into the original (3 +
1) variables.An idea of studying the relativistic electrodynamics using technique involv-ing a change of variables is useful from a pedagogical point of view. It is41 ig. 2. Transversely moving mirror with small aperture at normal incidence. Ac-cording to textbooks, there is no deviation of the energy transport for the reflectedlight beam. A monochromatic plane wave of light is falling normally on the smallmoving aperture mirror, and generates a reflected oblique beam. worth remarking that the absent of a dynamical explanation for wavefrontrotation in the Lorentz coordinatization has disturbed some physicists. Itshould be clear from the discussion in the Chapter 3 that a good way tothink of the wavefront rotation is to regard it as a result of transformationto a new time variable in the framework of the Galilean (”single frame”)electrodynamics.
It is generally believed that for a mirror moving tangentially to its surfacethe law of reflection which holds for the stationary mirror is preserved,as shown in Fig.2. In other words, the velocity of the energy transport isequal to the phase velocity. This statement, presented in most textbooks, isincorrect.First, we examine the reasoning presented in textbooks (2) . The reflectionfrom the mirror is analyzed in two Lorentz reference frames. The fixed (lab)frame K is at rest with respect to the plane-wave emitter. The moving frame K (cid:48) has velocity v . In this frame, the mirror is at rest. In both frames, we use aCartesian coordinate system in which x − y plane is tangent to the reflectionsurface. The x direction coincides with the direction of v . For simplicity,we consider the case in which light is incident from the z direction in thelab frame. Incident light is described by its four-dimensional wave vector,whose time like component is the angular frequency ω and whose space likecomponents define the direction of propagation. In the lab frame, ( t , x , y , z ),this vector has components k = ( ω, , , − ω/ c ), where the negative signindicates propagation towards the mirror (Fig. 3a). Our task is to determinethe wave vector for the reflected beam.42 ig. 3. The e ff ect of aberration of light is described in the language of relativistickinematics, in terms of the wavenumber four vector. Geometry of the reflectionas seen from (a) lab frame, (b) inertial frame moving with the same velocity asthe mirror. According to textbooks, there is no aberration for light reflected from atangentially moving mirror. The argument that there is no aberration for light reflected from a trans-versely moving mirror runs something like this. It is easiest to consider thereflection in the moving frame (Fig. 3b). In this frame, the surface is at rest, sothe usual laws of optical reflection apply. We will describe the e ff ect of aber-ration of light by working only up to the first order v / c . An observer mov-ing with the mirror surface sees the wave vector k (cid:48) = ( ω, − v ω/ c , , − ω/ c ).The e ff ect of reflection is to reverse the sign of the z (cid:48) component of thewave vector, k (cid:48) = ( ω, − v ω/ c , , ω/ c ). We now obtain the reflected wavevector in the lab frame by applying the inverse Lorentz transformation: k = ( ω, , , ω/ c ).This vector represents a light beam traveling away fromthe mirror, having the same frequency as the incoming beam. This showsthat the beam is reflected according to the usual geometrical optics laws,and the beam su ff ers no aberration.The concept of a plane wave and an infinite plane mirror is used in manytextbooks on electromagnetic theory. However, in dealing with a plane waveand infinite mirror one will have an incorrect model of aberration from atransversely moving mirror. It is impossible to know the energy transportdirection when one has deal with a plane wave. What authors of textbooksgenerally overlooked is the fact that the energy transport problem is well-defined only if the source and mirror apertures have already been specified.The solution of problems in theoretical physics begins with application of thequalitative methods. By ”qualitative methods” we mean the investigationof limiting cases where one can exploit the smallness of some parameter.In this book we demonstrate a method that allow us to find the definingcharacteristics of aberration measurements in the (practically important)43imit of very small aperture mirror. By very small aperture, we mean that thetransverse size of the moving mirror is very small relative to the transversesize of the ”plane wave” emitter. +
1) Space and Time
The aberration of light problem can be treated within the same ”singleinertial frame” description without reference to Lorentz transformations.We shall first discuss the situation where there is a finite aperture mirrormoving tangentially to its surface. For simplicity, we shall assume that thetransverse size of the moving mirror is very small relative to the transversesize of the ”plane-wave” emitter, as sketched in Fig. 2. It is worth notingthat we consider an aberration angle that is relatively large compared tothe divergence of the reflected radiation. In other words, (cid:111) / D e (cid:28) (cid:111) / D m (cid:28) v / c , where D e and D m are the transverse size of the emitter and mirror,respectively.When the illumination of the object originates from a monochromatic spa-tially coherent source there exists a method for calculating the reflectedintensity that has the special appeal of conceptual simplicity. It uses Fouriertransforms of spatial filtering theory that is the Abbe di ff raction theory.The essence of Abbe’s approach, in our case of interest, is that one regardsthe mirror as a kind of a di ff raction grating which breaks up the incidentbeam of the plane wave into a number of di ff racted beams constituted byplane waves. Each of these beams corresponds to one of the Fourier com-ponents into which the reflected power of the mirror can be resolved. Thefinite-aperture mirror is a non-periodic object. It gives an infinite number ofdi ff racted beams forming a continuum.A simple example of a di ff raction grating is shown in Fig.4. Let us as-sume that the reflectance of the grating varies according to the law R = g ( K ⊥ ) cos ( (cid:126) K ⊥ · (cid:126) r ). The reflectance is sinusoidally space-modulated. It shouldbe noted that the permanent reflectance distribution grating discussed hereis only our mathematical model and we do not need to discuss how it canbe created.The (cid:126) k vectors shown in Fig.4 represent the propagation vector of the inci-dent plane wave (cid:126) k i , which is assumed to be directed perpendicularly to thesurface. The vectors (cid:126) k s ( + ) and (cid:126) k s ( − ) are added to indicate the scattered light.The Bragg condition (cid:126) k s = (cid:126) k i ± (cid:126) K ⊥ shows how the direction of the incidentand scattered wave are related. The first-order maxima dominate due to thefact that light is being scattered from a sinusoidal grating, rather than a setof discrete planes (grooves). 44 ig. 4. The Bragg di ff raction grating at normal incidence. The reflectance is sinu-soidally space-modulated. We assume that the (cid:126) K ⊥ vector is directed parallel to the side of the di ff ractiongrating with the incident wave impinging on the grating perpendicularly,as shown in Fig.4. The length of the vectors (cid:126) k s and (cid:126) k i must, of course,be the same, but the vector diagram does not quite match up. The Braggconditions are then not satisfied precisely. For small angles, (cid:126) k s = (cid:126) k i ± (cid:126) K ⊥ stillholds approximately, so that we obtain the scattering angle θ = K ⊥ / k i .When the scatterer wave is a progressive wave rather than a fixed modula-tion, the frequency of the scattered wave is di ff erent from that of the incidentwave. This fact is interpreted as a Doppler e ff ect, since the reflection is froma moving, rather than a stationary, set of waves. In the case of a transverselymoving grating, light in the di ff raction maxima undergoes a Doppler shiftresulting from the fact that it has been reflected from moving waves withwavenumber vectors (cid:126) K ⊥ .Let us demonstrate that the new terms that have to put into the field equa-tions due to use of absolute time coordinatization lead to the prediction ofthe Doppler e ff ect. We recall that with a plane wave exp( i (cid:126) k · (cid:126) r − i ω t ) with thewavenumber vector (cid:126) k and the frequency ω dispersion equation in the abso-lute time coordinatization becomes: (1 − v / c ) k x + vk x ω/ c + k z − ω / c = k z = (cid:113) ω i / c − k x , k x = K ⊥ , where K ⊥ is the wavenumber ofsinusoidally space-modulated reflectance. From this dispersion equation,we find the requirement that the wavenumber K ⊥ and the frequency change ∆ ω are related by ∆ ω = K ⊥ v . 45he following important detail of such ”single inertial frame” descriptioncan hardly be emphasize enough. If the source of light is at rest and themirror is in motion, it is obvious that the electrodynamics equations mustbe identical for all electromagnetic waves. In other words, the dispersionequation in the absolute time coordinatization should be applied and keptin a consistent way for both incoming and scattered waves (Fig. 4). In ourprevious discussion of absolute time coordinatization we learned that theemitter at rest must be in the same time described by Maxwell’s electrody-namics. A dispersion equation in the case of Maxwell’s electrodynamics isreduced to k i − ω i =
0. From the initial conditions we find that (cid:126) k i = (cid:126) e z k z , ω i = ck z . The contradiction, however disappears if we perform geometri-cal analysis of light reflection. The peculiarity of the discussed geometry isthat even after the Galilean transformation along the x -axis the dispersionequation in the absolute time coordinatization will have the same (diagonal)form k z − ω = ff ect, we find an angular fre-quency dispersion of the light waves reflected o ff the moving mirror withfinite aperture. If (cid:126) n = (cid:126) k / | (cid:126) k | denotes a unit vector in the direction of the wavenormal, and (cid:126) v is the mirror velocity vector relative to the lab frame, we getthe equation ω s = ω i (1 + (cid:126) n · (cid:126) v / c ) = ω i + ( ω i v / c ) cos θ . The Doppler e ff ect isresponsible for angular frequency dispersion to the first order of v / c evenwhen (cid:126) n · (cid:126) v = θ = ω s / d θ = − ( ω i v / c ) sin θ = − ω i v / c at θ = π/
2. We can rewrite this equation in a di ff erent way. The di ff erentialof the scattered angle is given by d θ = − dk x / k i . With the help of this relationand account to that k i = ω i / c we have d ω s / dk x = v .One of the most important conclusions of the foregoing discussion is aremarkable prediction on the theory of the aberration of light, concerningthe deviation of the energy transport for light reflected from a mirror movingtransversely. Namely, when a plane wave of light is falling normally on themirror, there is a deviation of the energy transport for reflected light beam(see Fig. 5). This phenomenon can be regarded as a simple consequence ofthe Doppler e ff ect.The argument that in the process of reflection from a transversely mov-ing mirror the direction of propagation is not given by the normal to thewavefront is considered erroneous in literature (4) . This fact is ascribed to alack of understanding of the di ff erence between convention-dependent andconvention-invariant parts of the theory. The direction of the energy trans-port has an exact objective meaning i.e. is convention-invariant. However,the phase front orientation (i.e. the plane of simultaneity in the judgementof an observer) has no exact objective meaning since, due to the finitenessof the speed of light, we cannot specify any experimental method by whichthis orientation could be ascertained.46 ig. 5. Transversely moving mirror with a small aperture at normal incidence. Whena plane wave of light is falling normally on the mirror, there is a deviation of theenergy transport for the reflected light beam. This e ff ect is a consequence of the factthat the Doppler e ff ect is responsible for angular frequency dispersion of the lightwaves reflected from the mirror. As a result, the velocity of the energy transport isnot equal to the phase velocity. Since the phase front orientation does not exist as physical reality withinthe angular range v / c , a question arises: why do we need to account forthe exact phase front orientation in our electrodynamics calculations? Theanswer is that when the evolution of the radiation beam is treated accordingto the single inertial frame of reference, one will experience that phasefront orientation remains unvaried: this has no objective meaning but isused in the analysis of the electrodynamics problem. A comparison with agauge transformation in Maxwell’s electrodynamics might help here. Evenif the phenomena are quite di ff erent, the common mathematical formulationpermits us to draw this analogue. Physicists who try to understand the situation related to the use of thecovariant approach in the aberration of light phenomena, are often troubledby the fact that in the situation where there is a mirror moving normallyto its surface the reasoning presented in textbooks is correct. The reflectionfrom the mirror is analyzed in two Lorentz reference frames. It is interestingto note that, for this case, relativistic kinematics correctly predicts the lightfrequency variation on reflection from a moving mirror. At this point areasonable question arises: why the same method gives incorrect result inthe case of reflection of light from a mirror moving tangentially to its surface?There is a common mistake made in general physics connected with the47nergy transport direction in the case of an (infinite) plane wave. Whenan infinite plane wave travels, there is a uniform average energy densitythroughout the space. It is impossible to know the energy transport direc-tion when one has deal with a plane wave. All experimental methods formeasuring the aberration increment operate with light signals, and hencedo not measure the phase velocity (i.e. frequency and wavenumber vector)but the group velocity. It can be defined only if we consider the propagationof a peculiarity, that is change in amplitude impressed on a train of waves.The authors of textbooks did not make a computational mistake in theirtreatment of the aberration of light phenomena in an inertial frame of ref-erence, but rather a conceptual one. We must say that there is no objectionsto the moving frame transformation. It is easiest to consider the reflectionin the inertial frame moving with the same velocity as the mirror (Fig. 3b).An observer moving with the mirror surface sees the incoming wave vector k (cid:48) = ( ω, − v ω/ c , , − ω/ c ). Then, where does the mistake comes from? Thepresented above commonly accepted covariant treatment of reflection froma mirror moving transversely includes one delicate point. We state that thetypical textbook statement ”The e ff ect of reflection is to reverse the sign ofthe z (cid:48) component of the wave vector, k (cid:48) = ( ω, − v ω/ c , , ω/ c )” is incorrect. Infact, as we have already discussed in this section, the infinite plane mirrorcannot be used when we deal with aberration of light phenomena.The finite aperture mirror, that is treated as the source of reflected radiation,is usually modeled with the help of a physical optics approach. This iswell-known high-frequency approximation technique, often used in theanalysis of the electromagnetic waves scattered from large (relative to thewavelength) objects. The present approach to mirror reflection problemuses the Fourier transform methods. The beam is reflected according tousual physical optics laws and has the angular spectrum width ∆ θ (cid:39) (cid:111) / D m ,where D m is the characteristic mirror size. A reflected light beam in thecomoving frame traveling away from the mirror has the same frequency asthe incoming plane wave. So we must conclude that the Doppler e ff ect isabsent and the velocity of the energy transport in the x direction is equal tozero.From the initial conditions and the Lorentz transformation we find that in thelab frame ω = γ ( ω i + vK ⊥ ), k z = (cid:113) ω i / c − K ⊥ , k x = γ ( v ω i / c + K ⊥ ), where K ⊥ isthe transverse wavenumber of the plane wave in the Fourier decompositionof the reflected beam. As one of the consequences of the Doppler e ff ect inthe Lorentz coordinatization, we find an angular frequency dispersion ofthe light waves reflected from the moving mirror with finite aperture. TheDoppler shift, ∆ ω , of reflected light wave (in the first order approximation)is given by ∆ ω = (cid:126) K ⊥ · (cid:126) v . The last equation states that reflected light beam48ith finite transverse size moves along the x direction with group velocity d ω/ dk x = v . That is the reflection appears as shown in Fig. 5.Let us examine in a little more detail how group velocity comes about fromcovariant and non-covariant point of view. After the Galilean transforma-tion x (cid:48) = x − vt , t (cid:48) = t we would obtain the same group velocity as after theLorentz transformation x (cid:48) = x − vt , t (cid:48) = t − vx / c . The two approaches givethe same result for real observable e ff ect. First we want to rise the followinginteresting and important point. An acceleration of the mirror with respectto the inertial frame is absolute (i.e. is physical reality) and described in bothapproaches by the same coordinate transformation x (cid:48) = x − vt . This trans-formation (boost) in the x direction leads to angular frequency dispersionof the light waves reflected from the moving mirror with finite aperture, in-dependently of the coordinatization. On the other hand, if make a Lorentztransformation, we introduce a time transformation t (cid:48) = t − vx / c and thee ff ect of this transformation is just a rotation of the radiation wavefront.This rotation is not a real observable e ff ect. This is a good point to make a general remark about the emitter-mirrorproblem. The peculiarity of this problem with the viewpoint of relativistickinematics is that here the emitter is at rest in the lab inertial frame andthe mirror is moving with the constant speed with respect to the lab frameand interacts with the radiated light beam. How can we solve a probleminvolving the emitter-mirror relative velocity? Each physical phenomenonoccurs in space and time. A concrete method for representing space andtime is a frame of reference (coordinate-time grid), which requires carefuldescription.The Maxwell’s equations can be applied in the lab inertial frame only in thecase when Lorentz coordinates are assigned. It is incorrectly believed thatthe emitter-mirror electrodynamics problem can be treated within the same”single Lorentz frame” description. In other words, it is incorrectly believedthat the common Lorentz time coordinate axis for emitter and mirror canbe assigned. This is misconception. We can prepare, for mirror and emitter,a common set of synchronized clocks in the lab frame only in the case ofabsolute time coordinatization i.e in the case when simultaneity is absolute.As we already mentioned, the utilization of the electrodynamics in theabsolute time coordinatization becomes indispensable when we consideroptical phenomena associated with a relative motion of two or more bodies.Suppose that we assign the Lorentz time coordinate for the description ofthe emitter radiation. But this will be the absolute time coordinatization for49he boosted mirror and this boost will be described in such coordinatizationby the Galilean transformation. Suppose that we re-synchronize clocks inthe lab frame in order to assign the Lorentz time coordinate for the boostedmirror. In this coordinatization, we describe the reflection of light using theusual Maxwell’s equations. But this new time coordinate in the lab frameis interpreted by saying that Maxwell’s equations are not applicable to theemitter radiation description.So far we have considered the covariant way to solve the emitter-mirrorproblem. It is interesting to note that, the use of relativistic kinematics forthe calculation of the reflection from a transversely moving mirror doesnot necessarily leads to mistake. We would like to discuss the followingquestion: since the common Lorentz time coordinate axis for emitter andmirror cannot be assigned, how the relativistic kinematic method leads to thecorrect result if applied to computation of the reflection from a transverselymoving mirror? Above we demonstrated that the both (covariant and non-covariant) approaches give the same result for group velocity of the reflectedlight beam. The reason is that an acceleration of the mirror with respect tothe inertial frame is physical reality and described in both approaches bythe same boost x (cid:48) = x − vt . This transformation leads to the Doppler e ff ectof the light waves reflected from the moving mirror with finite apertureindependently of the coordinatization.Let us now discuss more about consequences of the Lorentz transforma-tions. If we rely on the relativistic kinematic method, the reflection resultsin a di ff erence between the direction reflected beam motion and the normalto the radiation wavefront. This is already a conflict result, because we nowconclude that, according to covariant approach, the direction of propagationafter the reflection is not perpendicular to the radiation wavefront. This iswhat we would get for the case when our analysis is based on the relativistickinematics and is obviously absurd from the viewpoint of Maxwell’s elec-trodynamics. In fact, we demonstrated that our assumption about existenceof common Lorentz time axis for emitter and mirror leads to logical incon-sistency. We conclude that this assumption is incorrect. Only the solutionof emitter-mirror problem in the absolute time coordinatization gives theconsistent description of the reflection from a mirror moving transversely. In this chapter our discussion is limited to the region of problem parameters,in which we forget about the emitter edges. Although this aberration of lighttheory is just an approximation, it is a very great importance practically. Weshall also discuss the situation where the transverse size of the emitter isvery small relative to the transverse size of the moving mirror. It should be50 ig. 6. Transversely moving mirror with a large aperture at normal incident. Whena beam of light is falling normally on the mirror, there is no deviation of the energytransport for the reflected light beam. The velocity of the energy transport is equalto the phase velocity. note that this situation is not realized in the stellar aberration measurements.We shall work out this case in order to understand all the physical principlesvery clearly. It is easy to show that the deviation of the energy transport isabsent in this case, Fig. 6. The direct approach to moving mirror problemuses the Fourier transform methods. We now consider another method ofcalculating the aberration of light e ff ect - we want to illustrate the greatvariety of possibilities. The way of thinking that made the law about thebehavior of light reflected from a large aperture mirror evident is based onBabinet’s principle. It is well known that, when light comes through a holeof a given shape, made in an opaque screen, the distribution of intensityafter the hole (i.e. the di ff raction pattern) is the same as in the case when thehole is replaced by sources (dipoles) uniformly distributed over the hole. Inother words, the di ff racted plane wave from a hole, or from a source withthe same shape of the hole are the same.This is a particular case of Babinet’s principle, which states that the sum ofdi ff raction fields behind two complementary opaque screens is the incidentwave. We know from this principle, that the solution we have found usingAbbe’s approach also corresponds to that for large aperture hole in a movingopaque screen. We see clearly that there is no electromagnetic interactionof a light beam with screen. Indeed light beam is not scattered by the holeedges. Does this discussion about large aperture hole have any meaning?To see whether it does, we should remember about the Babinet’s principle.Here we only wish to show how easy the law of reflection from a largeaperture mirror can be found with the help of the Babinet’s principle.51 .1.9 Analysis of Transmission through a Hole in a Opaque Screen Above we demonstrated that when one has a small aperture mirror movingtransversely and the plane wave of light is falling normally on the mirror,there is the aberration (deviation of the energy transport) for light reflectedfrom the mirror. The problem to be considered in this section is of morepractical importance. We now consider the case of a screen, in the lab frame,moving with velocity v along its surface. It is generally believed that thereis no deviation of the energy transport for light transmitted through a holein the moving opaque screen, Fig. 7.However, there is a common mistake made in relativistic optics, connectedwith aberration e ff ects from a transversely moving screen containing a hole.We describe the system using, again, a Fourier transform method similar tothat considered above. The screen containing a hole is a kind of di ff ractiongrating which breaks up the incident beam of the plane wave into a numberof di ff racted beams of plane waves. Each of these beams corresponds toone of the Fourier components into which a transmitted light beam can beresolved.The gratings discussed so far modulate the amplitude of the incident planewave by a periodic reflection function. However, we can immediately ex-tend the range of validity of our analysis to gratings that modulate theamplitude of the incident light by a periodic transmission function. Letus assume that the transmittance of the grating varies according to thelaw T = g ( K ⊥ ) cos ( (cid:126) K ⊥ · (cid:126) r ), Fig.8. The transmittance is sinusoidally space-modulated. All the equations that we derived so far hold immediately forthe forward scattered beams.According to our approach, there is a remarkable prediction of the theoryof aberration of light concerning the deviation of the energy transport forlight transmitted through a hole in a moving screen. Namely, when one hasa transversely moving screen with a hole in it and a plane wave of light isfalling normally on the screen, there is a deviation of the energy transportfor light transmitted through the hole (see Fig. 9). Let us suppose that transmitted light pulse propagates in the x − z plane.Now we are interested in the space-time intensity distribution in this plane.Spatiotemporal coupling arises naturally in transmitted radiation behindthe screen, because the transmission process involves the introduction ofan angular-frequency dispersion of the transmitted radiation. The emittedlight beam is represented with su ffi cient accuracy as the product of factors52 ig. 7. Aberration of light in an inertial frame of reference. Transversely movingscreen which has a hole in it. According to textbooks, a monochromatic plane waveof light is falling normally on the screen and generates a transmitted oblique beam.There is no deviation of the energy transport for the transmitted oblique light beam.The velocity of the energy transport is equal to the phase velocity.Fig. 8. The Bragg di ff raction grating at the normal incident. The transmittance issinusoidally space-modulated. separately depending on space and time. However, when the manipulationof the emitted light requires the transmission through a hole in a movingopaque screen, such assumption fails.We start by writing the field of an emitted pulse as E ( x , t ) = b i ( x ) exp[ i ω i ( z / c − t )]. The initial amplitude distribution b i ( x ) in front of the moving screen is theoptical replica of the emitter aperture. The electric field of the transmittedpulse expressed in the reciprocal domain as ¯ E ( ∆ k x , ∆ ω ) = ¯ E ( K ⊥ , K ⊥ d ω/ dk x ),which is the Fourier component of the electric field of a beam with angularfrequency dispersion and d ω/ dk x = v . The inverse Fourier transform to the53 ig. 9. Aberration of light in an inertial frame of reference. Transversely movingscreen which has a hole in it. A monochromatic plane wave of light is fallingnormally on the screen and generates a transmitted light beam. The Doppler e ff ectis responsible for the angular frequency dispersion of the light waves transmittedthrough the hole. As a result, the velocity of the energy transport is not equal to thephase velocity. space-time domain can be expressed as E = b ( x − vt ) exp[ i ω i ( z / c − t )]. Thisis the field immediately behind the moving screen. Consider a screen atrest position on the distance l behind the moving aperture. For simplicitywe assume here that Fresnel number is large, N F = D h / ( (cid:111) l ) (cid:29)
1, and wecan neglect the di ff raction e ff ects. Here D h is the characteristic aperturesize. We conclude that the light spot on the observer screen (which is theoptical replica of the moving aperture) moves with the same velocity v asthe moving screen (5) . Let us move on to consider the predictions of the existing aberration oflight theory in the case of a transversely moving screen which has a holein it (Fig. 7). According to conventional theory, the spot of the transmittedlight beam on the observer screen also moves with the same velocity as themoving screen. It is important at this point to emphasize that the electro-dynamics dictates that this would also lead to a consequent introduction ofan angular-frequency dispersion. However, such angular spectrum changewould mean a correction to a deviation of the energy transport direction ofthe transmitted light beam so that there is a glaring conflict with the predic-tion of the energy transport direction according to conventional aberrationof light theory. The absence of the group velocity along the moving directionis the prediction of conventional aberration of light theory and is obviously54 ig. 10. Transverse moving screen which has a hole in it. Light corpuscles are fallingnormally on the screen and, according to literature, generates an oblique light beam. absurd from the viewpoint of electrodynamics.This incorrect statement is a straightforward consequence of the generallyaccepted way of looking at the aberration of light phenomena of most au-thors of the texbooks. Today one is told that the phenomenon of aberrationof light could be interpreted, using corpuscular model of light. Light corpus-cles are falling normally on the moving screen and, according to literature,generate oblique light beam as Fig. 10 shows. This wrong argument persiststo this day. If the optical system is spatially coherently illuminated, thena satisfactory treatment of the aberration of light should be based on theelectromagnetic wave theory.Some experts believe that the applicability of corpuscular model in the the-ory of light should be reinterpreted as the applicability of ray optics. Let ussee what happens, according to the ray optics, in our case of interest. Lightrays are falling normally on the moving screen and generate oblique raybeam as Fig. 10 shows (6) . In this situation, we just treat light rays like littleparticles and the e ff ect is entirely familiar. We would like now to discussthe region of applicability of ray optics. The situation relating to use the rayoptics in the theory of aberration of light is complicated. One could naivelyexpect that the region of applicability of ray optics, following from the op-tics textbooks reasoning, should be identified with any spatially incoherentradiation. However, incorrect results are obtained by doing so. In particu-lar, a spatially completely incoherent source (e.g. an incandescent lamp ora star) is actually a system of elementary (statistically independent) pointsources with di ff erent o ff sets. Radiation field generated by a completelyincoherent source can be seen as a linear superposition of fields of individ-ual elementary point sources. An elementary source produces in front ofa hole aperture e ff ectively a plane wave. In other words, the transmissionprocess involves the introduction of an angular frequency dispersion of the55ransmitted radiation. It should be remarked that any linear superpositionof radiation fields from elementary point sources conserves single pointsource characteristics like a deviation of the energy transport direction. Thisargument gives reason why ray optics is not applicable in the theory ofaberration of light from (spatially) completely incoherent sources.We will illustrate the applicability of ray optics in the theory of aberrationof light for a particular class of spatially incoherent light beams. In fact, theray beam shown in Fig. 10 can be realized as follows. Such beam may beproduced by many commonly used lasers with a random spread of phases.One of the very useful properties of laser sources is their ability to producefields that are highly directional.The intensity of such fields is concentratedin a very narrow solid angle. A method can be proposed for generating raybeam from primary sources by the use of array of randomly phased lasers.Such planar source generates rays which are falling normally on the movingscreen (within the laser Rayleigh range) and generate oblique transmittedray beam, Fig. 10. Intuitively, a transversely moving screen containing a holeacts like a switcher for lasers. Surely, a luminous spot moving at velocity v can be realized simpler, so to speak, ”manually”. We can arrange thelaser-like sources along the x axis and switch them on one after another(independently) from left to right with a given time lag. Naturally, we canget a luminous spot moving at any velocity (even at v > c ). From thisexample it is seen that in this process no information can be transmitted(along the x axis) since each source radiates independently. Aberration of light theory describes the deviation of the energy transportfor transmitted light beam. But how to measure this deviation? A movingwith group velovity v transmitted light beam changes its position alongthe x axis in time. The question arises whether it is possible to give anexperimental interpretation of the aberration e ff ects. We illustrate the prob-lem of how to represent the deviation of the energy transport in the caseof a time-dependent aberration of light problem with a simple example.Let us imaging the practical situation in which emitter radiated pulse. Weconsider a pulse of nearly monochromatic radiation having a duration andbandwidth equal to T p and ∆ ω , respectively. The present approach to lighttransmission problem uses the Fourier transform methods. The essenceof Fourier approach, in our case of interest, is that the incident radiationpulse expanded into the superposition of incoming beams constituted byplane monochromatic waves. Each of these beams corresponds to one of theFourier components into which the incoming light pulse can be resolved.It is useful to calculate the transmission to each of these elementary beams,and to superimpose the individual responses to find the total response. One56f the important conclusions of the Fourier analysis is follows. When a lightpulse is falling normally on the moving screen there is a deviation of theenergy transport for the transmitted light pulse. Consider a light positiondetector in the rest position. The detector is placed at the distance l fromthe screen. It worth noting that we consider the aberration shift is relativelylarge compared to the hole size D h . In other words, D h (cid:28) vl / c . Also note that,in order to resolve aberration shift, we must require that cT p (cid:28) l . In smalldi ff raction angle approximation (cid:111) / D h (cid:28) v / c we also have a second smallproblem parameter D h / l (cid:28) v / c . Let us discuss interdependence of these twosmall parameters. A combination of these two parameters N F = D h / ( (cid:111) l ) canbe refereed to as the Fresnel number. It is worth noting that, in our caseof interest, there is no restriction on the parameter N F . At first glance, onecan determine the aberration shift v / c to any desired degree of accuracy byincreasing distance l . However, the measuring device produces the uncer-tainty. In fact, the direction of light pulse propagation cannot be ascertainedmore accurately than up to the finite angle of the hole aperture (cid:111) / D h . Let us now consider the case when a ”plane-wave” emitter in the lab inertialframe is accelerated from rest up to velocity v along the x axis. An emitterwith finite aperture is a kind of active medium which breaks up the radiatedbeam into a number of di ff racted beams of plane waves. Each of these beamscorresponds to one of the Fourier components into which an active mediumcan be resolved. We already know from our discussion from very beginningof this chapter that there is a deviation of the energy transport for thecoherent light radiated by the transversely moving emitter, which is nothingelse but well-known (from textbooks) result: there is the aberration of lightfrom the transversely moving emitter in the inertial frame of reference (Fig.11).The specific of our case of interest with the viewpoint of kinematics isthat here the screen is at rest in the lab frame and the emitter is movingtangentially to its surface with constant speed with respect to the lab frame.For simplicity we shall assume that the transverse size of the moving ”plane-wave” emitter is very large relative to the transverse size of the hole in thescreen. Suppose that an observer, which is at rest with respect to the screenperforms the direction of the energy transport measurement.The way of thinking that made the law about the behaviour of transmittedlight evident is called ”Abbe’s approach”. We call attention to the fact thatif the transverse size of the incoming light beam D e is much large thanthe transverse size of the hole D h , the group velocity of the transmittedbeam is dramatically reduced. This suppression is not surprising, if one57 ig. 11. Aberration of light in an inertial frame of reference. The large aperture”plane-wave” emitter moving tangentially to its surface. As one of the consequencesof the Doppler e ff ect, we find group velocity of the light waves radiated o ff a largeaperture moving emitter. The screen is at rest and we have actually the problem ofsteady-state transmission. The transmitted light beam is going vertically because ithas lost its horizontal (group) velocity component.Fig. 12. Aberration of light in an inertial frame of reference. A point source producesin front of a hole aperture e ff ectively a plane wave. If the motion of the point sourceis parallel to the screen, transmitted beam is going vertically. The aberration of lightphenomenon is absent in this situation. v x ) g = ∆ ω/ ∆ k x . In fact, theDoppler shift of a light wave radiated from the moving emitter is given by ∆ ω = (cid:126) K e ⊥ · (cid:126) v , where K e ⊥ ∼ / D e is the characteristic emitter wave number.But the transverse component of the transmitted wavenumber vector ∆ k x in our case of interest can be written as ∆ k x = ( k x ) i + K h ⊥ , where ( k x ) i ∼ / D e is the transverse component of the incoming wavenumber vector, K h ⊥ is thecharacteristic hole wavenumber. In the large aperture emitter case we have ∆ ω/ ∆ k x ∼ ( D h / D e ) v (cid:28) v .At close look at the physics of this subject shows that in the inertial lab frame,where the screen is at rest, we have actually the problem of steady-statetransmission. The Doppler e ff ect is absent and the transmitted beam is goingvertically because it has lost its horizontal (group velocity) component. Thatis the transmission appears as shown in Fig.11. We only wish to emphasizehere the following point. When the light passes through the small aperturehole we have a light beam whose fields have been perturbed by di ff raction,and now not include information about emitter motion.One of the important conclusions of the discussion presented above is thatthe aberration of point source is absent in this situation, Fig. 12. What are theconsequences of this? There are a number of remarkable e ff ects which are aconsequence of the fact that the information about a point source motion isnot included into the light beam transmitted through the hole. In fact, thisis the key to the binary star paradox discussed in the next chapter. In this section we reexamine the issue of light transmission in non-inertialframes with particular reference to the aberration of light phenomena. Wederive the aberration for a pulse of light traveling in the accelerated systemsusing the Langevin metric in general relativity (7) . It comes out naturallyif one writes the equation of the time transfer, from the inertial frame tothe accelerated frame, in a generally covariant context. We only wish toemphasize here the following point. From a mathematical standpoint, thereis no di ff erence between calculations in the framework of the general theoryof relativity and of the special theory of relativity in the absence of space-time curvature.The aberration of light problem is solved with discovery of the essentialasymmetry between the non-inertial and the inertial observers. Actually, inresent years it seems to be almost normally accepted in scientific community59hat the ”theory of relativity” is just a name, not to be taken literally. Onecan conclude that not all is relative in relativity, because this theory alsocontains some features which are absolute.Has all our talk about asymmetry violated the relativity principle? At a firstglance it might seem so, since the relativity principle is often interpreted asimplying perfect symmetry among moving frames. It is generally incorrectlybelieved that key feature distinguishing special relativity from the classicaltheory is its distinctive dependence on the unqualified motional symmetryimplied by the relativity of motion: therefore seemingly not restricted torelative velocity but applicable to all form of relative motion, includingthe higher time derivatives of separation distance. In that case accelerationdoes not spoil the motional symmetry between the non-inertial referenceframe and inertial reference frame, and the asymmetry ”paradox” (actuallya disagreement with experimental facts) persists.The principle of relativity denied the possibility for an observer partakingin a uniform motion relative to an inertial frame of discovering by anymeasurement such a motion, of course, that one does not look outside.The arguments concerning the relativity of motion in our case of interestcannot be applied, since the inertial and non-inertial reference systems arenot equitable. A typical resolution of the asymmetry paradox identifiesacceleration as the agency of asymmetry.However, there remains an intriguing puzzle to solve: how can the ob-servers tell which observer took the acceleration? The surprising fact is, thedetermination of which observer took the acceleration can be made only byobservation of the ”fixed stars”. The acceleration is in principle defined interms of motion relative to the fixed stars, and they must be consulted inorder to determine whether an acceleration occurred. Thus when we statethat the earth-base observer undergoes an acceleration, and the sun-basedobserver does not, there is a hidden assumption concerning the distribu-tion of mass in the universe. The implicit ”absolute” acceleration meansacceleration relative to the fixed stars. We will consider the problem of aberration of light in the accelerated systemson the basis of the theory of special relativity. Let us demonstrate that forexplanation of the optical e ff ects in the rotating frame of reference one doesnot need neither modify the special theory of relativity, nor apply the generaltheory of relativity. It is only necessary to strictly follow the special theoryof relativity.Suppose that a system S n (and an observer with his measuring instruments)60n the stationary inertial lab frame S is accelerated from the rest with respectto the fixed stars up to velocity v along the x -axis. In accelerated systems,only the theory maintaining an absolute simultaneity is logically consistentwith the natural behavior of clocks. The method of synchronization consistsin keeping, without changes, the same set of uniformly synchronized clocksused in the case when the system S n was at rest. It is well known that duringthe motion with acceleration (with respect to the fixed stars) the procedureof Einstein’s clock synchronization cannot be performed and the interval inthe accelerated reference system S n will, by the moment when the system S n starts moving with constant velocity, have the non-diagonal form.Absolute simultaneity can be introduced in special relativity without af-fecting neither the logical structure, no the (convention-independent) pre-dictions of the theory. We begin with the metric as the true measure ofspace-time intervals for an non-inertial observer S n with coordinates ( t n , x n ).Here we neglect the two perpendicular space components that do not enterin our reasoning. We transform coordinates ( t , x ) that would be coordinatesof an inertial observer S moving with velocity − v with respect to the ob-server S n , using a Galilean transformation: we substitute x n = x − vt , whileleaving time unchanged t n = t into the Minkowski metric ds = c dt − dx to obtain ds = c (1 − v / c ) dt n − vdx n dt n − dx n . (8)Inspecting Eq. (8) we can find the components of the metric tensor g µν inthe coordinate system ( ct n , x n ) of S n . We obtain g = − v / c , g = − v / c , g = −
1. Note that the metric in Eq. (8) is not diagonal, since, g (cid:44)
0, andthis implies that time is not orthogonal to space.The velocity of light emitted by a source at rest in the coordinate system ( t , x )for S is c . In that case the Minkowski metric Eq.(1) associated with inertialframe S predict a symmetry in the one-way speed of light. In the coordinatesystem ( t n , x n ), however, the speed of light emitted by the accelerated source(i.e. source which is at rest with respect to the accelerated frame S n ) cannotbe equal c anymore because ( t n , x n ) is related to ( t , x ) via a Galilean transfor-mation. This is readily verified if one recalls that the velocity of light in thereference system S is equal to c . If ds is the infinitesimal displacement alongthe world line of a ray of light, then ds = c = ( dx / dt ) . Inthe accelerated reference system, since x n = x − vt and t = t n , this expressiontakes the form c = ( dx n / dt n + v ) , which can be seen by setting ds = ct n , x n ) the velocity of light parallel to the x-axis, is dx n / dt n = c − v in thepositive direction, and dx n / dt n = − c − v in the negative direction as statedabove. The reason why it is di ff erent from the electrodynamics constant c isdue to the fact that the clocks are synchronized following the absolute time61onvention, which is fixed because ( t n , x n ) is related to ( t , x ) via a Galileantransformation. We discovery of the essential asymmetry between the inertial and acceler-ated frames, namely, the Maxwell’s equations are not applicable from theviewpoint of an observer at rest with respect to accelerated frame S n . In fact,the metric Eq.(8) associated with accelerated reference frame S n predicts anasymmetry in the one-way speed of light in the relative velocity direction.Accelerations (with respect to the fixed stars) have an e ff ect on the propa-gation of light. On accelerated system S n , the velocity of light emitted bya source at rest must be added to (or subtracted from) the speed due toacceleration and the velocity of light is di ff erent in opposite directions. Onthe contrary, the Maxwell’s equations continue to hold from the viewpointof an observer at rest with respect to inertial frame S . On inertial system S ,the velocity of light emitter by a source at rest is c . This asymmetry is of thesame nature as that of the well-known clock paradox (8) .Suppose that an observer in the accelerated frame S n performs an aberrationmeasurement. How shell we describe the aberration of light from the ”planewave” emitter which is at rest in the inertial frame S ? In order to predictthe result of the aberration measurement the accelerated observer shoulduse the non-diagonal metric Eq.(8) (i.e. anisotropic electrodynamics fieldequations).According to the asymmetry between the inertial and accelerated frames,there is a remarkable prediction on the theory of the aberration of light.Namely, if the opaque screen with hole were at rest relative to the fixedstars and the screen started from rest to motion relative to the fixed stars,then the apparent angular position of the ”plane-wave” emitter seen in theaccelerated frame through the aperture would jump by angle v / c . That isthe transmission through the hole in the opaque screen in the frame S n appears as shown in Fig 13. It could be said that the crossed term in metricEq.(8) generates anisotropy in the accelerated frame that is responsible forthe change of radiation direction (aberration).Above we considered the emitter-screen problem in the frame of reference S which is at rest with respect to the fixed stars. It is found that there is noaberration proceeding from the emitter, independently of their motion inthe case of the small aperture hole. It is important to emphasize that theaberration of the light beam transmitted through the small aperture hole isalso independent of the emitter motion in the accelerated system S n . Thepoint is that the crossed term in metric Eq.(8), which generates aberration in62 ig. 13. Aberration of light in an accelerated frame of reference S n . Radiation wave-front orientation and the anisotropy of speed of light presented in the absolute timecoordinatization ( t n = t ). The crossed term in metric Eq.(8) generates anisotropyin the accelerated frame that is responsible for the change of radiation direction(aberration). this particular case, depends only on the velocity of the accelerated framerelative to the fixed stars. It should be noted, however, that there is another satisfactory way of ex-plaining the e ff ect of aberration of light in the accelerated frame S n . Theexplanation consists in using a clock re-synchronization procedure. Wellknown that in their original form Maxwell’s equations are valid in the iner-tial frames. But Maxwell’s equations can be written down only if the Lorentzcoordinates has already been specified.When the system S n starts moving with constant velocity the standard pro-cedure of Einstein’s clock synchronization can be performed. The Einsteinsynchronization is defined in terms of light signals emitted by a source atrest assuming that light propagates with the same velocity c in all direc-tion. Using such synchronization procedure we actually select a Lorentzcoordinate system for the screen. In this synchronization, we describe thetransmission through the aperture using usual Maxwell’s equations. Theinterval in the accelerated reference system S n will have the diagonal formEq.(1) for the transmitted light beam.63 ig. 14. Aberration of light in the accelerated frame of reference S n . The plane wave-front rotates in the accelerated frame after the accelerated clock re-synchronization.The Maxwell’s equations can now be used to describe the transmitted (throughthe aperture) light beam. According to the Maxwell’s electrodynamics, the trans-mitted light beam is propagated at the angle v / c , yelding the phenomenon of theaberration of light. The time t (cid:48) n under the Einstein’s synchronization in the S n frame is readilyobtained by introducing the o ff set factor x n v / c and substituting t (cid:48) n = t n − x n v / c in the first order approximation. This time shift has the e ff ect ofrotation the plane of simultaneity (that is emitter radiation wavefront) onthe angle v / c in the first order approximation. As a consequence of this, theplane wavefront rotates in the accelerated frame after re-synchronization.The new time coordinate in the accelerated frame is interpreted by sayingthat Maxwell’s equations are applicable to the light transmission (throughthe aperture) description. Then, the transmitted light beam is propagated atthe angle v / c , yielding the phenomenon of the aberration of light: the twoapproaches give the same result.The choice between these two di ff erent clock synchronizations is a matterof pragmatics. By changing the (four-dimensional) coordinate system, onecannot obtain a physics in which new physical phenomena appear. But wecan obtain a more consistent description of these phenomena. Let us analyze the aberration of light radiated by a single emitter movingin the accelerated system. It is assumed that the detector for the direction of64he radiation is an energy propagation detector and the size of the detectoraperture is su ffi ciently large compared with the radiation beam size. In otherwords, it is assumed that there is no physical influence of the detector (e.g.aperture) on the measurement.Before we go on to analyze an observations of a non-inertial observer, weshould make one more remark about observations of an inertial observer. Inthis chapter we already emphasized that in the description of the aberrationof light in an inertial frame of reference there are two choices of (four-dimensional) coordinatizations useful to consider:(a) Non-standard (absolute time) coordinatization(b) Standard Lorentz coordinatizationWe should underline that we claim the ”single frame” approach to relativis-tic electrodynamics is actually based on the use of a not standard (absolutetime) clock synchronization assumption within the theory of relativity.When the light source in the inertial frame is accelerated from rest up tovelocity v along the x axis, the simplest ( absolute time) method of synchro-nization consists in keeping, without changes, the same set of uniformlysynchronized clocks used in the case when the light source was at rest, i.e.we still enforce the clock transport synchronization ( or Einstein synchro-nization which is defined in terms of light signals emitted by the light sourceat rest). This choice is usually the most convenient one from the viewpointof connection to laboratory reality.Now we are in position to assign Lorentz coordinates in the case when thelight source in the lab frame is accelerated from rest up to velocity v along the x -axis. In order to assign a Lorentz coordinate system in the inertial frameafter the Galilean boost x (cid:48) = x − vt , t = t (cid:48) , one needs to perform additionallya change scale of reference rules x → γ x , accounting for length contraction.After this, one needs to change the rhythm of all clocks t → t /γ , thusaccounting for time dilation. The transformation of the rule-clock structurecompletes with the distant clock resynchronization t → t + xv / c . This newspace-time coordinates in the lab frame are interpreted, mathematically, bysaying that the metric is now diagonal and the speed of light from themoving source is isotropic and equal to c .Now let us return to observations of a non-inertial observer. In the descrip-tion of the aberration of light in an accelerated frame of reference there arethree choices of coordinatizations useful to consider:(a) External non-standard (absolute time) coordinatization65 ig. 15. Aberration of light in an accelerated frame of reference S n . Emitter is atrest with respect to the fixed stars. Wave fronts orientation and the group velocitypresented in the internal absolute time coordinatization. The aberration increment θ a is connected with the physical parameters by the relation: θ a = v / c , where v isthe velocity of the fixed stars in the accelerated frame S n . (b) Internal non-standard (absolute time) coordinatization(c) Internal standard Lorentz coordinatization(a) When the system S n in the stationary inertial lab frame S is acceleratedfrom the rest with respect to the fixed stars up to velocity v along the x -axis,the simplest (external absolute time) method of synchronization consists inkeeping, without changes, the same set of uniformly synchronized clocksused in the case when the system S n was at rest. Acceleration with respectto the fixed stars have an e ff ect on the propagation of light. The velocity oflight from the source which is at rest in the accelerated frame S n will, bythe moment when the system S n starts moving with constant velocity, haveanisotropy along the x axis.(b) When the system S n starts moving with constant velocity the standardprocedure of Einstein’s clock synchronization can be performed. The Ein-stein synchronization is defined in terms of light signals emitted by a sourceat rest (in the accelerated frame) assuming that light propagates with thesame velocity c in all direction. Using such (internal absolute time) synchro-nization procedure we actually describe the light from the source at restusing usual Maxwell’s equations.(c) We now consider the case when the accelerated observer looks outside66 ig. 16. Aberration of light in an accelerated frame of reference S n . Emitter isat rest with respect to the fixed stars. Wave fronts orientation presented in theinternal Lorentz coordinatization. The aberration increment θ a is connected withthe physical parameters by the relation: θ a = v / c , where v is the velocity of thefixed stars in the accelerated frame S n . on the source which is at rest in the inertial frame S . In order to assign theinternal Lorentz coordinate system in the frame S n after the accelerationwith respect to the fixed stars one needs to perform additionally (to theintertal absolute time coordinatization) a change scale of reference rules x n → γ x n . After this, one needs to change the rhythm of all clocks t n → t n /γ .The transformation of the rule-clock structure completes with the distantclock resynchronization t n → t n − x n v / c . This new space-time coordinatesin the accelerated frame are interpreted, mathematically, by saying that thespeed of light from the moving source is isotropic and equal to c .Above we considered the single emitter problem in the frame of reference S which is at rest with respect to the fixed stars. It is found that there is adeviation of the energy transport for the light radiated by the transverselymoving emitter, which is nothing else but well-known (from textbooks)result: there is the aberration of light from the transversely moving emitterin the inertial frame of reference (Fig. 1).We now consider the case when the emitter is at rest in the lab inertial frame S (i.e. at rest with respect to the fixed stars) and the observer, which is at restwith respect to the accelerated frame of reference S n performs the directionof the energy transport measurement.It is important to emphasize that the aberration of light radiated by the single67 ig. 17. The aberration of light from stationary and moving sources. According tothe asymmetry between the inertial and accelerated frames, there is a remarkableprediction on the theory of the aberration of light. Namely, if the emitter is at restrelative to the fixed stars and the observer with measuring devices started from restto uniform motion relative to the fixed stars, then the apparent angular position ofthe emitter seen in the accelerated frame would jump by angle θ a = − v / c . Thissituation is not symmetrical with respect to the change of the reference frames. Ifthe observer is at rest relative to the fixed stars and the emitter started from restto uniform motion relative to the fixed stars, then the apparent angular position ofthe emitter seen in the inertial frame would jump by angle θ a = v / c . emitter is also dependent of the emitter motion in the accelerated system S n .According to the asymmetry between the inertial and accelerated frames,there is a remarkable prediction on the theory of the aberration of light.Namely, if the emitter is at rest relative to the fixed stars and the observerstarted from rest to motion relative to the fixed stars, then the apparentangular position of the ”plane-wave” emitter seen in the accelerated framewould jump by angle 2 v / c . That is the aberration of light in the frame S n appears as shown in Fig 15 - Fig. 16.Let us now consider the most general case when emitter in the inertialframe S is accelerated from rest up to velocity u along the x axis and thesystem S n in the inertial frame S accelerated from the rest up to velocity v along the same x axis. Suppose that an observer in the accelerated frame S n performs an aberration measurement. At close look at the physics of thissubject shows that the aberration increment is connected with the problemparameters by the relation θ a = v / c − u / c . The point is that the crossed termin metric Eq.(8), which generates the anisotropy in the accelerated frame,depends only on the velocity v of the accelerated frame relative to the fixed68 ig. 18. Reciprocity in a theory of the aberration of light. One must use the samesource phasing to demonstrate the reciprocity. After the re-phasing procedure theinertial observer would find that angular displacement is θ a = v / c . stars. In the present section we shall continue our discussion of the aberrationof light radiated by a single moving emitter. Imaging that there are twoidentical emitters. Let us consider the case when the first emitter is at restin an inertial frame and the second emitter is accelerated from rest up tovelocity v along the x axis. Suppose that an observer, which is at rest withrespect to the inertial frame of reference performs the direction of the energytransport measurement. Now we must be careful about initial phasing ofthese emitters. As example, we consider the case in which initially thevelocity component of the light beam along the x -axis is equal to zero.Then how does the light beam from the moving emitter looks? The inertialobserver would find that angular displacement is equal to θ a = v / c . That isthe radiation appears as shown in Fig. 17. This is example what called thephenomenon of aberration of light and it is well known.When the accelerated system starts moving with constant velocity the stan-dard procedure of Einstein’s clock synchronization can be performed. In thisinternal (absolute time) synchronization, the accelerated observer describesthe light beam from an emitter at rest using usual Maxwell’s equations.Let us describe what happens when accelerated observer performed there-phasing (re-directing) of the accelerated emitter. We consider also the69 ig. 19. Aberration of light in an accelerated frame of reference. Point source is at restin the accelerated frame. Wavefront orientation presented in the (external) absolutetime coordinatization. The crossed term in metric Eq.(8) generates anisotropy inthe accelerated frame that is responsible for the change of transmitted radiationdirection. The point source follows the same pattern as fixed stars under the sameaberration angle i.e. its apparent position changes with an angular displacementcommon to fixed stars. case in which finally the velocity component of the light beam along the x -axis is equal to zero. After this re-phasing procedure the inertial observerwould find that angular displacement is θ a = v / c . Suppose that the accel-erated observer also performs an aberration measurement. Fig. 18 showsthat the aberration increment is connected with the problem parameters bythe relation θ a = − v / c . The situation can be described quite naturally inthe following way. It is well known that the special relativity is a recipro-cal theory and we demonstrated that symmetry is a correct concept in theaberration of light problem. It should be note, however, that one must usethe same source phasing to demonstrate the reciprocity. Now let us return to observations of a non-inertial observer. Above weconsidered a single moving ”plane wave” emitter in a non-inertial frameof reference. In the description of the aberration of light in an acceleratedframe there are two choices of local sources useful to consider:(a) A ”plane wave” (i.e. laser-like) emitter70 ig. 20. Aberration of light in an accelerated frame of reference. Point source is atrest with respect to the fixed stars. The aberration of point source is consideredindependent of the source speed and to have just a local origin exclusively basedon the observer (with his measuring instruments) speed relative to the fixed stars. (b) A point-like ( or, more generally, spatially completely incoherent) sourceSource field di ff raction can be divided into categories - the Fresnel (near-zone) di ff raction and Fraunhofer (far-zone) di ff raction. In Fraunhofer di ff rac-tion, the phase of the wave is assumed to vary linearly across the detectoraperture. This would occur if, for example, a plane wave were incident onthe aperture at an angle with respect to the optical axis. In the Fresnel di ff rac-tion, we replace the assumption of a linear phase variation with quadraticphase variation.In the far zone, both types of sources produce in front of pupil detectione ff ectively a plane wave. In other words, there is always the physical influ-ence of the instrument on the measurement of the aberration of light in theFraunhofer zone. One of the specific properties of laser-like sources is theirability to produce fields that are highly directional. The intensity of suchfields is concentrated in a very narrow angle. In the Fresnel zone at largedetector size there is no influence of the detector on the measurement andwe have possibility to discuss about the aberration of light radiated by asingle ”plane wave” emitter. In contrast, the peculiarity of point-like sourcesis that radiation emitted at one instant form a sphere around the source andthe measuring instrument always influences the measured radiation, Fig.19-Fig. 20 .In the framework of the conventional theory of the aberration of light,71here is an outstanding puzzle concerning the stellar aberration. There aredouble star systems the components of which change their velocity on atime scale ranging from days to years. The components of such binarysystems at some times can have velocities relative to the earth very di ff erentfrom one another; nevertheless it is well known that these componentsexhibits always the same aberration angle. Rotating binary systems followthe same pattern as all fixed stars and are observed within a period of ayear under the same universal aberration angle, i.e. their apparent positionchanges with an annual period common to all distant stars. This argumentsuggests that results of the astronomical experiments confirm our predictionfor aberration of light from a point source in an non-inertial frame. In thenext chapter we discuss this experimental test in more detail. Now we wish to continue in our analysis a little further. We will look fora di ff erent way of calculating the aberration e ff ect. The second solution ofthe emitter-screen problem will be discussed going back to ether theory.We are going to demonstrate that the ether-related solution is simple andstraightforward.While the results presented above are fundamental, there is nothing un-expected about them, except perhaps that they can be derived using pre-relativistic theory only, and thus that they could have been proven long ago.Indeed, the e ff ect of light aberration in an accelerated frame of referencecan easily be explained on the basis of the pre-relativistic ether theory if wemay assume that terms of the second-order are below the accuracy of theexperiments.We note that the denial of the ether by the special relativity cannot be takenseriously anymore. We remark that, as shown by Lorentz, there is an agree-ment between the pre-relativistic ether theory and the theory of relativityas regards all optical e ff ects of the first order in v / c . How shall we changethe ether theory so that it will be completely equivalent to the theory of rel-ativity? As it turn out, the only requirement is that the length of any objectmoving in ether must be contracted. When this change is made, the ethertheory and the theory of special relativity will harmonize (9) . The presenttheory sustains the concept of a stationary ether with respect to the fixedstars, although the word ”stationary” can be a misnomer. It is importantto remark that a close examination of all experiments inside a uniformlymoving frame (relative to the fixed stars) from viewpoint of the ether the-72 ig. 21. Aberration of light in an accelerated frame of reference S n . According to theether theory, the ether wind generates anisotropy in the accelerated ( relative to theether) frame that is responsible for change of radiation direction. ory, however, shows that in reality one there never measures the absolutevelocity. All phenomena appear to be independent of the uniform motionrelative to the fixed stars. If we are to build such a theory of the ether, then,this ether is relativistic, the meaning of which is that it upholds the principleof relativity. The principle of relativity which we owe to Poincare, who firstcoined the term, denied the possibility for an observer partaking in a uni-form motion relative to the fixed stars of discovering, by any measurement,such a motion, under the assumption, of course, that one does not lookoutside. In our case, the peculiarity of the aberration of light measurementsis that the accelerated observer looks outside to the fixed stars. We accept the ether theory in the original form: there is ether, which rulesthe speed of light. Because the ether is immovable, it is causes anisotropyin every frames moving relative to the ether. Because the non-inertial frameis accelerated with respect to the fixed stars; consequently we must feelan ”ether wind” when measuring light propagation. Firstly we discuss thee ff ect of the ether wind on the light speed.According to the hypothesis of an ether at rest in the unaccelerated labora-tory, the velocity of light judged from an accelerated reference system wouldbe c + u for the beam propagating in the same direction as the acceleratedsystem and − c + u for the beam propagating in the opposite direction, where73 = − v is the ether velocity in the accelerated reference system.The aberration e ff ect is an e ff ect of the ether-wind on the speed of lightthat must be corrected for in transformations of accelerating coordinatesystems (Fig. 21). We remark again that there is an agreement between theether theory and the theory of relativity. In fact, the metric Eq.(8) associatedwith accelerated reference frame S n predicts an asymmetry in the one-wayspeed of light in the relative velocity direction. Accelerations (with respectto the fixed stars) have an e ff ect on the propagation of light in the theory ofrelativity. On accelerated system S n , the velocity of light must be added to(or subtracted from) the speed due to acceleration and the velocity of lightis di ff erent in opposite directions (10) .The fact that an ether theory is consistent with accelerated motion providesstrong evidence that an ether exists, but does not inevitably imply that uni-form motion relative to the ether is measurable. It is important to remarkthat a close examination in the framework of the ether theory of all ex-periments inside the uniformly moving (relative to the fixed stars) frame,however, shows that in reality all phenomena appeared to be independentof the uniform motion relative to the fixed stars. In this section we will collect a number of useful facts concerning the space-time measurements made by di ff erent observers moving with respect to eachother. It is important to stress at this point that our discussion of aberrationof light in an inertial and non-inertial frames would be a prototype for anyspecial relativity problem. Let us discuss the observations of an inertial observer. Imaging that thereare two identical emitters. The first emitter is at rest in the observer frameand the second emitter is accelerated up to velocity v along the x -axis. Inabsolute time coordinatization, the radiation of the first emitter is describedwith the help of the Maxwell’s equations in their usual form independentlyof the second emitter acceleration. Inertial observer can deduce electromag-netic field equations for an accelerated source by studying the form takenMaxwell’s equations under the coordinate transformation between the co-moving coordinate system and the coordinate system where the inertialobserver is at rest. The principle of relativity dictates that the Maxwell’sequations always valid in the comoving coordinate system. We begin withthe Minkowski metric as the true measure of space-time intervals in the co-74oving coordinate system with coordinates ( t (cid:48) , x (cid:48) ) and substitute x (cid:48) = x − vt , t (cid:48) = t into the Minkowski metric ds = c dt (cid:48) − dx (cid:48) to obtain ds = c (1 − v / c ) dt + vdxdt − dx . (9)Inspecting Eq. (9), we can find the components of the metric tensor g µν in thecoordinate system ( ct , x ) of S . We obtain g = − v / c , g = v / c , g = − g (cid:44)
0, and this impliesthat time is not orthogonal to space.The velocity of light in the comoving coordinate system ( t (cid:48) , x (cid:48) ) is c . In thecoordinate system ( t , x ), however, the speed of light cannot be equal c any-more because ( t , x ) is related to ( t (cid:48) , x (cid:48) ) via a Galilean transformation. If ds isthe infinitesimal displacement along the world line of a beam of light, then ds = c = ( dx (cid:48) / dt (cid:48) ) . In the moving reference system, since x (cid:48) = x − vt and t (cid:48) = t , this expression takes the form c = ( dx / dt − v ) , whichcan be seen by a trivial change of variable. This means that in the inertialreference system of coordinates ( ct , x ) the velocity of light parallel to thex-axis, is dx / dt = c + v in the positive direction, and dx / dt = − c + v in thenegative direction.We conclude that the speed of light emitted by a moving source measured inthe lab frame ( t , x ) depends on the relative velocity of source and observer,in our example v . In other words, the speed of light is compatible with theGalilean law of addition of velocities. The reason why it is di ff erent fromthe electrodynamics constant c is due to the fact that the clocks are synchro-nized following the absolute time convention, which is fixed because ( t , x )is related to ( t (cid:48) , x (cid:48) ) via a Galilean transformation. Note that from what wejust discussed follows the statement that the di ff erence between the speedof light and the electrodynamics constant c is convention-dependent andhas no direct physical meaning.One other point on terminology. We should say that coordinate transfor-mations usually are called ”passive” transformations or passive boosts ofvelocity. It should be clear that a good way to think of coordinate transfor-mations is to regard it as a result of change variables. At passive boost afour-vector of event is thought to be fixed and one system of coordinateschanges with respect to the other coordinate system. The Galilean or Lorentzcoordinate (passive) transformations within a single inertial frame are sim-ply another parametrization of the observations of the inertial observer.At active boost of velocity there is a one coordinate system and four-vectorof event changes. In the case of an active (physical) boost of velocity weconsider the e ff ect of interaction on motion which defined in terms of ac-celerating motion relative to the fixed stars. Thus when we state that the75econd emitter undergoes an acceleration, and the inertial observer (withmeasuring devices) does not, the acceleration means acceleration relativeto the fixed stars. Any change of velocity, or any acceleration relative to thefixed stars (i.e. any active boost of velocity) has an absolute meaning.It should be clear that the common Lorentz coordinatization for both emit-ters cannot be assigned. Suppose that we assigned the absolute time coordi-natization. This mean that we assign the diagonal metric for the descriptionof the first emitter radiation. But the radiation from the second emitterwill be described in such coordinatization by the non-diagonal metric. Sup-pose that we re-synchronize clocks in the inertial frame in order to assignthe Lorentz coordinates for the second emitter. In this coordinatization wedescribe the radiation from the second emitter using the usual Maxwell’sequations. But these new coordinates in the inertial frame is interpreted bysaying that Maxwell’s equations are not applicable to the description of theradiation from the first emitter. This is no problem when these two sourcesare independent. The possibility to introduce Lorentz coordinates in this sit-uation consists in introducing individual coordinate system (i.e. individualset of clocks) for each source. Suppose now that the second source interactswith the light beam radiated from the first source. For example, the firstsource is used for phasing of the second source. The peculiarity of this prob-lem with the viewpoint of the special relativity is that here we can preparefor sources a common set of synchronized clocks in the single inertial frameonly in the case of absolute time synchronization. In the general case, the problem to assigning Lorentz coordinates in an iner-tial frame is complicated. Let us consider a source, arbitrary accelerating inthe inertial frame, and let us analyze its evolution with a Lorentz coordinatesystem. The permanent rest frame of the source is obviously non inertial. Toget around that di ffi culty, one introduces an infinite sequence of comovingcoordinate systems. At each instant, the comoving coordinate system is aLorentz coordinate system centered on the source and moving with it. As thesource velocity changes to its new value at an infinitesimally latter instant,a new Lorentz coordinate system centered on the source and moving withit at the new velocity is used to observe the source. The source trajectory,which follows from this approach is viewed from the inertial frame as theresult of successive Lorentz transformations.We should make one further remark about this covariant algorithm. Anopinion is sometimes expressed that described above algorithm includesa hidden postulate. It seems necessary a dynamical assumption to justifyattributing to an accelerated clock the same rate as a clock in inertial motion76n relation to which it is momentary at rest. This is, in view of some authors,an extra condition that a clock must satisfy. It is assumed that the e ff ectof the motion on the clock depends only on it instantaneous speed, not itsacceleration. This condition often refereed to as a ”clock hypothesis” (11) . Westate that the clock hypothesis does not have status of independent hypoth-esis is not needed as an independent postulate in the theory of relativity.As discussed, the passive Lorentz transformations within a single inertialframe are simply another parametrization of the observations of the inertialobserver. In other words, when we perform coordinate transformations (i.e.change variables) the problem of accelerated clock does not exist at all. Common textbook presentations of the special relativity use the approachwhich deals only with observations of an inertial observer and, conse-quently, only with a coordinate transformations within a single inertialframe. The fact that in the real process of observer transmission to a co-moving reference frame (i.e in the process of an observer accelerating withrespect to the fixed stars) the metric of the observer is changed, is not consid-ered in textbooks. Clearly, the passive Galilean or Lorentz transformationswithin a single inertial frame is quite distinct from the active transforma-tions (actually acceleration with respect to the fixed stars) of an observerwith his measuring devices from one inertial frame to another.So far in our discussion of measurements, we have considered measuringdevices that go make up inertial frames. However, one often has occasionto make measurements with non-inertial devices; for example, aberrationof light measurements made in a laboratory rotating with the earth. In thissection we will discuss the observations of a non-inertial observer.We begin with the metric as the true measure of space-time intervals foran non-inertial observer S n with coordinates ( t n , x n ). We transform coordi-nates ( t , x ) that would be coordinates of an inertial observer S moving withvelocity − v with respect to the accelerated observer S n , using a Galileantransformation: we substitute x n = x − vt , while leaving time unchanged t n = t into the Minkowski metric ds = c dt − dx to obtain Eq.(8).The velocity of light emitted by a source at rest in the coordinate system ( t , x )for S is c . In that case the Minkowski metric Eq.(1) associated with inertialframe S predict a symmetry in the one-way speed of light. In the coordi-nate system ( t n , x n ), however, the speed of light emitted by the acceleratedsource (i.e. source which is at rest with respect to the accelerated frame S n )cannot be equal c anymore because ( t n , x n ) is related to ( t , x ) via a Galileantransformation. As a result, the speed of light in the direction parallel to the77 n axis is equal to c − v in the positive direction, and − c − v in the negativedirection.According to the asymmetry between the inertial and accelerated frames,there is a remarkable prediction on the theory of the aberration of light.Namely, if the emitter is at rest relative to the fixed stars and the observerwith measuring devices started from rest to uniform motion relative to thefixed stars, then the apparent angular position of the emitter seen in theaccelerated frame would jump by angle θ a = − v / c , Fig. 16. This situationis not symmetrical with respect to the change of the reference frames. Ifthe observer is at rest relative to the fixed stars and the emitter startedfrom rest to uniform motion relative to the fixed stars, then the apparentangular position of the emitter seen in the inertial frame would jump byangle θ a = v / c , Fig. 1.The di ff erence between these two situations, ending with a final uniformmotion is very interesting. Many people who learn theory of relativity in theusual way find this disturbing. For this seems to contradict the very prin-ciple of relativity. Indeed, the principle of relativity denied the possibilityfor an observer partaking in a uniform motion relative to the fixed stars ofdiscovering, by any measurement, such a motion. The contradiction, how-ever, disappears if we identify the principle of relativity with the concept ofreciprocity. It is important to emphasize that one must use the same sourcephasing to demonstrate the reciprocity.When the accelerated system starts moving with constant velocity, theMaxwell’s equations are not applicable from the viewpoint of an observerat rest with respect to accelerated frame. Acceleration have an e ff ect onthe propagation of light. When the reference frame moving with constantvelocity the standard procedure of Einstein’s clock synchronization can beperformed. The Einstein synchronization is defined in terms of light signalsemitted by a source at rest assuming that light propagates with the same ve-locity c in all direction. Using such synchronization procedure non-inertialobserver actually selects the Lorentz coordinate system for the source at restin the accelerated frame. A close examination of all experiments inside theuniformly moving accelerated frame shows that all phenomena appearedto be independent of the uniform motion relative to the fixed stars. Whereis the information about the observer acceleration recorded in the case of in-ternal (absolute time) clock synchronization? This information is recordedin the phase front orientation (with respect to the coordinate axes of theaccelerated frame) of the light beam radiated from the source at rest withrespect to the fixed stars, Fig. 15. In fact, when the non-inertial observerperformed the standard procedure of clock synchronization the time shifthas the e ff ect of rotation the plane of simultaneity, that is source radiationwavefront, on the angle v / c . 78 .4.4 Inertial Frame View of Observations of the Noninertial Observer First we want to rise the following interesting and important point. Thelaws of physics in any inertial reference frame should be able to account forall physical phenomena, including the observations made by non-inertialobservers. For example, when viewed from the inertial lab system, the inter-pretation of the Sagnac e ff ect is simple and the phase di ff erence (attributedto the Sagnac e ff ect) between counter-propagating waves may be derivedfrom the relativistic law of velocity composition. In other words, Sagnace ff ect in the rotating frame, as viewed from the inertial lab frame, presents akinematic e ff ect (Einstein’s velocity addition) of special theory of relativity.In the first order approximation, the Galilean law of velocity compositionmay be used. Surprisingly, classical kinematic method yields correct re-sult accurate to within small (quadratic) relativistic corrections. In contrast,the Sagnac e ff ect is not easy to calculate in a frame of reference attendingrotation. In this case, authors of textbook used a metric tensor (Langevinmetric) in a plane four-dimensional Minkowski space-time to calculate thepropagation time di ff erence between counter-running waves.Can we not look at the aberration of light e ff ect in the same way? Here alsowe used the Langevin metric in a non-inertial frame of reference. Let us firstconsider the application of the classical kinematic method to the computa-tion of the aberration of light e ff ect in a non-inertial frame, as viewed froman inertial frame. In our first order approximation, the Galilean law of ve-locity composition may be used. However, the classical kinematics methodleads to a serious mistake if applied to the computation of the aberrationof light e ff ect. Let us consider the Fig. 9 and the Fig. 14. Based only on the(Galilean or Einstein’s) velocity addition, we arrive at the conclusion thatthere is no aberration of light e ff ect in the accelerated frame. This result is notsurprising. A close look at the physics of the Sagnac e ff ect and the aberrationof light e ff ect shows aspects that are not common to these phenomena. Thedi ff erence is given by the global nature of the Sagnac e ff ect. When compar-ing the local e ff ects with the global ones, we found that the time coordinatedefined by the standard isotropic synchronization convention can not beused as global coordinate because of a time-lag associated with the roundtravel.Errors inherent in the classical kinematics method applied to the computa-tion of the aberration of light e ff ect are due to the use of the velocity of lightsimply as a classical velocity of c = / s. Relativistic e ff ects do nothave a place in this description. According to classical approach, there is noprincipal di ff erence between the aberration of light and the aberration ofraindrops. We first notice that one of the postulate of special relativity statesthat if two distinct events cannot be connected by a causal signal that trav-els no faster than a light signal, then they cannot be connected by a causal79ignal at all. We will show that the aberration of light e ff ect is a corollaryto the relativistic kinematic e ff ects. The appearance of relativistic e ff ects inradiation phenomena does not depends on a large speed of the radiationsources. Light is always a relativistic object. In particular the relativity ofsimultaneity is responsible for aberrations to the first order of v / c .Now, it is very interesting to show that the geometric e ff ects in our ordinaryspace world are closely associated with the relativity of simultaneity. In thecase of the relativity of simultaneity we have a mixture - of positions andtime. In other words, in the space measurement of one observer there is amixed a little bit of the time as seen by the other.In Fig. 14 the transmitted light beam is propagated at the angle v / c , yieldingthe phenomenon of the aberration of light. The question cannot be avoidedrelative to what a light beam propagated in the accelerated frame with an-gular displacement v / c ? Suppose that an observer in the accelerated frameperforms the direction of the light beam measurement and the plane wave-front of transmitted light beam is imaged by a lens to a di ff raction spotwhich lies in the focal plane on the optical axis. Measurement of the direc-tion of the optical axis with respect to the frame axes is equivalent to thedetermination the angular displacement. In order to detect the aberration oflight e ff ect inside the accelerated frame, it is obvious that some coordinatesystem with reference direction is needed. We must inquire in detail by whatmethod we assign coordinates. This method involves some sort of physicalprocedure; eventually it must be such that it will be give us coordinates inboth (inertial and accelerated) frame of reference.In ordinary space we find that the accelerated frame moves with respect tothe inertial frame along the line motion and the inertial frame moves withrespect to the accelerated frame along the same line motion. In other words,it follows that the line motion is the same in the accelerated frame as inthe inertial frame. The angle between the axis of the observer’s coordinatesystem and the line motion is a simple ordinary space geometric parameter.Using the line motion as a reference x -axis, the accelerated observer can thendefine the second reference axis. We need to give a ”practical”, ”operational”answer to the question of how to assign an axis perpendicular to the x -axis.Clearly, it is possible to define a reference direction using a light beam.We will define the second reference direction in the following way. Letus suppose that the aberration direction inside the accelerated frame isdetermined with reference to the fixed direction of light beam from a ”plane-wave” emitter which is at rest in the accelerated frame. In other words, acoordinate system is formed here by electromagnetic axis and line motion.The motion of the aberrated light beam are assumed for simplicity, to lie inthe same plane and the angular position of the aberrated beam is describedby one angle. 80 ig. 22. Aberration of light in the accelerated frame of reference. In the internalabsolute time synchronization, the accelerated observer describes the transmittedlight beam and the reference beam from an emitter at rest using usual Maxwell’sequations. The x n -axis is parallel to the wavefront of the reference beam. The aber-ration increment may be defined as an angle between the reference and transmittedlight beams. When the accelerated system starts moving with constant velocity the stan-dard procedure of Einstein’s clock synchronization can be performed. TheEinstein synchronization is defined in terms of light signals emitted by asource at rest assuming that light propagates with the same velocity c inall direction. In this internal (absolute time) synchronization, the acceler-ated observer describes the light beam from an emitter at rest using usualMaxwell’s equations. According to Maxwell’s electrodynamics, light is al-ways emitted in the direction normal to the radiation wavefront. We con-sider the case in which the velocity component of the (reference) light beamalong the x -axis is equal to zero. In other words, the x -axis is parallel to thewavefront of the reference ”plane wave”, Fig. 22. The number that specifiesthe aberration increment of the light beam transmitted through the hole maybe defined as an angle between the reference and transmitted light beams.Let us try to get an understanding of the relationship between the referencedirections inside the inertial frame and the reference directions inside theaccelerated frame. Such approach in our case of interest is based on theuse of local (reference) light sources. We determine the reference directionsperpendicular to the line motion in the inertial frame and the acceleratedframe using two local light sources. In e ff ect, one source should be stay atrest in the accelerated frame (Fig. 22) while the second should be at rest in81 ig. 23. Aberration of light in the inertial frame of reference. In the internal absolutetime synchronization, the inertial observer describes the reference beam from anemitter at rest using usual Maxwell’s equations. The x -axis is parallel to the wave-front of the reference beam. The aberration increment may be defined as an anglebetween the reference and transmitted light beams. the accelerated frame (Fig. 23). In other words, the reference electromagneticaxis in each frame is formed by an individual light beam. Of course, it ispossible to find the space description on a more practical level than that oflight beams. For instance, the reference axis in the earth-based frame may beformed by the gravitation field vector (e.g. a standard direction of plumb-line). The equivalence of all local physical frames of reference underlies thetheory of relativity, so when the aberration angle emerges, it emerges in thesame manner in all local physical reference systems. A comparison with alight clock might help here. When inertial observer looks at the light clockinside the accelerated frame, he sees that clock run slowly. Not only doesthis particular kind of clock run more slowly, but if the theory of relativityis correct, any other clock, operating on any principle whatsoever, wouldalso appear to run slower.Now let us return to observations of an accelerated observer, as viewedfrom an inertial frame. From what has preceded it is easy to see that inthe accelerated system, from the point of view of the inertial observer, theelectromagnetic axis, assigned by the accelerated observer, will not orthog-onal to the common x -axis (i.e. line motion), Fig. 24. Based on the Galileanvelocity addition, we arrive at the conclusion that in the accelerated frametransmitted light beam propagates along the z -axis of the inertial frame ofreference. But in the Lorentz coordinatization there is an angular displace-82 ig. 24. Inertial frame view of observations of the non-inertial observer. Whenviewed from an inertial Lorentz reference system, the aberration of light e ff ectinside a non-inertial frame is a corollary to the relativistic kinematic e ff ects. Thetransformation of observations from inertial observer with Lorentz internal coor-dinatization to the accelerated frame is described by a Lorentz boost. If make aLorentz boost, we introduce a time transformation t n = t − xv / c and the e ff ect ofthis transformation is just a rotation of the electromagnetic axis in the acceleratedframe. In the accepted coordinatization (according to Maxwell’s equations), thereference light beam in the accelerated frame is emitted in the direction normal tothe wavefront. From the viewpoint of inertial observer, the angular displacementbetween reference direction and the transmitted light beam may be derived fromthe relativity of simultaneity. ment, v / c , between the inertial and accelerated electromagnetic referencedirections. In other words, aberration of light e ff ect in the accelerated frame,as viewed from the inertial frame, presents a kinematic e ff ect (relativity ofsimultaneity) of special theory of relativity. The transformation of observa-tions from the inertial frame with Lorentz coordinates to the acceleratedframe is described by a Lorentz boost. On the one hand, the wave equationremains invariant with respect to Lorentz transformations. On the otherhand, if make a Lorentz boost, we automatically introduce a time transfor-mation t n = t − xv / c and the e ff ect of this transformation is just a rotationof the radiation phase front (and,consequently, the electromagnetic axis) inthe accelerated frame on the angle v / c .We introduce new approach to the aberration of light theory in non-inertialframes of reference, finding another way in which this complicated prob-lem can be solved. For example, when viewed from the inertial frame, theaberration of light e ff ect in the accelerated frame (Fig. 14) is easy to calculate83n the framework of the special theory of relativity taking advantage of therelativity of simultaneity. In contrast, in order to compute the aberration oflight e ff ect in non-inertial systems we used a metric tensor.The analysis of the aberration of light e ff ect in inertial frames within specialrelativity is based on the fact that this theory allows relativistic kinematictransformations to be considered not only for a source making a uniformmotion but also for a source undergoing acceleration. It is widely believedthat all phenomena in non-inertial (e.g. rotating) reference system should beconsidered only in the framework of general relativity. However, in the ab-sence of the gravitation field, when there is no space curvature, the analysisof physical phenomena in non-inertial frames of reference can be describedin an inertial frame within standard special relativity taking advantage ofwell known relativistic kinematic e ff ects.We derived the results for observations of the non-inertial observer with thehelp of Lorentz transformations. At first site, if reference axes were orthog-onal to the common line motion they must be parallel to each other. Sincethere exist the angular displacement v / c , the situation seems paradoxical.We have already discussed that the orientation of the radiation wavefront isnot a real observable e ff ect. The relativistic kinematic e ff ects (e.g. relativityof simultaneity) are convention-dependent e ff ects and have no exact objec-tive meaning. The statement that the wavefront orientation has objectivemeaning to within a certain accuracy can be visualized by the picture ofwavefront in the proper orientation with angle extension (blurring) givenby ∆ θ = v / c . The wavefront orientation has no objective meaning since, dueto the finiteness of the speed of light, we cannot specify any experimentalmethod by which this orientation could be ascertained.To continue our discussion of inertial frame view of observations of the non-inertial observer, let us consider the opposite situation. We determine thereference directions in the inertial frame and the accelerated frame using twolocal light sources. This has interesting consequences. For example, supposethat an observer in the inertial frame performs the direction of transmittedlight measurement relative to the reference light beam from the emitterwhich is at rest with respect to the inertial frame (Fig. 23). Now we must becareful about sign of an angular displacement. The inertial observer wouldfind that the angular displacement is positive and equal to θ a = v / c . Thedefinition of the sing that we use for the angular displacement is analogousto this: we suppose that the both light beams are imaged by a lens to thetwo di ff raction spots which lie in the focal plane and the spot of transmittedlight is shifted in positive direction along the x -axis relative to the spot ofreference beam.Now we ask about the angular displacement of transmitted light with84 ig. 25. Aberration of light in an accelerated frame of reference S n . Emitter is atrest with respect to the fixed stars. The aberration of light might be calculatedfrom classical theory, in which the light beam may be treated, say, as a corpuscular(raindrops) beam. On the basis of classical theory one gets a light aberration anglethat is only one-half as big as that predicted by special relativity. respect to the reference beam inside the accelerated frame. On the onehand, we know that angular displacement of the transmitted light withrespect to the reference light beam from the emitter which is at rest withrespect to the accelerated frame is θ a (transmitted beam) = − v / c (Fig. 22).On the other hand, the angular displacement of the inertial reference lightbeam is θ a (inertial reference beam) = − v / c (Fig. 16). We come to the con-clusion that the accelerated observer can directly measure the angulardisplacement of the transmitted light with respect to the reference lightbeam of the inertial frame. In fact, the relative angular displacement is θ a (transmitted beam) − θ a (inertial reference beam) = − v / c + v / c = v / c . Thisresult is consistent with the inertial observer measurements, as must be. We have presented theoretical evidence of an aberration increment 2 v / c concerning the single emitter problem in a non-inertial frame of reference.First, we notice that there is an extra factor 2. On the basis of classical theoryone gets aberration increment that is only one-half as big as that predictedby special relativity, Fig. 25. According to classical approach, there is no85 ig. 26. The deflection of starlight by the field of the sun. The light deflection mightbe calculated from classical theory, in which the light beam may be treated, say, asa comet approaches with the velocity of light. On the basis of classical theory onegets a deflection of light that is only one-half as big as that predicted by generalrelativity. principle di ff erence between the aberration of light and the aberration ofraindrops.There is a very interesting analogy between the aberration of light e ff ect andthe deflection of light in the general relativity. A ray of light which, comingfrom fixed star, passes close by the sun will thus be attracted to it and willdescribe a somewhat concave orbit with respect to the sun. This deflectionmight be calculated from Newton’s theory, in which the ray of light may betreated, say, as a comet which approaches with the velocity of light. We thenget a formula similar to that of Einstein, but giving only half the value ofthe deflection, Fig. 26. Errors inherent in the classical method applied to thecomputation of the deflection of light are due to the use of the velocity oflight simply as a classical velocity of c = / s. According to classicalapproach, there is no principal di ff erence between the deflection of light andthe deflection of comet. A methodological analogy with the calculation ofthe aberration increment from classical (raindrops) theory emerges.86 .4.6 Examination of Reflection and Refraction in the Context of Special Relativity We are now in a position to understand what is happening when lightradiated by a moving emitter goes through a stationary material like glass.We see that what we have to do is to calculate the interference between theincident radiation and the radiation due to the material. At first glance, ifone wants to calculate the refraction and the reflection in a single inertialframe one should take into account that the electrodynamics equations aredi ff erent for a moving emitter and a stationary glass. So the possibility ofcalculating interference e ff ects is not clear.We must now discuss a certain feature of the phenomenon of interference.Because of our usage of Galilean transformations within electrodynamicswe have some apparent paradoxes. One finds many books which say thata Galilean transformations of the velocity of light is not consistent with theelectron-theoretical explanation of refraction and reflection (12) . It is widelyaccepted that if we consider a moving source and a stationary glass, theincident light wave and wave scattered by the dipoles of the glass cannotinterfere as required by the electron theory of dispersion since their velocityare di ff erent. This is misconception. It is clear that an incident wave witha certain frequency, no matter what its velocity, excites the electrons of aglass into oscillations of the same frequency. They then emit radiation withthe same frequency. Thus, the incident and scattered wave at any givenpoint have the same frequency and can interfere. The e ff ect of the di ff erentvelocities is to produce a relative phase which varies with position in space.This, according to well known ideas, a ff ects the velocity and amplitudeenvelope of the single wave which results from the superposition of the twoseparate waves.It is interesting to note that, for the case of light reflected by a mirror, therefraction and reflection is usually analyzed in a single Lorentz frame andthe use of relativistic kinematics does not leads to mistake in the situationwhere there is an emitter moving normally to its surface. We would liketo discuss the following question: since the common Lorentz time coordi-nate axis for a stationary mirror and a moving emitter cannot be assigned,how the relativistic kinematics method leads to the correct computation ofinterference e ff ect?If the mirror is at rest and the emitter is in motion, it is obvious that the elec-trodynamics equations must be identical for all electromagnetic waves. Inother words, the electrodynamics equations in the absolute time coordina-tization (i.e. Galilean transformed Maxwell’s equations) should be appliedand kept in a consistent way for both incident and scattered waves. Therequirement that the electrodynamics equations must be identical for bothincident and reflected waves appear to be a paradox here. We demonstrated87hat the contradiction disappear in the aberration of light problem. The pe-culiarity of the aberration of light (perpendicular) geometry is that evenafter the Galilean transformation the electrodynamics equations in the ab-solute time coordinatization will have Maxwell’s (diagonal) form for theincident wave. Let us now see what happens in the collinear geometry.Nature apparently doesn’t see any paradox, however, because we discussinterference e ff ects. The point is that all methods to measure the interference,indeed, the standing wave (i.e. round-trip) measurements. In contrast, thedeviation of the energy transport direction is a geometrical e ff ect. The centralprinciple of special relativity is the Lorentz covariance of all the fundamentallaws of physics. It is important to emphasize that, consistently with thisLorentz covariance, the di ff erence in velocity of light in opposite directionsbecomes dependent on the direction and on speed of the source relativeto the single inertial frame, while the average value of the speed of lightin closed path is invariant. Because we have empirical access only to theround-trip average speed of light the value of the one-way speed of lightis just a matter of convention without physical meaning. In contrast to this,the two-way speed of light, directly measurable along a round-trip, hasphysical meaning.The following simple analysis confirms these ideas. Let exp i ( ω t − kx ) rep-resent an incoming wave whose velocity is ω/ k = c + v . Similarly, letexp i ( ω t + k (cid:48) x + φ ) represent another out-coming (scattered radiation) wave ofthe same amplitude and the same frequency, a di ff erent velocity ω/ k (cid:48) = c − v and di ff erent phase. The superposition of these two waves is representedby exp i ( ω t − kx ) + exp i ( ω t + k (cid:48) x + φ ) = k + k (cid:48) ) x / + φ/ i [ ω t − ( k − k (cid:48) ) x / + φ/ ff erence in the propagating constants k and k (cid:48) of the two componentwaves. This (at the level of the first order approximation) can be written ina simpler form 2[cos[ ω x / c + φ/ i [ ω t − xv ω/ c + φ/ ω . In the lab frame after theGalilean transformation the velocity of incoming wave is c + v . Thus if ω is the natural frequency, the observed in the lab frame frequency would be ω = ω (1 + v / c ). The shift in frequency observed in the above situation is thewell known Doppler e ff ect. Our equation for superposition of two wavesnow looks like 2[cos[ ω (1 + v / c ) x / c + φ/ i [ ω (1 + v / c )( t − xv / c ) + φ/ xv ω (1 + v / c ) / c in exp i [ ω (1 + v / c )( t − xv / c ) + φ/
2] and the88ssue of distant clock synchrony. Note that the scale of time (frequency) isalso unrecognizable from physical viewpoint.Suppose we took an ordinary atom, which had a natural frequency ω at restand we moved it toward the observer in the lab frame at speed v . In order tomeasure the velocity of the atom within the lab frame, the observer first hasto specify frequency (time) standard and length standard and then has tosynchronize distant clock. Let us suppose that the same atom at rest whichhas natural frequency ω is used as frequency standard. If we organize astanding wave by using (dipole) radiation from atoms at rest with standardfrequency, we can use the standing wavelength as a standard of length.Suppose that distant clocks are synchronized by light signals by using dipole(atom) radiation source at rest. It is also assumed that light from the sourceat rest propagate with the same velocity c in all direction in the lab frame.Let us go back to our calculations of the speed of light from the movingsource when the clocks in the lab frame are synchronized according to theprocedure described above i.e. according to the absolute time convention.When coordinates are assigned in the lab frame, the laboratory observercan directly measure the one-way speed of light. The result he observesis that the speed of light emitted by the moving source is consistent withthe Galilean law of addition of velocities. In particular, when the source ismoving with velocity v along the z -axis, the velocity of light in the directionparallel to the z -axis, is equal to c + v in the positive, and − c + v in thenegative orientations. The principle of relativity assures that no physical (i.e.convention-invariant) observable can depend on the value of v . In particular,the principle of relativity requires that the two-way speed of light is equal to c in any given inertial frame. Our next objective is to understand the resultsof a measurement of the two-way speed of light from the moving sourcedescribed above.Suppose that the laboratory observer performs a measurement of the wave-length of the standing wave. Then, when the measured data is analyzed,the laboratory observer finds that the speed of light is equal to c . We nowgive derivation of this interesting and important result. If we analyze thegeometry of the situation, we find that from the standing wave measure-ment we can only extract information about two-way speed of light. Thewavenumber observed in the above situation is ω (1 + v / c ) / c . So if ω / c isthe wavenumber of light emitted by the same atom at rest in the lab frame,the observer finds that the wavelength of radiation from moving source(the source moves towards the observer) is decreased by the factor (1 + v / c ).We see that it is the same factor that we can obtain by assuming that thevelocity of light from the moving source in the lab frame is c . Due to theGalilean vectorial velocities addition, the laboratory observer will measurethe same two-way speed of light, irrespective of the source velocity. In other89ords, the measurement of the two-way speed of light is universal and thelaboratory observer actually verifies the principle of relativity.
1. For a general discussion of the Fourier transform methods of spatialfiltering theory or Abbe di ff raction theory we suggest reading the book [22].2. It is generally believed that there is no aberration for light reflected frommirrors moving transversely. To quote e.g. Sommerfeld [23]: ”Thus, for amirror moving tangentially to its surface the law of reflection which holdsfor the stationary mirror is preserved.” Similar statements can also be foundin other textbooks. To quote e.g. Ugarov [24]: ” Hence, when the mirrormoves parallel to itself the frequency of incident light is equal to that ofreflected light, and the angle of incidence is equal to the angle of reflection.”3. There is another interpretation that we can give to the scattering process.We note the very close correspondence between angular and frequencyresponse of moving di ff raction gratings and the Raman scattering [25]. Sincethe theory of Bragg di ff raction applies to light scattering by sound wavesin liquids and solids, it is not surprising that we can obtain our resultsfrom the quantum theory of light scattering by phonon. This scatteringprocess is also known as Brillouin scattering. The quantum theory leads toequation (cid:126) k s = (cid:126) k i ± (cid:126) K ⊥ by the requirement that the momentum is conservedbetween interacting photon and phonon. The process of Brillouin scatteringis a special case of Raman scattering. The momentum balance for scatteringinvolving the emission or absorption of phonon leads to the Bragg condition.The requirement of conservation of energy leads to the equation ω s = ω i ± ω phon with ω s , ω i , and ω phon being the radian frequencies, of the scatteredphoton, the incident photon, and the phonon. The frequency shift that occursfor light scattering from sound waves comes about as a result of the Dopplere ff ect. Quantum mechanically, it is a consequence of the conservation ofenergy between the participating particles. We come to the conclussion that,according to the hidden choice of absolute time coordinatization in non-relativistic quantum mechanics, the lab observer actually sees the radiationafter reflection as a result of a Galilean boost rather than a Lorentz boost.4. There is a common misconception that the radiation wavefront orienta-tion has objective meaning. To quote Norton [26]: ”One might try to escapethe problem by supposing that the direction of propagation is not alwaysgiven by the normal to the wavefront. We might identify the direction ofpropagation with the direction of energy propagation, supposing the latterto transform di ff erently from the wave normal under Galilean transforma-90ion. Whatever may be the merits of such proposals, they are unavailableto some trying to implement a principle of relativity. If the direction ofpropagation of a plane wave is normal to the wavefronts in one inertialframe then that must be true in all inertial frame.” This incorrect statementis a straightforward consequence of the generally accepted way of look-ing at special relativity of most physicists. Accepting the postulate on theconstancy of the speed of light one also automatically assumes Lorentz coor-dinates. According to such limiting understanding of the theory of relativity,it is assumed that only Lorentz coordinatization must be used to map thecoordinates of events.5. Note that spatiotemporal coupling is discussed in literature usually in re-lation with ultrashort laser pulse propagation through a grating monochro-mator. Ultrashort laser pulses are usually represented as a products of elec-tric field factors separately dependent on space and time. However, whenthe manipulation of ultrashort laser pulses requires the propagation througha grating monochromator, such assumption fails. In this situation one hasto consider in addition (to phase fronts), planes of constant intensity, thatis pulse fronts. A pulse-front tilt can be present in the beam due to prop-agation through an optical setup incorporating dispersive optical element.In the grating monochromator case, the di ff erent spectral components ofthe out-coming pulse travel in di ff erent directions. The electric field of apulse including angular dispersion can be expressed in the Fourier domain[ k x , ω ] as ¯ E ( k x − p ω, ω ), while the inverse Fourier transform from the [ k x , ω ]to the space-time domain [ x , t ] can be expressed as E ( x , t + px ), which is theelectric field of pulse with a pulse-front tilt. The tilt angle θ tilt is given bytan θ tilt = cp . More specifically p = dk x / d ω = kd θ D / d ω = λ/ ( c θ D d ), where λ = π c /ω , θ D is the di ff racted angle, and d is the groove spacing. Thedi ff racted angle θ D is a function of frequency, according to the well-knownplane grating equation. Assuming di ff raction into the first order, one has λ = (cos θ i − cos θ D ) d , where θ i is the incident angle. By di ff erentiating thisequation one obtains d θ D / d λ = / ( θ D d ), where we assume for simplisitygrazing incidence geometry, θ i (cid:28) θ D (cid:28)
1. The physical meaning ofthe former equation is that di ff erent spectral components of the out-comingpulse travel in di ff erent directions. Therefore one concludes that the pulse-front tilt is invariably accompanied by angular dispersion. It follows thatany device like a grating monochromator, that produces an angular disper-sion, also introduces significant pulse-front tilt. In our case of interest wehave deal with light di ff racted by a (transversely) moving set of gratings andthe Doppler e ff ect is responsible for frequency dispersion d ω/ dk x = v . Thusthe spatitemporal coupling due to the light beam transmission through ahole in a moving screen and the usual pulse-front tilt distortion are quitedi ff erent.6. To quote Brillouin [27]: ” Fig. 5 explain the situation assuming a simplified91evice consisting of parallel plate moving with uniform velocity v in thehorizontal direction. Monochromatic light is falling normally on the plateand generates an oblique ray.” This oblique e ff ect is demonstrated in ourFig. 10.7. The aberration of light problem in the accelerated systems is solved withdiscovery of the essential asymmetry between the non-inertial and the in-ertial observers. This asymmetry is of the same nature as that of the well-known Sagnac e ff ect [28–30]. For instance Langevin’s 1921 explanation ofthe Sagnac e ff ect rested upon the assertion that ”any change of velocity, orany acceleration has an absolute meaning.” [31].8. Accelerations (with respect to the fixed stars) have an e ff ect on the prop-agation of light. This e ff ect is of the same nature as that of the well-knownclock paradox. Let us illustrate this statement by a concrete computation.Suppose we have two identical clocks at one and the same point of theinertial reference system S . Consider their readings to coincide at the initialmoment t =
0. Let first of these clocks always be at rest in the frame S . Atmoment t = v along the x axis. Now consider the reading of the clocks in theaccelerated reference system, where the second clock is always at rest. Thesystem S n is not inertial, since it accelerated with respect to the fixed stars.In the accelerated frame the first clock moves with velocity dx n / dt n = − v .Taking account to Eq.(8), we obtain d τ = ds / c = dt n , i.e. the time, shownby the first clock coincidence with the time t = t n . Since the second clockis at rest, its reading of its proper time is d τ = √ − v / c dt n . The slowlydown of the second clock, as compared to the first, is an absolute e ff ect anddoes not depend on the choice of reference system, in which this e ff ect iscomputed.9. As a way out of disagreement between the ether theory and the principleof relativity in the second and higher order in v / c Fitzgerald (1889) andindependently Lorentz (1892) proposed that the length of bodies movingin the ether is reduced in the direction of their motion; the amount of thislength reduction was assumed to be such as to explain the absence of anye ff ect due to the motion of the earth in Michelson’s experiment. The abolitionof the ether concept is often credited to Einstein. On the contrary, Einsteinhas stated the absolute necessity of the ether. To quote Einstein [32]: ”Thenegation of ether is not necessarily required by the principle of relativity.We can admit the existence of ether but we have to give up attributing it toa particular motion. The hypothesis of the ether as such does not contradictthe theory of special relativity.”10. The presented explanation of the aberration of light e ff ect in a rotatingframe of reference is based on the concept of the immovable ether. There is92 certain degree of analogy between the aberration of light and the Sagnace ff ect. The latter was first proposed and knowingly measured by GeorgeSagnac, and was then interpreted as the proof of existence of the immovableether and as a measurement of rotation relative to it [28]. It has been shownthat the Sagnac e ff ect can be understood to result from a rotational ethermotion, reveling a close relationship between the light transmission in arotating frame of reference and the Sagnac e ff ect. A close look at the physicsof these two subjects shows things which are common to these phenomena:In both situations these are experiments in the first order in v / c and caneasily be described in classical (pre-relativistic physics) terms. It is foundthat the non-standard (absolute) time coordinate in special relativity bettersuited for the description of both e ff ects in the rotating system. If we lookmore closely at the physics, we would see aspects that are not commonto these phenomena. The di ff erence is given by the global nature of theSagnac e ff ect. When comparing the local e ff ects with the global ones, wefound that the time coordinate defined by the standard (Einstein’s) isotropicsynchronization convention can not be used as global coordinate becauseof a time-lag associated with the round travel.11. To quote Brown [16]: ”If the accelerating forces are small in relationto the internal restorative forces of the clock, then the clock’s proper timewill be proportional to the Minkowski distance along its world line. ... Thiscondition is often referred to as the clock hypothesis, and its justification,as we have seen, rests on accelerative forces being small in the appropriatesense.” We state that this problem with accelerating clocks does not existat all when we discuss an observations of an inertial observer. In this caseobserver makes measurements with devices (and also clocks) that are atrest. Inertial observer can deduce electromagnetic field equations based onthe postulates of the theory of special relativity. The principle of relativitydictates that the Maxwell’s equations always valid in the comoving coor-dinate system. It should be clear that this boost to comoving coordinatesystem is simply change variables in the description of the measurementswith devices that are at rest.12. The peculiarity of the kinematic consequence of using Galilean transfor-mations is that the speed of light emitted by a moving source depends onthe relative velocity between source and observer. A widespread theoreticalargument used to support the incorrectness of Galilean transformations isthe conclusion that a Galilean transformation of the velocity of light is notconsistent with the explanation of reflection and refraction. This idea is apart of the material in well-known books. To quote Pauli [20] ”[...] it is es-sential that the spherical waves emitted by the dipoles in the body shouldinterfere with the incident wave. If we now think of the body as at rest,and the light source moving relative to it, then [...] the wave emitted by thedipoles will have velocity di ff erent from that of the incident wave. Interfer-93nce is therefore not possible.” This conclusion is incorrect. It is clear thatthe incident and scattered wave at any given point have the same frequencyand can interfere [21]. 94 Stellar Aberration
It is generally believed that the phenomenon of aberration of light could beinterpreted, using the corpuscular model of light, as being analogous of theobservation of the oblique fall of raindrops by a moving observer. This is aclassical kinematics method to the computation of the stellar aberration usedin astronomy for about three hundred years (1) . What had to be added in the20th century, was that the dynamical laws of Newton for light were foundto be all wrong, and electromagnetic wave theory had to be introduced tocorrect them.As well-known, according to textbooks the physical basis of the stellar aber-ration is the fact that the velocity of light is finite and changes its directionwhen seen from another reference frame. It is a consequence of the for-mula for addition of velocities applied to a light beam when the observer ischanging its reference frame.It is su ffi cient to describe the e ff ect of stellar aberration by working only upto the first order v / c . For an observer on the earth, it is, with respect to thesolar referential frame, of about 30 km / s, corresponding to the earth motionaround the sun. Clearly, in the theory of stellar aberration, we consider thesmall expansion parameter v / c (cid:39) − neglecting terms of order of v / c .According to the conventional approach, the study of stellar aberration isintimately connected with the old (Newtonian) kinematics: the Galileanvectorial law of addition of velocities is actually used. ff ect of the Measuring Instruments It is generally accepted that it is much more di ffi cult to describe the aber-ration phenomenon based on wave optics then based on the corpusculartheory of light. It can be easily demonstrated that the light produced by adistant star is approximately coherent over a circular area whose diameter,in all practical cases, is much larger than the telescope diameter. Hence, wesample such a tiny portion of the coherent area of the starlight with ourtelescopes that the waveforms are e ff ectively (flat) plane waves (2) . Mostauthors treat light propagating through the telescope barrel as raindrops,and not as a plane wave. Questioning the validity of standard reasoning weargue that a satisfactory treatment of stellar aberration should be based onthe coherent wave optics. 95he method used to explain the aberration phenomenon in the frameworkof wave theory is based on the belief that there is no aberration for light re-flected from mirrors moving transversely. According to the Babinet’s prin-ciple, this prediction of standard theory should be correct also for lighttransmitted through a hole punched in a moving opaque screen or, conse-quently, through the moving open end of a telescope barrel.In Chaper 4, we presented a critical reexamination of the textbook statementthat wavefronts and raindrops are to have the same aberration. We usedthe theory of relativity to show that when one has a transversely movingmirror and a plane wave of light is falling normally on the mirror, thereis a deviation of the energy transport for light reflected from the mirror.This e ff ect is a consequence of the fact that the Doppler e ff ect is responsiblefor angular frequency dispersion of light waves reflected from the movingmirror with finite aperture. As a result, the velocity of the energy transportis not equal to the phase velocity. This remarkable prediction of our theory iscorrect also for light transmitted through the moving open end of a telescopebarrel.It is generally believed that the theory of relativity appears to conform to thephenomenon of stellar aberration discovered by Bradley by claiming it is aconsequence of the motion of observers relative to light sources, but binarystars do not exhibit such shifts when traversing our direction of view (3) .Spectroscopic binaries have velocities exceeding the earth’s velocity roundthe sun. They revolve around their common center of gravity within days,a period during which the motion of the earth is practically constant. Thecomponents of the binary system should be easily separable, when theirchanging velocities are comparable to the earth’s velocity round the sun.This is, however, not observed (Fig.27-Fig.28) (4) .In principle the binary star paradox is resolved by noting that when lightpasses through the end of a telescope barrel we have a light beam whosefields have been perturbed by di ff raction, and, therefore, do not includeinformation about star motion relative to the fixed stars. If the telescopewere at rest relative to the fixed stars and the star started to move from rest,then the apparent position of the star seen in the telescope would neverjump by any angle.However, other di ffi culties arise in the explanation of the earth-based ob-servations, i.e the change in an apparent positions of the fixed stars, whichhappens when the earth-based telescope changes of its motion relative tothe fixed stars. It should be stressed that it is the telescope and not the starthat must change its velocity (relative to the fixed stars) to cause aberration.Though stellar aberration behaves asymmetrically, it does not contradict96 ig. 27. Aberration shift as inferred from astronomical observations. The plane wavefronts of starlight entering the telescope are imaged by the lens to a di ff raction spotwhich lies in the focal plane. Two cases are selected with di ff erent velocities of aearth-based telescope and a star. In the first case (left) the star is at rest with respectto the fixed stars. In the second case (right) the star moves with the same velocityas the earth.Fig. 28. Aberration shift as predicted by the conventional theory of the stellaraberration. Two cases are selected with di ff erent velocities of a earth-based telescopeand a star. In the first case (left) the star is at rest with respect to the fixed stars.In the second case (right) the star moves with the same velocity as the earth. Theconventional theory predicts no aberration of the image in this case. We will come to the conclusion that the standard analysis of the stellaraberration does not take into account the fundamental di ff erence betweenvelocity of light and velocity of the raindrops. The main di ff erence is thelimiting character of the velocity of light. No ”causal” signal can propagatewith velocity greater than that of light. Light is always a relativistic object,no matter how small the ratio v / c may be.Where does the relativistic kinematics in the aberration of light physicscomes from? It is immediately understood that, in contrast to the classicaltheory, the light wavefront (i.e. the plane of simultaneity) orientation doesnot exist as a physical reality within the angle interval v / c since, due to thelimiting character of the speed of light, we cannot specify any experimentalmethod by which this orientation could be ascertained. The relativistic kine-matics enters aberration of light physics in a most fundamental way throughthe rotating of the wavefront which, in the case of light, is associated to thelowest order ( v / c ) relativistic kinematics e ff ect (relativity of simultaneity).An objective of this chapter is to consider non-relativistic interpretations ofthe stellar aberration, and clearly demonstrate that they are incorrect. Classi-cal kinematics e ff ects leads to serious mistakes if applied to the computationof stellar aberration as seen on the earth rotating around the sun (i.e. in anon-inertial geocentric frame of reference). The problem of the earth-basedmeasurements is solved with the discovery of the essential asymmetry be-tween the earth-based and the sun-based observers, namely, the accelerationof the traveling earth-based observer relative to the fixed stars (5) . We derivethe aberration for a pulse of light traveling on the surface of the earth usingthe Langevin metric in general relativity. Above, we demonstrated that when one has some hole in the opaque screenat rest with respect to the fixed stars and the size of the hole is very smallrelative to the size of the transversely moving ”plane-wave” emitter, thereis no aberration (deviation of the energy transport) for light transmittedthrough the hole. The absence of the e ff ects of moving (relative to the fixedstars) source in this setup automatically implies the same problem for stellaraberration theory in the heliocentric frame of reference. How shell we solve98t? It is like a hole-emitter problem with the end of the telescope barrel as ahole.Suppose that an observer, which is at rest relative to the telescope, performsthe direction of the energy transport measurement. At close look at thephysics of this subject shows that in the heliocentric frame of reference,where the telescope is at rest, we have actually the problem of steady-statetransmission. Then how does the transmitted light beam looks? It looks asthough the transmitted beam is going along the telescope axis because ithas lost its horizontal group velocity component. That is the transmissionappears as shown in Fig 11. It takes the case of a telescope positionedperpendicular to the plane phase front. In other words, the telescope pointeddirectly at the star. If the motion of the star is parallel to the phase front(i.e. perpendicular to the telescope axis), starlight entering the end of thetelescope would be able to pass its full length.One of the most important conclusions of the discussion presented aboveis that the aberration of starlight phenomenon is absent in this situation.In particular, the binary components remains unresolved which means thattheir velocity has no influence on aberration. The analogy between the obliquity of raindrops and the stellar aberration isincorrect. Only on the basis of the theory of relativity and the wave optics,we are able to describe all earth-based experimental observations of stellaraberration. Our theory predicts an e ff ect of stellar aberration in completeagreement to the Bradley’s results, Fig. 29. According to the asymmetrybetween the inertial and rotating frames, there is a remarkable predictionon the theory of the aberration of light. Namely, if the telescope is at restrelative to the earth and the earth rotating relative to the fixed stars, thenthe direction of a star as seen from the earth is not the same as the directionwhen viewed by a hypothetical sun-based observer. Apparent angle is lessthan the actual angle. The di ff erence between the actual angle and apparentangle θ a is connected with the physical parameters by the relation: θ a = v / c ,where v is the velocity of the earth in its orbit around the sun. It could be saidthat the crossed term in metric Eq.(8) generates anisotropy in the rotatingframe that is responsible for the change of radiation direction (aberration).That is the transmission through the telescope aperture in the rotating frameappears as shown in Fig. 13. The stellar aberration in the geocentric frameof reference is considered independent of the star speed and to have just99 ig. 29. The direction of a star as seen from the earth is not the same as the directionwhen viewed by a hypothetical observer at the sun center. Apparent angle φ is lessthan the actual angle θ . The di ff erence between the actual angle and apparent angleis connected with the physical parameters by the relation: θ − φ = v / c , where v isthe velocity of the earth in its orbit around the sun. a local origin exclusively based on the observer speed with respect to thefixed stars.The main facts which a theory of stellar aberration in the earth-based frameof reference must explain are (1) the annual apparent motion of the fixedstars about their locations and (2) the null apparent aberration of rotatingbinary systems. We presented here a theory which accounts for all these,and in addition gives new results. We demonstrated that the aberration oflight is a complex phenomenon which must be branched out into a numberof varieties according to their origin. According to our interpretation, thereare many kinds of aberration and the stellar aberration in the earth-basedframe is only one of these. The aberration of light is the geometric phenomenon. In order to detect theaberration e ff ect inside the earth-based frame, it is obvious that some coor-dinate system with reference direction is needed. A conventional approachto the aberration of light e ff ect is forcefully based on a definite assumption ofreference direction, but this is actually a hidden assumption. Traditionallythe physical interpretation of the aberration of light e ff ect in terms of mea-surements performed with rods that are at rest in the observer’s frame. This(local frame of reference) convention is self-evident and this is the reasonwhy it is never discussed in the aberration of light theoryFortunately, it is possible to find the space description on a more fundamen-100al level than that of measuring rods. The reference axis in the earth-basedframe is formed by gravitation field vector. The plumb-line direction isknown as the nadir, leading to the earth’s center. This is the most funda-mental local earth-based coordinate system. For example, Bradley used thevertically-mounted telescope. The star he chose was Draconis because ittransited almost exactly in zenith. The traditional plumb line provided asu ffi ciently accurate zenith-point for observations of the stellar aberration. The region of applicability of our stellar aberration theory is more widerthan one might think. Above we assumed for simplicity that a star image isactually a point spread function in the image plane of a telescope. In otherwords, it is assumed that the input signal is e ff ectively a plane wave.Suppose that a telescope is able to distinguish detail in the star image. Acomplete understanding of the relation between object and image can beobtained if the e ff ect of di ff raction are included. The e ff ect of di ff raction isto convolve that ideal image with the Fraunhofer di ff raction pattern of thetelescope pupil.The star is a spatially completely incoherent source. This means that suchsource is actually a system of elementary (statistically independent) pointsources with di ff erent o ff sets. An elementary source with a given o ff set pro-duces in front of a telescope pupil e ff ectively a plane wave. An elementarysource o ff set tilts the far zone field. Radiation field generated by an com-pletely incoherent source can be seen as a linear superposition of fieldsof individual elementary point sources. The image of an elementary pointsource is a point spread function. In other words, there is always the physicalinfluence of the telescope on the measurement of a completely incoherentsource. It should be remarked that any linear superposition of radiationfields from elementary point sources conserves single point source charac-teristics like independence on source motion. This argument gives reasonwhy our theory of stellar aberration is correct also for imaging of arbitrarycompletely incoherent sources.
1. The phenomenon of the annual apparent motion of celestial objects abouttheir locations, named stellar aberration, was discovered by Bradley in 1727,who also explained it employing the corpuscular model of light [33].101. A star is a completely incoherent source. The character of the mutualintensity function produced by an incoherent source is fully described bythe Van Cittert-Zernike theorem [34]. Any star can be considered as very faraway from the sun. In all cases of practical interest, telescopes are situated inthe far-zone of the (distant star) source. Under these circumstances, the VanCittert-Zernike theorem takes its simplest form. It is the source linear di-mension d (star diameter) that determines the coherent area of the observedwave zc / ( ω d ), where ω is the frequency of the wave, and z is the distancebetween source and observer. Consider a star like Sirius, which is the near-est object. The coherent area of light observed from Sirius has a diameter ofabout 6 m. This correlation was observed by Brown and Twiss in 1956 [35].The star Bradley chose was Draconis, which is also one of the nearest stars.In this case the diameter of the coherent area on the earth surface can beestimated to be about 100 m.3. It is widely believed that stellar aberration depends on the relative velocityof the source (star) and observer. In the paper on the theory of relativityEinstein deduced the aberration formula from the idea that the velocity of v is the relative velocity of the star-earth system. The idea was representedby many authors of textbooks. To quote Moeller [12]: ”This phenomenon,which is called aberration, was observed ... by Bradley who noticed that thestars seem to perform a collective annual motion in the sky. This apparentmotion is simply due to the fact that the observed direction of a light raycoming from a star depends on the velocity of the earth relative to the star.”4. In 1950 Ives [36] stressed for the first time that the presence of binariesin the sky gave rise to an important di ffi culty for the theory of relativity. Itis stated that the idea that aberration may be described in terms of relativemotions of the bodies concerned is immediately refuted by the existence ofspectroscopic binaries with velocities comparable with that of the Earth inits orbit. Still this exhibit aberrations not di ff erent from other stars. For ex-ample, a spectroscopic binary, Mizar A, has well-known orbital parameters,from which can be calculated an observable angular separation of 1’10” ifaberration were due to relative velocity. The empirical value is less than0.01”, clearly incompatible with authors of textbooks point of view [37,38].There is no available explanation for the fact that, while the observationaldata on stellar aberration are compatible with moving earth, the symmetricdescription, when the star possesses the relative transverse motion, doesnot apparently lead to observations compatible with predictions.5. Aberration exist as observable phenomenon only in the presence of chang-ing states of motion (i.e. acceleration). The problem of the earth-based mea-surements is related with the essential asymmetry between the earth-basedand the sun-based observers, namely, the acceleration of the traveling earth-based observer relative to the fixed stars. This has been recognized by some102xpects, perhaps most explicitly by Selleri who states [38] that ”Thus a com-plete explanation of the aberration e ff ect is given in terms of variations ofthe earth absolute velocity due to orbital motion, while the star / earth rela-tive velocity is irrelevant. Thus acceleration (of planet, this time) enters oncemore into a game.” 103 Relativistic Dynamics and Electrodynamics
In previous chapters we considered the kinematics of the theory of rela-tivity, which concerns the study of the four vectors of positions, velocityand acceleration. Kinematics studies trajectories as geometrical objects, in-dependently of their causes. This means that it is not possible to predict thetrajectory of a particle evolving under a given dynamical field using just akinematic treatment. In dynamics we consider the e ff ect of interaction onmotion. Dynamics equations can be expressed as tensor equations in Minkowskispace-time. When coordinates are chosen, one may work with compo-nents, instead of geometric objects. Relying on the geometric structure ofMinkowski space-time, one can define the class of inertial frames and canadopt a Lorentz frame with orthonormal basis vectors for any given inertialframe. In any Lorentz coordinate system the law of motion becomes m d x µ d τ = eF µν dx ν d τ , (10)where here the particle’s mass and charge are denoted by m and e respec-tively. The electromagnetic field is described by a second-rank, antisymmet-ric tensor with components F µν . The coordinate-independent proper time τ is a parameter describing the evolution of physical system under therelativistic laws of motion, Eq. (10).The covariant equation of motion for a relativistic charged particle under theaction of the four-force K µ = eF µν dx ν / d τ in the Lorentz lab frame, Eq.(10),is a relativistic ”generalization” of the Newton’s second law. The three-dimensional Newton second law md (cid:126) v / dt = (cid:126) f can always be used in theinstantaneous Lorentz comoving frame. Relativistic ”generalization” meansthat the previous three independent equations expressing Newton secondlaw are be embedded into the four-dimensional Minkowski space-time (1) .The immediate generalization of md (cid:126) v / dt = (cid:126) f to an arbitrary Lorentz frameis Eq.(10), as can be checked by reducing to the rest frame. In Lorentzcoordinates there is a kinematics constraint u µ u µ = c for the four-velocity u µ = dx µ / d τ . Because of this constraint, the four-dimensional dynamics104aw, Eq.(10), actually includes only three independent equations of motion.Using explicit expression for Lorentz force we find that the four equationsEq.(10) automatically imply the constraint u µ u µ = c as it must be. To provethis, we calculate the scalar product between both sides of the equation ofmotion and u µ . Using the fact that F µν is antisymmetric (i.e. F µν = − F νµ ), wefind u µ du µ / d τ = eF µν u µ u ν =
0. Thus, for the quantity Y = ( u − c ) we find dY / d τ = Having written down the motion equation in a 4-vector form, Eq.(10), anddetermined the components of the 4-force, we satisfied the principle ofrelativity for one thing, and, for another, we obtained the four componentsof the equation of particle motion. This is covariant relativistic generalizationof the three dimensional Newton’s equation of motion which is based onparticle proper time as the evolution parameter.We next wish to describe a particle motion in the Lorentz lab frame usingthe lab time t as evolution parameter. Let us determine the first three spatialcomponents of the 4-force. We consider for this the spatial part of the dy-namics equation, Eq.(10): (cid:126) Q = ( dt / d τ ) d ( m γ(cid:126) v ) / dt = γ d ( m γ(cid:126) v ) / dt . The prefactor γ arises from the change of the evolution variable from the proper time τ ,which is natural since (cid:126) Q is the space part of a four-vector, to the lab frametime t , which is needed to introduce the usual force three-vector (cid:126) f : (cid:126) Q = γ (cid:126) f .Written explicitly, the relativistic form of the three-force is ddt (cid:32) m (cid:126) v √ − v / c (cid:33) = e (cid:32) (cid:126) E + (cid:126) vc × (cid:126) B (cid:33) . (11)The time component is ddt (cid:32) mc √ − v / c (cid:33) = e (cid:126) E · (cid:126) v . (12)The evolution of the particle is subject to these four equations, but also tothe constraint E / c − | (cid:126) p | = mc . (13)According to the non-covariant (3 +
1) approach we seek for the initial value105olution to these equations. Using explicit expression for Lorentz force wefind that the three equations Eq.(11) automatically imply the constraintEq.(13), once this is satisfied initially at t =
0. In the (3 +
1) approach, the fourequations of motion ”split up” into (3 +
1) equations and we have no mixtureof space and time parts of the dynamics equation Eq.(10). This approach torelativistic particle dynamics relies on the use of three independent equa-tions of motion Eq.(11) for three independent coordinates and velocities,”independent” meaning that equation Eq.(12) (and constraint Eq.(13)) areautomatically satisfied.One could expect that the particle’s trajectory in the lab frame, followingfrom the previous reasoning (cid:126) x ( t ), should be identified with (cid:126) x cov ( t ). However,paradoxical result are obtained by doing so. In particular, the trajectory (cid:126) x ( t )does not include relativistic kinematics e ff ects. In the non-covariant (3 + ff ectsdo not have a place in this description. In conventional particle tracking aparticle trajectory (cid:126) x ( t ) can be seen from the lab frame as the result of suc-cessive Galileo boosts that track the motion of the accelerated (in a constantmagnetic field) particle. The usual Galileo rule for addition of velocities isused to determine the Galileo boosts tracking a particular particle, instantafter instant, along its motion along the curved trajectory.The old kinematics is especially surprising, because we are based on theuse of the covariant approach. Where does it comes from? The previouscommonly accepted derivation of the equations for the particle motion inthe three dimensional space from the covariant equation Eq.(10) includesone delicate point. In Eq.(11) and Eq.(12) the restriction (cid:126) p = m (cid:126) v / √ − v / c has already been imposed. One might well wonder why, because in theaccepted covariant approach, the solution of the dynamics problem for themomentum in the lab frame makes no reference to the three-dimensionalvelocity. In fact, equation Eq.(10) tells us that the force is the rate of changeof the momentum (cid:126) p , but does not tell us how momentum varies with speed.The four-velocity cannot be decomposed into u = ( c γ, (cid:126) v γ ) when we dealwith a particle accelerating along a curved trajectory in the Lorentz labframe.Actually, the decomposition u = ( c γ, (cid:126) v γ ) comes from the relation u µ = dx µ / d τ = γ dx µ / dt = ( c γ, (cid:126) v γ ). In other words, the presentation of the timecomponent as the relation d τ = dt /γ between proper time and coordinatetime is based on the hidden assumption that the type of clock synchro-nization, which provides the time coordinate t in the lab frame, is basedon the use of the absolute time convention. In fact, the calculation carried106ut in the case of constant magnetic field shows that t /γ = τ and one cansee the connection between this dependence and the absolute simultaneityconvention. Here we have a situation where the temporal coincidence oftwo events has absolute character: ∆ τ = ∆ t = τ = dt /γ to an Arbitrary Motion Authors of textbooks are dramatically mistaken in their belief about theusual momentum-velocity relation. From the theory of relativity follows thatthe equation (cid:126) p cov = m (cid:126) v cov / (cid:112) − v cov / c does not hold for a curved trajectoryin the Lorentz lab frame. Many experts who learned the theory of relativityusing textbooks will find this statement disturbing at first sight.How can such an unusual momentum-velocity relation come about? Weknow that the components of momentum four-vector p µ = ( E / c , (cid:126) p ) behaveunder transformations from one Lorentz frame to another, exactly in thesame manner as the component of the four-vector event x = ( x , (cid:126) x ). Surprisescan surely be expected when we return from the four-vectors language tothe three-dimensional velocity vector (cid:126) v , which can be represented in termsof the components of four-vector as (cid:126) v = d (cid:126) x / dx . In contrast with the pseudo-Euclidean four-velocity space, the relativistic three-velocity space is a three-dimensional space with constant negative curvature, i.e. three-dimensionalspace with Lobachevsky geometry.It is well known that for rectilinear accelerated motion the usual momentum-velocity relation holds. In fact, for the rectilinear motion the combinationof the usual momentum-velocity relation and the covariant three-velocitytransformation (according to Einstein’s law of velocity addition) is consis-tent with the covariant three-momentum transformation and both (non-covariant and covariant) approaches produce the same trajectory.We can see why by examine the transformation of the three velocity in thetheory of relativity. For a rectilinear motion, this transformation is performedas v = ( v (cid:48) + V ) / (1 + v (cid:48) V / c ). The relativistic factor 1 / √ − v / c is given by:1 / √ − v / c = (1 + v (cid:48) V / c ) / ( √ − v (cid:48) / c √ − V / c ). The new momentum isthen simply mv times the above expression. But we want to express the newmomentum in terms of the primed momentum and energy, and we note that p = ( p (cid:48) + E (cid:48) V / c ) / √ − V / c . Thus, for a rectilinear motion, the combinationof Einstein addition law for parallel velocities and the usual momentum-velocity relation is consistent with the covariant momentum transformation.This result was incorrectly extended to an arbitrary trajectory.We already know that the collinear Lorentz boosts commute. This meansthat the resultant of successive collinear Lorentz boosts is independent of107he transformation order. On the contrary, Lorentz boosts in di ff erent direc-tions do not commute. A comparison with the three-dimensional Euclideanspace might help here. Spatial rotations do not commute either. However,also for spatial rotations there is a case where the result of two successivetransformations is independent of their order: that is, when we deal withrotation around the same axis.As well-known, the composition of non-collinear boosts is equivalent to aboost followed by a spatial rotation. This rotation is relativistic e ff ect thatdoes not have a non-covariant analogue. One of the consequences of non-commutativity of non-collinear Lorentz boosts is the unusual momentum-velocity relation (cid:126) p cov (cid:44) m (cid:126) v cov / (cid:112) − v cov / c , which also does not have anynon-covariant analogue.The theory of relativity shows that the unusual momentum-velocity relationdiscussed above is related with the acceleration along curved trajectories. Inthis case there is a di ff erence between covariant and non-covariant particletrajectories. Only the solution of the dynamics equations in covariant formgives the correct coupling between the usual Maxwell’s equations and par-ticle trajectories in the lab frame. A closer analysis of the concept of velocity,i.e. a discussion of the methods by which a time coordinate can actually beassigned in the lab frame, opens up the possibility of a description of suchphysical phenomena as radiation from a relativistic electron acceleratingalong a curved trajectory in accordance with the theory of relativity. In the non-covariant (3 +
1) approach, the solution of the dynamics prob-lem in the lab frame makes no reference to Lorentz transformations. Thismeans that, for instance, within the lab frame the motion of particles inconstant magnetic field looks precisely the same as predicted by Newtoniankinematics: relativistic e ff ects do not have a place in this description. Inconventional particle tracking a particle trajectory (cid:126) x ( t ) can be seen from thelab frame as the result of successive Galileo boosts that track the motion ofthe accelerated (in a constant magnetic field) particle. The usual Galileo rulefor addition of velocities is used to determine the Galileo boosts tracking aparticular particle, instant after instant, along its motion along the curvedtrajectory.In order to obtain relativistic kinematics e ff ects, and in contrast to conven-tional particle tracking, one actually needs to solve the dynamics equationin manifestly covariant form by using the coordinate-independent propertime τ to parameterize the particle world-line in space-time. Relying onthe geometric structure of Minkowski space-time, one defines the class of108nertial frames and adopts a Lorentz frame with orthonormal basis vec-tors. Within the chosen Lorentz frame, Einstein’s synchronization of distantclocks and Cartesian space coordinates are enforced. In the Lorentz labframe (i.e. the lab frame with Lorentz coordinate system) one thus has acoordinate representation of a particle world-line as ( t ( τ ) , x ( τ ) , x ( τ ) , x ( τ )).These four quantities basically are, at any τ , components of a four-vectordescribing an event in space-time. Therefore, if one chooses the lab time t as a parameter for the trajectory curve, after inverting the relation t = t ( τ ),one obtains that the space position vector of a particle in the Lorentz labframe has the functional form (cid:126) x cov ( t ). The trajectory (cid:126) x cov ( t ) is viewed from thelab frame as the result of successive Lorentz transformations that dependon the proper time. In this case relativistic kinematics e ff ects arise. In viewof the Lorentz transformation composition law, one will experience e.g. theEinstein’s rule of addition of velocities applies. Attempts to solve the dynamics equation Eq.(10) in manifestly covariantform can be found in literature. The trajectory which is found does notinclude relativistic kinematics e ff ects. Therefore, it cannot be identified with (cid:126) x cov ( t ) even if, at first glance, it appears to be derived following covariantprescription.First, we examine the reasoning presented in textbooks. Consider, for exam-ple, the motion of a particle in a given electromagnetic field. The simplestcase, of great practical importance, is that of an uniform electromagnetic fieldmeaning that F µν is constant on the whole space-time region of interest. Inparticular we consider the motion of a particle in a constant homogeneousmagnetic field, specified by tensor components F µν = B ( e µ e ν − e ν e µ ) where e µ and e µ are orthonormal space like basis vectors e = e = − e · e =
0. In thelab frame of reference where e µ is taken as the time axis, and e µ and e µ arespace vectors the field is indeed purely magnetic, of magnitude B and paral-lel to the e axis. Let us set the initial four-velocity u µ (0) = γ ce µ + γ ve µ , where v is the initial particle’s velocity relative to the lab observer along the axis e at the instant τ =
0, and γ = / √ − v / c . The components of the equa-tion of motion are then du (0) / d τ = du (1) / d τ = du (2) / d τ = − eBu (3) / ( mc ), du (3) / d τ = eBu (2) / ( mc ). We seek for the initial value solution to these equa-tions as done in the existing literature (2) . A distinctive feature of the initialvalue problem in relativistic mechanics, is that the dynamics is always con-strained. In fact, the evolution of the particle is subject to mdu µ / d τ = eF µν u ν ,but also to the constraint u = c . However, such a condition can be weak-ened requiring its validity at certain values of τ only, let us say initially, at τ =
0. Therefore, if Y ( τ ) vanishes initially, i.e. Y (0) =
0, then Y ( τ ) = τ .In other words, the di ff erential Lorentz-force equation implies the constraint109 = c once this is satisfied initially. Integrating with respect to the propertime we have u µ ( τ ) = γ e µ + γ v [ e µ cos( ωτ ) + e µ sin( ωτ )] where ω = eB / ( mc ). Wesee that γ is constant with time, meaning that the energy of a charged par-ticle moving in a constant magnetic field is constant. After two successiveintegrations we have X µ ( τ ) = X µ (0) + γ c τ e µ + R [ e µ sin( ωτ ) − e µ cos( ωτ )] where R = γ v /ω . This enables us to find the time dependence [0 , X (2) ( t ) , X (3) ( t )] ofthe particle’s position since t /γ = τ . From this solution of the equation ofmotion we conclude that the motion of a charged particle in a constantmagnetic field is a uniform circular motion.One could expect that the particle’s trajectory in the lab frame, followingfrom the previous reasoning [0 , X (2) ( t ) , X (3) ( t )], should be identified with (cid:126) x cov ( t ). However, paradoxical result are obtained by doing so. In particular,the trajectory [0 , X (2) ( t ) , X (3) ( t )] does not include relativistic kinematics ef-fects. In fact, the calculation carried out above shows that t /γ = τ and onecan see the connection between this dependence and the absolute simultane-ity convention. Here we have a situation where the temporal coincidence oftwo events has the absolute character: ∆ τ = ∆ t = ff ects and the Galilean vectorial law ofaddition of velocities is actually used. The old kinematics is especially sur-prising, because we are based on the use of the covariant approach. So wemust have made a mistake. We did not make a computational mistake inour integrations, but rather a conceptual one. We must say immediately thatthere is no objection to the first integration of Eq.(10) from initial conditionsover proper time τ . With this, we find the four-momentum. The momentumhas exact objective meaning i.e. it is convention-invariant. What must berecognized is that the concept of velocity is only introduced in the secondintegration step. However, in accepted covariant approach, the solution ofthe dynamics problem for the momentum in the lab frame makes no ref-erence to three-dimensional velocity. In fact, the initial condition which weused is u µ (0) = γ ce µ + γ ve µ and includes γ c and γ v , which are actually nota-tions for the time and space parts of the initial four-momentum. The three-dimensional trajectory and respectively velocity, which are convention-dependent, are only found after the second integration step. Then, wheredoes the old kinematics comes from? The second integration was performedusing the relation d τ = dt /γ . It is only after we have made those replacementfor d τ that we obtain the usual formula for conventional (non-covariant) tra-jectory for an electron in a constant magnetic field.We should then expect to get results similar to those obtained in the case ofthe (3 +
1) non-covariant particle tracking. In fact, based on the structure of110he four components of the equation of motion Eq.(10), we can arrive to an-other mathematically identical formulation of the dynamical problem. Thefact that the evolution of the particle in the lab frame is subject to a constrainthas already been mentioned. This means that the mathematical form of thedynamics law includes only three independent equations of motion. It iseasy to see from the initial set of four equations, du (0) / d τ = du (1) / d τ = du (2) / d τ = − eBu (3) / ( mc ), du (3) / d τ = eBu (2) / ( mc ), that the presentation of thetime component simply as the relation d τ = dt /γ between proper time andcoordinate time is just a simple parametrization that yields the correctedNewton’s equation Eq.(11) as another equivalent form of these four equa-tions in terms of absolute time t instead of proper time of the particle. Thisapproach to integrating dynamics equations from the initial conditions relieson the use of three independent spatial coordinates and velocities withoutconstraint and is intimately connected with old kinematics. The presenta-tion of the time component simply as the relation d τ = dt /γ between propertime and coordinate time is based on the hidden assumption that the typeof clock synchronization, which provides the time coordinate t in the labframe, is based on the use of the absolute time convention. So far we have considered the motion of a particle in three-dimensionalspace using the vector-valued function (cid:126) x ( t ). We have a prescribed curve(path) along which the particle moves. The motion along the path is de-scribed by l ( t ), where l is a certain parameter (in our case of interest thelength of the arc). Note the di ff erence between the notions of path andtrajectory. The trajectory of a particle conveys more information about itsmotion because every position is described additionally by the correspond-ing time instant. The path is rather a purely geometrical notion. Completepaths or their parts may consist of, e. g., line segments, arcs, circles, helicalcurves. If we take the origin of the (Cartesian) coordinate system and weconnect the point to the point laying on the path and describing the motionof the particle, then the creating vector will be a position vector (cid:126) x ( l ). Thederivative of a vector is the vector tangent to the curve described by theradius vector (cid:126) x ( l ). The sense of the d (cid:126) x ( l ) / dl is determined by the sense of thecurve arc l .We already know from our discussion in Introduction that the path (cid:126) x ( l ) hasexact objective meaning i.e. it is convention-invariant. The components ofthe momentum four vector mu = ( E / c , (cid:126) p ) have also exact objective meaning.In contrast to this, and consistently with the conventionality intrinsic inthe velocity, the trajectory (cid:126) x ( t ) of the particle in the lab frame is conventiondependent and has no exact objective meaning.111e want now to describe how to determine the position vector (cid:126) x ( l ) cov incovariant particle tracking. We consider the motion in a uniform magneticfield with zero electric field. Using the Eq.(10) we obtain d (cid:126) pd τ = emc (cid:126) p × (cid:126) B , d E d τ = . (14)From d E / d τ = E / c −| (cid:126) p | = mc we have dp / d τ = p = | (cid:126) p | = m | d (cid:126) x cov | / d τ . The unit vector (cid:126) p / p can be described by theequation (cid:126) p / p = d (cid:126) x cov / | d (cid:126) x cov | = d (cid:126) x cov / dl , where | d (cid:126) x cov | = dl is the di ff erential ofthe path length. From the foregoing consideration follows that d (cid:126) x cov dl = d (cid:126) x cov dl × e (cid:126) Bpc . (15)These three equations corresponds exactly to the equations for the com-ponents of the position vector that can be found using the non-covariantparticle tracking approach, and (cid:126) x ( l ) cov is exactly equal to (cid:126) x ( l ) as it must be.The point is that both approaches describe correctly the same physical re-ality and since the curvature radius of the path in the magnetic field, andconsequently the three-momentum, has obviously an objective meaning (i.e.is convention-invariant), both approaches yield the same physical results. In order fully to understand the meaning of the embedding of the Newton’sdynamics law in the Minkowski space-time, one must keep in mind that,above, we characterized Newton’s equation in the Lorentz comoving frameas a phenomenological law. The microscopic interpretation of the inertialmass of a particle is not given. In other words, it is generally acceptedthat Newton’s second law is a phenomenological law and the rest mass isintroduced in an ad hoc manner. The system of coordinates in which theequations of Newton’s mechanics are valid can be defined as Lorentz restframe. The relativistic generalization of the Newton’s second law to anyLorentz frame permits us to make correct predictions.We are in the position to formulate the following general statement: anyphenomenological law, which is valid in the Lorentz rest frame, can beembedded in the four dimensional space-time only by using Lorentz co-ordinatization (i.e. Einstein synchronization convention). Suppose we donot know why a muon disintegrates, but we know the law of decay in112he Lorentz rest frame. This law would then be a phenomenological law.The relativistic generalization of this law to any Lorentz frame allows us tomake a prediction on the average distance traveled by the muon. In partic-ular, when a Lorentz transformation of the decay law is tried, one obtainsthe prediction that after the travel distance γ v τ , the population in the labframe would be reduced to 1 / τ to γτ . In contrast, in the non covariant(3 +
1) space and time approach there is no time dilation e ff ect, since forGalilean transformations the time scales do not change. Therefore, in the(3 +
1) non covariant approach, there is no kinematics correction factor γ tothe travel distance of relativistically moving muons. The two approachesgive, in fact, a di ff erent result for the travel-distance, which must be, how-ever, convention-invariant. This glaring conflict between results of covariantand non covariant approaches can be explained as follows: it is a dynamicalline of arguments that explains this paradoxical situation with the relativis-tic γ factor. In fact, there is a machinery behind the muon disintegration.Its origin is explained in the framework of the Lorentz-covariant quantumfield theory. In the microscopic approach to muon disintegration, Einsteinand absolute time synchronization conventions give the same result forsuch convention-invariant observables like the average travel distance, andit does not matter which transformation (Galilean or Lorentz) is used. In the non covariant (3 +
1) space and time approach, there is no time dilationnor length contraction, because for Galilean transformations time and spa-tial coordinates scales do not change. Moreover, it can easily be verified thatNewton’s second law keeps its form under Galilean transformations. There-fore, in the (3 +
1) non covariant approach, there is no kinematics correctionfactor γ to the mass in Newton’s second law. However, in contrast to kine-matics e ff ects like time dilation and length contraction, the correction factor γ to the mass in the Newton’s second law has direct objective meaning. Infact, if we assign space-time coordinates to the lab frame using the absolutetime convention, the equation of motion is still given by Newton’s secondlaw corrected for the relativistic dependence of momentum on velocity eventhough, as just stated, it has no kinematical origin. Understanding this resultof the theory of relativity is similar to understanding previously discussedresults: at first we use Lorentz coordinates and later the (3 +
1) non covariantapproach in terms of a microscopic interpretation that must be consistentwith the principle of relativity.It is well-known from classical electrodynamics that the electromagneticfield of an electron carries a momentum proportional to its velocity for113 (cid:28) c , while for an arbitrary velocity v , the momentum is altered by therelativistic γ factor in the case when the absolute time convention is used.Many attempts have been made to explain the electron mass as fully orig-inating from electromagnetic fields. However, these attempts have failed.In fact, it is impossible to have a stationary non-neutral charge distributionheld together by purely electromagnetic forces. In other words, mass andmomentum of an electron cannot be completely electromagnetic in originand in order to grant stability there is a necessity for compensating elec-tromagnetic forces with non electromagnetic fields. From this viewpoint,Newton’s second law is an empirical phenomenological law where the rel-ativistic correction factor γ to the mass is introduced in an ad hoc manner.From a microscopic viewpoint, today accepted explanation of how struc-tureless particles like leptons and quarks acquire mass is based on thecoupling to the Higgs field, the Higgs boson having been recently experi-mentally observed at the LHC. This mechanism can be invoked to explainNewton’s second law from a microscopic viewpoint even for structurelessparticles like electrons. However, at larger scales, an interesting and intuitiveconcept of the origin of physical inertia is illustrated, without recurring tothe Higgs field, by results of Quantum Chromodynamics (QCD) for protonsand neutrons, which are not elementary and are composed of quarks andgluon fields. If an initial, unperturbed nuclear configuration is disturbed,the gluon field generates forces that tend to restore this unperturbed config-uration. It is the distortion of the nuclear field that gives rise to the force inopposition to the one producing it, in analogy to the electromagnetic case.But in contrast to the electromagnetic model of an electron, the QCD modelof a nucleon is stable, and other compensation fields are not needed. Now,the gluon field mass can be computed from the total energy (or momen-tum) stored in the field, and it turns out that the QCD version in whichquark masses are taken as zero provides a remarkably good approxima-tion to reality. Since this version of QCD is a theory whose basic buildingblocks have zero mass, the most of the mass of ordinary matter (more than90 percent) arises from pure field energy. In other words, the mass of a nu-cleon can be explained almost entirely from a microscopic viewpoint, whichautomatically provides a microscopic explanation of Newton’s second lawof motion. In order to predict, on dynamical grounds, the inertial mass ofa relativistically moving nucleon one does not need to have access to thedetailed dynamics of strong interactions. It is enough to assume Lorentzcovariance (i.e. Lorentz form-invariance of field equations) of the completeQCD dynamics involved in nucleon mass calculations.The previous discussion, results in a most general statement: it is enough toassume Lorentz covariance of the quantum field theory involved in micro-particle (elementary or not elementary) mass calculations in order to obtainthe same result for the relativistic mass correction from the two synchro-114ization conventions discussed here, and it does not matter which transfor-mation (Galilean or Lorentz) is used. It is important to stress at this point that the dynamical line of argumentdiscussed here explains what the Minkowski geometry physically means.The pseudo-Euclidean geometric structure of space-time is only an interpre-tation of the behavior of the dynamical matter fields in the view of di ff erentobservers, which is an observable, empirical fact. It should be clear that therelativistic properties of the dynamical matter fields are fundamental, whilethe geometric structure is not. Dynamics, based on the field equations, isactually hidden in the language of kinematics. The Lorentz covariance of theequations that govern the fundamental interactions of nature is an empiricalfact, while the postulation of the pseudo-Euclidean geometry of space-timeis a mathematical interpretation of it that yields the laws of relativistic kine-matics: at a fundamental level this postulate is, however, based on the wayfields behave dynamically. The di ff erential form of Maxwell’s equations describing electromagneticphenomena in the Lorentz lab frame is given by Eq.(4). To evaluate radia-tion fields arising from an external sources in Eq. (4), we need to know thevelocity (cid:126) v and the position (cid:126) x as a function of the lab frame time t . As discussedabove, it is generally accepted that one should solve the usual Maxwell’sequations in the lab frame with current and charge density created by par-ticles moving along non-covariant trajectory like (cid:126) x ( t ). The trajectory (cid:126) x ( t ),which follow from the solution of the corrected Newton’s second law underthe absolute time convention, does not include, however, relativistic e ff ects.We argue that this algorithm for solving usual Maxwell’s equations in the labframe, which is considered in all standard treatments as relativistically cor-rect, is at odds with the principle of relativity. However, the usual Maxwell’sequations in the lab frame, Eq. (4), are compatible only with covariant tra-jectories calculated by using Lorentz coordinates, therefore including rela-tivistic kinematics e ff ects.The covariant particle trajectory (cid:126) x cov ( t ) is calculated by projecting the cor-responding world line to the lab frame basis and using the lab time t as aparameter for the trajectory curve. The charge and current densities Eq. (5),must be written as 4-vector current by representing charge world line inLorentz lab frame 115 µ ( τ ) = [ t ( τ ) , x ( τ ) , x ( τ ) , x ( τ )] , (16)and integrating over proper time with an appropriate additional delta func-tion. Thus j µ ( x ) = ec (cid:90) d τ u µ ( τ ) δ ( x − x ( τ )) , (17)where charge 4-velocity u µ ( τ ) = dx µ / d τ . The integration over the propertime of τ leads to j µ ( (cid:126) x , t ) = eu µ ( t ) δ ( (cid:126) x − (cid:126) x cov ( t )) , (18)Thus we obtain ρ ( (cid:126) x , t ) = e δ ( (cid:126) x − (cid:126) x cov ( t )) ,(cid:126) j ( (cid:126) x , t ) = e (cid:126) v cov ( t ) δ ( (cid:126) x − (cid:126) x cov ( t )) , (19)where (cid:126) v cov = d (cid:126) x cov / dt .It is generally believed that the usual momentum-velocity relation (cid:126) p cov = m (cid:126) v cov / (cid:112) − v cov / c holds for any arbitrary world-line x ( τ ) (3) . We state thatthis incorrect and misleading. In fact, as we have already discussed, thefour-velocity cannot be decomposed into u = ( c γ, (cid:126) v γ ) when we deal with aparticle accelerating along a curved trajectory in the Lorentz lab frame.One of the consequences of non-commutativity of non-collinear Lorentzboosts is the unusual momentum-velocity relation. In this case there is adi ff erence between covariant and non covariant particle trajectories. Onecan see that this essential point has never received attention by the physicalcommunity. As a result, a correction of the conventional radiation theory isrequired. The di ff erence between covariant and non-covariant particle trajectorieswas never understood. So, physicists did not appreciate that there wasa contribution to the radiation from relativistic kinematics e ff ects. At thispoint, a reasonable question arises: why the error in radiation theory shouldhave so long remained undetected?116or an arbitrary parameter v / c covariant calculations of the radiation processis very di ffi cult. There are, however, circumstances in which calculations canbe greatly simplified. As example of such circumstance is a non-relativisticradiation setup. The non-relativistic asymptote provides the essential sim-plicity of the covariant calculation. The reason is that the non-relativisticassumption implies the dipole approximation which is of great practicalsignificance. In accounting only for the dipole part of the radiation we ne-glect all information about the electron trajectory That means that the dipoleradiation does not show any sensitivity to the di ff erence between covariantand non-covariant particle trajectories.We want now to solve electrodynamics equations mathematically in a gen-eral way and consider the radiation associated with the succeeding termsin (multi-pole) expansion of the field in powers of the ratio v / c . Radiationtheory is naturally developed in the space-frequency domain, as one is usu-ally interested in radiation properties at a given position in space and at acertain frequency. In this book we define the relation between temporal andfrequency domain via the following definition of Fourier transform pair:¯ f ( ω ) = ∞ (cid:90) −∞ dt f ( t ) exp( i ω t ) ↔ f ( t ) = π ∞ (cid:90) −∞ d ω ¯ f ( ω ) exp( − i ω t ) . (20)Suppose we are interested in the radiation generated by an electron andobserved far away from it. In this case it is possible to find a relativelysimple expression for the electric field [55]. We indicate the electron velocityin units of c with (cid:126)β , the electron trajectory in three dimensions with (cid:126) R ( t ) andthe observation position with (cid:126) R . Finally, we introduce the unit vector (cid:126) n = (cid:126) r − (cid:126) r ( t ) | (cid:126) r − (cid:126) r ( t ) | (21)pointing from the retarded position of the electron to the observer. In thefar zone, by definition, the unit vector (cid:126) n is nearly constant in time. If theposition of the observer is far away enough from the charge, one can makethe expansion (cid:12)(cid:12)(cid:12) (cid:126) r − (cid:126) r ( t ) (cid:12)(cid:12)(cid:12) = r − (cid:126) n · (cid:126) r ( t ) . (22)We then obtain the following approximate expression for the the radiationfield in the space-frequency domain (see Appendix I):117 ¯ E ( (cid:126) r , ω ) = − i ω ecr exp (cid:20) i ω c (cid:126) n · (cid:126) r (cid:21) ∞ (cid:90) −∞ dt (cid:126) n × (cid:104) (cid:126) n × (cid:126)β ( t ) (cid:105) exp (cid:34) i ω (cid:32) t − (cid:126) n · (cid:126) r ( t ) c (cid:33)(cid:35) (23)where ω is the frequency, ( − e ) is the negative electron charge and we makeuse of Gaussian units.First we will limit our consideration to the case of sources moving in a non-relativistic fashion. According to the principle of relativity, usual Maxwell’sequations can always be used in any Lorentz frame where sources are atrest. The same considerations apply where sources are moving in non-relativistic manner. In particular, when oscillating, charge particles emitradiation, and in the non-relativistic case, when the velocities of oscillatingcharges v n (cid:28) c , dipole radiation will be generated and described with thehelp of the Maxwell’s equations in their usual form, Eq. (4).Let’s examine in a more detail how the dipole radiation term comes about.The time (cid:126) r ( t ) · ( (cid:126) n / c ) in the integrands of the expression for the radiation fieldamplitude, Eq. (23), can be neglected in the cases where the trajectory of thecharge changes little during this time. It is easy to find the conditions forsatisfying this requirement. We have said earlier in the Chapter 3 that thedimensions of the system must be small compared to radiation wavelength.This condition can be written in the in still another form v (cid:28) c , where v isof the order of magnitude of the velocities of the charges.We consider the radiation associated with the first order term in the expan-sion of the Eq. (23) in power of (cid:126) r ( t ) · ( (cid:126) n / c ). In doing so, we neglected allinformation about the electron trajectory (cid:126) r ( t ). In this dipole approximationthe electron orbit scale is always much smaller than the radiation wave-length and Eq. (23) gives fields very much like the instantaneous theory.So we are satisfied using the non-covariant approach when considering thedipole radiation theory.But that is only the first and most practically important term. The otherterms tell us that there are higher order corrections to the dipole radiationapproximation. The calculation of this correction requires detailed infor-mation about the electron trajectory. Obviously, in order to calculate thecorrection to the dipole radiation, we will have to use the covariant trajec-tory and not be satisfied with the non-covariant approach.118 .3 An Illustrative Example In the next chapter we present a critical reexamination of existing syn-chrotron radiation theory. But before the discussion of this topic it wouldbe well to illustrate error in standard coupling fields and particles in ac-celerator and plasma physics by considering the relatively simple example,wherein the essential physical features are not obscured by unnecessarymathematical di ffi culties. This illustrative example is mainly addressed toreaders with limiting knowledge of accelerator and synchrotron radiationphysics. Fortunately, the error in standard coupling fields and particles canbe explained in a very simple way.Let us try out our algorithm for reconstructing (cid:126) x cov ( t ) on some example, tosee how it works. An electron kicker setup is a practical case of study forillustrating the di ff erence between covariant and non-covariant trajectories.Let us consider the simple case when an ultrarelativistic electron movingwith the velocity v along z -axis in the lab frame is kicked by a weak dipolemagnetic field directed along x -axis. We assume for simplicity that the kickangle is small compare with 1 /γ , where γ = / √ − v / c is the relativisticfactor. This means that we take the limit γ (cid:29) (cid:29) γ v y / v . Let us start withnon-covariant particle tracking calculations. The trajectory of the electron,which follows from the solution of the corrected Newton’s second law underthe absolute time convention, does not include relativistic e ff ects. Therefore,as usual for Newtonian kinematics, Galilean vectorial law of addition ofvelocities is actually used. Non-covariant particle dynamics shows that theelectron direction changes after the kick, while the speed remains unvaried(Fig. 30). According to non-covariant particle tracking, the magnetic field B (cid:126) e x is only capable of altering the direction of motion, but not the speedof the electron. After the kick, the beam velocity components are (0 , v y , v z ),where v z = (cid:113) v − v y . Taking the ultrarelativistic limit v (cid:39) c and using thesecond order approximation we get v z = v [1 − v y / (2 v )] = v [1 − v y / (2 c )].In contrast, covariant particle tracking, which is based on the use of Lorentzcoordinates, yields di ff erent results for the velocity of the electron. Let usconsider a composition of Lorentz transformations that track the motion ofthe relativistic electron accelerated by the kicker field. Let the S be the labframe of reference and S (cid:48) a comoving frame with velocity (cid:126) v relative to S .Upstream of the kicker, the particle is at rest in the frame S (cid:48) . In order tohave this, we impose that S (cid:48) is connected to S by the Lorentz boost L ( (cid:126) v ),with (cid:126) v parallel to the z axis, which transforms a given four vector event X in a space-time into X (cid:48) = L ( (cid:126) v ) X . Let us analyze the particle evolution within S (cid:48) frame. Our particle is at rest and the kicker is running towards it withvelocity − (cid:126) v . The moving magnetic field of the kicker produces an electric119 ig. 30. A setup for illustration the di ff erence between covariant and non-covarianttrajectories. The motion of a relativistic electron accelerated by a kicker field. It isassumed for simplicity that γ (cid:29) (cid:29) γ v y / v . According to the non-covariant particletracking, the magnetic field B (cid:126) e x is only capable altering the direction of motion, butnot the speed of the electron. field orthogonal to it. When the kicker interacts with the particle in S (cid:48) wethus deal with an electron moving in the combination of perpendicularelectric and magnetic fields.We consider the small expansion parameter γ v y / c (cid:28)
1, neglecting terms oforder ( γ v y / c ) , but not of order ( γ v x / c ) . In other words, we use the second-order kick angle approximation. It is easy to see that the acceleration in thecrossed fields yields a particle velocity v (cid:48) y = γ v y parallel to the y -axis and v (cid:48) z = − v ( γ v y / c ) / z -axis. If we neglect terms in ( γ v y / c ) , therelativistic correction in the composition of velocities does not appear in thisapproximation.Let S ” be a frame fixed with respect to the particle downstream the kicker. Asis known, the non collinear Lorentz boosts does not commute. In our secondorder approximation we can neglect the di ff erence between the γ v y / c and γ z v y / c , where γ z = / (cid:112) − v z / c . Here v z = v (1 − θ k /
2) and θ k = v y / v = v y / c in our (ultrarelativistic) case of interest. Therefore we can use a sequenceof two commuting non-collinear Lorentz boosts linking X (cid:48) in S (cid:48) to X (cid:48)(cid:48) in S (cid:48)(cid:48) as X ” = L ( (cid:126) e y v (cid:48) y ) L ( (cid:126) e z v (cid:48) z ) X (cid:48) = L ( (cid:126) e z v (cid:48) z ) L ( (cid:126) e y v (cid:48) y ) X (cid:48) in order to discuss the beammotion in the frame S (cid:48) after the kick. Here (cid:126) e y and (cid:126) e z are unit vectors directed,120espectively, along the x and z axis. Note that as observed by an observer on S (cid:48) , the axes of the frame S (cid:48)(cid:48) are parallel to those of S (cid:48) , and the axes of S (cid:48) areparallel to those of S . The relation X ” = L ( (cid:126) e y v (cid:48) y ) L ( (cid:126) e z v (cid:48) z ) L ( (cid:126) e z v ) X presents a step-by-step change from S to S (cid:48) and then to S ”. For the simple case of parallelvelocities, the addition law is L ( (cid:126) e z v (cid:48) z ) L ( (cid:126) e z v ) = L ( (cid:126) e z v z ). The resulting boostcomposition can be represented as X ” = L ( (cid:126) e x v (cid:48) y ) L ( (cid:126) e z v z ) X = L ( (cid:126) e z v z ) L ( (cid:126) e y v y ) X .In the ultrarelativistic approximation one finds the simple result v = v z , sothat a Lorentz boost with non-relativistic velocity v y leads to a rotation ofthe particle velocity v z of the angle v y / c .Let us now return to our consideration of the motion of a relativistic electronaccelerated by the kicker field and let us analyze the resynchronizationprocess of the lab distant clocks during the acceleration of the electron. Thiswill allow us to demonstrate a direct relation between the decrease of theelectron speed after the kick in Lorentz coordinates and the time dilationphenomenon. As we already remarked, the Lorentz coordinate system isonly a mental construction: manipulations with non existing clocks are onlyneeded for the application of the usual Maxwell’s equations for synchrotronradiation calculations.Suppose that upstream the kicker we pick a Lorentz coordinates in the labframe. Then, an instant after entering the magnetic field, the electron ve-locity changes of the infinitesimal value d (cid:126) v along the y -axis. At this firststep, Eq.(11) allows us to express the di ff erential d (cid:126) v through the di ff erential dt in the Lorentz coordinate system assigned upstream the kicker. If clocksynchronization is fixed, this is equivalent to the application of the absolutetime convention. In order to keep Lorentz coordinates in the lab frame, asdiscussed before, we need to perform a clock resynchronization by intro-ducing an infinitesimal time shift. The simplest case is when the kick angle θ k is very small, and we evaluate transformations, working only up to theorder ( θ k γ ) . The restriction to this order provides an essential simplicity ofcalculations in our case of interest for two reasons. First, relativistic correc-tion to compositions of non-collinear velocity increments does not appear inthis expansion order, but only in the order ( γθ k ) . Second, the time dilationappears in the highest order we use. Thus, Eq.(11) allows us to express thesmall velocity change ∆ (cid:126) v after the kick in the initial Lorentz coordinates sys-tem, and to perform clock resynchronization only downstream the kicker.Therefore, after the kick we can consider the composition of two Lorentzboosts along the perpendicular x and z directions. The first boost imparts thevelocity v θ k (cid:126) e y to the electron along the y -axis and the second boost impartsthe additional velocity − ( v θ k / (cid:126) e z along the z axis, while the restriction tosecond order assures that the boosts commute.In order to keep a Lorentz coordinates system in the lab frame after the kick,that is equivalent to describe the kicker influence on the electron trajectory121 ig. 31. The motion of a relativistic electron accelerated by the kicker field. In orderto describe the kicker influence on the electron trajectory as Lorentz transforma-tion, one needs to perform a clock resynchronization by introducing a time shiftand change the scale of time, that is the rhythm of all clocks, from t to γ y t , with γ y = + θ k /
2. It follows that the total electron speed in the Lorentz lab framedownstream the kicker decreases from v to v (1 − θ k / as Lorentz transformation, we need to perform a clock resynchronization byintroducing a time shift and change the scale of time, that is the rhythm ofall clocks, from t to γ y t , with γ y (cid:39) + θ k /
2. It is immediately understood thatthe speed of electron downstream the kicker is no longer independent ofthe electron motion in the magnetic field (Fig. 31). No relativistic correctionto the velocity component along the y -axis appears in the second order, buta correction of the longitudinal velocity component, changing v z to v z /γ y with v z = v (1 − θ k /
2) and v z /γ y = v (1 − θ k ). It follows that the total electronspeed in the lab frame, after clock resynchronization downstream the kicker,decreases from v to v (1 − θ k / (cid:126) v = v (cid:126) e z .Note that we discuss particle tracking in the limit of a small kick angle γ v y / c (cid:28)
1. However, even in this simple case and for a single electron weare able to demonstrate the di ff erence between non-covariant and covariantparticle trajectories. The electron speed decreases from v to v (1 − θ k / ig. 32. Geometry for radiation production from a bending magnet. The motion ofa relativistic electron accelerated by a kicker field. According to the non-covariantparticle tracking , the magnetic field B (cid:126) e x of the kicker is only capable altering thedirection of motion, but not the speed of the electron. In our relativistic but non-covariant study of electron motion in a givenmagnetic field, the electron has the same velocity and consequently thesame relativistic factor γ upstream and downstream of the kicker. Supposewe now put the electron through a bending magnet (i.e. a uniform mag-netic field directed along the y -axis ), Fig. 32. The motion in the bendingmagnet we obtained is practically the same as in the case of non-relativisticdynamics, the only di ff erence being the appearance of the relativistic factor γ in the determination of cyclotron frequency ω c = eB / ( m γ ). The curvatureradius R of the trajectory is derived from the relation v ⊥ / R = ω c , where v ⊥ = v (1 − θ k /
2) is the component of the velocity normal to the field of thebending magnet (cid:126) B = B (cid:126) e y . According to non-covariant particle tracking, afterthe kick, the correction to the radius R is only of order θ k .One could naively expect that according to covariant particle tracking, sincethe total speed of electron in the lab frame downstream of the kicker de-creases from v to v (1 − θ k / | (cid:126) p | from m γ v to m γ v (1 − γ θ k /
2) in our approximation.However, such a momentum change would mean a correction to the radius R of order γ θ k so that there is a glaring conflict with the calculation of theradius according to non covariant tracking. Since the curvature radius of123he trajectory in the bending magnet has obviously an objective meaning,i.e. it is convention-invariant, this situation seems paradoxical. The para-dox is solved taking into account the fact that in Lorentz coordinates thethree-vector of momentum (cid:126) p is transformed, under Lorentz boosts, as thespace part of the four vector p µ . Let us consider a composition of Lorentzboosts that track the motion of the relativistic electron accelerated by thekicker field. Under this composition of boosts the longitudinal momentumcomponent remains unchanged (with accuracy θ k ).Let us verify that this assertion is correct. We have p µ = [ E / c , (cid:126) p ]. We considerthe Lorentz frame S (cid:48) fixed with respect to the electron upstream the kicker,and in the special case when electron is at rest p (cid:48) µ = [ mc , (cid:126) S (cid:48) . Acceleration in the crossed kicker fields gives rise to anelectron velocity v (cid:48) y = γ v y parallel to the y -axis and v (cid:48) z = − v ( γ v y ) / z -axis. Downstream of the kicker the transformed four-momentum is p (cid:48) µ = [ mc + mv (cid:48) y / (2 c ) , , mv (cid:48) y , mv (cid:48) z ], where we evaluate the transformation onlyup to the order ( γ v y / c ) , as done above. We note that, due to the transverseboost, there is a contribution to the time-like part of the four-momentumvector i.e. to the energy of the electron. In fact, the energy increases from mc to mc + m ( γ v y ) /
2. We remind that S (cid:48) is connected to the lab frame S bya Lorentz boost. Now, with a boost to a frame moving at velocity (cid:126) v = − v (cid:126) e z ,the transformation of the longitudinal momentum component, normal tothe magnetic field of the bend, is p z = γ ( p (cid:48) z + vp (cid:48) / c ) = γ mv . Therefore we cansee that the momentum component along the z -axis remains unchangedin our approximation of the Lorentz transformation. We also have, fromthe transformation properties of four-vectors, that the time component p = γ ( p (cid:48) + vp (cid:48) z ) = γ mc .Let us now return to our consideration on the covariant electron trajec-tory calculation in the Lorentz lab frame when a constant magnetic fieldis applied. We analyzed a very simple (but very practical) kicker setupand we noticed that, in fact, the three-momentum is not changed; so wehave already verified that this transformation is the same as the non co-variant transformation for the three-momentum, i.e. (cid:126) p cov = (cid:126) p . We also foundthat there is a di ff erence between covariant and non covariant output ve-locities, v cov < v . In these transformations we therefore demonstrated that (cid:126) p cov (cid:44) m (cid:126) v cov / (cid:112) − v cov / c for curved trajectory in ultrarelativistic asymptotic.It is interesting to discuss what it means that there are two di ff erent (covari-ant and non covariant) approaches that produce the same particle three-momentum. The point is that both approaches describe correctly the samephysical reality and the curvature radius of the trajectory in the magneticfield (and consequently the three-momentum) has obviously an objectivemeaning, i.e. is convention-invariant. In contrast to this, the velocity of theparticle has objective meaning only up to a certain accuracy, because thefiniteness of velocity of light takes place.124ext we discuss the interesting problem of emission of synchrotron radia-tion in a bending magnet with and without kick. Let us consider the setuppictured in Fig. 32. Suppose that an ultrarelativistic electron moving alongthe z -axis in the lab frame is kicked by a weak dipole field directed along the x -axis before entering a uniform magnetic field directed along the y -axis, i.e.a bending magnet. An accelerated electron traveling on a curved trajectoryemits radiation. When moving at relativistic speed, this radiation is emittedas a narrow cone tangent to the path of the electron. Moreover, the radiationamplitude becomes very large in this direction. This phenomenon is knownas Doppler boosting. Synchrotron radiation is generated when a relativisticelectron is accelerated in a bending magnet. Without going into the detailsof computation, it is possible to present intuitive arguments explaining whythe characteristics of the spectrum of synchrotron radiation only depend, inthe ultrarelativistic limit, on the di ff erence between electron and light speed.An electromagnetic source propagates through the system as a function oftime following a certain trajectory (cid:126) x ( t (cid:48) ). However, an electromagnetic signalemitted at time t (cid:48) at a given position (cid:126) x ( t (cid:48) ) arrives at the observer position at adi ff erent time t , due to the finite speed of light. As a result, an observer seesthe motion of the electromagnetic source as a function of t . Let us discussthe case when the source is heading towards the observer. If disregard theuninteresting constant delay, which just means change the origin of t by aconstant, then it says that ct = ct (cid:48) + z ( t (cid:48) ). Now we need to find x as a function of t , not t (cid:48) , and we can do this in the following way. Using the fact that c − v (cid:28) c we obtain the well-known relation dt / dt (cid:48) = ( c − v cos θ ) / c (cid:39) (1 − v / c + θ / (cid:39) (1 / /γ + θ ), where θ is the observation angle (Fig. 33). The observer seesa time compressed motion of the source, which go from point A to point B in an apparent time corresponding to an apparent distance 2 R θ dt / dt (cid:48) . Let usassume (this assumption will be justified in a moment) θ > /γ . In this caseone has 2 R θ dt / dt (cid:48) (cid:39) R θ . Obviously one can distinguish between radiationemitted at point A and radiation emitted at point B only when compresseddistance R θ (cid:29) (cid:111) , i.e. for θ (cid:29) ( (cid:111) / R ) / . This means that, as concerns theradiative process, we cannot distinguish between point A and point B onthe bend such that R θ < ( R (cid:111) ) / . It does not make sense at all to talk aboutthe position where electromagnetic signals are emitted within L f = ( R (cid:111) ) / (here we assuming that the bend is longer than L f ). This characteristic lengthis called the formation length for the bend. The formation length can also beconsidered as a longitudinal size of the single electron source (in the space-frequency domain). Note that a single electron always produces di ff raction-limited radiation. The limiting condition of spatially coherent radiation isa space-angle product θ d (cid:39) (cid:111) , where d being the transverse size and θ thedivergence of the source. Since d (cid:39) L f θ it follows that the divergence angle θ is strictly related to L f and (cid:111) : θ (cid:39) (cid:112) (cid:111) / L f . One may check that using L f = ( R (cid:111) ) / , one obtains θ (cid:39) ( (cid:111) / R ) / as it must be. In particular, at θ (cid:39) /γ one obtains the characteristic wavelength (cid:111) cr (cid:39) R /γ as is well known for125 ig. 33. Geometry for synchrotron radiation from a bending magnet. Radiationfrom an electron passing through the setup is observed through a spectral filter bya fixed observer positioned on the tangent to the bend at point P , as shown in Fig.(a). Electromagnetic source propagates through the system, as a function of time,as shown in Fig. (b). However, electromagnetic signal emitted at time t (cid:48) at a givenposition x ( t (cid:48) ) arrives at observer position at a di ff erent time t , due to finite speed oflight. As a result, the observer in Fig. (a) sees the electromagnetic source motion asa function of t . The apparent motion is a hypocycloid, and not the real motion x ( t (cid:48) ).The observer sees a time-compressed motion of the sources, which go from point A to point B in an apparent time corresponding to an apparent distance 2 R θ dt / dt (cid:48) . bending magnet radiation (Fig. 34).It is clear from the above that, according to conventional synchrotron ra-diation theory, if we consider radiation, the introduction of the kick onlyamounts to a rigid rotation of the angular distribution along the new direc-tion of the electron motion. This is plausible, if one keeps in mind that afterthe kick the electron has the same velocity and emits radiation in the kickeddirection owing to the Doppler e ff ect.According to the correct coupling of fields and particles, there is a remark-able prediction of synchrotron radiation theory concerning the setup de-scribed above. Namely, there is a red shift of the critical frequency of thesynchrotron radiation in the kicked direction. To show this, let us considerthe covariant treatment, which makes explicit use of Lorentz transforma-tions. When the kick is introduced, covariant particle tracking predicts anon-zero red shift of the critical frequency, which arises because in Lorentz126 ig. 34. Formation length for bend. Formation length L f can also be considered asa longitudinal size of a single-electron source. It does not make sense at all to talkabout the position where electromagnetic signals are emitted within L f . coordinates the electron velocity decreases from v to v − v θ k /
2, while thevelocity of light is unvaried and equal to the electrodynamics constant c .The red shift in the critical frequency can be expressed by the formula ∆ ω cr /ω cr (cid:39) − γ v x / c = − γ θ k . We now see a second order correction θ k thatis, however, multiplied by a large factor γ .It should be note, however, that there is another satisfactory way of ex-plaining the red shift. We can reinterpret this result with the help of a non-covariant treatment, which deals with non- covariant particle trajectories,and with Galilean transformations of the electromagnetic field equations.According to non-covariant particle tracking the electron velocity is unvar-ied. However, Maxwell’s equations do not remain invariant with respectto Galilean transformation, and the velocity of light has increased from c , without kick, to c (1 + θ k /
2) with kick. The reason for the velocity oflight being di ff erent from the electrodynamics constant c is due to the factthat, according to the absolute time convention, the clocks after the kickare not resynchronized. The ratio of the electron velocity to that of light isconvention independent i.e. it does not depend on the distant clock syn-chronization or on the rhythm of the clocks. Our calculations show thatcovariant and non-covariant treatments (at the correct coupling fields andparticles) give the same result for the red shift prediction, which is obviouslyconvention-invariant and depends only on the (dimensionless) parameter127 (1 − θ k / / c = v / [ c (1 + θ k / ffi cientlynarrow bandwidth for our purposes. They cause the electron beam to fol-low a periodic undulating trajectory with the consequence that interferencee ff ects occur. Undulators have typically many periods. The interference ofradiation produced in di ff erent periods results in a bandwidth that scalesas the inverse number of periods. Therefore, the use of insertion devicesinstalled at third generation synchrotron radiation facilities would allow usto realize a straightforward increase in the sensitivity to the red shift at arelatively small kick angle, θ k < /γ . In the next chapter we discuss thisexperimental test in more detail.
1. Let us try to get a better understanding of the geometric restatement ofNewton’s second law. To derive the covariant form of relativistic dynamics,we should embed the three-dimensional vector relation md (cid:126) v / dt = (cid:126) f intothe four-dimensional geometry of Minkowski space [39]. The idea of em-bedding is based on the principle of relativity i.e. on the fact that the usualNewton’s second law can always be used in any Lorentz frame where theparticle, whose motion we want to describe, is at rest. In other words, ifan instantaneously comoving Lorentz frame is given at some instant, onecan precisely predict the evolution of the particle in this frame during aninfinitesimal time interval. In geometric language, the Newton law is stricton a hyperplane perpendicular to the world line. However, the hyperplanetilts together with its normal u µ as one moves along the world line. Forthe embedding we need an operator ˆ P ⊥ that continually projects vectors ofMinkowski space on hyperplanes perpendicular to world line. The desiredoperator is ( ˆ P ⊥ ) µν = η µν − u µ u ν / u [39]. In the instantaneously comovingframe one can unambiguously construct a four-force K µ = [0 , (cid:126) f ]. Then, inan arbitrary Lorentz frame, the components K µ can be found through theappropriate Lorentz transformation. In the rest frame obviously u µ K µ =
0. Itfollows that, since u µ K µ is an invariant, the four-force K µ is perpendicular tothe four-velocity u µ in any Lorentz frame. The desired embedding of New-128on’s second law in hyperplanes perpendicular to the world line is found byimposing ( ˆ P ⊥ ) µν ( mdu ν / d τ − K ν ) =
0. This is a tensor equation in Minkowskispace-time that relates geometric objects and does not need coordinates tobe expressed. The evolution of a particle can be described in terms of worldline σ ( τ ), and the 4-velocity by u = d σ/ d τ , having a meaning independentlyof any coordinate system. Similarly in geometric language, the electromag-netic field is described by the second-rank, antisymmetric tensor F , whichalso requires no coordinates for its definition. This tensor produces a 4-forceon any charged particle given by ˆ P ⊥ · ( mdu / d τ − eF · u ) = P ⊥ suggests that we have only three inde-pendent equations. In the case of Maxwell’s equation we are able to rewritethe equations in the relativistic form without any change in the meaning atall, just with a change notations. It is important to noticed that the situationwith dynamics equations is more complicated. In order fully to understandthe meaning of the embedding of the dynamics law in the hyperplanesperpendicular to the world line, one must keep in mind that, above, wecharacterized the Newton’s equation in the Lorentz comoving frame as aphenomenological law. The microscopic interpreting of the inertial mass ofa particle is not given. In other words, it is generally accepted that the New-ton’s second law is an phenomenological law and rest mass is introducedin an ad hoc manner. The system of coordinates in which the equations ofNewton’s mechanics are valid can be defined as Lorentz rest frame. Therelativistic generalization of the Newton’s second law to any Lorentz framepermits us to make correct predictions. The projector operator guaranteesthat this coordinate system restriction will be satisfied.2. In general, the covariant equation of motion can be solved only by nu-merical methods; however, it is always attractive to find instances whereexact solutions can be obtained. The simplest case of great practical impor-tance is that the motion of a particle in a constant homogeneous magneticfield. From the solution of covariant equation of motion authors of text-books [39–41] conclude that the covariant motion of a charge particle in aconstant magnetic field is a uniform circular motion. The trajectory of theparticle does not include relativistic kinematics e ff ects and the Galilean vec-torial law of addition of velocities is actually used. Then when does the oldkinematics comes from? The derivation in [39–41] has one delicate point.The restriction d τ = dt (cid:48) /γ has already been imposed. This relation betweenpropper time and coordinate time is based on the hidden assumption thatthe type of clock synchronization, which provides the time coordinate in thelab frame, is based on the use of the absolute time convention. It is only afterthe authors of textbooks have make those replacement for d τ that authorsobtain the usual formula for non-covariant trajectory.129. Let us present a typical textbook statement [42] concerning the projectionof an arbitrary world line onto the Lorentz lab frame basis: ”A chargedpoint particle moving along the world line x ( τ ), τ being proper time, withinthe framework of Special Relativity has the velocity u ( τ ) = dx ( τ ) / d τ = ( γ c , γ(cid:126) v ). The four-velocity is normalized such that its invariant squarednorm equals c , u = c γ (1 − β ) = c . While x ( τ ) and u ( τ ) are coordinate-freedefinitions the decomposition u = ( c γ, (cid:126) v γ ) presupposes the choice of a frameof reference K . The particle, which is assumed to curry the charge e , createsthe current density j ( x ) = ec (cid:82) d τ u ( y ) δ ( y − x ( τ )). This is a Lorentz vector.[...] Furthermore, in any frame of reference K , one recovers the expectedexpressions for the charge and current densities by integrating over τ bymeans of relation d τ = dt (cid:48) /γ between proper time and coordinate timeand using the formula δ ( y − x ( τ )) = δ ( ct − ct (cid:48) ) = δ ( t − t (cid:48) ) / c , j ( t , y ) = ce δ (3) ( y − x ( t )) ≡ c ρ ( t , y ), j i ( t , y ) = ev i ( t ) δ (3) ( y − x ( t )), i = , , d τ = dt (cid:48) /γ . We statethat this incorrect and misleading. In fact, as we already discussed, thisrestriction cannot be imposed when we deal with a particle acceleratingalong a curved trajectory in the Lorentz lab frame.130 Synchrotron Radiation
Accelerator physics was always thought in terms of the old (Newtonian)kinematics that is not compatible with Maxwell’s equations. At this point,a reasonable question arises: since storage rings are designed without ac-counting for the relativistic kinematics e ff ects, how can they actually op-erate? In fact, electron dynamics in storage ring is greatly influenced bythe emission of radiation. Due to synchrotron radiation, electron motionbecomes dumped. However, dumping is counterbalanced in storage ringby quantum e ff ects. These two radiation e ff ects determine transverse elec-tron beam size, energy spread and bunch length. We would like now to useour ideas about dynamics and electrodynamics to consider in some greaterdetail the question: ”Why did the error in the synchrotron radiation theoryremain so long undetected?”In this chapter, we begin our more detailed study of the di ff erent aspectsof relativistic electrodynamics. For an arbitrary setup covariant calculationsof the radiation process is very di ffi cult. There are, however, circumstancesin which calculations can be greatly simplified. As example of such cir-cumstance is a synchrotron radiation setup. Similar to the non-relativisticasymptote, the ultrarelativistic asymptote also provides the essential sim-plicity of the covariant calculation. The reason is that the ultrarelativisticassumption implies the paraxial approximation. Since the formation lengthof the radiation is much longer than the wavelength, the radiation is emittedat small angles of order 1 /γ or even smaller, and we can therefore enforcethe small angle approximation. We assume that the transverse velocity issmall compared to the velocity of light. In other words, we use a secondorder relativistic approximation for the transverse motion. Instead of small(total) velocity parameter ( v / c ) in the non-relativistic case, we use a smalltransverse velocity parameter ( v ⊥ / c ). The next step is to analyze the lon-gitudinal motion, following the same method. We should remark that theanalysis of the longitudinal motion in a synchrotron radiation setup is verysimple. If we evaluate the transformations up to second order ( v ⊥ / c ) , therelativistic correction in the longitudinal motion does not appear in thisapproximation.According to covariant approach, the various relativistic kinematics e ff ectsconcerning to the synchrotron radiation setup, turn up in successive ordersof approximation.In the first order ( v ⊥ / c ). - relativity of simultaneity. Wigner rotation, which131n the ultrarelativistic approximation appears in the first order already, andresults directly from the relativity of simultaneity.In the second order ( v ⊥ / c ) . - time dilation. Relativistic correction in law ofcomposition of velocities, which already appears in the second order, andresults directly from the time dilation.The first order kinematics term ( v ⊥ / c ) plays an essential role only in thedescription of the coherent radiation from a modulated electron beam. Ina storage ring the distribution of the longitudinal position of the electronsin a bunch is essentially uncorrelated. In this case, the radiated fields dueto di ff erent electrons are also uncorrelated and the average power radiatedis a simple sum of the radiated power from individual electrons; that is wesum intensities, not fields. A motion of the single ultrarelativistic electronin a constant magnetic field, according to the theory of relativity, influencesthe kinematics terms of the second order ( v ⊥ / c ) only. We call z the observation distance along the optical axis of the system, while (cid:126) r fixes the transverse position of the observer. Using the complex notation,in this and in the following sections we assume, in agreement with Eq. (20),that the temporal dependence of fields with a certain frequency is of theform: (cid:126) E ∼ (cid:126) ¯ E ( z , (cid:126) r , ω ) exp( − i ω t ) . (24)With this choice for the temporal dependence we can describe a plane wavetraveling along the positive z -axis with (cid:126) E = (cid:126) E exp (cid:18) i ω c z − i ω t (cid:19) . (25)In the following we will always assume that the ultra-relativistic approxi-mation is satisfied, which is the case for SR setups. As a consequence, theparaxial approximation applies too. The paraxial approximation implies aslowly varying envelope of the field with respect to the wavelength. It istherefore convenient to introduce the slowly varying envelope of the trans-verse field components as (cid:126) (cid:101) E ( z , (cid:126) r , ω ) = (cid:126) ¯ E ( z , (cid:126) r , ω ) exp ( − i ω z / c ) . (26)132ote that with ultra relativistic accuracy one has (cid:126) n × ( (cid:126) n × (cid:126)β ) (cid:39) (cid:126)β − (cid:126) n . Intro-ducing angles θ x = x / z and θ y = y / z , the transverse components of theenvelope of the field in Eq. (23) in the far zone and in paraxial approximationcan be written as (cid:126) (cid:101) E ( z , (cid:126) r , ω ) = − i ω ec z ∞ (cid:90) −∞ dz (cid:48) exp [ i Φ T ] (cid:34)(cid:32) v x ( z (cid:48) ) c − θ x (cid:33) (cid:126) e x + (cid:32) v y ( z (cid:48) ) c − θ y (cid:33) (cid:126) e y (cid:35) (27)where the total phase Φ T is Φ T = ω (cid:34) s ( z (cid:48) ) v − z (cid:48) c (cid:35) + ω c (cid:104) z ( θ x + θ y ) − θ x x ( z (cid:48) ) − θ y y ( z (cid:48) ) + z (cid:48) ( θ x + θ y ) (cid:105) . (28)Here v x ( z (cid:48) ) and v y ( z (cid:48) ) are the horizontal and the vertical components ofthe transverse velocity of the electron, x ( z (cid:48) ) and y ( z (cid:48) ) specify the transverseposition of the electron as a function of the longitudinal position, (cid:126) e x and (cid:126) e y are unit vectors along the transverse coordinate axis. Finally, s ( z (cid:48) ) = vt (cid:48) ( z (cid:48) )is the longitudinal coordinate along the path. The electron is moving withvelocity (cid:126) v , whose magnitude is equal to v = ds / dt (cid:48) . Let us now discuss the case of the radiation from a single electron withan arbitrary angular deflection (cid:126)η and an arbitrary o ff set (cid:126) l with respect toa reference orbit defined as the path through the origin of the coordinatesystem, that is x (0) = y (0) = B = B ( z ), an initial o ff set x (0) = l x , y (0) = l y shifts the path ofan electron of (cid:126) l . Similarly, an angular deflection (cid:126)η = ( η x , η y ) at z = η x (cid:28) η y (cid:28) (cid:126) r ( z ) = (cid:126) r r ( z ) + (cid:126)η z + (cid:126) l ,(cid:126) v ( z ) = (cid:126) v r ( z ) + v (cid:126)η , (29)133here the subscript ‘r’ refers to the reference path. The pair ( (cid:126) r ( z ) , z ) givesa parametric description of the path of a single electron with o ff set (cid:126) l anddeflection (cid:126)η . The curvilinear abscissa on the path can then be written as s ( z ) = z (cid:90) dz (cid:48) + (cid:32) dxdz (cid:48) (cid:33) + (cid:32) dydz (cid:48) (cid:33) / (cid:39) z (cid:90) dz (cid:48) + (cid:32) dx r dz (cid:48) (cid:33) + (cid:32) dy r dz (cid:48) (cid:33) + (cid:16) η x + η y (cid:17) + η x dx r dz (cid:48) + η y dy r dz (cid:48) = s r ( z ) + η z + (cid:126) r r ( z ) · (cid:126)η , (30)where we expanded the square root around unity in the first passage, wemade use of Eq. (29), and of the fact that the curvilinear abscissa along thereference path is s r ( z ) (cid:39) z + (cid:82) z | d (cid:126) r r / dz (cid:48) | / (cid:126) (cid:101) E ( z , (cid:126) r , ω ) = − i ω ec z ∞ (cid:90) −∞ dz (cid:48) exp [ i Φ T ] × (cid:34)(cid:32) v x ( z (cid:48) ) c − ( θ x − η x ) (cid:33) (cid:126) e x + (cid:32) v y ( z (cid:48) ) c − ( θ y − η y ) (cid:33) (cid:126) e y (cid:35) , (31)where the total phase Φ T is Φ T = ω (cid:34) s r ( z (cid:48) ) v + η z (cid:48) v + v (cid:126) r r ( z (cid:48) ) · (cid:126)η − z (cid:48) c (cid:35) + ω c (cid:104) z ( θ x + θ y ) − θ x x r ( z (cid:48) ) − θ x η x z (cid:48) − θ x l x − θ y y ( z (cid:48) ) − θ y η y z (cid:48) − θ y l y + z (cid:48) ( θ x + θ y ) (cid:105) , (32)which can be rearranged as Φ T (cid:39) ω (cid:34) s r ( z (cid:48) ) v − z (cid:48) c (cid:35) − ω c ( θ x l x + θ y l y ) + ω c (cid:104) z ( θ x + θ y ) − θ x − η x ) x r ( z (cid:48) ) − θ y − η y ) y r ( z (cid:48) ) + z (cid:48) (cid:16) ( θ x − η x ) + ( θ y − η y ) (cid:17)(cid:105) . (33)134 .3 Undulator Radiation To generate specific synchrotron radiation characteristics, radiation is oftenproduced from special insertion devices called undulators. The resonanceapproximation, that can always be applied in the case of undulator radiationsetups, yields simplifications of the theory. This approximation does not re-place the paraxial one, but it is used together with it. It takes advantage ofanother parameter that is usually large, i.e. number of undulator periods N w (cid:29)
1. In this approximation, all undulator radiation is emitted within anangle much smaller than 1 /γ . This automatically selects observation anglesof interest. In fact, if we consider observation angles outside the di ff rac-tion angle, we obtain zero intensity with accuracy 1 / N w . In working out thecorresponding formula for the radiation field in the far zone using the limi-tation for the observation angles described above, we find that observationangles in the Fraunhofer phase factor can be taken to be zero and that thetransverse constrained electron trajectory does not a ff ect the undulator ra-diation. So, we are satisfied using the conventional approach for describingthe undulator radiation into the central cone, that is the practical situationof interest. Eq. (133) can be used to characterize the far field from an electron movingon any path. In this section we present a simple derivation of the frequencyrepresentation of the radiated field produced by an electron in the planarundulator. The magnetic field on the undulator axis has the form (cid:126) B ( z ) = (cid:126) e y B w cos( k w z ) , (34)The Lorentz force is used to derive the equation of motion of the electron inthe presence of a magnetic field. Integration of this equation gives v x ( z ) = − c θ s sin( k w z ) = − c θ s i (cid:2) exp( ik w z ) − exp( − ik w z ) (cid:3) . (35)Here k w = π/λ w , and λ w is the undulator period. Moreover, θ s = K /γ , where K is the deflection parameter defined as K = e λ w B w π mc , (36)135 being the electron mass at rest and H w being the maximal magnetic fieldof the undulator on axis.In this case the electron path is given by x ( z ) = r w cos( k w z ) , (37)where r w = θ s / k w is the oscillation amplitude.We write the undulator length as L = N w λ w , where N w is the number ofundulator periods. With the help of Eq. (133) we obtain an expression, validin the far zone: (cid:126) (cid:101) E = i ω ec z L / (cid:90) − L / dz (cid:48) exp [ i Φ T ] exp (cid:34) i ωθ z c (cid:35) (cid:34) K γ sin ( k w z (cid:48) ) (cid:126) e x + (cid:126)θ (cid:35) . (38)Here Φ T = (cid:32) ω c ¯ γ z + ωθ c (cid:33) z (cid:48) − K θ x γ ω k w c cos( k w z (cid:48) ) − K γ ω k w c sin(2 k w z (cid:48) ) , (39)where the average longitudinal Lorentz factor ¯ γ z is defined as¯ γ z = γ √ + K / . (40)The choice of the integration limits in Eq. (38) implies that the referencesystem has its origin in the center of the undulator.Usually, it does not make sense to calculate the intensity distribution fromEq. (38) alone, without extra-terms (both interfering and not) from the otherparts of the electron path. This means that one should have complete in-formation about the electron path and calculate extra-terms to be added toEq. (38) in order to have the total field from a given setup. Yet, we can findparticular situations for which the contribution from Eq. (38) is dominantwith respect to others. In this case Eq. (38), alone, has independent physicalmeaning.One of these situations is when the resonance approximation is valid. Thisapproximation does not replace the paraxial one, based on γ (cid:29)
1, but it is136 ig. 35. Geometry for radiation production from an undulator used together with it. It takes advantage of another parameter that is usuallylarge, i.e. the number of undulator periods N w (cid:29)
1. In this case, the integralin dz (cid:48) in Eq. (38) exhibits simplifications, independently of the frequency ofinterest due to the long integration range with respect to the scale of theundulator period.A well known expression for the angular distribution of the first harmonicfield in the far zone (see Appendix II for a detailed derivation) can be ob-tained from Eq. (38). Such expression is axis-symmetric, and can, therefore,be presented as a function of a single observation angle θ , where θ = θ x + θ y .One obtains the following distribution for the slowly varying envelope ofthe electric field: (cid:126) (cid:101) E = − K ω e c z γ A JJ exp (cid:34) i ωθ z c (cid:35) L / (cid:90) − L / dz (cid:48) exp (cid:34) i (cid:32) C + ωθ c (cid:33) z (cid:48) (cid:35) (cid:126) e x , (41)Here ω = ω r + ∆ ω , C = k w ∆ ω/ω r and137 r = k w c ¯ γ z , (42)is the fundamental resonance frequency. Finally A JJ is defined as A JJ = J (cid:32) K + K (cid:33) − J (cid:32) K + K (cid:33) , (43) J n being the n -th order Bessel function of the first kind. The integration overlongitudinal coordinate can be carried out leading to the well-known finalresult: (cid:126) (cid:101) E ( z , (cid:126)θ ) = − K ω eL c z γ A JJ exp (cid:34) i ωθ z c (cid:35) sinc (cid:34) L (cid:32) C + ωθ c (cid:33)(cid:35) (cid:126) e x , (44)where sinc( · ) ≡ sin( · ) / ( · ). Therefore, the field is horizontally polarized andazimuthal symmetric. Eq. (44) describes a field with spherical wavefrontcentered in the middle of the undulator. We have seen that in all generality the expression for the undulator field inthe far zone and in the ultrarelativistic (i.e. paraxial) approximation can bewritten as Eq. (114). Within the resonance approximation ( N w (cid:29)
1) for thefrequencies around the first harmonic it can be simplified to the well-knownexpression Eq. (44) where the field is horizontally polarized and azimuthalsymmetric. The divergence of this radiation is much smaller compared tothe angle 1 / ¯ γ z . The mathematical reason stems from the fact that the factorsin( · ) / ( · ) represents the well-known resonance character of the undulatorradiation. If we are interested in the angular width of the peak around theobservation angle θ =
0, we can introduce an angular displacement ∆ θ .Taking the first zero of the sin( · ) / ( · ) function at C = θ c . The cone with aperture θ c is usually called central cone. It can be foundthat θ c = / (2 N w ¯ γ z ) (cid:28) / ¯ γ z .Now we would like to understand what is the characteristic transverse sizeof the field distribution at the exist of the undulator. The radiation frommagnetic poles always interferes coherently at zero angle with respect toundulator axis. This interference is constructive within an angle of about (cid:112) c / ( ω L w ). We can estimate the interference size at the undulator exit asabout √ cL w /ω . On the other hand, the electron oscillating amplitude is given138y r w = c θ s / k w = cK / ( γ k w ). It follows that r w / ( cL w /ω ) = K ω/ ( L w k w γ ) = K / [ π N w (1 + K / (cid:28)
1, where we use the fact that γ = (1 + K /
2) ¯ γ z . Thisinequality holds independently of the value of K , because N w (cid:29)
1. Thus,the electron oscillating amplitude is always much smaller than the radiationdi ff raction size at the undulator exit.We consider the radiation associated with the first order term in the expan-sion of the Eq. (117) in power of v = K θ x ω/ ( γ k w c ). But in doing so we missall information about transverse electron trajectory in the phase factor Eq.(39) since the term K θ x ω cos( k w z (cid:48) ) / ( γ k w c ) is neglected. In this approximationthe electron orbit scale is always much smaller than the radiation di ff ractionsize and Eq. (44) gives fields very much in agreement with the dipole ra-diation theory. So we are satisfied using the non covariant approach whenconsidering the transverse electron motion.There are several points to be made about the above result. We have justexplained that in accounting only for the radiation in the central cone, wemiss all information about the transverse electron motion. To be completewe must add an analysis of the accelerated motion along the z -direction (i.e.along the undulator axis). We assume that the transverse velocity v ⊥ ( z ) issmall compared to the velocity of light c . We consider the small expansionparameter v ⊥ / c , neglecting terms of order ( v ⊥ / c ) , but not of order ( v ⊥ / c ) . Inother words we use a second order relativistic approximation for transversemotion. We should remark that the analysis of the longitudinal motion in theultrarelativistic approximation is much simpler than in the case of transversemotion. It is easy to see that the acceleration in the constant magnetic fieldyields an transverse electron velocity v ⊥ and ∆ v z = − v ( v ⊥ / c ) / z -axis.If we evaluate the transformations up to the second order ( v ⊥ / c ) , the rela-tivistic correction in the longitudinal motion does not appear. So one shouldnot be surprised to find that, in this approximation, there is no influence ofthe di ff erence between the non-covariant and covariant constrained electrontrajectories on the undulator radiation in the central cone. Eq. (44) can be generalized to the case of a particle with a given o ff set (cid:126) l and deflection angle (cid:126)η with respect to the longitudinal axis, assuming thatthe magnetic field in the undulator is independent of the transverse coordi-nate of the particle. Although this can be done using Eq. (31) directly, it issometimes possible to save time by getting the answer with some trick. Forexample, in the undulator case one takes advantage of the following geo-metrical considerations, which are in agreement with rigorous mathematical139erivation. First, we consider the e ff ect of an o ff set (cid:126) l on the transverse plane,with respect to the longitudinal axis z . Since the magnetic field experiencedby the particle does not change, the far-zone field is simply shifted by aquantity (cid:126) l . Eq. (44), can be immediately generalized by systematic substitu-tion of the transverse coordinate of observation, (cid:126) r with (cid:126) r − (cid:126) l . This meansthat (cid:126)θ = (cid:126) r / z must be substituted by (cid:126)θ − (cid:126) l / z , thus yielding (cid:101) E (cid:16) z ,(cid:126) l , (cid:126)θ (cid:17) = − K ω eL c z γ A JJ exp i ω z c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:126) lz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinc ω L (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:16) (cid:126) l / z (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) c . (45)Let us now discuss the e ff ect of a deflection angle (cid:126)η . Since the magnetic fieldexperienced by the electron is assumed to be independent of its transversecoordinate, the path followed is still sinusoidal, but the e ff ective undulatorperiod is now given by λ w / cos( η ) (cid:39) (1 + η / λ w . This induces a relative redshift in the resonant wavelength ∆ λ/λ ∼ η /
2. In practical cases of interestwe may estimate η ∼ /γ . Then, ∆ λ/λ ∼ /γ should be compared withthe relative bandwidth of the resonance, that is ∆ λ/λ ∼ / N w , N w being thenumber of undulator periods. For example, if γ > , the red shift due tothe deflection angle can be neglected in all situations of practical relevance.As a result, the introduction of a deflection angle only amounts to a rigidrotation of the entire system. Performing such rotation we should accountfor the fact that the phase factor in Eq. (45) is indicative of a sphericalwavefront propagating outwards from position z = · ) function in Eq. (45),instead, is modified because the rotation maps the point ( z , ,
0) into thepoint ( z , − η x z , − η y z ). As a result, after rotation, Eq. (45) transforms to (cid:101) E (cid:16) z , (cid:126)η,(cid:126) l , (cid:126)θ (cid:17) = − K ω eLA JJ c z γ exp i ω z c (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:126) lz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sinc ω L (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:16) (cid:126) l / z (cid:17) − (cid:126)η (cid:12)(cid:12)(cid:12)(cid:12) c (46)Finally, in the far-zone case, we can always work in the limit for l / z (cid:28)
1, thatallows one to neglect the term (cid:126) l / z in the argument of the sinc( · ) function,as well as the quadratic term in ω l / (2 cz ) in the phase. Thus Eq. (46) can befurther simplified, giving the generalization of Eq. (44) in its final form: (cid:101) E (cid:16) z , (cid:126)η,(cid:126) l , (cid:126)θ (cid:17) = − K ω eLA JJ c z γ exp (cid:34) i ω c (cid:32) z θ − (cid:126)θ · (cid:126) l (cid:33)(cid:35) sinc ω L (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:126)η (cid:12)(cid:12)(cid:12)(cid:12) c . (47)140t is clear from the above that, according to conventional synchrotron radi-ation theory, if we consider radiation from one electron at detuning C fromresonance, the introduction of a kick only amounts to a rigid rotation of theangular distribution along the new direction of the electron motion. This isplausible, if one keeps in mind that after the kick the electron has the samevelocity and emits radiation in the kicked direction owing to the Dopplere ff ect. After such rotation, Eq. (44) transforms into Eq. (47) According to the correct coupling of fields and particles, there is a remark-able prediction of undulator radiation theory concerning to the undulatorradiation from the single electron with and without kick. Namely, when akick is introduced, there is a red shift in the resonance wavelength of the un-dulator radiation in the velocity direction. To show this, let us consider thecovariant treatment, which makes explicit use of Lorentz transformations.When the kick is introduced, covariant particle tracking predicts a non-zero red shift of the resonance frequency, which arises because in Lorentzcoordinates the electron velocity decreases from v to v − v θ k / c .Now the formula Eq. (33) is not quite right, because we should have used notthe velocity of electron v but v − v θ k /
2. The shift in the total phase Φ T underthe integral Eq. (31) can be expressed by the formula ∆Φ T = ωθ k z (cid:48) / (2 c ),where we account to that v (cid:39) c in ultrarelativistic approximation.Suppose that without kick the electron moves along the constrained trajec-tory parallel to the undulator axis. The field which produces this electron inthe far zone is given by Eq. (44). Referring back to the Eq. (47), we see thatthe conventional undulator radiation theory gives the following expressionfor radiation field after the kick (cid:126) (cid:101) E = − K ω eL c z γ A JJ exp (cid:34) i ωθ z c (cid:35) sinc L C + ω (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:126)θ k (cid:12)(cid:12)(cid:12)(cid:12) c (cid:126) e x . (48)The covariant equations say that, when the kick is introduced, the radiationfield in question is given by the formula141 (cid:101) E = − K ω eL c z γ A JJ exp (cid:34) i ωθ z c (cid:35) sinc L C + ωθ k c + ω (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:126)θ k (cid:12)(cid:12)(cid:12)(cid:12) c (cid:126) e x , (49)This formula has nearly, but not quite the same form as Eq. (48), the dif-ference consisting in the term ωθ k / (2 c ) in the argument of sinc function.Attention must be called to the di ff erence in resonance frequency betweenthe undulator radiation setup with and without the kick. Remembering thedefinition of the detuning parameter C = k w ∆ ω/ω r , we can write the redshift in resonance frequency as ∆ ω/ω r = − ω r θ k / (2 k w c ). With this we alsopointed out that the red shift can be written as ∆ ω/ω r = − γ θ k / (1 + K / θ k that is, however, multiplied by thelarge factor γ .We are now ready to investigate, more generally, what form the field ex-pression takes under the introduction of a kick. Suppose that, without kick,the electron moves along the trajectory with angle (cid:126)η with respect to theundulator axis. The field produced by this electron is given by Eq. (47). Welet (cid:126)θ k be the kick angle of the electron with respect to its initial motion. Theconventional approach gives the following expression for the field after thekick (cid:126) (cid:101) E = − K ω eL c z γ A JJ exp (cid:34) i ωθ z c (cid:35) sinc L C + ω (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:126)η − (cid:126)θ k (cid:12)(cid:12)(cid:12)(cid:12) c (cid:126) e x . (50)In contrast, the covariant approach gives (cid:126) (cid:101) E = − K ω eL c z γ A JJ exp (cid:34) i ωθ z c (cid:35) sinc L C + ωθ k c + ω (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)θ − (cid:126)η − (cid:126)θ k (cid:12)(cid:12)(cid:12)(cid:12) c (cid:126) e x , (51)Now this all leads to an interesting situation. According to the conventionaltheory, the resonance wavelength depends only on the observation anglewith respect to the electron velocity direction. Equation (50) says that for anykick angle (cid:126)θ k and for any angle (cid:126)η between the undulator axis and the initialelectron velocity direction, the radiation along the velocity direction has no142ed shift. We would like to emphasize a very important di ff erence betweenconventional and covariant theory. The result of the covariant approachEq. (51) clearly depends on the absolute value of the kick angle θ k and theradiation along the velocity direction has the red shift only when the kickangle has nonzero value.We must conclude that when we accelerate the electron in the lab frameupstream the undulator, the information about this acceleration is includedinto the covariant trajectory. One way to demonstrate incompatibility between the standard approachto relativistic electrodynamics, which deals with the usual Maxwell’s equa-tions, and particle trajectories calculated by using non-covariant particletracking, is to make a direct laboratory test of synchrotron radiation theory.In other words, we are stating here that, despite the many measurementsdone during decades, synchrotron radiation theory is not an experimentallywell-confirmed theory.Let us analyze the potential for exploiting synchrotron radiation sources inorder to confirm the predictions of corrected synchrotron radiation theory.The emittance of the electron beam in new generation synchrotron radia-tion sources is small enough, so that one can neglect finite electron beamsize and angular divergence in the soft X-ray wavelength range, and suchsynchrotron radiation source can be examined under the approximation ofa filament electron beam. This allows us to take advantage of analyticalpresentations for single electron synchrotron radiation fields.The basic setup for a test experiment is sketched in Fig 36. The soft X-ray un-dulator beam line should be tuned to a minimum photon energy (typicallythis limit is related with the so called ”water window” wavelength range).The radiation pulse goes through a monochromator filter F and its energyis subsequently measured by the detector. No precise monochromatizationof the undulator radiation is required in this case: a monochromator linewidth ∆ ω/ω (cid:39) − is su ffi cient. In order for proposed test experiment tobe curried out, it is necessary to control the beam kicking e.g. by correctormagnet. In the case of no kick the maximum pulse energy registered bythe detector will coincide with the monochromator line tuned to resonance.When the kick is introduced the conventional synchrotron radiation theorystill predicts a zero red shift in the resonance wavelength. In contrast to this,one of the immediate consequences of the corrected theory is the occurrenceof a non-zero red shift of the resonance wavelength.The proposed experimental procedure is relatively simple, because is based143 ig. 36. Basic setup for the proposed critical experimental test of synchrotron radi-ation theory with third generation light source. Top: the case for an electron beamwithout kick. Bottom: the case for the electron beam kicked by an angle η . In bothcases, the X-ray pulse is filtered by a monochromator and the total energy recordedby a detector as a function of the undulator detuning. on relative measurements in the (electron beam) velocity direction with andwithout transverse kick. Such a measurement is critical, in the sense that theprediction of conventional theory is the absence of red shift, and has neverbeen performed to our knowledge (1) . Consider a single relativistic electron moving on a circular orbit. It is worthto underline the di ff erence between the geometry which we use and thegeometry used in most synchrotron radiation textbooks for the treatmentof bending magnet radiation. The observer in the standard treatment isassumed to be located in a vertical plane tangent to the circular trajectory atthe origin, at an angle θ above the level of the orbit. In other words, in thisgeometry the z axis is not fixed, but depends on the observer’s position. Notethat the geometry of the electron motion has a cylindrical symmetry, withthe vertical axis going through the center of the circular orbit. Because of thissymmetry, in order to calculate spectral and angular photon distributions, itis not necessary to consider an observer at a more general location. However,since the wavefront is not spherical, this way of proceeding can hardly help144o obtain the phase of the field distribution on a plane perpendicular to afixed z axis. We can use Eq. (133) to calculate the far zone field of radiation from arelativistic electron moving along an arc of a circle. Assuming a geometrywith a fixed z we can write the transverse position of the electron as afunction of the curvilinear abscissa s as (cid:126) r ( s ) = − R (1 − cos( s / R )) (cid:126) e x (52)and z ( s ) = R sin( s / R ) (53)where R is the bending radius.Since the integral in Eq. (133) is performed along z we should invert z ( s ) inEq. (53) and find the explicit dependence s ( z ): s ( z ) = R arcsin( z / R ) (cid:39) z + z R (54)so that (cid:126) r ( z ) = − z R (cid:126) e x , (55)where the expansion in Eq. (54) and Eq. (55) is justified, once again, in theframework of the paraxial approximation.With Eq. (133) we obtain the radiation field amplitude in the far zone: (cid:126) (cid:101) E = i ω ec z ∞ (cid:90) −∞ dz (cid:48) e i Φ T (cid:18) z (cid:48) + R θ x R (cid:126) e x + θ y (cid:126) e y (cid:19) (56)where Φ T = ω θ x + θ y c z + γ c + θ x + θ y c z (cid:48) (cid:18) θ x Rc (cid:19) z (cid:48) + (cid:18) R c (cid:19) z (cid:48) (cid:21) . (57)One can easily reorganize the terms in Eq. (57) to obtain Φ T = ω θ x + θ y c z − R θ x c (cid:32) γ + θ x + θ y (cid:33) + (cid:32) γ + θ y (cid:33) ( z (cid:48) + R θ x )2 c + ( z (cid:48) + R θ x ) R c (cid:35) . (58)With redefinition of z (cid:48) as z (cid:48) + R θ x under integral we obtain the final result: (cid:126) (cid:101) E = i ω ec z e i Φ s e i Φ ∞ (cid:90) −∞ dz (cid:48) (cid:18) z (cid:48) R (cid:126) e x + θ y (cid:126) e y (cid:19) × exp (cid:40) i ω (cid:34) z (cid:48) γ c (cid:16) + γ θ y (cid:17) + z (cid:48) R c (cid:35)(cid:41) , (59)where Φ s = ω z c (cid:16) θ x + θ y (cid:17) (60)and Φ = − ω R θ x c (cid:32) γ + θ x + θ y (cid:33) . (61)In standard treatments of bending magnet radiation, the phase term exp( i Φ )is absent. In fact, the horizontal observation angle θ x is always equal tozero. The reason for this is that most textbooks focus on the calculation ofthe intensity radiated by a single electron in the far zone, which involvesthe square modulus of the field amplitude but do not analyze, for instance,situations like source imaging. Our case of interest is an ultrarelativistic electron accelerating in a circle.As already remarked, in conventional (non-covariant) particle tracking thedescription of the dynamical evolution in the lab frame is based on theuse of the absolute time convention. In this case simultaneity is absolute,and we only need one set of synchronized clocks in the lab frame, to be146sed for the description of the accelerated motion. However, the use ofthe absolute time convention automatically implies the use of much morecomplicated field equations, and these equations are di ff erent for each valueof the particle velocity i.e. for each point along its path. This is the reason toprefer the covariant approach within the framework of both dynamics andelectrodynamics.We want to solve the electrodynamics problem based on Maxwell’s equa-tions in their usual form. In this case we should analyze the particle evo-lution within the framework of special relativity, where the problem ofassigning Lorentz coordinates to the lab frame in the case of acceleratingmotion is complicated. The only possibility to introduce Lorentz coordinatesin this situation consists in introducing individual coordinate systems (i.e.individual rule-clock structure) for each point of the path.We start by considering an electron moving along a circular trajectory thatlies in the ( x , z )-plane and tangent to the z axis. Because of cylindrical sym-metry, in order to calculate spectral and angular photon distributions, it isnot necessary to consider an observer at general location. The observer isassumed to be located in the vertical plane tangent to the circular trajec-tory at the origin. In ultrarelativistic (paraxial) approximation we evaluatetransformations working only up to the order of v x / c . The restriction tothis order provides an essential simplicity of calculations. We can interpretmanipulation with rule-clock structure in the lab frame simply as a changeof variables according to the transformation x d = γ x x , t d = t /γ x + γ x xv x / c .We are dealing with a second order approximation and γ x = + v x / (2 c ).The overall combination of Galilean transformation and variable changesactually yields to the transverse Lorentz transformation (see section 3.3.5for more detail). Since the Galilean transformation, completed by the in-troduction of the new variables, is mathematically equivalent to a Lorentztransformation, it obviously follows that transforming to new variablesleads to the usual Maxwell’s equations.In order to keep Lorentz coordinates in the lab frame, as discussed before,we need only to perform a clock resynchronization by introducing the timeshift ∆ t = t d − t = − [ v x / (2 c )] t + xv x / c . The relativistic correction to the par-ticle’s o ff set ” x ” does not appear in this expansion order, but only in orderof v x / c and x d = x in our case of interest. Although we have only shownthat time shift in one rather special case, the result is right for any o ff set and(transverse) velocity direction: ∆ t = t d − t = − [ | (cid:126) v ⊥ | / (2 c )] z (cid:48) / c + (cid:126) r ⊥ · (cid:126) v ⊥ / c .To finish our analysis we need only find a relativistic correction to the lon-gitudinal motion. We remark again that if we evaluate the transformationsup to the second order ( v ⊥ / c ) , the relativistic correction in the longitudinalmotion does not appear in this approximation. We have demonstrated thecovariant method that can be used for any trajectory - a general way of147unding what happens directly in space-frequency domain and in paraxialapproximation.Let us now see how to apply this covariant method to a special situation.Let’s use our knowledge of the relativistically correct method for calculatingsynchrotron radiation emission to find the photon angular-spectral densitydistributions from a bending magnet. In the ultrarelativistic approximation,we have a uniform acceleration of the electron a = v / R = c / R in thetransverse direction. We can, then, write velocity and o ff set of the electronas follows v x = at = az (cid:48) / v = az (cid:48) / c , x = at / = az (cid:48) / (2 c ). We have nowall quantities we wanted. Let us put them all together in relativistic timeshift: ∆ t = t d − t = − a z (cid:48) / (2 c ) + a z (cid:48) / (2 c ) =
0. There is no di ff erence! Wedo not need to use covariant particle tracking for derivation of the bendingmagnet radiation. Why should that be? Usually, such a beautiful cancellationis found to stem from a deep underlying principle. Nevertheless, in thiscase there does not appear to be any such profound implication. This is acoincidence. It is because we have deal with uniform acceleration in thetransverse direction using a second order (paraxial) approximation whenan electron is moving along an arc of a circle.This cancellation is not surprising, if one analyzes the general expressionfor the radiation field from bending magnet in the far zone Eq.(59). In ourprevious discussion of undulator radiation, we learned that the relativisticcorrection appears only when the transverse electron trajectory is includedin the total phase Φ T under the integral Eq.(133). Referring back to Eq.(28) forthe phase factor Φ T , we see that the term which depends on the transverseposition of the electron can be written as exp i ( ω/ c )[ θ x x ( z (cid:48) ) + θ y y ( z (cid:48) )]. Weconclude that the observation angle in the total phase factor under theintegral must be related with the contribution of the transverse electrontrajectory. Now look at Eq.(59). This equation includes only the observationangle θ y in the phase factor under the integral. This means that the transverseconstraint motion of the electron in the bending magnet does not a ff ectsynchrotron radiation. So we are justified using a non-covariant approachfor considering the constrained electron motion along the nominal orbit in( x , z )-plane.We point out that the cancellation in relativistic time shift and the indepen-dence of the Fraunhofer propagator (to be more precise, in space-frequencydomain we are dealing with a paraxial approximation of Green’s function ofnonhomogeneous Helmholtz equation) on the observation angle θ x in thefar zone can be regarded as the two sides of the same coin: they are manifes-tation of the cylindrical symmetry when an electron is moving along an arcof a circle. Because of cylindrical symmetry, in order to calculate spectral andangular photon distributions in the far zone, it is not necessary to consideran observer at a general location. The observer is assumed to be located in148he vertical plane tangent to the circular trajectory at the origin. In this caseobservation angle θ x = θ y is above the levelof the orbit. In other words, in this very special geometry the z -axis is notfixed, but depends on the observer position. However, this way of proceed-ing can hardly help to obtain radiation fields in the near zone. Indeed, inthe near zone we are dealing with the Fresnel propagator, which obviouslydepends on the constrained motion of the electron. We use far-zone argu-ments only to show that there is no influence of the di ff erence between thenon-covariant and covariant trajectories on the synchrotron radiation frombending magnets. The cancellation in the relativistic time shift leads to thesame outcome in the near zone as it must be. Up to this point we considered an electron moving along a circular trajectorythat lies in the ( x , z )-plane and tangent to the z axis. The phase di ff erencein the fields will be determined by the position of the observer and by theelectron trajectory. Let us now discuss the bending magnet radiation froma single electron with arbitrary angular deflection and o ff set with respect tothe nominal orbit.Approximation for the electron path Eq. (29), Eq. (30) can be used to char-acterize the field from an electron moving on any trajectory. Using Eq. (54)and Eq. (55) an approximated expression for s ( z ) can be found: s ( z ) = z + z R + z η x R + z η x + z η y (cid:126) v ( z ) = (cid:18) − vzR + v η x (cid:19) (cid:126) e x + (cid:16) v η y (cid:17) (cid:126) e y (63)and (cid:126) r ( z ) = (cid:32) − z R + η x z + l x (cid:33) (cid:126) e x + (cid:16) η y z + l y (cid:17) (cid:126) e y . (64)It is evident that the o ff sets l x and l y are always subtracted from x and y respectively: a shift in the particle trajectory on the vertical plane isequivalent to a shift of the observer in the opposite direction. With this inmind we introduce angles ¯ θ x = θ x − l x / z and ¯ θ y = θ y − l y / z to obtain149 (cid:101) E = i ω ec z ∞ (cid:90) −∞ dz (cid:48) e i Φ T (cid:32) z (cid:48) + R ( ¯ θ x − η x ) R (cid:126) e x + ( ¯ θ y − η y ) (cid:126) e y (cid:33) (65)and Φ T = ω ¯ θ x + ¯ θ y c z + ω c (cid:32) γ + (cid:0) ¯ θ x − η x (cid:1) + (cid:16) ¯ θ y − η y (cid:17) (cid:33) z (cid:48) + (cid:32) ω ( ¯ θ x − η x )2 Rc (cid:33) z (cid:48) + (cid:18) ω R c (cid:19) z (cid:48) . (66)One can easily reorganize the terms in Eq. (66) to obtain Φ T = ω ¯ θ x + ¯ θ y c z − ω R ( ¯ θ x − η x )2 c × (cid:32) γ + ( ¯ θ y − η y ) + ( ¯ θ x − η x ) (cid:33) + (cid:32) γ + ( ¯ θ y − η y ) (cid:33) ω (cid:0) z (cid:48) + R ( ¯ θ x − η x ) (cid:1) c + ω (cid:0) z (cid:48) + R ( ¯ θ x − η x ) (cid:1) R c . (67)Redefinition of z (cid:48) as z (cid:48) + R ( ¯ θ x − η x ) gives the result (cid:126) (cid:101) E = i ω ec z e i Φ s e i Φ ∞ (cid:90) −∞ dz (cid:48) (cid:18) z (cid:48) R (cid:126) e x + ( ¯ θ y − η y ) (cid:126) e y (cid:19) × exp (cid:40) i ω (cid:34) z (cid:48) γ c (cid:16) + γ ( ¯ θ y − η y ) (cid:17) + z (cid:48) R c (cid:35)(cid:41) , (68)where Φ s = ω z c (cid:16) ¯ θ x + ¯ θ y (cid:17) (69)and Φ = − ω R ( ¯ θ x − η x )2 c (cid:32) γ + ( ¯ θ y − η y ) + ( ¯ θ x − η x ) (cid:33) . (70)In the far zone we can neglect terms in l x / z and l y / z , which leads to150 (cid:101) E = i ω ec z e i Φ s e i Φ ∞ (cid:90) −∞ dz (cid:48) (cid:18) z (cid:48) R (cid:126) e x + (cid:16) θ y − η y (cid:17) (cid:126) e y (cid:19) × exp (cid:40) i ω (cid:34) z (cid:48) γ c (cid:18) + γ (cid:16) θ y − η y (cid:17) (cid:19) + z (cid:48) R c (cid:35)(cid:41) , (71)where Φ s = ω z c (cid:16) θ x + θ y (cid:17) (72)and Φ o (cid:39) − ω R ( θ x − η x )2 c (cid:32) γ + ( θ y − η y ) + ( θ x − η x ) (cid:33) − ω c ( l x θ x + l y θ y ) . (73)It is clear from the above that the field distribution in the far zone dependsonly on the observation angle with respect to the electron velocity direction.According to the conventional (incorrect) coupling of fields and particles,there is a prediction of radiation theory concerning to the bending magnetradiation from a single electron with and without kick. Namely, when a kickis introduced, there is a rigid rotation of the angular distribution in the farzone. Let us discuss the covariant treatment, which makes explicit use of Lorentztransformations. Consider the bending magnet radiation from a single elec-tron with a kick with respect to the nominal orbit in ( x , z )-plane. In thiscase, we additionally have a translation along the y -axis with constantvelocity v y = v θ k . We can, then, write the o ff set of the electron as fol-lows y = θ k z (cid:48) . Let’s put velocity and o ff set in the relativistic time shift: ∆ t = t d − t = − θ k z (cid:48) / (2 c ) + θ k z (cid:48) / c = θ k z (cid:48) / (2 c ). So, the shift in the totalphase under the integral along the path can be expressed by the formula ∆Φ T = ωθ k z (cid:48) / (2 c ). The result agrees with our red shift calculation in theundulator case when the kick is introduced, as it must be.We would like to make a historical note. The di ff erence between covariantand non-covariant particle trajectories was never understood. So, accel-erator physicists did not appreciate that there was a contribution to thesynchrotron radiation from relativistic kinematics e ff ects. The question nowarises how can storage rings actually operate. The point is that this exam-ple deals with a situation where electron beam kinetics is determined by151he emission of synchrotron radiation from bending magnets. However, be-cause of the cylindrical symmetry, covariant and non-covariant solutionsfor the electron motion along an arc of a circle yield similar properties ofsynchrotron radiation except the following modification. The covariant ap-proach predicts a non-zero red shift of the critical frequency, which ariseswhen there are perturbations of the electron motion in the vertical direction.But synchrotron radiation from bending magnets is emitted within a widerange of frequencies, and the output intensity is not sensitive on the redshift. We shall now discuss the situation where there are n arbitrary spaced kick-ers, all di ff erent from one another in terms of the rotation angle introduced.Let us consider how we may apply covariant particle tracking in this cir-cumstance, and try to understand what is happening when we have forexample an undulator downstream of the kicker setup. One might say thatthis is getting ridiculous. If one wants to calculate the radiation from theundulator one should take into account all kicks in the electron trajectory,from the generation of the electron. However, this situation is not surprising,if one analyzes the general expression for the radiation field from a singleelectron Eq.(23). In fact, we should note that, in general, one needs to knowthe entire history of the electron from t (cid:48) = −∞ to t (cid:48) = ∞ since the integra-tion in Eq.(23) is performed between these limits. However, this statementshould be interpreted physically, depending on the situation under study:integration should in fact be performed from and up to times when theelectron does not contribute to the field anymore.We should pointed out that it is the electrodynamics theory, which ultimatelydecides what part of the particle trajectory is important for calculating undu-lator radiation and what part can be neglected. The most important, generalstatement concerning the relevant part of the particle trajectory, is that itmust be calculated according to the covariant method (if one wants to usethe usual Maxwell’s equations).Let us consider the ultrarelativistic assumption 1 /γ (cid:28)
1, which is verifiedfor synchrotron radiation setups. In general, the introduction of a smallparameter in any theory brings simplifications. The ultrarelativistic approx-imation implies a paraxial approximation and Eq.(23) can be simplified toEq.(133). Suppose that we take a situation in which the rotation angle of thefirst bending magnet upstream of the undulator is much larger than 1 /γ . Inother words, we now consider an electron moving along a standard syn-chrotron radiation setup. The electron enters the setup via a bending magnet,152asses through a straight section, an undulator, and another straight sec-tion. Finally, it leaves the setup via another bend. Note that, although theintegration in Eq.(133) is performed from −∞ to ∞ , the only (edge) part ofthe trajectory into the bending magnets contributing to the integral is oforder of the radiation formation length L f . Mathematically, it is reflectedin the fact that Φ T ( z (cid:48) ) in Eq.(133) exhibits more and more rapid oscillationsas z (cid:48) becomes larger than the formation length. At the critical wavelengththe formation length is simply of order of R /γ , R being the radius of thebend. That simply corresponds to an orbiting angular interval ∆ θ (cid:39) /γ .Typically, the critical wavelength of the radiation from a bending magnet insynchrotron radiation source is about 0.1 nm and the formation length inthis case is only few millimeters.Note that for ultrarelativistic systems in general, the formation length isalways much longer than the radiation wavelength. This counterintuitiveresult follows from the fact that for ultrarelativistic systems one cannotlocalize sources of radiation within a macroscopic part of the trajectory.The formation length can be considered as the longitudinal size of a singleelectron source. It does not make sense at all to talk about the position whereelectromagnetic signals are emitted within the formation length. This meansthat, as concerns the radiative process in the bending magnet, we cannotdistinguish between radiation emitted at point A and radiation emitted atpoint B when the distance between these two points is shorter than theformation length L f . Let us now consider the case of a straight section oflength L inserted between the bending magnet and the undulator. One canstill use the same reasoning considered for the bend to define a region ofthe trajectory where it does not make sense to distinguish between di ff erentpoints. As in bending magnet case, the observer sees a time compressedmotion of the source and in the case of straight motion the apparent timecorresponds to an apparent distance (cid:111) γ . At the critical wavelength thebending magnet formation length L f (cid:39) R /γ is simply order of the straightline formation length (cid:111) γ .Intuitively, bending magnets act like switchers for the ultrarelativistic elec-tron trajectory. We consider the case when switchers are presented in theform of bending magnets, but other setups can be considered where switch-ers have di ff erent physical realizations. The only feature that these di ff erentrealizations must have in common, by definition of switcher, is that theswitching process must depends exponentially on the distance from thebeginning of the process. Then, a characteristic length d s can be associatedto any switcher. Consider, for example, a plasma accelerator where an elec-tron is accelerated with high-gradient fields. In this case it is the acceleratoritself that switches on the relativistic electron trajectory, since accelerationin the GeV range takes place within a few millimeters only. In the (soft)X-ray range the acceleration distance d a is shorter than the formation length153 ig. 37. Standard undulator radiation setup. When the electron passes througha bending magnet there is the synchrotron radiation, washing out the (Frank-Ginzburg) fields of the fast moving charge. At the exit of the bending magnetwe have ”naked” electron. There is a process of formation of the field-dressedelectron within the formation length from the very beginning of the straight sectiondownstream the bending magnet. The field-dressed ultrarelativistic electron has avisible transverse size of order a few microns for third generation synchrotronradiation sources. (cid:111) γ for the following straight section. In this particular case length d a playsthe role of the characteristic length of the switcher d s , which switch on theultrarelativistic electron trajectory.Let us now return to our consideration of the standard synchrotron radiationsetup and let us analyze the radiation process in an insertion device (un-dulator). We have actually the ”creation” of the relativistic electron withina distance of order (cid:111) γ from the very beginning of the straight section up-stream the undulator. It is assumed that the length of the straight section L is much longer than the formation length (cid:111) γ that is clearly always thecase in the X-ray range. When the switching distance d s (cid:46) (cid:111) γ (cid:28) L , thenature of the switcher is not important for describing the radiation from theundulator installed within the straight section (Fig. 37).Downstream of the switcher we have a uniformly moving electron. Thefields associated to an electron with a constant velocity exhibit an interest-154ng behavior when the speed of the charge approaches that of light. Namely,in the space-frequency domain there is an equivalence of the fields of a rel-ativistic electron and those of a beam of electromagnetic radiation. In fact,for a rapidly moving electron we have nearly equal transverse and mutu-ally perpendicular electric and magnetic fields. These are indistinguishablefrom the fields of a beam of radiation. This virtual radiation beam has amacroscopic transverse size of order (cid:111) γ (see Appendix III). At the exit of theswitcher we have a ”naked” (or ”field-free”) electron i.e. an electron that isnot accompanied by virtual radiation fields. There is a process of formationof the ”field-dressed” electron (i.e. the formation of the fields from a fastmoving charge) within the distance of order (cid:111) γ from the very beginning ofthe straight section downstream of the switcher.The electron trajectory being divided into two essentially di ff erent parts:before and after the switcher. If we accelerate the electron in the lab frameupstream of the switcher, the information about this acceleration is includedinto the first part of the covariant trajectory. But this acceleration prehistory(together with the fields of the ultrarelativistic electron) is washed out dur-ing the switching process and at the entrance of the straight section we havea ”naked” electron.We start with the description of the field formation process along the straightsection downstream of the switcher, based on the covariant approach. Firstof all we have to synchronize distant clocks within the lab frame. The syn-chronization procedure that follows is the usual Einstein synchronizationprocedure. It is assumed that in the Lorentz lab frame the electron proceedsfollowing a rectilinear trajectory with velocity v . This assumption is used asinitial condition. Then we can analyze situation downstream the switcherby using the usual Maxwell’s equations.When one analyzes the process of ”field-dressed” electron formation fromthe viewpoint of the non covariant approach, one assumes the same initialconditions (rectilinear trajectory with velocity v ) for the electron motion.Then one solves the electrodynamics problem of fields formation by us-ing the usual Maxwell’s equations. We already mentioned that the type ofclock synchronization which results in time coordinate t in an electron tra-jectory (cid:126) x ( t ) is never discussed in accelerator physics. However, we knowthat the usual Maxwell’s equations are only valid in the Lorentz frame. Thenon covariant approach is obviously based on a definite synchronizationassumption, but this is actually a hidden assumption. In our case of in-terest, within the lab frame the Lorentz coordinates are then automaticallyenforced.So one should not be surprised to find that in this simple case of rectilinearmotion (i.e. in the situation when we have only deal with the description155nitial conditions) there is no di ff erence between covariant and non covariantcalculations of the initial conditions at the undulator entrance.Because of the characteristics of undulator radiation, in order to calculatethe radiation field within the central cone, we only need to account for thelongitudinal accelerated motion. So we are satisfied using a non covariantapproach for considering the constrained motion along the undulator. Weconclude that it does not matter which approach is used to describe the stan-dard synchrotron radiation setup. The two approaches, treated accordingto Einstein’s or absolute time synchronization conventions give the sameresult for the radiation within the central cone.Let us now see what happens with a weak dipole magnet (a kicker), which isinstalled in the straight section upstream of the undulator and is character-ized by a small kick angle ( γθ k ) (cid:28)
1. What do we expect for the undulatorradiation? At first glance the situation is similar to the switcher setup andthe electron trajectory is again divided into two parts: before and after thekicker. The most important di ff erence, however, is that electrodynamicsnow dictates that both trajectories are important for the calculation of theundulator radiation. When the electron passes through the kicker there isno synchrotron radiation (to be more precise, in this case radiation is indis-tinguishable from the self-electromagnetic fields of the electron), washingout the virtual radiation fields like in the switcher case. We expect that anelectron that passes through a kicker is still ”field-dressed”, but we havean electron whose fields has been perturbed, and now include informationabout the acceleration.According to the conventional theory, as usual for Newtonian kinematics,the Galilean vectorial law of addition of velocities is actually used. Non-covariant particle dynamics shows that the direction of the electron trajec-tory changes after the kick, while its speed remains unvaried. In contrast,covariant particle tracking, which is based on the use of Lorentz coordinates,yields di ff erent results for the trajectory of the electron. The electron speeddecreases from v to v (1 − θ k / θ k .According to the correct coupling of fields and particles, there is a remark-able prediction of synchrotron radiation theory concerning the setup de-156cribed above. Namely, there is a red shift of the resonance frequency of theundulator radiation in the kicked direction. To show this, let us first con-sider the covariant treatment, which makes explicit use of Lorentz transfor-mations. When the kick is introduced, covariant particle tracking predictsa non-zero red shift of the resonance frequency, which arises because inLorentz coordinates the electron velocity decreases from v to v − v θ k / c . The red shift in the resonance frequency can be expressed by theformula ∆ ω r /ω r = − γ θ k / (1 + K / The presence of red shift in bending magnet radiation automatically impliesthe same problem for conventional cyclotron radiation theory. In the ultra-relativistic limit, there are well-known analytical formulas that describe thespectral and angular distribution of cyclotron radiation emitted by an elec-tron moving in a constant magnetic field having a non-relativistic velocitycomponent parallel to the field, and an ultrarelativistic velocity componentperpendicular to it. According to the conventional approach, exactly as forthe bending magnet case, the angular-spectral distribution of radiation is afunction of the total velocity of the particle due, again, to the Doppler e ff ect.In contrast, the covariant approach predicts a non-zero red shift of the criticalfrequency, which arises when there are perturbations of the electron motionin the magnetic field direction. It should be note that cyclotron-synchrotronradiation emission is one of the most important processes in plasma physicsand astrophysics and our corrections are important for a much wider partof physics than that of synchrotron sources. Let us discuss in some detail the relativistic cyclotron radiation. Here weshall only give some final results and discuss their relation with the conven-tional synchrotron radiation theory from bending magnet. In the case of anuniform translation motion with non-relativistic velocity along the magneticfield direction (Fig. 38), a widely accepted (in astrophysics) expression forthe angular and spectral distributions of radiation from an ultra-relativisticelectron on a helical orbit is given by (2) (cid:126) (cid:101) E ( χ, α ) ∼ (cid:40) (cid:126) e x ( ξ + ψ ) K / ω ω c (cid:32) + ψ ξ (cid:33) / ig. 38. Geometry for radiation production from helical motion. − i (cid:126) e y ( ξ + ψ ) / ψ K / ω ω c (cid:32) + ψ ξ (cid:33) / (cid:41) , (74)where K / and K / are the modified Bessel functions, ξ = /γ , ψ = χ − α ,( χ is the angle between (cid:126) v and (cid:126) B and α that between (cid:126) n and (cid:126) B ); the angle ψ isclearly the angular distance between the direction of the electron velocity (cid:126) v and the direction of observation (cid:126) n . Here the ω c is defined by 3 eB γ / (2 mc ).Actually we have already discussed radiation from an ultrarelativistic elec-tron on a helical orbit in the previous section. Equation Eq. (71) is the resultwe worked out above for the bending magnet radiation from a single elec-tron with angular deflection with respect to nominal orbit. Eq. ( 74) doesnot look the same as Eq. (71). It will, however, if we now define the smalldeflection angle η y = π/ − χ and the observation angle θ y = π/ − α (theobserver is also assumed to be located in the vertical plane tangent to thetrajectory i.e. θ x , η x = ∞ (cid:90) x sin[(3 / α ( x + x / dx = (1 / √ K / ( α ) , ∞ (cid:90) cos[(3 / α ( x + x / dx = (1 / √ K / ( α ) . (75)Then, making the necessary variable changes, the formula reduces to Eq.(74). 158he calculation leading to Eq. ( 74) is rather elaborate. It is therefore desir-able to have an independent derivation. The simplest way of analyzing theradiation for an ultrarelativistic helical motion makes use of the theory ofrelativity and involves practically no calculations. The way for computingthe radiation in the case of uniform translation is simple. One describes acomplicated situation by finding a reference system where the analysis is al-ready done (radiation in the case of circular motion) and transforms back tothe old reference frame. The reference frame S (cid:48) in which the electron movesin circular motion can be transformed to a frame S in which the electron pro-ceeds following a helical trajectory. Eq. ( 74) holds, indeed, in the frame S fora particle whose velocity is ( v x , v y , v z ) = ( v sin χ sin φ, v cos χ, v sin χ cos φ ).The Lorentz transformation, which leads to the value v y = v cos χ for the y -component of the velocity yields ( v x , v y , v z ) = ( v (cid:48) sin φ (cid:48) /γ y , v y , v (cid:48) cos φ (cid:48) /γ y ),where γ y = / (cid:113) − v y / c , v (cid:48) is the velocity of the electron in the frame S (cid:48) andthe phase angle φ (cid:48) = φ is invariant. This means that, in order to end up in S with a transverse (to the magnetic field direction) velocity v ⊥ = v sin χ , onemust start in S (cid:48) with v (cid:48) = γ y v sin χ . In the ultrarelativistic approximation γ ⊥ = / (1 − v ⊥ / c ) (cid:29)
1, and one finds the simple result v = v (cid:48) , so thata Lorentz boost with non-relativistic velocity v y leads to a rotation of theparticle velocity (cid:126) v of the angle η = π/ − χ (cid:39) v y / c (cid:28) η is smalland v (cid:39) c , we would write γ y sin χ (cid:39) S (cid:48) , one obtains the resultthat the e ff ect of a boost amounts to a rigid rotation of the angular-spectraldistribution of the radiation emitted by the electron moving with velocity v on a circle that is, once more, Eq. ( 74) (3) .From above argument, one could naively expect that the covariant approachpredicts a zero red shift of the critical frequency. But when the situation isdescribed as we have done it here, there does not seem to be any paradoxat all; it comes out quite naturally that the covariant way of analyzing theradiation for helical motion considered above is based on the Lorentz trans-formation. In other words, within the lab frame the Lorentz coordinates areautomatically enforced. It assumed that in the Lorentz lab frame the electronproceeds following a helical trajectory with velocity v . This is employedas initial condition. In the ultrarelativistic approximation a Lorentz boostalong the field direction with non relativistic velocity v y leads to the circularmotion of the electron with the same velocity v . Thus the boost will leavethe radiation properties unchanged. Now what about the value of the elec-tron velocity on a helical orbit in the Lorentz lab frame? How this velocityis defined? It is generally believed that (cid:126) x ( t ) = (cid:126) x ( t ) cov and this is the reasonwhy in the literature there is no distinction between the two (non covariantand covariant) approaches to describe the electron motion on a helix. If wewill keep the Lorentz coordinate system in the lab frame downstream of thekicker, we will find that the covariant velocity on the helical orbit after the159ick decreases from v to v − v θ k / v = v − v θ k / We now want to point out that there are two di ff erent sets of initial conditionsresulting in the same uniform translation along the magnetic field directionin the Lorentz lab frame. We start by considering an electron moving alonga circular trajectory that lies in the ( x , z )-plane. We then rotate the magneticfield vector (cid:126) B in the ( y , z )-plane by an angle θ , assuming that rotation angleis small. We consider a situation in which the electron is in uniform motionwith velocity v θ along the magnetic field direction. It is clear that if weconsider the radiation from an electron moving on a circular orbit, theintroduction of the magnetic field vector rotation will leave the radiationproperties unchanged. This is plausible, if one keeps in mind that afterrotating the bending magnet, the electron has the same velocity and emitsradiation in the velocity direction owing to the Doppler e ff ect. After therotation, correction to the curvature radius R is only of order θ and can beneglected.Now we consider another situation. Let us see what happens with a kicker,which is installed in the straight section upstream of the bending magnetand is characterized by a small kick angle ( γθ ) (cid:28)
1. When the kick in the y direction is introduced, there is a red shift of the critical wavelength whicharise because, according to Einstein’s addition velocities law, the electronvelocity decreases from v to v − v θ / ω c can be expressed by the formula ∆ ω c /ω c = − (3 / γ θ . We seea second order correction θ that is, however, multiplied by a large factor γ .The result of the covariant approach clearly depends on the absolute valueof the kick angle θ and the radiation along the velocity direction has a redshift only when the kick angle has nonzero value.The di ff erence between these two situations, ending with a final uniformtranslation along the direction of the magnetic field is very interesting. Itcomes about as the result of the di ff erence between two Lorentz coordinatesystems in the lab frame. By trying to accelerate the electron upstream thebending magnet we have changed Lorentz coordinates for that particularsource. We know that in order to keep a Lorentz coordinates system in thelab frame after the kick we need to perform a clock resynchronization. So we160 ig. 39. Two sets of initial conditions resulting in the same uniform motion alongthe magnetic field direction in the case of absolute time coordinatization in the labframe. The magnitude of the electron velocity and the orientation of the velocityvector with respect to the magnetic field vector are identical in both setups. should expect the electron velocity to be changed. The di ff erence betweenthe two setups is understandable. When we do not perturb the electronmotion upstream of the bending magnet, no clock resynchronization takesplace, while when we do perturb the motion, clock resynchronization isintroduced.We would now like to describe an apparent paradox. A paradox is a state-ment that is seemingly contradictory, but, in reality, expresses truth withoutcontradiction. Because of our usage of Galilean transformations within elec-trodynamics we have some apparent paradoxes. An analysis of paradoxesleads to a better understanding of the four-dimensional geometrical signif-icance of the concepts of space and time in the theory of relativity.When the situation is described as we have done it here, there doesn’t seemto be any paradox at all. The argument that the di ff erence between these twosituations, ending with a final uniform translation along the magnetic fieldsdirection, is paradoxical can be summarized in the following way: in the caseof absolute time coordinatization in the lab frame, the initial conditions at thebending magnet entrance are apparently identical. In fact, the magnitude ofthe electron velocity and the orientation of the velocity vector with respectto the magnetic field vector are identical in both setups. We must conclude161hat when we accelerate the electron in the lab frame upstream the bendingmagnet, the information about this acceleration is not included into the non-covariant trajectory. Where is the information about the electron accelerationrecorded in the case of absolute time coordinatization? Since an electron isa structureless particle, the situation seems indeed paradoxical.Electrodynamics deals with observable quantities. Let us consider the mea-surement of the red shift in the bending magnet radiation from our kickedelectron. We can measure the accurate value of the red shift using a spec-trometer in the lab frame, and this leads to a description of the setup in thespace-frequency domain.Suppose we have a uniformly moving electron. The fields associated toan electron with constant velocity exhibit an interesting behavior when thespeed of the charge approaches that of light. Namely, in the space-frequencydomain there is an equivalence of the fields of a relativistic electron and thoseof a beam of electromagnetic radiation. In fact, for a rapidly moving electronwe have nearly equal transverse and mutually perpendicular electric andmagnetic fields. These are indistinguishable from the fields of a beam of radi-ation. This virtual radiation beam has a macroscopic transverse size of order (cid:111) γ . An ultrarelativistic electron at synchrotron radiation facilities, emittingat nanometer-wavelengths (we work in the space-frequency domain) hasindeed a macroscopic transverse size of order of 1 µ m. The field distributionof the virtual radiation beam is described by the Ginzburg-Frank formula(see Appendix III).When the electron passes through a kicker, its fields are perturbed, and nowinclude information about the acceleration. According to the old kinematics,the orientation of the virtual radiation phase front is unvaried. However,Maxwell’s equations do not remain invariant with respect to Galilean trans-formations and, as discuss throughout this paper, the choice of the oldkinematics implies using anisotropic field equations. As a result, the phasefront remains plane but the direction of propagation is not perpendicu-lar to the phase front. In other words, the radiation beam motion and theradiation phase front normal have di ff erent directions. Then, having cho-sen the absolute time synchronization, electrodynamics predicts that thevirtual radiation beam propagates in the kicked direction with the phasefront tilt θ k . This is the key to the ”paradox” discussed here. The informa-tion about the electron acceleration is recorded in the perturbation of theself-electromagnetic fields of the electron. Mathematically information isrecorded in the phase front tilt of the virtual radiation beam.162 .7 Bibliography and Notes
1. It should be note that results of the beam splitting experiment at LCLSconfirm our correction for spontaneous undulator emission [48]. It appar-ently demonstrated that after a modulated electron beam is kicked on a largeangle compared to the divergence of the XFEL radiation, the modulationwavefront is readjusted along the new direction of the motion of the kickedbeam. Therefore, coherent radiation from the undulator placed after thekicker is emitted along the kicked direction practically without suppression(see the Chapter 8 for more detail). In the framework of the conventionaltheory, there is a second outstanding puzzle concerning the beam splittingexperiment at the LCLS. In accordance with conventional undulator radia-tion theory, if the modulated electron beam is at perfect resonance withoutkick, then after the kick the same modulated beam must be at perfect reso-nance in the velocity direction. However, experimental results clearly showthat when the kick is introduced there is a red shift in the resonance wave-length. The maximum power of the coherent radiation is reached when theundulator is detuned to be resonant to the lower longitudinal velocity afterthe kick [48]. It should be remarked that any linear superposition of a givenradiation field from single electrons conserves single-particle characteris-tics like parametric dependence on undulator parameters and polarization.Consider a modulated electron beam kicked by a weak dipole field beforeentering a downstream undulator. Radiation fields generated by this beamcan be seen as a linear superposition of fields from individual electrons. Nowexperimental results clearly show that there is a red shift in the resonancewavelength for coherent undulator radiation when the kick is introduced.It follows that the undulator radiation from the single electron has red shiftwhen the kick is introduced as well. This argument suggests that results ofthe beam splitting experiment in reference [48] confirm our correction forspontaneous undulator emission.2. A widely accepted expression for the angular and spectral distributions ofradiation from an ultra-relativistic electron on a helical orbit were calculatedin [43,44]. At present, relativistic cyclotron radiation results are textbookexamples (see e.g. [45]) and do not require a detail description.3. The covariant way of analyzing the radiation for helical motion wasconsidered in [46]. It is generally believed that (cid:126) x ( t ) = (cid:126) x ( t ) cov and this isthe reason why in the [46] there is no distinction between the two (noncovariant and covariant) approaches to describe the electron motion on ahelix downstream of the kicker setup.163 Relativity and X-Ray Free Electron Lasers
In the previous chapter we attempted to answer the question of why theerror in radiation theory should have so long remained undetected. Ac-cording to covariant approach, the various relativistic kinematics e ff ectsconcerning to the synchrotron radiation setup, turn up in successive ordersof approximation. Instead of small (total) velocity parameter ( v / c ) in thenon-relativistic case, we use a small transverse velocity parameter ( v ⊥ / c ). Amotion of the single ultrarelativistic electron in a constant magnetic field,according to the theory of relativity, influences the kinematics terms of thesecond order ( v ⊥ / c ) only. It is demonstrated that due to a combination ofthe ultrarelativistic (i.e. paraxial) approximation and a very special symme-try of the conventional synchrotron radiation setup there is a cancellation ofthe second order relativistic kinematics e ff ects. That means that the sponta-neous synchrotron radiation does not show any sensitivity to the di ff erencebetween covariant and non-covariant particle trajectories.But in the 21st century with the operation XFELs this situation changes.An XFEL is an example where the first order kinematics term ( v ⊥ / c ) playsan essential role in the description of the XFEL radiation and, in this case,covariant coupling of fields and particles predicts an e ff ect in completecontrast to the conventional treatment.In this chapter we present a critical reexamination of existing XFEL theory.It would be well to illustrate error in standard coupling fields and particlesin XFEL physics by considering the relatively simple example, wherein theessential physical features are not obscures by unnecessary mathematicaldi ffi culties. The main emphasize of this chapter is on coherent undulatorradiation from the modulated electron beam. This chapter mainly addressedto readers with limiting knowledge of accelerator and XFEL physics.The usual XFEL theory based on the use of old Newtonian kinematics forparticle dynamics and the Einstein’s kinematics for the electrodynamics. Infact, the usual theoretical treatment of relativistic particle dynamics involvesonly a corrected Newton’s second law and is based on the use Galileanedition of velocities. For rectilinear motion of the modulated electron beam,non-covariant and covariant approaches produce the same trajectories, andMaxwell’s equations are compatible with the result of conventional particletracking. However, one of the consequences of the relativity of simultaneity(i.e. mixture of positions and time) is a di ff erence between covariant andnon-covariant kinematics of a modulated electron beam in a given magnetic164 ig. 40. A well-known result of conventional (non-covariant) particle tracking. Amicro-bunching electron beam passing through a weak dipole magnet (kicker) andundergoes a kick of an angle θ k . The propagation axis of the electron beam isdeflected, while the wavefront orientation is preserved. field. The theory of relativity shows that discussed above di ff erence relatedwith the acceleration along curved trajectories.There are several cases where the first order relativistic e ff ect can occur inXFELs, mainly through the introduction of an trajectory kick (1) . The mostelementary of the e ff ect that represents a crucial test of the correct couplingfields and particles is a problem involves the production of coherent un-dulator radiation by modulated ultrarelativistic electron beam kicked by aweak dipole field before entering a downstream undulator.It would be well to begin with bird’s view of some of the main results. Letus now move on to consider the predictions of the existing XFEL theoryin the case of non-collinear electron beam motion. As well-known result ofconventional particle tracking states that after an electron beam is kicked bya weak dipole magnet there is a change in the trajectory of the electron beam,while the orientation of the modulated wavefront remains as before (Fig.40). In other words, the kick results in a di ff erence between the directionsof the electron motion and the normal to the modulation wavefront (i.e. ina wavefront tilt). In existing XFEL theory the wavefront tilt is consideredas real. According to conventional treatment, a transverse kick does notchange the orientation of a modulation wavefront, and hence suppresses165 ig. 41. A result of covariant particle tracking. In the ultrarelativistic limit, the ori-entation of the modulation wavefront, i.e. the orientation of plane of simultaneity,is always perpendicular to the electron beam velocity when the evolution of themodulation electron beam is treated using Lorentz coordinates. The theory of rel-ativity dictates that a modulated electron beam in the ultrarelativistic limit has thesame kinematics, in Lorentz coordinates, as laser beam. According to Maxwell’sequations, the wavefront of a laser beam is always orthogonal to the propagationdirection. the radiation emitted in the direction of the electron motion (2) .The covariant approach within the framework of both mechanics and elec-trodynamics predicts an e ff ect in complete contrast to the conventionaltreatment. Namely, in the ultrarelativistic limit, the wavefront of modula-tion, that is a plane of simultaneity, is always perpendicular to the electronbeam velocity (Fig. 41). As a result, the Maxwell’s equations predict strongemission of coherent undulator radiation from the modulated electron beamin the kicked direction. Experiments show that this prediction is, in fact, true (3) . The results of XFEL experiments demonstrated that even the direction ofemission of coherent undulator radiation is beyond the predictive power ofthe conventional theory (4) .It is worth remarking that the absent of a dynamical explanation for themodulation wavefront readjusting in the Lorentz coordinatization has dis-turbed some XFEL experts. We only wish to emphasize that a good wayto think of the modulation wavefront readjusting is to regard it as a resultof transformation to a new time variable in the framework of the Galilean166”single frame”) electrodynamics. Let us suppose that a modulated electron beam moves along the z -axis of aCartesian ( x , y , z ) system in the lab frame. As an example, suppose that themodulation wavefront is perpendicular to the velocity v z . How to measurethis orientation? A moving electron bunch changes its position with time.The natural way to do this is to answer the question: when does each electroncross the x -axis of the reference system? If we have adopted a method fortiming distant events (i.e. a synchronization convention), we can also spec-ify a method for measuring the orientation of the modulation wavefront:if electrons located at the position with maximum density cross the x -axissimultaneously at certain position z , then the modulation wavefront is per-pendicular to z -axis. In other words, the modulation wavefront is definedas a plane of simultaneous events (the events being the arrival of particleslocated at maximum density): in short, a ”plane of simultaneity”.Let us formulate the initial conditions in the lab frame in terms of orientationof the modulation wavefront and beam velocity. Suppose that v z is thevelocity of the comoving frame R ( τ ) with respect to the lab frame K ( τ ) alongthe common z -axis in positive direction. In the lab frame we select a specialtype of coordinate system, a Lorentz coordinate system to be precise. Withina Lorentz frame (i.e. inertial frame with Lorentz coordinates), Einstein’ssynchronization of distant clocks and Cartesian space coordinates ( x , y , z )are enforced. In order to have this, we impose that R is connected to K bythe Lorentz boost L ( (cid:126) v z ), with (cid:126) v z , which transforms a given four vector event X in space-time into X R = L ( (cid:126) v z ) X .We now consider the acceleration of the beam in the lab frame up to velocity v x along the x -axis. The question arises how to assign synchronization inthe lab frame after the beam acceleration. Before acceleration we picked aLorentz coordinate system. Then, after the acceleration, the beam velocitychanges of an small value v x along the x -axis. Without changing synchroniza-tion in the lab frame after the particle acceleration we have a complicatedsituation as concerns electrodynamics of moving charges. As a result of suchboost, the transformation of time and spatial coordinates has the form of aGalilean transformation. In order to keep a Lorentz coordinate system in thelab frame after acceleration, one needs to perform a clock resynchronizationby introducing the time shift t → t + xv x / c . This form of time transformationis justified by the fact that we are dealing with a first order approximation.Therefore, v x / c is so small that v x / c can be neglected and one arrives at thecoordinate transformation x → x + v x t , t → t + xv x / c . The Lorentz trans-167ormation just described di ff ers from a Galilean transformation just by theinclusion of the relativity of simultaneity, which is only relativistic e ff ect thatappearing in the first order in v x / c . The relation X R = L ( (cid:126) v z ) L ( (cid:126) v x ) X presents astep-by-step change from the lab reference frame K ( τ + d τ ) to K ( τ ) and thento the proper reference frame R . The shift in the time when electrons locatedat the position with maximum density cross the x -axis of the lab frame ∆ t = xv x / c has the e ff ect of a rotation the modulation wavefront on theangle v z ∆ t / x = v z v x / c in the first order approximation. In ultrarelativisticlimits, v z (cid:39) c , and the modulation wavefront rotates exactly as the velocityvector (cid:126) v .What does this wavefront readjustment mean in terms of measurements? Inthe absolute time coordinatization the simultaneity of a pair of events hasabsolute character. The absolute character of the temporal coincidence oftwo events is a consequence of the absolute time synchronization conven-tion. According to this old kinematics, the modulation wavefront remainsunvaried. However, according to the covariant approach we establish a cri-terion for the simultaneity of events, which is based on the invariance of thespeed of light. It is immediately understood that, as a result of the motionof electrons along the x axis (i.e. along the plane of simultaneity before theboost) with the velocity v x , the simultaneity of di ff erent events is no longerabsolute, i.e. independent of the kick angle θ = v x / c . This reasoning is inanalogy with Einstein’s train-embankment thought experiment.The wavefront orientation has no exact objective meaning, because the rela-tivity of simultaneity takes place. The statement that the wavefront orienta-tion has objective meaning to within a certain accuracy can be visualized bythe picture of wavefront in the proper orientation with approximate angleextension (blurring) given by ∆ θ (cid:39) v z ( v x / c ). This relation specifies the lim-its within which the non relativistic theory can be applied. In fact, it followsthat for a very non relativistic electron beam for which v z / c is very small,the angle ”blurring” becomes very small too. In this case angle of wavefronttilt θ = v x / v z is practically sharp ∆ θ/θ (cid:39) v z / c (cid:28)
1. This is a limiting caseof non-relativistic kinematics. The angle ”blurring” is a peculiarity of rela-tivistic beam motion. In the ultrarelativistic limit when v z (cid:39) c , the wavefronttilt has no exact objective meaning at all since, due to the finiteness of thespeed of light, we cannot specify any experimental method by which thistilt could be ascertained. The most elementary of the e ff ect that represents a crucial test of the cor-rect coupling fields and particles is a problem involves the production of168oherent undulator radiation by modulated ultrarelativistic electron beamkicked by a weak dipole field before entering a downstream undulator. Wewant to study the process of emission of coherent undulator radiation fromsuch setup.The key element of a XFEL source is the udulator, which forces the electronsto move along curved periodical trajectories. There are two popular undu-lator configurations: helical and planar. To understand the basic principlesof undulator source operation, let us consider the helical undulator. Themagnetic field on the axis of the helical undulator is given by (cid:126) B w = (cid:126) e x B w cos( k w z ) − (cid:126) e y B w sin( k w z ) , (76)where k w = π/λ w is the undulator wavenumber and (cid:126) e x , y are unit vectorsdirected along the x and y axes. We neglected the transverse variation of themagnetic field. It is necessary to mention that in XFEL engineering we dealwith a very high quality of the undulator systems, which have a su ffi cientlywide good-field-region, so that our studies, which refer to a simple modelof undulator field nevertheless yields a correct quantitative description inlarge variety of practical problems. The Lorentz force (cid:126) F = − e (cid:126) v × (cid:126) B w / c is usedto derive the equation of motion of electrons with charge − e and mass m inthe presence of magnetic field m γ dv x dt = ec v z B y = − ec v z B w sin( k w z ) , m γ dv y dt = − ec v z B x = − ec v z B w cos( k w z ) . (77)Introducing ˜ v = v x + iv y , dz = v z dt we obtain m γ d ˜ vdz = − i ec ( B x + iB y ) = − i ec B w exp( − ik w z ) . (78)Integration of the latter equation gives˜ vc = θ w exp( − k w z ) , (79)where θ w = K /γ and K = eB w / ( k w mc ) is the undulator parameter. The ex-plicit expression for the electron velocity in the field of the helical undulatorhas the form 169 v = c θ w [ (cid:126) e x cos( k w z ) − (cid:126) e y sin( k w z )] , (80)This means that the reference electron in the undulator moves along theconstrained helical trajectory parallel to the z axis. As a rule, the electronrotation angle θ w is small and the longitudinal electron velocity v z is closeto the velocity of light, v z = (cid:112) v − v ⊥ (cid:39) v (1 − θ w / (cid:39) c .Let us consider a modulated ultrarelativistic electron beam moving alone the z axis in the field of the helical undulator. In the present study we introducethe following assumptions. First, without kick the electrons move alongconstrained helical trajectories in parallel with the z axis. Second, electronbeam density at the undulator entrance is simply n = n ( (cid:126) r ⊥ )[1 + a cos ω ( z / v z − t )] , (81)where a = const . In other words we consider the case in which there are novariation in amplitude and phase of the density modulation in the trans-verse plane. Under these assumptions the transverse current density maybe written in the form (cid:126) j ⊥ = − e (cid:126) v ⊥ ( z ) n ( (cid:126) r ⊥ )[1 + a cos ω ( z / v z − t )] . (82)Even through the measured quantities are real, it is generally more conve-nient to use complex representation, starting with real (cid:126) j ⊥ , one defines thecomplex transverse current density: j x + i j y = − ec θ w n ( (cid:126) r ⊥ ) exp( − ik w z )[1 + a cos ω ( z / v z − t )] . (83)The transverse current density has an angular frequency ω and two wavestraveling in the same direction with variations exp i ( ω z / v z − k w z − ω t ) andexp − i ( ω z / v z + k w z − ω t ) will add to give a total current proportional toexp( − ik w z )[1 + a cos ω ( z / v z − t )]. The factor exp i ( ω z / v z − k w z − ω t ) indicates afast wave, while the factor exp − i ( ω z / v z + k w z − ω t ) indicates a slow wave. Theuse of the word ”fast” (”slow”) here implies a wave with a phase velocityfaster (slower) than the beam velocity.Having defined the sources, we now should consider the electrodynamicsproblem. Maxwell equations can be manipulated mathematically in manyways in order to yield derived equations more suitable for certain appli-cations. For example, from Maxwell equations Eq.(4) we can obtain an170quation which depends only on the electric field vector (cid:126) E (in Gaussianunits): c (cid:126) ∇ × ( (cid:126) ∇ × (cid:126) E ) = − ∂ (cid:126) E /∂ t − π∂(cid:126) j /∂ t . (84)With the help of the identity (cid:126) ∇ × ( (cid:126) ∇ × (cid:126) E ) = (cid:126) ∇ ( (cid:126) ∇ · (cid:126) E ) − ∇ (cid:126) E (85)and Poisson equation (cid:126) ∇ · (cid:126) E = πρ (86)we obtain the inhomogeneous wave equation for (cid:126) Ec ∇ (cid:126) E − ∂ (cid:126) E /∂ t = π c (cid:126) ∇ ρ + π∂(cid:126) j /∂ t . (87)Once the charge and current densities ρ and (cid:126) j are specified as a function oftime and position, this equation allows one to calculate the electric field (cid:126) E ateach point of space and time. Thus, this nonhomogeneous wave equationis the complete and correct formula for radiation. However we want toapply it to still simpler circumstance in which second term (or, the currentterm) in the right-hand side provides the main contribution to the valueof the radiation field. It is relevant to remember that our case of interestis the coherent undulator radiation and the divergence of this radiation ismuch smaller compared to the angle 1 /γ . It can be shown that when thiscondition is fulfilled the gradient term, 4 π c (cid:126) ∇ ρ , in the right-hand side of thenonhomoheneous wave equation can be neglected. Thus we consider thewave equation c ∇ (cid:126) E − ∂ (cid:126) E /∂ t = π∂(cid:126) j ⊥ /∂ t . (88)We wish to examine the case when the phase velocity of the current wave isclose to the velocity of light. This requirement may be met under resonancecondition ω/ c = ω/ v z − k w . This is the condition for synchronism between thetransverse electromagnetic wave and the fast transverse current wave withthe propagation constant ω/ v z − k w . With the current wave traveling withthe same phase speed as electromagnetic wave, we have the possibility ofobtaining a spatial resonance between electromagnetic wave and electrons.171f this the case, a cumulative interaction between modulated electron beamand transverse electromagnetic wave in empty space takes place. We aretherefore justified in considering the contributions of all the waves exceptthe synchronous one to be negligible as long as the undulator is made of alarge number of periods.Here follows an explanation of the resonance condition which is elemen-tary in the sense that we can see what is happening physically. The fieldof electromagnetic wave has only transverse components, so the energy ex-change between the electron and electromagnetic wave is due to transversecomponent of the electron velocity. For e ff ective energy exchange betweenthe electron and the wave, the scalar product − e (cid:126) v ⊥ · (cid:126) E should be kept nearlyconstant along the whole undulator length. We see that required synchro-nism k w + ω/ c − ω/ v z = λ w / v z = λ/ ( c − v z ), where λ = π/ω is the radiation wavelength. This tells us that the angle betweenthe transverse velocity of the particle (cid:126) v ⊥ and the vector of the electric field (cid:126) E remains nearly constant. Since v z (cid:39) c this resonance condition may bewritten as λ (cid:39) λ w / (2 γ z ) = λ w (1 + K ) / (2 γ ).We will use an adiabatic approximation that can be taken advantage of, inall practical situations involving XFELs, where the XFEL modulation wave-length is much shorter than the electron bunch length σ b , i.e. σ b ω/ c (cid:29) c ∇ (cid:126) ¯ E + ω (cid:126) ¯ E = − π i ω(cid:126) ¯ j ⊥ , (89)where (cid:126) ¯ j ⊥ ( (cid:126) r , ω ) is the Fourier transform of the current density (cid:126) j ⊥ ( (cid:126) r , t ). Thesolution can be represented as a weighted superposition of solutions cor-responding to a unit point source located at (cid:126) r (cid:48) . The Green function for theinhomogeneous Helmholtz equation is given by (for unbounded space andoutgoing waves)4 π G ( (cid:126) r , (cid:126) r (cid:48) , ω ) = | (cid:126) r − (cid:126) r (cid:48) | exp (cid:20) i ω c | (cid:126) r − (cid:126) r (cid:48) | (cid:21) , (90)172ith | (cid:126) r − (cid:126) r (cid:48) | = (cid:112) ( x (cid:48) − x ) + ( y (cid:48) − y ) + ( z (cid:48) − z ) . With the help of this Greenfunction we can write a formal solution for the field equation as: (cid:126) ¯ E = (cid:90) d (cid:126) r (cid:48) G ( (cid:126) r , (cid:126) r (cid:48) ) (cid:20) − π i ω c (cid:126) ¯ j ⊥ (cid:21) . (91)This is just a mathematical description of the concept of Huygens’ secondarysources and waves, and is of course well-known, but we still recalled how itfollows directly from the Maxwell’s equations. We may consider the ampli-tude of the beam radiated by plane of oscillating electrons as a whole to bethe resultant of radiated spherical waves. This is because Maxwell’s theoryhas no intrinsic anisotropy. The electrons lying on the plane of simultaneitygives rise to spherical radiated wavelets, and these combine according toHuygens’ principle to form what is e ff ectively a radiated wave. If the planeof simultaneity is the xy -plane (i.e. beam modulation wavefront is perpen-dicular to the z - axis), then the Huygens’ construction shows that planewavefronts will be emitted along the z -axis.In summary: according to Maxwell’s electrodynamics, coherent radiationis always emitted in the direction normal to the modulation wavefront.We already stressed that Maxwell’s equations are valid only in a Lorentzreference frame, i.e. when an inertial frame where the Einstein synchroniza-tion procedure is used to assign values to the time coordinates. Einstein’stime order should be applied and kept in consistent way in both dynamicsand electrodynamics. Our previous description implies quite naturally thatMaxwell’s equations in the lab frame are compatible only with covarianttrajectories (cid:126) x cov ( t ), calculated by using Lorentz coordinates and, therefore,including relativistic kinematics e ff ects.Let us go back to the modulated electron beam, kicked transversely withrespect to the direction of motion, that was discussed before. Conventionalparticle tracking shows that while the electron beam direction changes afterthe kick, the orientation of the modulation wavefront stays unvaried. Inother words, the electron motion and the wavefront normal have di ff erentdirections. Therefore, according to conventional coupling of fields and par-ticles that we deem incorrect, the coherent undulator radiation in the kickeddirection produced in a downstream undulator is expected to be dramati-cally suppressed as soon as the kick angle is larger than the divergence ofthe output coherent radiation.In order to estimate the loss in radiation e ffi ciency in the kicked directionaccording to the conventional coupling of fields and particles, we make theassumption that the spatial profile of the modulation is close to that of theelectron beam and has a Gaussian shape with standard deviation σ . A mod-173lated electron beam in an undulator can be considered as a sequence ofperiodically spaced oscillators. The radiation produced by these oscillatorsalways interferes coherently at zero angle with respect to the undulator axis.When all the oscillators are in phase there is, therefore, strong emission inthe direction θ =
0. If we have a triangle with a small altitude r (cid:39) θ z andlong base z , than the diagonal s is longer than the base. The di ff erence is ∆ = s − z (cid:39) z θ /
2. When ∆ is equal to one wavelength, we get a minimum inthe emission. This is because in this case the contributions of various oscilla-tors are uniformly distributed in phase from 0 to 2 π . In the limit for a smallsize of the electron beam, σ →
0, the interference will be constructive withinan angle of about ∆ θ (cid:119) (cid:112) c / ( ω L w ) = / ( √ π N w γ z ) (cid:28) /γ , where L w = λ w N w is the undulator length. In the limit for a large size of the electron beam, theangle of coherence is about ∆ θ (cid:119) c / ( ωσ ) instead. The boundary betweenthese two asymptotes is for sizes of about σ dif (cid:119) √ cL w /ω . The parameter ωσ / ( cL w ) can be referred to as the electron beam Fresnel number. It is worthnoting that, for XFELs, the transverse size of electron beam σ is typicallymuch larger than σ dif (i.e electron beam Fresnel number is large). Thus, wecan conclude that the angular distribution of the radiation power in thefar zone has a Gaussian shape with standard deviation σ θ (cid:119) c / ( √ ωσ ).However, still according to the conventional treatment, after the electronbeam is kicked we have the already-mentioned discrepancy between di-rection of the electron motion and wavefront normal. Then, the radiationintensity along the new direction of the electron beam can be approximatedas I (cid:119) I exp[ − θ k / (2 σ θ )], where I is the on-axis intensity without kick and θ k is the kick angle. The exponential suppression factor is due to the tiltof the modulation wavefront with respect to the direction of motion of theelectrons.We presented a study of very idealized situation for illustrating the di ff er-ence between conventional and covariant coupling of fields and particles.We solved the dynamics problem of the motion of a relativistic electrons inthe prescribed force field of weak kicker magnet by working only up to theorder of γθ k . This approximation is of particular theoretical interest becauseit is relatively simple and at the same time forms the basis for understandingrelativistic kinematic e ff ects such as relativity of simultaneity.Let us discuss the region of validity of our small kick angle approxima-tion θ k γ (cid:28)
1. Since in XFELs the Fresnel number is rather large, we canalways consider a kick angle which is relatively large compared to thedivergence of the output coherent radiation, and, at the same time, it is rel-atively small compared to the angle 1 /γ . In fact, from ωσ / ( cL w ) (cid:29)
1, withsome rearranging, we obtain σ θ (cid:39) c / ( ω σ ) (cid:28) c / ( ω L w ). Then we recall that (cid:112) c / ( ω L w ) = / ( √ π N w γ z ) (cid:28) /γ . Therefore, the first order approximationused to investigate the kicker setup in this chapter is of practical interest inXFEL engineering. 174 ig. 42. Basic setup for the experimental test of XFEL theory. The correct couplingfields and particles predicts an e ff ect in complete contrast to the conventionaltreatment. According to the covariant approach, in the ultrarelativistic limit, thewavefront of modulation is always perpendicular to the electron beam velocity.As a result, the Maxwell’s equations predict strong emission of coherent undulatorradiation from the modulated electron beam in the kicked direction. Experimentsshow that this prediction is, in fact, true. It is one of the aims of this chapter is to demonstrate the kind of experimen-tal predictions we are expecting from our corrected radiation theory. Weworked out a very simple case in order to illustrate all the essential physi-cal principles very clearly. Surprisingly, the first order approximation usedto investigate the kicker setup in this section has also important practicalapplications.Above we have shown that our correct coupling of fields and particles pre-dicts an e ff ect in complete contrast to the conventional treatment. Namely,in the ultrarelativistic limit, the plane of simultaneity, that is wavefront ori-entation of the modulation, is always perpendicular to the electron beamvelocity. As a result, we predict strong emission of coherent undulator radi-ation from the modulated electron beam in the kicked direction.XFEL experts actually witnessed an apparent wavefront readjusting due tothe relativistic kinematics e ff ect, but they never drew this conclusion. Inthis book, we are actually first in considering the idea that results of theconventional theory of radiation by relativistically moving charges are notconsistent with the principle of relativity. In previous literature, identifica-175ion of the trajectories in the source part of the usual Maxwell’s equationswith the trajectories calculated by conventional particle tracking in the (”sin-gle”) lab frame has always been considered obvious. Now everything fitstogether, and our theory shows the existence of coherent radiation in thekicked direction. In existing literature theoretical analysis is presented, of an XFEL drivenby an electron beam with wavefront tilt, and this analysis is based on theexploitation of usual Maxwell’s equations and standard simulation codes.Using only a kicker setup (i.e. without undulator radiation setup) we candemonstrate that the coupling fields and particles in the conventional XFELtheory is intrinsically incorrect.The existing XFEL theory based on the use of the absolute time convention(i.e. old kinematics) for particle dynamics. Here we will give a simple proofof the conflict between conventional particle tracking and Maxwell’s elec-trodynamics. The purpose is to show how one can demonstrate in a simpleway that the conventional XFEL theory is absolutely incapable of correctlydescribing the distribution of the electromagnetic fields from a fast movingmodulated electron beam downstream the kicker.Under the Maxwell’s electrodynamics, the fields of a modulated electronbeam moving with a constant velocity exhibit an interesting behavior whenthe velocity of charges approaches that of light: namely, in the space-timedomain they resemble more and more close of a laser beam (see AppendixIII). In fact, for a rapidly moving modulated electron beam we have nearlyequal transverse and mutually perpendicular electric and magnetic fields:in the limit v → c they become indistinguishable from the fields of a laserbeam, and according to Maxwell’s equations, the wavefront of the laserbeam is always perpendicular to the propagation direction. This is indeedthe case for virtual laser-like radiation beam in the region upstream thekicker.Let us now consider the e ff ect of the kick on the electron modulation wave-front. If we rely on the conventional particle tracking, the kick results in adi ff erence between the directions of electron motion and the normal to themodulation wavefront, i.e. in a tilt of the modulation wavefront.This is already a conflict result, because we now conclude that, accordingto the conventional ”single frame” approach, the direction of propagationafter the kick is not perpendicular to the radiation beam wavefront. In otherwords, the radiation beam motion and the radiation wavefront normal have176i ff erent directions. The virtual radiation beam (which is indistinguishablefrom a real radiation beam in the ultrarelativistic asymptote) propagates inthe kicked direction with a wavefront tilt. This is what we would get for thecase when our analysis is based on the conventional particle tracking, andis obviously absurd from the viewpoint of Maxwell’s electrodynamics.In existing literature theoretical analysis is presented, of an XFEL driven byan electron beam with wavefront tilt, and this analysis is based on the ex-ploitation of usual Maxwell’s equations and standard simulation codes. Us-ing only a kicker setup (i.e. without undulator radiation setup) we demon-strated that the coupling fields and particles in the conventional XFEL theoryis intrinsically incorrect.The di ffi culty above is a part of the continual problem of XFEL physics,which started with coherent undulator radiation from an ultrarelativisticmodulated electron beam in the kicked direction, and now has been focusedon the wavefront tilt of the self-electromagnetic fields of the modulatedelectron beam.In the ultrarelativistic domain the wavefront tilt has no exact objective mean-ing. The angle of wavefront tilt depends on the choice of a procedure forclock synchronization in the lab frame, as a result of which it can be givenany preassigned values within the interval (0 , θ k ). For instance, in the ultra-relativistic domain, the orientation of the modulation wavefront is alwaysperpendicular to the electron beam velocity (i.e. θ tilt =
0) when the evolu-tion of the modulated electron beam is treated using Lorentz coordinates.No physical e ff ects may depends on an arbitrary constant or an arbitraryfunction (5) .
1. An angular kick is often an essential part of many XFEL related diag-nostic or experimental procedures. The standard gain length measurementprocedure in XFELs employs such kicks. Other applications include ”beam-splitting” schemes where di ff erent polarization components are separatedby means of an angular kick to the modulated electron beam [47,48].2. In a typical configuration for an XFEL, the orbit of a modulated electronbeam is controlled to avoid large excursions from the undulator axis. Allexisting XFEL codes are based on a model in which the modulated electronbeam moves only along the undulator axis. However, random errors in thefocusing system can cause angular trajectory errors (or ”kicks”). The discrep-ancy between directions of the electron motion and wavefront normal afterthe kick have been discussed in the literature. One particular consequence177hat received attention following the [49] is the e ff ect of the trajectory er-ror (single kick error) on the XFEL amplification process. It was pointed outthat coherent radiation is emitted towards the wavefront normal of the beammodulation. Thus, according to conventional coupling of fields and particles(which we claimed incorrect), the discrepancy between the two directionsdecreases the radiation e ffi ciency [49]. Analysis of the trajectory errors onthe XFEL amplification process showed that any XFEL undulator magneticfield must satisfy stringent requirements. However, semi-analytical stud-ies of this critical aspect in the design of a XFEL sources are based on anincorrect coupling of fields and particles. The pleasant surprise is that thetolerances predicted are more stringent than they need be according to thecorrected XFEL theory. This can be considered as one of the reason for theexceptional progress in XFEL developments over last decades.3. The fact that our theory predicts reality in a satisfactory way is well-illustrated by comparing the prediction we just made with the results of anexperiment involving ”X-ray beam splitting” of a circularly-polarized XFELpulse from the linearly-polarized XFEL background pulse, a technique usedin order to maximize the degree of circular polarization. The XFEL exper-iments apparently demonstrated that after a modulated electron beam iskicked on a large angle compared to the divergence of the XFEL radiation,the modulation wavefront is readjusted along the new direction of motion ofthe kicked beam. This is the only way to justify coherent radiation emissionfrom the short undulator placed after the kicker and along the kicked direc-tion, see Fig. 14 in [50]. These results came unexpectedly, but from a practicalstandpoint, the ”apparent wavefront readjusting” immediately led to the re-alization that the unwanted, linearly-polarized radiation background couldbe fully eliminated without extra-hardware.4. In existing literature a theoretical analysis of XFELs driven by an elec-tron beam with wavefront tilt was presented in [51,52], based on the usethe usual Maxwell’s equations. In fact, the Maxwell solver was used as apart of the standard simulation code. We state that this approach is funda-mentally incorrect. In ultrarelativistic asymptote a modulation wavefronttilt is absurd with the viewpoint of Maxwell’s electrodynamics. In the caseof an XFEL we deal with an ultrarelativistic electron beam and within theLorentz lab frame (i.e. within the validity of the Maxwell’s equations) thetilted modulation wavefront is at odds with the principle of relativity.5. In existing XFEL theory the wavefront tilt is considered as real. Let usconsider one example. One finds some papers (see e.g. [49]) which saythat a wavefront tilt leads to significant degradation of the electron beammodulation in XFELs. First, suppose that modulation wavefront is perpen-dicular to the beam velocity v . The e ff ect of betatron oscillations, whichcan influence the operation of the XFEL, has its origin in an additional lon-178itudinal velocity spread. Particles with equal energies, but with di ff erentbetatron angles, have di ff erent longitudinal velocities. In other words, ontop of the longitudinal velocity spread due to the energy spread, there isan additional source of velocity spread. To estimate the importance of thelast e ff ect, we should calculate the dispersion of the longitudinal velocitiesdue to both e ff ects. The deviation of the longitudinal velocity from nominalvalue is ∆ v z = v ∆ γ/γ − v ∆ θ /
2. The finite angular spread of the electronbeam results in a di ff erence in time when each electron arrives at the samelongitudinal position, and this spoils the phase coherence. This is so callednormal debunching e ff ect. From the viewpoint of the existing XFEL theory,the time di ff erence is enhanced by the kick angle θ k . In this case, accordingto conventional (non-covariant) particle tracking, the angle of wavefront tiltis θ tilt = θ k . It is a widespread belief that the wavefront tilt has physicalmeaning, and that the deviation of the longitudinal velocity component(i.e. velocity component which is perpendicular to the modulation wave-front within the framework of Galilean kinematics) is now given by theexpression ∆ v z = − v | ∆ (cid:126)θ + (cid:126)θ k | /
2. If such picture is correct, the crossed term v (cid:126)θ k · ∆ (cid:126)θ leads to a significant degradation of the modulation amplitude. Thismechanism is called smearing of modulation and should be distinguishedfrom the normal debunching [49]. Many experts would like to think thatany debunching process obviously has objective meaning. The theory ofrelativity says, however, that normal debunching has objective meaning,but smearing e ff ects not exist at all. The explanation of the new debunchingmechanism clearly demonstrates the essential dependence of the smear-ing e ff ect on the choice of the coordinate system in the four-dimensionalspace, which from the physical point of view is meaningless. Now let usunderstand physically why the new debunching mechanism does not ex-ist in framework of Galilean kinematics. In this old kinematics the crossedterm v (cid:126)θ k · ∆ (cid:126)θ leads to a degradation of modulation amplitude in the for-ward direction. Our Galilean transformed electrodynamics says, however,that by making a measurement on the coherent radiation, one can observeonly radiation in the kicked direction. But the crossed term is absent in theexpression for the deviation of the velocity component along the kickeddirection. It comes out quite naturally that the smearing e ff ect is not a realphenomenon. 179 Relativistic Spin Kinematics
Relativistic e ff ects start to be important when velocities of objects get closerto the speed of light. Usually, only elementary particle velocities may be asubstantial fraction of the speed of light. Consider a particle with its ownangular momentum (spin). Suppose that this particle is moving along acurved trajectory. In the theory of relativity, the Lorentz transformations arenot commutative in the general case. This leads to a rotation of the spinunder changes the particle velocity relative to the inertial lab frame. Thisso called ”Wigner rotation” does not occur due to the action of some forcesresponsible for variation in the angular position of a spinning particle, andtherefore has a purely kinematic origin (1) .It is known that Wigner rotation represents a kinematics e ff ect, due to thefact that the successive Lorentz transformations with non-collinear relativevelocities are accompanied by an additional spatial rotation of coordinateaxes of corresponding reference frames. The Wigner rotation is a purelyrelativistic kinematics e ff ect but it influences the dynamics of the spin asobserved in the Lorentz lab frame. The covariant equation of spin motionfor a relativistic particle under the action of the four-force in the Lorentzlab frame is a relativistic generalization of the phenomenological equationof motion for a particle angular momentum in its rest frame. This ”gener-alization” naturally involves the Wigner rotation, since this is the purelykinematic addition to the Larmor rotation which, in turn, is a consequenceof interaction of the intrinsic magnetic moment with the external magneticfield.Lorentz transformations are essential to the further mathematical develop-ment of the Wigner rotation theory, so the next two sections detail the usualapplications together with some physical discussion. Let us now consider a relativistic particle, accelerating in the lab frame,and let us analyze its evolution within Lorentz coordinate systems. Thepermanent rest frame of the particle is obviously not inertial and any trans-formation of observations in the lab frame, back to the rest frame, cannotbe made by means of Lorentz transformations. To get around that di ffi cultyone introduces an infinite sequence of comoving frames. At each instant,the rest frame is a Lorentz frame centered on the particle and moving with180t. As the particle velocity changes to its new value at an infinitesimally laterinstant, a new Lorentz frame centered on the particle and moving with it atthe new velocity is used to observing the particle.Let us denote the three inertial frames by K , R ( τ ) , R ( τ + d τ ). The lab frame is K , R ( τ ) is the rest frame with velocity (cid:126) v = (cid:126) v ( τ ) relative to K , and R ( τ + d τ ) is therest frame at the next instant of proper time τ + d τ , which moves relative to R ( τ ) with infinitesimal velocity d (cid:126) v (cid:48) . All inertial reference frames are assumedto be Lorentz reference frames. In order to have this, we impose that R ( τ )is connected to K by the Lorentz boost L ( (cid:126) v ), with (cid:126) v , which transforms agiven four vector event X in a space-time into X R = L ( (cid:126) v ) X . The relation X R = L ( d (cid:126) v (cid:48) ) L ( (cid:126) v ) X presents a step-by-step change from K to R ( τ ) and then to R ( τ + d τ ).There is another composition of reference-frame transformations which de-scribes the same particle evolution in the Minkowski space-time. Let K ( τ )be an inertial frame with velocity d (cid:126) v relative to the lab frame K ( τ + d τ ). Weimpose that K ( τ ) is connected to K ( τ + d τ ) by the Lorentz boost L ( d (cid:126) v ). TheLorentz rest frame R is supposed to move relative to the Lorentz frame K ( τ )with velocity (cid:126) v . The relation X R = L ( (cid:126) v ) L ( d (cid:126) v ) X presents a step-by-step changefrom K ( τ + d τ ) to K ( τ ) and then to the rest frame R .Let us examine the transformation of the three-velocity in the theory ofrelativity. For a rectilinear motion along the z axis it is performed in ac-cordance with the following equation: v z ( τ + d τ ) = ( dv z + v z ) / (1 + v z dv z / c ).Like it happens with the composition of Galilean boosts, collinear Lorentzboosts commute: L ( dv z ) L ( v z ) = L ( v z ) L ( dv z ). This means that the resultant ofsuccessive collinear Lorentz boosts is independent of which transformationapplies first. In contrast with the case of Lorentz boosts in collinear directions, Lorentzboosts in di ff erent directions do not commute. A comparison with the three-dimensional Euclidean space might help here. Spatial rotations do not com-mute either. However, also for spatial rotations there is a case where theresult of two successive transformations is independent of their order: thatis, when we deal with rotation around the same axis. While the successiveapplication of two Galilean boosts is Galilean boost and the successive ap-plication of two rotations is a rotation, the successive application of twonon-collinear Lorentz boosts is not a Lorentz boost. The composition ofnon-collinear boosts will results to be equivalent to a boost, followed byspatial rotation, the Wigner rotation.181et us compare the succession K → R ( τ ) → R ( τ + d τ ) with the succession K ( τ + d τ ) → K ( τ ) → R in the case when the acceleration in the rest frameis perpendicular to the line of flight of the lab frame in the rest frame. Theframe R ( τ + d τ ) is supposed to move relative to R ( τ ) with velocity d (cid:126) v (cid:48) x .Because of time dilation in the moving frame, the velocity increment inthe lab frame dv x corresponds to a velocity γ dv x and − γ dv x in the frames R ( τ ) and R ( τ + d τ ) respectively. The resulting boost compositions can berepresented as X R = L ( d (cid:126) v (cid:48) x ) L ( (cid:126) v z ) X = L ( (cid:126) v z ) L ( d (cid:126) v x ) X . In other words, Lorentzboosts in di ff erent direction do not commute: L ( (cid:126) v z ) L ( d (cid:126) v x ) (cid:44) L ( d (cid:126) v x ) L ( (cid:126) v z ).Now, since we can write the result in terms of succession L ( (cid:126) v z ) L ( d (cid:126) v x ) aswell as in terms of succession L ( γ d (cid:126) v x ) L ( (cid:126) v z ), there is a need to clarify a num-ber of questions associated with these compositions of Lorentz frames. Wecan easily understand that the operational interpretation of the succession L ( (cid:126) v z ) L ( d (cid:126) v x ) is particular simple, involving physical operation used in themeasurement of the particle’s velocity increment d (cid:126) v x in the lab frame. Weshould be able to understand the operational interpretation of the succession L ( γ d (cid:126) v x ) L ( (cid:126) v z ). We begin by making an important point: the laws of physicsin any one reference frame should be able to account for all physical phe-nomena, including the observations made by moving observers. The labobserver sees the time dilation in the Lorentz frame which moves with re-spect to the lab frame with velocity (cid:126) v z : dt /γ = d τ . What velocity increment d (cid:126) v R is measured by moving observer? As viewed from the lab frame themoving observer measures the increment d (cid:126) v R = γ d (cid:126) v x . Consider the succession of inertial frame systems K → R ( τ ) → R ( τ + d τ ).As viewed from the lab frame the observer in the proper frame measuresthe velocity increment d (cid:126) v R ( τ + d τ ) = − γ d (cid:126) v x . The corresponding rotation of thevelocity direction in the proper frame R ( τ + d τ ) is γ dv x / v z . In the lab frame thevelocity rotation angle would be dv x / v z . The di ff erence of these two velocityrotation angles γ dv x / v z − dv x / v z is the Wigner rotation angle of the lab frameaxises in the proper frame R ( τ + d τ ). One way to see this is follows.In 3D space we find that the proper frame moves with respect to the labframe along the line motion and the lab frame moves with respect to theproper frame along the same line motion. In other words, it follows that theline motion is the same in the proper frame as in the lab frame. The anglebetween the axis of the observer’s coordinate system and the line motion isa simple 3D space geometric parameter. The lab observer is able to account182 ig. 43. The interpretation of the Wigner rotation about the proper frame axes. Labframe view of the observation of the proper observer. The lab observer is able toaccount for the observation the velocity rotation angle in the proper frame ( γ dv x / v z )and the observation of the rotation angle made in the lab frame ( dv x / v z ). In 3D spacethe proper frame moves with respect to the lab frame along the line motion andthe lab frame moves with respect to the proper frame along the same line motion.Using the line motion as reference line, the lab observer sees that the observer inthe proper frame measures the rotating angle of the lab frame axes with respect toproper frame axes δθ w = δθ ( γ − for the observation of the rotation angle made in the proper frame andthe observation of the rotation angle made in the lab frame. Using the linemotion as a reference line, the lab observer can then calculate the di ff erencebetween these angles to find the rotation angle of the lab frame in theproper frame. We have found that the lab observer sees that the observerin the proper frame measures the rotation angle of the lab frame axes withrespect to proper frame axes γ dv x / v z − dv x / v z (Fig. 43) . In vector form this isseen to be d (cid:126) Φ R = ( γ − (cid:126) v × d (cid:126) v / v . We wish to remark that the expression forthe Wigner rotation angle in the proper frame can be presented in the form d (cid:126) Φ R = (1 − /γ ) (cid:126) v R × d (cid:126) v R / v R . Thus, the Wigner rotation angle in the properframe is expressed in terms of the lab frame velocity (cid:126) v R and its increment d (cid:126) v R . The Wigner rotation (similar to the time dilation and length contraction)is a symmetric phenomenon as it must be from kinematic consideration. Wecan now write expression for the Wigner rotation in the lab frame in thesame form d (cid:126) Φ = (1 − /γ ) (cid:126) v × d (cid:126) v / v .183he Wigner rotation is a relativistic kinematic e ff ect, which consists in thatthe spin of an elementary particle or as well as a coordinate axes of areference frame, moves along a curvalinear trajectory rotating about theaxes of a Lorentz lab frame. In the Lorentz lab frame, for infinitely smalltransformations (due to acceleration) we obtained the formula d (cid:126) Φ = (1 − /γ ) (cid:126) v × d (cid:126) v / v = (cid:32) − γ (cid:33) (cid:126)δθ , (92)where d (cid:126) v is the vector of small velocity change due to acceleration, Φ is theWigner rotation angle of the spatial coordinate axes of the system comovingwith an elementary particle relative to Lorentz lab frame, and θ is the orbitalangle of the particle in the lab frame.We note that owing to the relativistic e ff ect of time dilation in the referenceframe that moves to the lab frame, the Wigner rotation angle in the referenceframe comoving with spinning particle is always γ time higher than in theLorentz lab frame. Wigner rotation theory describes the rotation of the axes of a moving ref-erence frame which is observed in the lab frame. But how to measure thisorientation? A moving coordinate system changes its position in time. Thequestion arises whether it is possible to give an experimental interpretationof the rotation of a moving coordinate system. We illustrate the problem ofhow to represent orientation of the moving coordinate system with a simpleexample.The execution of successive transformations from K ( τ + d τ ) to K ( τ ) at velocity d (cid:126) v x and from from K ( τ ) to R at velocity (cid:126) v z equivalent to the compositionof boost L ( (cid:126) v + d (cid:126) v ) and rotation. The Wigner rotation Eq.(92) is performedadditionally to the Lorentz boost at velocity (cid:126) v z + d (cid:126) v x . The interpretation ofthis rotation about the lab frame of reference is closely associated with thelength contraction.Suppose that the lab observer after the Lorentz boost at velocity v z rotatesthe coordinate system on the angle (cid:126)δθ = (cid:126) v z × d (cid:126) v x / v z and now x r locatesorthogonally to the vector (cid:126) v z + d (cid:126) v x . Similarly, the axis z r is parallel to thevector (cid:126) v z + d (cid:126) v x . Consider a rod directed along x -axis in the comoving frame.The motion takes place in the plane ( x , z ) and the rod located perpendicularlyto the velocity (cid:126) v z . After the rotation of the lab frame axes, the projection ofthe rod on the z r axes will be simply l δθ , where l is the rod length in the R ig. 44. The interpretation of the Wigner rotation about the lab frame. A rod directedalong x -axis in the comoving frame. After the first boost the motion takes place alongthe z -axis. The lab frame ( x r , z r ) coordinate system rotated with respect to the initiallab frame ( x , z ) coordinate system on the angle δθ . The projection of the moving rodon the z r -axis is simply L δθ . After the second boost at velocity dv x along the x -axisof the lab frame this projection will be contracted down to L δθ/γ . It is assumedthat infinitesimal angle δθ = dv x / v z . According to the contracted projection thecomoving frame is rotated with respect to the initial lab frame axes by the Wignerangle equal to δθ (1 − /γ ). frame and also in the lab frame after the first boost along the z -axis. Afterthe second Lorentz boost at velocity d (cid:126) v x this projection will be contracteddown to L δθ/γ (Fig. 44). Let the observer in the lab frame fix the position ofthe axes of the comoving frame. In ultrarelativistic limit γ → ∞ these axeswill be parallel to the rotated lab frame axes ( x r , z r ). In fact, projection of therod on the z r axis will be zero. In the case of an arbitrary velocity, axes of thecomoving frame turn out to not only parallel to the rotated lab frame axes( x r , z r ). According to contracted projection, the angle will be − (cid:126)δθ/γ . And onecan verify directly that the axes of the comoving frame are actually rotatedwith respect to the initial lab frame axes ( x , z ) by the angle equal to δθ − δθ/γ ,which is just the Wigner rotation angle in accordance with equation Eq.(92).Here we only wished to show how naturally Lorentz transformations leadto the Wigner rotation phenomenon. We have come to the conclusion thatwhat are usually considered advanced parts of the theory of relativity are, infact, quite simple. Indeed, we demonstrated that the Wigner rotation results185irectly from the length contraction. Why our derivation of the expressionfor the Wigner rotation is so simple? The reason is that we employed anew method that is very useful in this kind of problem. What we did was toanalyze the physical operations used in the measurement of Wigner rotation,which has never been done before. In fact, the operations for performingmeasurements on a moving object are not the same as those for measuringan object at rest, and an absolute significance has been attributed to theconcept of simultaneity.We derived the exact relation, Eq.(92), using only rudimentary knowledgeof special relativity. The rotation of axes e ff ect can, of course, be describedalgebraically in terms of the transformation matrices for four-vector compo-nents. In textbooks on the theory of relativity, the spatial rotation associatedwith the composition of two Lorentz boosts in non-parallel directions isoften introduced using the algebraic approach. This is one of the reasonwhy authors of textbooks obtained an incorrect expression for the Wignerrotation. They describe the rotation of a moving object without operational(geometrical) interpretation of such rotation and encounter serious di ffi cul-ties in the interpretation of the applied calculations and of the results.
1. As known, a composition of noncollinear Lorentz boosts does not resultsin a di ff erent boost but in a Lorentz transformation involving a boost and aspatial rotation, the latter being known as Wigner rotation [53,54]. Wignerrotation is sometimes called Thomas rotation (see e.g. [12,55]).2. The correct expression for the Thomas precession was first obtained by V.Ritus [56]. In deriving expressions for the Thomas precession, the majorityof authors (see e.g. [55]) were supposedly guided by the incorrect expres-sion for Thomas precession from Moeller’s monograph [12]. The expressionobtained by Moeller is given by (cid:126)δ Φ = (1 − γ ) (cid:126) v × d (cid:126) v / v = (1 − γ ) (cid:126)δθ (and sub-sequently Ω T = (1 − γ ) ω ). It should be note that, in his monograph, Moellerstated several times that this expression valid in the lab Lorentz frame.Clearly, this expression and correct result Eq. (92) di ff er both in sign andin magnitude. An analysis of the reason why Moeller obtained an incorrectexpression for the Thomas precession in the lab frame is the focus of Rituspaper [57]. As shown in [57], the Moeller’s mistake is not computational,but conceptual in nature. In review [58] it is shown that the correct resultwas obtained in the works of several authors, which were published morethan half century ago but remained unnoticed against the background ofnumerous incorrect works. 186. The authors of some papers believe that the incorrect result for Wignerrotation in the lab frame presented in textbooks d (cid:126) Φ = − ( γ − (cid:126) v × d (cid:126) v / v is only incorrectly interpreted with the understanding that it should bereinterpreted as a Wigner rotation of the lab frame in the proper frame. Wenote that such reinterpreted expression for Wigner rotation in the properframe d (cid:126) Φ = − ( γ − (cid:126) v × d (cid:126) v / v → d (cid:126) Φ R = − ( γ − (cid:126) v × d (cid:126) v / v is also incorrect insign.4. We note that in 1986, M. Stranberg obtained an expression for the Wignerrotation correct both in the lab inertial frame and the reference frame co-moving with a spinning particle [59]. It is noteworthy that [59] is one of thefew papers that explicitly states that the angle of the Wigner rotation in thecomoving reference frame is γ times higher than in the lab frame.187 In 1959, a paper by Bargmann, Michel, and Telegdi was published, whichdealt with the motion of elementary charged spinning particles with ananomalous magnetic moment in electromagnetic field (1) . The extremelyprecise measurements of the magnetic-moment anomaly of the electronmade on highly relativistic electrons are based on the BMT equation. Theanomalous magnetic moment can be calculated by use of quantum electro-dynamics. The theoretical result agrees with experiments to within a veryhigh accuracy. This can be regarded as a direct test of BMT equation.The existing textbooks suggest that the experimental test of the BMT equa-tion is a direct test of what we consider the incorrect expression for Wignerrotation in the Lorentz lab frame. We claim that the inclusion of this incorrectexpression as an integral part of the BMT equation in most texts is based onan incorrect physical argument. In this chapter we will investigate in detailthe reason why this is the case (2) . Let us consider at first the spin precession for a non relativistic charge par-ticle. The proportionality of magnetic moment (cid:126)µ and angular momentum (cid:126) s has been confirmed in many ”gyromagnetic” experiments on many dif-ferent systems. The constant of proportionality is one of the parameterscharactering a particular system. It is normally specified by giving the gyro-magnetic ratio or g factor, defined by (cid:126)µ = ge (cid:126) s / (2 mc ). This formula says thatthe magnetic moment is parallel to the angular momentum and can haveany magnitude. For an electron g is very nearly 2.Suppose that a particle is at rest in an external magnetic field (cid:126) B R . The equa-tion of motion for the angular momentum in its rest frame is d (cid:126) s / d τ = (cid:126)µ × (cid:126) B R = eg (cid:126) s × (cid:126) B R / (2 mc ) = (cid:126)ω s × (cid:126) s . In other words, the spin precesses around the direc-tion of magnetic field with the frequency ω s = − eg (cid:126) B R / (2 mc ). In the same nonrelativistic limit the velocity processes around the direction of (cid:126) H R with thefrequency ω p = − ( e / mc ) (cid:126) B R : d (cid:126) v / d τ = ( e / mc ) (cid:126) v × (cid:126) B R . Thus, for g = Spin dynamics equations can be expressed as tensor equations in Minkowskispace-time. We shall limit ourselves to the case of a particle with a mag-netic moment (cid:126)µ in a microscopically homogeneous electromagnetic field.Evidently the torque a ff ects only the spin and the force a ff ects only the mo-mentum. It follows that the motion of the system as a whole in any frame isdetermined entirely by its charge, independent of magnetic dipole moment.This part of the motion has been treated in the Chapter 6. We need now onlyconsider the spin motion.In seeking the equation for the spin motion, we shall be guided by the knowndynamics in the rest frame and the known relativistic transformation laws.We emphasize that spin is defined in a particular frame (the rest frame).Therefore, to form expressions with known transformation behavior, weneed to introduce a four-quantity related to the spin. A convenient choice isa four- (pseudo)-vector S defined by the requirement that in the rest frameits space-like components are the spin components, while the time-likecomponent is zero. We shall call S four-spin; when normalized by dividingby its invariant length, it will be called polarization four-vector. It is space-like, and therefore in no frame does it space-like part vanish.Let the spin of the particle be represented in the rest frame by (cid:126) s . The four-vector S α is by definition required to be purely spatial at time τ in an in-stantaneous Lorentz rest frame R ( τ ) of the particle and to coincide at thistime with the spin (cid:126) s ( τ ) of the particle; that is S α R ( τ ) = (0 , (cid:126) S R ( τ )) = (0 , (cid:126) s ( τ )). Ata later instant τ + ∆ τ in an instantaneous inertial rest frame R ( τ + ∆ τ ), wehave similarly S α R ( τ + ∆ τ ) = (0 , (cid:126) S R ( τ + ∆ τ )) = (0 , (cid:126) s ( τ + ∆ τ )).The BMT equation is manifestly covariant equation of motion for a four-vector spin S α in an electromagnetic field F αβ : dS α d τ = ge mc (cid:20) F αβ S β + c u α (cid:16) S λ F λµ u µ (cid:17)(cid:21) − c u α (cid:32) S λ du λ d τ (cid:33) , (93)where u µ = dx µ / d τ is the four-dimensional particle velocity vector. WithEq.(10), one has (1) dS α d τ = emc (cid:34) g F αβ S β + g − c u α (cid:16) S λ F λµ u µ (cid:17)(cid:35) , (94)The BMT equation is valid for any given inertial frame, and consistently189escribes, together with the covariant-force law, the motion of a chargedparticle with spin and magnetic moment. If F µν (cid:44)
0, even with g =
0, wesee that dS µ / d τ (cid:44)
0. Thus, a spinning charged particle will precess in anelectromagnetic field even if it has no magnetic moment. This precession ispure relativistic e ff ect.The covariant equation of spin motion for a relativistic particle under theaction of the four-force Q µ = eF µν u ν in the Lorentz lab frame, Eq.(93), is arelativistic ”generalization” of the equation of motion for a particle angularmomentum in its rest frame. Relativistic ”generalization” means that thethree independent equations expressing the Larmor spin precession are beembedded into the four-dimensional Minkowski space-time. The idea ofembedding is based on the principle of relativity i.e. on the fact that theclassical equatuion of motion for particle angular momentum d (cid:126) s / d τ = eg (cid:126) s × (cid:126) B R / (2 mc ) can always be used in any Lorentz frame where the particle, whosemotion we want to describe, is at rest. In other words, if an instantaneouslycomoving Lorentz frame is given at some instant, one can precisely predictthe evolution of the particle spin in this frame during an infinitesimal timeinterval.In Lorentz coordinates there is a kinematics constraint S µ u µ =
0, whichis orthogonality condition of four-spin and four-velocity. Because of thisconstraint, the four-dimensional dynamics law, Eq.(93), actually includesonly three independent equations of motion. Using the explicit expressionfor Lorentz force we find that the four equations Eq.(93) automatically implythe constraint S µ u µ = S = (cid:126) S · (cid:126) v . While S vanishes in the rest frame, dS / d τ need not. In fact d ( S µ u µ ) / d τ = dS / d τ = (cid:126) S · d (cid:126) v / d τ . Theimmediate generalization of d (cid:126) s / d τ = eg (cid:126) s × (cid:126) B R / (2 mc ) and dS / d τ = (cid:126) S · d (cid:126) v / d τ to arbitrary Lorentz frames is Eq.(93) as can be checked by reducing to therest frame. A methodological analogy with the relativistic generalization ofthe Newton’s second law emerges.In order to fully understand the meaning of embedding of the spin dynamicslaw in the Minkowski space-time, one must keep in mind that, above, wecharacterized the spin dynamics equation in the Lorentz comoving frameas a phenomenological law. The microscopic interpretation of the magneticmoment of a particle is not given. In other words, it is generally acceptedthat the spin dynamics law is a phenomenological law and the magneticmoment is introduced in an ad hoc manner. The system of coordinates inwhich the classical equations of motion for particle angular momentum arevalid can be defined as Lorentz rest frame. The relativistic generalization ofthe three-dimensional equation d (cid:126) s / d τ = eg (cid:126) s × (cid:126) B R / (2 mc ) to any Lorentz framepermits us to make correct predictions.190 The equation Eq.(94) is more complex than one might think. In fact, it iscomposed by a set of coupled di ff erential equations. To find solution directlyfrom the system seems quite di ffi cult, even for a very symmetric, uniformmagnetic field setup.In order to apply Eq.(94) to specific problems it is convenient to introduce athree vector (cid:126) s by the equation (cid:126) s = (cid:126) S + S (cid:126) pc / ( E + mc ). With the help of thisrelation one can work out the equation of motion for (cid:126) s . In the importantcase of a uniform magnetic field with no electric field in the lab frame onehas, after a somewhat lengthly calculations: d (cid:126) sd τ = − e m (cid:34)(cid:32) g − + γ (cid:33) γ(cid:126) B − ( g − γγ + (cid:126) vc (cid:32) (cid:126) vc · γ(cid:126) B (cid:33)(cid:35) × (cid:126) s , (95)What must be recognized is that in the accepted covariant approach (in-deed, Eq.(94) is obviously manifestly covariant), the solution of the dy-namics problem for the spin in the lab frame makes no reference to thethree-dimensional velocity. In fact, the Eq.(95) includes relativistic factor γ and vector (cid:126) v / c , which are actually notations: γ = E / ( mc ), (cid:126) v / c = (cid:126) pc / E . Allquantities E , (cid:126) p , (cid:126) B are measured in the lab frame and have exact objectivemeaning i.e. they are convention-independent. The evolution parameter τ is also measured in the lab frame and has exact objective meaning . Forinstance, it is not hard to demonstrate that d τ = mdl / | (cid:126) p | , where dl is thedi ff erential of the path length.Spin vector (cid:126) s is not part of a four-vector, and depends on both (cid:126) S and S . Whilenot being a four-vector, it is e ff ectively a three-dimensional object (havingzero time component in the inertial frame in question) and the spatial partof this object undergoes pure rotation with constant rate for the example ofmotion along a circle in special relativity. If we perform an arbitrary velocitymapping, (cid:126) s will have to be recomputed from the transformed values S µ and p µ . However, this new (cid:126) s will satisfy an equation of the form Eq.(95), with (cid:126) B computed from the transformed F µν .Let us restrict our treatment of spinning particle dynamics to purely trans-verse magnetic fields. This means that the magnetic field vector (cid:126) H is orientednormal to the particle line motion. If the field is transverse, then equationEq.(95) is reduced to 191 (cid:126) sd τ = (cid:126) Ω × (cid:126) s = − e mc (cid:34)(cid:32) g − + γ (cid:33) γ(cid:126) B (cid:35) × (cid:126) s , (96)Now we have an equation in the most convenient form to be solved. Supposewe let the charged spinning particle in the lab frame through a bendingmagnet with the length dl . We know that d θ = − eBdl / ( | (cid:126) p | c ) is the orbitalangle of the particle in the lab frame. Note that d τ = mdl / | (cid:126) p | . Then, Eq.(96)tells us that we may write the spin rotation angle with respect to the labframe axes Ω d τ as Ω d τ = [( g / − γ d θ + d θ ].This tell us that in the lab frame the spin of a particle (cid:126) s changes the angle φ with its line motion. The rate of change of the angle φ with the orbital angleis ( g / − γ , so we can write d φ = ( g / − γ d θ .We would like to discuss the following question: Is the vector (cid:126) s merely adevice which is useful in making calculations - or is it a real quantity ( i.e. aquantity which has direct physical meaning)? Knowing that there is a simplealgebraic relation between (cid:126) s and the standard spin vector, the spin vector (cid:126) s can be used as an intermediate step to easily find the standard spin vector S µ . There is, however, also a direct physical meaning to the spin vector (cid:126) s .The spin vector (cid:126) s directly gives the spin as perceived in a comoving system.The approach in which we deal with the proper spin is much preferred inthe experimental practice due to mathematical simplicity and clear physicalmeaning of the vector (cid:126) s . Unlike momentum, which has definite componentsin each reference frame, angular momentum is defined only in one particularreference frame. It does not transform. Any statement about it refers to therest frame as of that instant. If we say that in the lab frame the spin of aparticle makes the angle φ with its velocity, we mean that in the particle’srest frame the spin makes this angle with the line motion of the lab frame. When Bargman, Michel and Telegdi first discovered the correct laws of spindynamics, they wrote a manifestly covariant equation in Minkowski space-time, Eq.(93), which describes the motion of the four spin S µ . The derivationof this equation was very similar to the four-tensor equations that werealready known to relativistic particle dynamics. How to solve this four-tensor equation is an interesting question. In relativistic spin dynamics it isdone in one particular way, which is very convenient. In order to apply four-tensor equation Eq.(93) to specific problems it is convenient to transform192his equation to the rest frame as of that instant. Should one be surprisedthat the starting point of Bargman, Michel and Telegdi was the particle restframe and the classical equation of motion for particle angular momentum,which they generalized to the Lorentz lab frame and then transformed backto the rest frame?We want to emphasize that the equation Eq.(96) for the proper spin (cid:126) s andthe BMT equation Eq.(93) for the four spin S µ are completely equivalent,they both determine the behaviour of the spin from the point of view ofthe lab frame. With Eq.(96) have what we need to know - the evolution ofthe proper spin vector (cid:126) s with respect to the lab frame axes. Starting fromthe classical equation d (cid:126) s / d τ = eg (cid:126) s × (cid:126) B R / (2 mc ), which describes the Larmorprecession with respect to the proper frame axes, we have derived theequation Eq.(95), which describes the spin motion with respect to the labframe axes in the proper frame and reduced to Eq.(96) in the case of purelytransverse magnetic fields. That means that we know the orientation of theproper spin with respect to the lab coordinate system which is moving withvelocity − (cid:126) v and acceleration − γ d (cid:126) v / d τ in the proper frame.Above we described the BMT equation, Eq.(96), in the standard manner. Ituses a spin quantity defined in the proper frame but observed with respectto the lab frame axes. Let’s look at what the equation Eq.(96) says in a littlemore detail. It will be more convenient if we rewrite this equation as d (cid:126) s = (cid:126) Ω d τ × (cid:126) s = − eg γ(cid:126) Bd τ/ (2 mc ) × (cid:126) s + e ( γ − (cid:126) Bd τ/ ( mc ) × (cid:126) s . (97) Now let’s see how we can write the right-hand side of Eq.(97) . The firstterm is that we would expect for the spin rotation due to a torque withrespect to the proper frame axes d (cid:126)φ L = − eg γ(cid:126) Bd τ/ (2 mc ) = ( g / γ d (cid:126)θ . Here d (cid:126)θ = − eBdl / ( | (cid:126) p | c ) is the angle of the velocity rotation in the lab frame. It hasalso been made evident by our analysis in the previous Chapter 9 that angleof rotation d (cid:126)φ W = − e ( γ − (cid:126) Bd τ/ ( mc ) = ( γ − d (cid:126)θ corresponds to the Wignerrotation of the lab frame axes with respect to the proper frame axes. Withthis definitions, we have d (cid:126) s = (cid:126) Ω d τ × (cid:126) s = d (cid:126)φ L × (cid:126) s − d (cid:126)φ W × (cid:126) s , (98)which begins to look interesting. 193 Now we introduce our new approach to the BMT theory, finding anotherway in which our complicated problem can be solved. We know that d (cid:126)φ L and d (cid:126)φ W are the rotations with respect to the proper frame axes. Actuallywe only need to find the spin motion with respect to the lab frame axes.Now we must be careful about signs of rotations.There is a good mnemonic rule to learn the signs of di ff erent rotations. Therule says, first, that the direction of the velocity rotation in the proper frameis the same as the direction of the velocity rotation in the lab frame. Second,the direction of the lab frame rotation in the proper frame is the same asthe direction of the velocity rotation in the proper frame. Third, the signof the spin rotation due to a torque at g > g is positive and very nearly 2) means that the direction ofthe rotation in the proper frame is the same as the direction of the velocityrotation in the proper frame.We now ask about the proper spin rotation with respect to to the lab frameaxes. This is easy to find. The relative rotation angle is d (cid:126)φ L − d (cid:126)φ W . So webegin to understand the basic machinery behind spin dynamics. We see whythe Wigner rotation of the lab frame axes in the proper frame must be takeninto account if we need to know the proper spin dynamics with respect tothe lab frame axes.Why the new derivation of the BMT equation is so simple? The reason isthat the splitting of the particle spin motion with respect to the lab frameaxes into the dynamic (Larmor) and kinematic (Wigner) parts can only berealized in the proper frame. In the proper frame, we do not need to knowany more about a relativistic ”generalization” of the (phenomenological)classical equation of motion for the particle angular momentum. In thiscase, it is possible to separate the spin dynamics problem into the trivialdynamic problem and into the kinematic problem of Wigner rotation of thelab frame in the proper frame. Having written down the spin motion equation in a 4-vector form, Eq.(94),and determined the components of the 4-force, we satisfied the principle ofrelativity for one thing, and, for another, we obtained the four componentsof the equation for the spin motion. This is a covariant relativistic generaliza-tion of the usual three dimensional equation of magnetic moment motion,which is based on the particle proper time as the evolution parameter. We194ext wish to describe the spin motion with respect to the Lorentz lab frameusing the lab time t as the evolution parameter. When going from the proper time τ to the lab time t , the frequency of spinprecession with respect to the lab frame can be obtained using the well-known formula d τ = dt /γ . We then find d (cid:126) sdt = (cid:126)(cid:36) × (cid:126) s = − e mc (cid:34)(cid:32) g − + γ (cid:33) (cid:126) B (cid:35) × (cid:126) s . (99)The frequency of spin precession can be written in the form (cid:36) = ω [1 + γ ( g / − , (100)where ω is the particle revolution frequency. Now the time-like part of thefour-velocity is decomposed to c γ = c / √ − v / c and the trajectory doesnot include relativistic kinematics e ff ects. In particular, the Galilean vectoriallaw of addition of velocities is actually used. So we must have made a jumpto the absolute time coordinatization.The previous commonly accepted derivation of the equations for the spinprecession in the lab frame from the covariant equation Eq.(93) has the samedelicate point as the derivation of the equation of particle motion from thecovariant equation Eq.(10). The four-velocity cannot be decomposed into u = ( c γ, (cid:126) v γ ) when we deal with a particle accelerating along a curved trajectoryin the Lorentz lab frame. One of the consequences of non-commutativity ofnon-collinear Lorentz boosts is the unusual momentum-velocity relation. Inthis case there is a di ff erence between covariant and non-covariant particletrajectories.The old kinematics comes from the relation d τ = dt /γ . The presentation ofthe time component simply as the relation d τ = dt /γ between proper timeand coordinate time is based on the hidden assumption that the type ofclock synchronization that provides the time coordinate t in the lab frame isbased on the use of the absolute time convention.195 In the Chapter 6 we saw that the particle path (cid:126) x ( l ) has an exact objectivemeaning i.e. it is convention-invariant. The spin orientation (cid:126) s at each pointof the particle path (cid:126) x ( l ) has also exact objective meaning. In contrast to this,and consistently with the conventionality of the three-velocity, the function (cid:126) s ( t ) describing the spinning particle in the lab frame has no exact objectivemeaning.We now want to describe how to determine the spin orientation along thepath (cid:126) s ( l ) in covariant spin tracking. Using the covariant equation Eq.(93) weobtain Eq.(96). If we use the relation d τ = mdl / | (cid:126) p | our convention-invariantequation of spin motion reads d (cid:126) sdl = − e E m | (cid:126) p | c (cid:34)(cid:32) g − + mc E (cid:33) (cid:126) B (cid:35) × (cid:126) s = (cid:20)(cid:18) g − (cid:19) E mc + (cid:21) d (cid:126)θ dl × (cid:126) s , (101)which is based on the path length l as the evolution parameter. These threeequations corresponds exactly to the equations for components of the properspin vector that can be found from the non-covariant spin tracking equationEq.(99). So everything comes out all right. We want to emphasize that thereare two di ff erent (covariant and non covariant) approaches that produce thesame spin orientation (cid:126) s ( l ) along the path. The point is that both approachesdescribe correctly the same physical reality and the orientation of the properspin (cid:126) s at any point of particle path in the magnetic field has obviously anobjective meaning, i.e. is convention-invariant.Now we take an example, so it can be seen that we do not need to askquestions about the function (cid:126) s ( t ) of a spinning particle experimentally. Justthink of experiments related with accelerator physics. Suppose we wantto calibrate the beam energy in a storage ring based on measurement ofspin precession frequency of polarized electrons. To measure the precessionfrequency (cid:36) , a method of beam resonance depolarization by an oscillatingelectromagnetic field can be used (2) . There are many forms of depolarizers,but we will mention just one, which especially simple. It is a depolarizerwhose operation depends on the radio-frequency longitudinal magneticfield which is produced by a current-curring loop around a ceramic sectionof the vacuum chamber.Suppose the observer in the lab frame performs the beam energy measure-ment. We should examine what parts of the measured data depends on thechoice of synchronization convention and what parts do not. Clearly, physi-cally meaningful results must be convention-invariant. One might think thatthis is a typical time-depending measurement of function (cid:126) s ( t ). However, we196tate that the precession frequency (cid:36) has no intrinsic meaning - its meaningis only being assigned by a convention. It is not possible to determine theprecession frequency (cid:36) uniquely, because there is always some arbitrarinessin the (cid:126) s ( t ). For instance, it is always possible to make an arbitrary changein the rhythm of the clocks (i.e. scale of the time). But our problem is todetermine the energy for an electron beam. So one needs to measure alsothe revolution frequency ω by using the same space-time grid. What thisall means physically is very interesting. The ratio (cid:36)/ω is convention in-dependent i.e it does not depend on the distant clocks synchronization oron the rhythm of the clocks. It means, for example, that if we observe thedimensionless frequency (cid:36)/ω , we can find out the value of the convention-invariant beam energy E . The ( g / −
1) factor can be calculated by use ofquantum electrodynamics.Let us now return to our examination of the measured data in experimentsrelated with the calibration of the beam energy in a storage ring. The spin (cid:126) s of a particle makes the angle φ with it velocity. From Eq.(101) we have beenable to write the angle φ in therm of orbital angle θ ( l ) in a form φ = φ ( θ ).We thus use the orbital angle θ as evolution parameter. Suppose that thedepolarizer is placed at an azimuth θ . During a period of velocity rotation,the spin will rotate through an angle of ∆ φ = φ ( θ + π ) − φ ( θ ). The point isthat depolarizer measurements are made to determine the observable ∆ φ .Let us see how equation Eq.(101) gives the observable ∆ φ . It can be writtenin integral form ∆ φ = (cid:82) d θ [( g / − E / ( mc )] = π [( g / − E / ( mc )]. We canalready conclude something from these results. The convention-invariantobservation ∆ φ is actually a geometric parameter. It comes quite naturallythat in experiments related with spin dynamics in a storage ring we do notneed to ask question about the function (cid:126) s ( t ) experimentally. → as Dynamics E ff ect10.7.1 Spin Tracking at g → g →
0. The BMT equation for a particle withsmall g factor is dS α d τ = − c u α (cid:32) S λ du λ d τ (cid:33) = − emc u α (cid:16) S λ F λµ u µ (cid:17) . (102)It is often more convenient to write this equation as the equation of motion197or (cid:126) s . If the field is transverse, then the equation Eq.(102) is reduced to d (cid:126) sd τ = (cid:20)(cid:18) E mc − (cid:19) emc (cid:126) B (cid:21) × (cid:126) s , (103)Note that the equation Eq.(103) for the proper spin (cid:126) s and the BMT equationEq.(102) for four spin S µ are completely equivalent. Eq.(103) is the result oftransformation to new spin variables.Conventional spin tracking treats the space-time continuum in a non rela-tivistic format, as a (3 +
1) manifold. In the conventional spin tracking, weassign absolute time coordinate and we have no mixture of positions andtime. This approach to relativistic spin dynamics relies on the use of threeequations for the spin motion d (cid:126) sdt = (cid:34)(cid:32) − γ (cid:33) emc (cid:126) B (cid:35) × (cid:126) s = − (cid:2)(cid:0) γ − (cid:1) (cid:126)ω (cid:3) × (cid:126) s , (104)which are based on the use of the absolute time t as the evolution parameter.Here, (cid:126)ω = − e (cid:126) B / ( mc γ ) is the particle angular frequency in the lab frame. Nowthe time-like part of the four-velocity is decomposed to c γ = c / √ − v / c .This decomposition is a manifestation of the absolute time convention.There are two di ff erent (covariant and non covariant) approaches that pro-duce the same spin orientation (cid:126) s ( l ) along the path. Using the Eq.(103) orEq.(104) we obtain d (cid:126) sdl = − (cid:20) E mc − (cid:21) d (cid:126)θ dl × (cid:126) s , (105)Both approaches describe correctly the same physical reality, and the ro-tation of the proper spin (cid:126) s with respect to the lab frame axes at g → ff ects such as Wigner rotation, Lorentz-Fitzgeraldcontraction, time dilation and relativistic corrections to the law of composi-tion of velocities are coordinate (i.e. convention-dependent) e ff ects and haveno exact objective meaning. In the case of the Lorentz coordinatization, onewill experience e.g. the Wigner rotation phenomenon. In contrast to this, inthe case of absolute time coordinatization there are no relativistic kinematicse ff ects, and no Wigner rotation will be found (3) .198owever, the spin orientation at each point of the particle path has exactobjective meaning. In fact, Eq.(105) is convention-invariant i.e includes onlyquantities which have exact objective meaning. Understanding this resultof the theory of relativity is similar to understanding the previously dis-cussed result for relativistic mass correction. We find that the evolution ofa particle along its path is still given by the corrected Newton’s second laweven though the relativistic mass correction has no kinematical origin. Amethodological analogy with the spin dynamics equation Eq.(105) emergeby itself. The spin rotation in the lab frame at g → ff ect (asthe relativistic mass correction) but it has no kinematical origin. The expression for the Wigner rotation in the lab frame obtained by authorsof textbooks is given by (cid:126)δ
Φ = (1 − γ ) (cid:126) v × d (cid:126) v / v = (1 − γ ) (cid:126)δθ , which often pre-sented as (cid:126)ω Th = d (cid:126) Φ / dt = (1 − γ ) (cid:126) v × d (cid:126) v / dt / v . In other words, the proper framecoordinate performs a precession relative to the lab frame with the velocityof precession (cid:126)ω Th , where d (cid:126) v / dt is the acceleration of the spinning particlein the lab frame. This precession phenomenon is called Thomas precession.From the viewpoint of the generally accepted terminology, Thomas preces-sion is actually a manifestation of the Wigner (Thomas) rotation. Accordingto expression for Thomas precession in the lab frame presented in textbooks,the comoving frame precesses in the opposite direction with respect to thedirection of the angular velocity of the precession (cid:126)ω = (cid:126) v × d (cid:126) v / dt / v and ω Th → −∞ in the limit γ −→ ∞ . The theory of relativity shows us that thetextbook expression for the Thomas precession in the lab frame and correctresult (cid:126)ω Th = (1 − /γ ) (cid:126) v × d (cid:126) v / dt / v actually di ff er both in sign and magnitude. The existence of the usual incorrect expression for the Thomas precessionin the lab frame has led to incorrect interpretation of the BMT result and, inparticular, of the spin dynamics equation Eq.(104). Using the incorrect resultfor the Thomas precession, the BMT result for a small g factor, Eq.(104), isusually presented as d (cid:126) sdt = − (cid:2)(cid:0) γ − (cid:1) (cid:126)ω (cid:3) × (cid:126) s = (cid:126)ω Th × (cid:126) s , (106)199requently, the first stumbling blocks in the process of understanding andaccepting the correct Wigner (Thomas) rotation theory is a widespread beliefthat the experimental test of the BMT equation is a direct test of the incorrectexpression for Thomas precession. There are many physicists who havealready received knowledge about the Thomas precession from well-knowntextbooks and who would say, ”The extremely precise measurements ofthe magnetic-moment anomaly of the electron made on highly relativisticelectrons are based on the BMT equation, of which the Thomas precessionis an integral part, and can be taken as experimental confirmations of thestandard expression for the Thomas precession.” This misconception aboutexperimental test of the incorrect expression for the Thomas precession inthe lab frame is common and pernicious.The interpretation of Eq.(104) as the Thomas precession Eq.(106) is presentedin textbooks as alternative approach to the already developed BMT theory (3) . Authors of textbooks got the correct BMT result by using the incorrectexpression for the Thomas precession and an incorrect physical argument.This wrong argument is an assumption about the splitting of spin precessionin the lab frame into dynamics (Larmor) and kinematics (Thomas) parts. Inthis chapter we demonstrated that this splitting can not be obtained in thelab frame. It is possible to perform this splitting only in the Lorentz properframe where the spinning particle is at rest and the Lorentz lab referenceframe moves with velocity − (cid:126) v and acceleration − γ d (cid:126) v / d τ with respect to theproper frame axes.Let us now review the subjects discussed in this chapter. We considered thewidespread misconception that if a particle with spin has no magnetic mo-ment ( g → (4) . We discussed how authors of textbooks got the correct BMT result by theincorrect expression for the Thomas precession and an incorrect physicalargument. This wrong argument is an assumption about the splitting ofthe spin precession in the lab frame into dynamics (Larmor) and kinematics(Thomas) parts. This splitting can not be realized in the lab frame for the fol-lowing reason: the starting point of the BMT theory is the phenomenologicaldynamics law d (cid:126) s / d τ = eg (cid:126) s × (cid:126) B R / (2 mc ), which is the equation of motion forthe angular momentum in its rest frame (i.e. with respect to the rest frameaxes). It is phenomenological because the microscopic interpretation of the(anomalous) magnetic moment of a particle is not given. The BMT equationis a relativistic generalization of the phenomenological dynamics law. It isvalid for any given Lorentz frame, for example for the Lorentz lab frame. Inthe lab frame, the BMT equation is a phenomenological dynamics equationfor the spin motion with respect to the lab frame axis even at g → g → ff ects are coordinate (i.e.200onvention-dependent) e ff ects and have no exact objective meaning. In thecase of Lorentz coordinatization, one will experience e.g. the Thomas pre-cession phenomenon. In contrast to this, in the case of absolute time co-ordinatization there are no relativistic kinematics e ff ects, and therefore noThomas precession will be found. However, the spin orientation at g → ff erent result for spin rotation withrespect to the lab frame axes.At this point a reasonable question arises: why in the lab frame the spinorientation with respect to the lab frame axes has physical meaning, butthe same orientation in the proper frame does not? The answer is that inthe lab frame an observer who performs spin orientation measurement isat rest with respect to the lab reference frame. This situation is symmetricalwith respect to a change of the reference frames. In fact, the spin orientationmeasurement with respect to the proper frame axes in the proper frame hasexact objective meaning (i.e. it has dynamical origin) and we observe thesame result no matter how the lab frame rotates with respect to the properframe.The argument that the result of spin orientation measurements with respectto the lab frame axes in the proper frame is paradoxical runs somethinglike this: the laws of physics in any one reference frame should be ableto account for all physical phenomena, including the observations made bymoving observers. Suppose that an observer in the lab frame performs a spinrotation measurement. Viewed from the proper frame, the two proper framecoordinatizations give a di ff erent result for the spin rotation with respect to201he lab frame axes in the lab frame which must be convention-invariant.Nature doesn’t see a paradox, however, because the proper observer seesthat the lab polarimeter is moving on an accelerated motion, and the lab ob-server, moving with the polarimeter, performs the spin direction measure-ment. In order to predict the result of the moving polarimeter measurementone does not need to have access to the detailed dynamics of the particleinto the polarimeter. It is enough to assume the Lorentz covariance of thefield theory involved in the description of the polarimeter operation.In the Lorentz proper frame the field theory involved in the description ofthe (lab) polarimeter operation is isotropic. Clearly, in the case of Lorentzcoordinatization we can discuss in the proper frame about the spin orienta-tion with respect to the lab frame axes as a prediction of the measurementmade by the lab observer.Now the question is, what is the prediction of the proper observer in the caseof absolute time coordinatization? How shell we describe the polarimeteroperation after the Galilean transformations? After the Galilean transfor-mations we obtain the complicated (anisotropic ) field equations. The newterms that have to be put into the field equations due to the use of Galileantransformations lead to the same prediction as concerns experimental re-sults: the spin of the particle is rotated with respect to the lab frame axesaccording to the Lorentz coordinatization prediction. Let us examine in alittle more detail how this spin rotation comes about. As usual, in the case ofabsolute time coordinatization, we are going to make a mathematical trickfor solving the field equations with anisotropic terms. In order to eliminatethese terms we make a change of the variables. Using new variables weobtain the phenomenon of spin rotation with respect to the lab frame axesin the lab frame.
1. The motion of the classical spin in an external electromagnetic field ispresented by the Bargmann-Michel-Telegdi (BMT) equation [60]. The BMTequation is manifestly covariant and can be used in any inertial frame. It isthe law of motion of the four-spin for a particle in a uniform electric andmagnetic fields.2. A method for measuring the particle energy in an electron-positron stor-age ring by means of resonance depolarization by a radio-frequency longi-tudinal magnetic field is described in [61].3. The results in the Bargmann-Michel-Telegdi paper [60] were obtained202y the method of semi-classical approximation of the Dirac equation. TheWigner rotation was not considered in [60] at all, because the Dirac equationallow obtaining the solution for the total particle’s spin motion without anexplicit splitting it into the Larmor and Wigner parts.4. It is generally believed that ”If the particle with spin has no magneticmoment ( g = U nucleus has a rather small g factor ( g = − . ff ect dominates over the dynamics one.”[62]. Thisstatement presented in most published papers and textbooks is misleading.The reason is that a splitting of particle spin motion into the dynamic andkinematic (Wigner) parts cannot be performed in the Lorentz lab frame.In the Lorentz lab frame, Eq.(102), is a relativistic ”generalization” of theequation of motion for a particle angular momentum in its rest frame. Inother words, Eq.(102) is a dynamics equation. The relativistic kinematicse ff ects, such as Wigner rotation, are coordinate e ff ects and have no exactobjective meaning. However, the spin orientation with respect to the labframe axes at each point of the particle path in the lab frame has exactobjective meaning. 203 eferences [1] D. Bohm ”The Special Theory of Relativity” W. A. Benjamin Inc., 1965[2] S. Drake and A. Purvis, Am. J. Phys. 82, 52 (2014)[3] L. Landau and E. Lifshitz, ”The Classical Theory of Fields” Pergamon, 1975[4] A. French ”Special Relativity” W.W. Norton Company Inc., 1968[5] R. Henriksen ”Practical Relativity” John Wiley and Son Ltd., 2011[6] R. Baierlein, Am. J. Phys. 74, 193 (2006).[7] C. Cristodoulides ”The Special Theory of Relativity” Springer InternationalPublishing, 2016[8] C. R. Anderson, I. Vetharaniam, and G. Stedman, Phys. Rep. 295, 93 (1998)[9] J. Awrejcewicz, ”Classical Mechanics” Springer, 2012[10] C. Leubner, K. Auflinger, and P. Krumm, Eur. J. Phys. 13, 170 (1992)[11] G. Ferrarese and D. Bini ” Introduction to Relativistic Continuum Mechanics”Springer-Verlag Berlin, 2008[12] C. Moeller, ”The Theory of relativity”, Clarendon, 1952[13] H. Reichenbach, ”The Philosophy of Space and Time”, Dover Pub. Inc. , 1958[14] M. Friedman, ”Foundation of Space Time Theories”, Princenton Universitypress, (1983)[15] A. Logunov, ”Lectures on the Relativity and Gravity Theory”, M. Nauka (1987)[16] C. H. Brown, ”Physical relativity”, Clarendon Press (2005)[17] A. Eddington ” The Mathematics Theory of Relativity”, Cambridge UniversityPress, 1923[18] H. Erlichson , Am. J. Phys. 53, 1 (1985).[19] P. Hrasko, ””Basic Relativity” Springer, 2011[20] W. Pauli, ”Theory of Relativity” Pergamon Press, 1958[21] J. Fox, American Journal of Physics, 33,1, 1965[22] J. Goodman ” Introduction to Fourier Optics” McGraw-Hill Comp., ()1996)[23] A. Sommerfeld ”Optics” (Academic Press Inc., 1954)[24] V. Ugarov ”Special theory of relativity”, Mir Publishers, 1979[25] D. Marcuse ”Light Transmission Optics”, Van Nostrand Reinhold Comp.,(1972)
26] J. Norton, Archive for history of exact science, November 2004 , V 59, pp 45-105[27] L. Brillouin ”Wave propagation and Group Velocity”, Academic Press, 1960[28] M. G. Sagnac, C.R. Acad. Sci. 157 (1913) 1410[29] E. Post, Rev. Mod. Phys. 39 (1967) 475[30] G. Malykin, Physics- Uspechi 43 (2000) 1229[31] P. Langevin, C. R. Hebt, Seances Acad. Sci. Paris 173 (1921) 831[32] A. Einstein, Reflecsions sur l’Electrodynamique l’Ether la Geometrie et laRelativite, Collection Discours de la Methode (Gauthier-Villars, Paris, 1972)pp. 68[33] J. Bradley, Phil Trans. 35(1728) 637[34] J. Goodman, ”Statistical Optics”, Jon Wiley and Sons, 1985[35] R. Brown and R. Twiss, Nature, 178 (1956)1046[36] H. Ives, J. Opt. Soc. Am., 40 (1950)185[37] G. Puccini and F. Selleri, Nuovo Cimento, 117 B (2002)283[38] F. Selleri, Foundation of Physics, 36 (2006)443[39] B. Kosyakov ”Introduction to the Classical theory of Particles and Fields”Springer-Verlag Berlin, 2007[40] J. Rafelski ”Relativity Matter” Springer International Publishing, 2017[41] E. Gourgoulhon ”Special Relativity in General Frames” Springer-Verlag BerlinHeidelberg, 2013[42] F. Scheck, Classical field theory” Springer-Verlag (2012)[43] K. Westfold, Astrophysical Journal 130, 241[44] R. Epstein and P. Feldman, Astrophysical Journl 150, 109[45] V. Ginzburg, ”Application of Electrodynamics in Theoretical Physics andAstrophysics” Gordon and Breach Science Publisher, 1989[46] L. Oster, Phys. Rev. 121 p 961 (1961)[47] Y. Li et al., Phys. Rev. ST AB 13, 080705 (2010)[48] A. Lutman et al., Nature Photonics 10, 468 (2016)[49] T. Tanaka, H. Kitamura and T. Shintake, Nucl. Instr. and Meth. A 528, 172 (2004)[50] H.-D. Nuhn et al., ‘Commissioning of the Delta polarizing undulator at LCLS’,in Proceedings of the 2015 FEL Conference, Daejeon, South Korea, WED01(2015).
51] P. Baxevanis, Z. Huang, and G. Stupakov Phys. Rev. ST AB 20, 040703 (2017)[52] J. MacArthur, et al., Phys. Rev. X 8, 041036 (2018)[53] E. Wigner, Z. Phys. 124, 665 (1948)[54] E. Wigner, Rev. Mod. Phys. 29, 255 (1957)[55] J. Jackson, ”Classical Electrodynamics”, 3rd ed., Wiley, New York (1999)[56] V. Ritus, Sov. Phys. JETP 13, 240 (1961)[57] V. Ritus, Phys. Usp. 50, 95-101 (2007)[58] G. Malykin, Phys. Usp. 49, 37 (2006)[59] M. Stranberg, Am. J. Phys. 54, 321 (1986)[60] V. Bargmann, L. Michel, V. Telegdi, Phys. Rev. Lett. 2, 435, (1959)[61] Ya. Derbenev at al., Particle Accelerators 10, 177 (1980)[62] S. Stepanov, Physics of Particles and Nuclei, 2012, V43, p128[63] G. Geloni et al., Optics Commun. 276, 167 (2007)[64] H. Onuki and P. Elleaume, ”Undulators, Wigglers and their Applications”,Tailor Francis, 2003[65] V. Ginzburg and I. Frank, Soviet Phys. JETP 16, 15 (1946) ppendix I. Radiation by Moving Charges
We start with the solution of Maxwell’s equation in the space-time domain,the well-known Lienard-Wiechert expression, and we subsequently applya Fourier transformation. The Lienard-Wiechert expression for the electricfield of a point charge ( − e ) reads (see, e.g. [55]): (cid:126) E ( (cid:126) r o , t ) = − e (cid:126) n − (cid:126)βγ (1 − (cid:126) n · (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) | − ec (cid:126) n × [( (cid:126) n − (cid:126)β ) × ˙ (cid:126)β ](1 − (cid:126) n · (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) | . (107) R = | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | denotes the displacement vector from the retarded positionof the charge to the point where the fields are calculated. Moreover, (cid:126)β = (cid:126) v / c , (cid:126) ˙ β = (cid:126) ˙ v / c , while (cid:126) v and (cid:126) ˙ v denote the retarded velocity and accelerationof the electron. Finally, the observation time t is linked with the retardedtime t (cid:48) by the retardation condition R = c ( t − t (cid:48) ). As is well-known, Eq.(107) serves as a basis for the decomposition of the electric field into asum of two quantities. The first term on the right-hand side of Eq. (107)is independent of acceleration, while the second term linearly depends onit. For this reason, the first term is called ”velocity field”, and the second”acceleration field” [55]. The velocity field di ff ers from the acceleration fieldin several respects, one of which is the behavior in the limit for a very largedistance from the electron. There one finds that the velocity field decreaseslike R − , while the acceleration field only decreases as R − . Let us apply aFourier transformation: (cid:126) ¯ E ( (cid:126) r o , ω ) = − e ∞ (cid:90) −∞ dt (cid:48) (cid:126) n − (cid:126)βγ (1 − (cid:126) n · (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) | exp (cid:34) i ω (cid:32) t (cid:48) + | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | c (cid:33)(cid:35) − ec ∞ (cid:90) −∞ dt (cid:48) (cid:126) n × [( (cid:126) n − (cid:126)β ) × ˙ (cid:126)β ](1 − (cid:126) n · (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) | exp (cid:34) i ω (cid:32) t (cid:48) + | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | c (cid:33)(cid:35) . (108)As in Eq. (107) one may formally recognize a velocity and an accelerationterm in Eq. (108) as well. Since Eq. (108) follows directly from Eq. (107), thatis valid in the time domain, the magnitude of the velocity and accelerationparts in Eq. (108), that include terms in 1 / R and 1 / R respectively, do notdepend on the wavelength λ . It is instructive to take advantage of integrationby parts. With the help of1 c ddt (cid:48) | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | = − (cid:126) n · (cid:126)β and d (cid:126) ndt (cid:48) = c | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | (cid:104) − (cid:126)β + (cid:126) n (cid:16) (cid:126) n · (cid:126)β (cid:17)(cid:105) , (109)Eq. (108) can be written as 207 ¯ E ( (cid:126) r o , ω ) = − e ∞ (cid:90) −∞ dt (cid:48) (cid:126) n | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | exp (cid:34) i ω (cid:32) t (cid:48) + | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | c (cid:33)(cid:35) + ec ∞ (cid:90) −∞ dt (cid:48) ddt (cid:48) (cid:126)β − (cid:126) n (1 − (cid:126) n · (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | exp (cid:34) i ω (cid:32) t (cid:48) + | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | c (cid:33)(cid:35) . (110)Eq. (110) may now be integrated by parts. When edge terms can be droppedone obtains [63] (cid:126) ¯ E ( (cid:126) r o , ω ) = − i ω ec ∞ (cid:90) −∞ dt (cid:48) (cid:126)β − (cid:126) n | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | − ic ω (cid:126) n | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | × exp (cid:40) i ω (cid:32) t (cid:48) + | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | c (cid:33)(cid:41) . (111)The only assumption made going from Eq. (108) to Eq. (111) is that edgeterms in the integration by parts can be dropped. This assumption canbe justified by means of physical arguments in the most general situationaccounting for the fact that the integral in dt (cid:48) has to be performed over theentire history of the particle and that at t (cid:48) = −∞ and t (cid:48) = + ∞ the electrondoes not contribute to the field anymore. Let us give a concrete example foran ultra-relativistic electron. Imagine that bending magnets are placed at thebeginning and at the end of a given setup, such that they deflect the electrontrajectory of an angle much larger than the maximal observation angle ofinterest for radiation from a bending magnet. This means that the magnetswould be longer than the formation length associated with the bends, i.e. L fb ∼ ( c ρ /ω ) / , where ρ is the bending radius. In this way, intuitively, themagnets act like switches: the first magnet switches the radiation on, thesecond switches it o ff . Then, what precedes the upstream bend and whatfollows the downstream bend does not contribute to the field detected atthe screen position. With these caveat Eq. (111) is completely equivalent toEq. (108).The derivation of Eq. (111) is particularly instructive because shows thateach term in Eq. (111) is due to a combination of velocity and accelerationterms in Eq. (108). In other words the terms in 1 / R and in 1 / R in Eq. (111)appear as a combination of the terms in 1 / R (acceleration term) and 1 / R (velocity term) in Eq. (108). As a result, one can say that there exist contri-butions to the radiation from the velocity part in Eq. (108). The presentationin Eq. (111) is more interesting with respect to that in Eq. (108) (althoughequivalent to it) because the magnitude of the 1 / R -term in Eq. (111) can208irectly be compared with the magnitude of the 1 / R -term inside the integralsign.The bottom line is that physical sense can be ascribed only to the integral inEq. (108) or Eq. (111). The integrand is, in fact, an artificial construction. Inthis regard, it is interesting to note that the integration by parts giving Eq.(111) is not unique. First, we find that [63] (cid:126) n × [( (cid:126) n − (cid:126)β ) × (cid:126) ˙ β ] | (cid:126) r o − (cid:126) r (cid:48) | (1 − (cid:126) n · (cid:126)β ) = | (cid:126) r o − (cid:126) r (cid:48) | ddt (cid:48) (cid:126) n × ( (cid:126) n × (cid:126)β )(1 − (cid:126) n · (cid:126)β ) − (cid:126) ˙ n ( (cid:126) n · (cid:126)β ) + (cid:126) n ( (cid:126) ˙ n · (cid:126)β ) − (cid:126) ˙ n ( (cid:126) n · (cid:126)β ) − (cid:126)β ( (cid:126) ˙ n · (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) | (1 − (cid:126) n · (cid:126)β ) . (112)Note that Eq. (112) accounts for the fact that (cid:126) n = ( (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) )) / | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | is nota constant in time. Using Eq. (112) in the integration by parts, we obtain (cid:126) ¯ E ( (cid:126) r o , ω ) = − i ω ec ∞ (cid:90) −∞ dt (cid:48) − (cid:126) n × ( (cid:126) n × (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | + ic ω (cid:126)β − (cid:126) n − (cid:126) n ( (cid:126) n · (cid:126)β ) | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | × exp (cid:40) i ω (cid:32) t (cid:48) + | (cid:126) r o − (cid:126) r (cid:48) ( t (cid:48) ) | c (cid:33)(cid:41) . (113)Similarly as before, the edge terms have been dropped. Eq. (108), Eq. (111)and Eq. (113) are equivalent but include di ff erent integrands. This is nomistake, as di ff erent integrands can lead to the same integral.If the position of the observer is far away enough from the charge, one canmake the expansion Eq. (22). Using Eq. (113), we obtain Eq. (23).209 ppendix II. Undulator Radiation in Resonance Approximation. Far Zone Calculations pertaining undulator radiation are well established see e.g.[64]. In all generality, the field in Eq. (38) can be written as (cid:126) (cid:101) E = exp (cid:34) i ωθ z c (cid:35) i ω ec z × L / (cid:90) − L / dz (cid:48) (cid:40) K i γ (cid:2) exp (2 ik w z (cid:48) ) − (cid:3) (cid:126) e x + (cid:126)θ exp ( ik w z (cid:48) ) (cid:41) × exp (cid:34) i (cid:32) C + ωθ c (cid:33) z (cid:48) − K θ x γ ω k w c cos( k w z (cid:48) ) − K γ ω k w c sin(2 k w z (cid:48) ) (cid:35) . (114)Here ω = ω r + ∆ ω , C = k w ∆ ω/ω r and ω r = k w c ¯ γ z , (115)is the fundamental resonance frequency.Using the Anger-Jacobi expansion:exp (cid:2) ia sin( ψ ) (cid:3) = ∞ (cid:88) p = −∞ J p ( a ) exp (cid:2) ip ψ (cid:3) , (116)where J p ( · ) indicates the Bessel function of the first kind of order p , to writethe integral in Eq. (114) in a di ff erent way: (cid:126) (cid:101) E = exp (cid:34) i ωθ z c (cid:35) i ω ec z ∞ (cid:88) m , n = −∞ J m ( u ) J n ( v ) exp (cid:20) i π n (cid:21) × L / (cid:90) − L / dz (cid:48) exp (cid:34) i (cid:32) C + ωθ c (cid:33) z (cid:48) (cid:35) × (cid:40) K i γ (cid:2) exp (2 ik w z (cid:48) ) − (cid:3) (cid:126) e x + (cid:126)θ exp ( ik w z (cid:48) ) (cid:41) exp [ i ( n + m ) k w z (cid:48) ] , (117)210here u = − K ω γ k w c and v = − K θ x ωγ k w c . (118)Up to now we just re-wrote Eq. (38) in a di ff erent way. Eq. (38) and Eq. (117)are equivalent. Of course, definition of C is suited to investigate frequenciesaround the fundamental harmonic but no approximation is taken besidesthe paraxial approximation.Whenever C + ωθ c (cid:28) k w , (119)the first phase term in z (cid:48) under the integral sign in Eq. (117) is varyingslowly on the scale of the undulator period λ w . As a result, simplificationsarise when N w (cid:29)
1, because fast oscillating terms in powers of exp[ ik w z (cid:48) ]e ff ectively average to zero. When these simplifications are taken, resonanceapproximation is applied, in the sense that one exploits the large parameter N w (cid:29)
1. This is possible under condition (119). Note that (119) restricts therange of frequencies for positive values of C independently of the obser-vation angle θ , but for any value C < (cid:111) r = c /ω r ) there is always some range of θ such that Eq. (119) can be applied.Altogether, application of the resonance approximation is possible for fre-quencies around ω r and lower than ω r . Once any frequency is fixed, (119)poses constraints on the observation region where the resonance approxima-tion applies. Similar reasonings can be done for frequencies around higherharmonics with a more convenient definition of the detuning parameter C .Within the resonance approximation we further select frequencies such that | ∆ ω | ω r (cid:28) , i . e . | C | (cid:28) k w . (120)Note that this condition on frequencies automatically selects observationangles of interest θ (cid:28) /γ z . In fact, if one considers observation anglesoutside the range θ (cid:28) /γ z , condition (119) is not fulfilled, and the inte-grand in Eq. (117) exhibits fast oscillations on the integration scale L . As aresult, one obtains zero transverse field, (cid:126) (cid:101) E =
0, with accuracy 1 / N w . Underthe constraint imposed by (120), independently of the value of K and for Here the parameter v should not be confused with the velocity. θ (cid:28) /γ z , we have | v | = K | θ x | γ ω k w c = (cid:18) + ∆ ωω r (cid:19) √ K √ + K ¯ γ z | θ x | (cid:46) ¯ γ z | θ x | (cid:28) . (121)This means that, independently of K , | v | (cid:28) J n ( v ) inEq. (117) according to J n ( v ) (cid:39) [2 − n / Γ (1 + n )] v n , Γ ( · ) being the Euler gammafunction Γ ( z ) = ∞ (cid:90) dt t z − exp[ − t ] . (122)Similar reasonings can be done for frequencies around higher harmonicswith a di ff erent definition of the detuning parameter C . However, aroundodd harmonics, the before-mentioned expansion, together with the appli-cation of the resonance approximation for N w (cid:29) ik w z (cid:48) ] e ff ectively average to zero), yields extra-simplifications.Here we are dealing specifically with the first harmonic. Therefore, theseextra-simplifications apply. We neglect both the term in cos( k w z (cid:48) ) in thephase of Eq. (114) and the term in (cid:126)θ in Eq. (114). First, non-negligible termsin the expansion of J n ( v ) are those for small values of n , since J n ( v ) ∼ v n , with | v | (cid:28)
1. The value n = J ( v ) ∼
1. Then,since the integration in dz (cid:48) is performed over a large number of undulatorperiods N w (cid:29)
1, all terms of the expansion in Eq. (117) but those for m = − m = K /γ , and can be traced back tothe term in (cid:126) e x only, while the term in (cid:126)θ in Eq. (117) averages to zero for n = n = ± J ± ( v ) ∼ v .Then, the term in (cid:126) e x in Eq. (117) is v times the term with n = m . The term in (cid:126)θ wouldsurvive averaging when n = , m = − n = − , m =
0. However,it scales as (cid:126)θ v . Now, using condition (120) we see that, for observation anglesof interest θ (cid:28) /γ z , | (cid:126)θ | | v | ∼ ( √ K / √ + K ) ¯ γ z θ (cid:28) K /γ . Therefore, theterm in (cid:126)θ is negligible with respect to the term in (cid:126) e x for n =
0, that scales as K /γ . All terms corresponding to larger values of | n | are negligible.Summing up, all terms of the expansion in Eq. (116) but those for n = m = − m = A JJ = J (cid:32) ω K k w c γ (cid:33) − J (cid:32) ω K k w c γ (cid:33) , (123)212hat can be calculated at ω = ω r since | C | (cid:28) k w , we have (cid:126) (cid:101) E = − K ω e c z γ A JJ exp (cid:34) i ωθ z c (cid:35) L / (cid:90) − L / dz (cid:48) exp (cid:34) i (cid:32) C + ωθ c (cid:33) z (cid:48) (cid:35) (cid:126) e x . (124)213 ppendix III. Self-Electromagnetic Fields of the Modulated Electron Beam The transverse field (cid:126) ¯ E ⊥ can be treated in terms of Paraxial Maxwell’s equa-tions in the space-frequency domain (see e.g. [63]). From the paraxial ap-proximation follows that the electric field envelope (cid:126) (cid:101) E ⊥ = (cid:126) ¯ E ⊥ exp [ − i ω z / c ]does not vary much along z on the scale of the reduced wavelength (cid:111) = λ/ (2 π ). As a result, the following field equation holds: D (cid:20) (cid:126) (cid:101) E ⊥ ( z , (cid:126) r ⊥ , ω ) (cid:21) = (cid:126) g ( z , (cid:126) r ⊥ , ω ) , (125)where the di ff erential operator D is defined by D ≡ (cid:32) ∇ ⊥ + i ω c ∂∂ z (cid:33) , (126) ∇ ⊥ being the Laplacian operator over transverse cartesian coordinates. Eq.(125) is Maxwell’s equation in paraxial approximation. The source-termvector (cid:126) g ( z , (cid:126) r ⊥ ) is specified by the trajectory of the source electrons, and canbe written in terms of the Fourier transform of the transverse current density, (cid:126) ¯ j ⊥ ( z , (cid:126) r ⊥ , ω ), and of the charge density, ¯ ρ ( z , (cid:126) r ⊥ , ω ), as (cid:126) g = − π exp (cid:20) − i ω zc (cid:21) (cid:18) i ω c (cid:126) ¯ j ⊥ − (cid:126) ∇ ⊥ ¯ ρ (cid:19) . (127) (cid:126) ¯ j ⊥ and ¯ ρ are regarded as given data. We will treat (cid:126) ¯ j ⊥ and ¯ ρ as macroscopicquantities, without investigating individual electron contributions. In thetime domain, we may write the charge density ρ ( (cid:126) r , t ) and the current density (cid:126) j ( (cid:126) r , t ) as ρ ( (cid:126) r , t ) = v ρ ⊥ ( (cid:126) r ⊥ ) f (cid:18) t − zv (cid:19) (128)and (cid:126) j ( (cid:126) r , t ) = v (cid:126) v ρ ⊥ ( (cid:126) r ⊥ ) f (cid:18) t − zv (cid:19) , (129)where v denote the velocity of the electron. The quantity ρ ⊥ has the meaningof transverse electron beam distribution, while f is the longitudinal chargedensity distribution. 214n the space-frequency domain, Eq. (128) and Eq. (129) transform to:¯ ρ ( (cid:126) r ⊥ , z , ω ) = ρ ⊥ (cid:0) (cid:126) r ⊥ (cid:1) ¯ f ( ω ) exp [ i ω z ) / v ] , (130)and (cid:126) ¯ j ( (cid:126) r ⊥ , z , ω ) = (cid:126) v ρ ⊥ (cid:0) (cid:126) r ⊥ (cid:1) ¯ f ( ω ) exp [ i ω z / v ] . (131)It should be remarked that ¯ ρ and (cid:126) ¯ j = ¯ ρ(cid:126) v satisfy the continuity equation. Inother words, one can find (cid:126) ∇ · (cid:126) ¯ j = i ω ¯ ρ .We find an exact solution of Eq. (126) without any other assumption aboutthe parameters of the problem. A Green’s function for Eq. (126), namely thesolution corresponding to the unit point source can be written as (see e.g.[63]): G ( z − z (cid:48) ; (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ ) = − π ( z − z (cid:48) ) exp i ω | (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | c ( z − z (cid:48) ) , (132)assuming z − z (cid:48) >
0. When z − z (cid:48) < z − z (cid:48) < z − z (cid:48) >
0, i.e. we can neglect contributions from sources located at z − z (cid:48) < (cid:126) (cid:101) E ⊥ ( z , (cid:126) r ⊥ ) = − i ω c ¯ f ( ω ) z (cid:90) dz (cid:48) (cid:90) d (cid:126) r (cid:48)⊥ exp (cid:40) i ω (cid:34) | (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | c ( z − z (cid:48) ) (cid:35) + i ω z (cid:48) c γ (cid:41) × z − z (cid:48) ρ ⊥ (cid:16) (cid:126) r (cid:48)⊥ (cid:17) (cid:32) (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ z − z (cid:48) (cid:33) . (133)Eq. (133) describes the field at any position z .First, we make a change in the integration variable from z (cid:48) to ξ ≡ z − z (cid:48) . Inthe limit for z −→ ∞ , corresponding to the condition z (cid:29) γ (cid:111) , we can writefor the transverse field 215 (cid:101) E ⊥ ( z , (cid:126) r ⊥ ) = − i ω ¯ f ( ω ) c (cid:90) d (cid:126) r (cid:48)⊥ ρ ⊥ (cid:16) (cid:126) r (cid:48)⊥ (cid:17) exp (cid:34) i ω z c γ (cid:35) (cid:40) ic ω (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | · dd (cid:104) | (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | (cid:105) × ∞ (cid:90) d ξξ exp (cid:34) + i ω | (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | c ξ − i ωξ c γ (cid:35) (cid:41) (134)We now use the fact that, for any real number α > ∞ (cid:90) d ξ exp [ i ( − ξ + α/ξ )] /ξ = K (cid:16) √ α (cid:17) , (135)where K is the zero order modified Bessel function of the second kind.Using Eq. (135) we can write Eq. (134) as (cid:126) (cid:101) E ⊥ ( z , (cid:126) r ⊥ ) = i ω ¯ f ( ω ) c exp (cid:34) i ω z c γ (cid:35) (cid:90) d (cid:126) r (cid:48)⊥ ρ ⊥ (cid:16) (cid:126) r (cid:48)⊥ (cid:17) × (cid:40) (cid:34) ic ω (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | (cid:35) γ z (cid:111) K (cid:32) | (cid:126) r ⊥ − (cid:126) r (cid:48)⊥ | γ (cid:111) (cid:33) (cid:41) , (136)where K ( · ) is the modified Bessel function of the first order.Let us assume a Gaussian transverse charge density distribution of theelectron bunch with rms size σ i.e. ρ ⊥ = (2 πσ ) − exp[ − r ⊥ / (2 σ )]. Within thedeep asymptotic region when the transverse size of the modulated electronbeam σ (cid:28) (cid:111) γ the Ginzburg-Frank formula can be applied [65] (cid:126) (cid:101) E ⊥ ( z , (cid:126) r ⊥ ) = − ω ec γ exp (cid:34) i ω z c γ (cid:35) (cid:126) r ⊥ r ⊥ K (cid:32) ω r ⊥ c γ (cid:33) . (137)Analysis of Eq.(137) shows a typical scale related to the transverse fielddistribution of order (cid:111) γ in dimensional units. Here λ is the modulationwavelength. In this asymptotic region radiation can be considered as virtualradiation from a filament electron beam (with no transverse dimensions).However, in XFEL practice we only deal with the deep asymptotic re-gion where σ (cid:29) (cid:111) γγ