Generalized Principle of limiting 4-dimensional symmetry.Relativistic length expansion in accelerated system revisited
GGeneralized Principle of limiting 4-dimensionalsymmetry. Relativistic length expansion in acceleratedsystem revisited.
Jaykov FoukzonIsrael Institute of [email protected] Scientific-Research Institutefor Optical and Physical Measurements,Moscow 119361, Russia
Abstract:
In this article, Generalized Principle of "limiting 4-dimensionalsymmetry":
The laws of physics in non-inertial frames must display the4-dimensional symmetry of the Generalized Lorentz-Poincare group in the limit ofzero acceleration, is proposed. Classical solution of the relativistic length expansionin general accelerated system revisited.
I.Introduction
In his famous 1905 paper, Einstein proposed that all physical theories shouldsatisfy the (now well-known) two postulates of special relativity:
Axiom The physical laws of nature and the results of all experimentsare independent of the particular inertial frame of the observer (in which theexperiment is performed); and
Axiom The speed of light is independent of the motion of the source.
Remark
Consider the Galilean group: x x v t ; t t ,
1. 1 or in equivalent infinitesimal form dx dx v dt ; dt dt ,
1. 2 From the above two postulates
Galilean group (0.1) was changed by Lorentzgroup: v x v t , t v t v c x , y y , z z , v v c .
1. 3 or in equivalent infinitesimal form dx v dx v dt , dt v dt v c dx , dy dy , dz dz .
1. 4 Axiom The correct implementation of postulates 1 and 2 is torepresent time as a fourth coordinate and constrain the relationship betweencomponents so as to satisfy the natural invariance induced by the Lorentz group(of electromagnetism). As is well known, this procedure leads to theconcept of Minkowski space M . Notation The third postulate was made by Minkowski [1],[2], awell-known mathematician, and was embraced by many.
Remark
In other words, we postulate that in the whole space there is aphysical frame of reference (FR) called an inertial (Galilee) one in which the intervalbetween events of this space is written as ds c dt dx dy dz . . In general, in the Minkowski space-time any FR in which the interval (1.1) has thegeneral form ds g ik dx i dx k ; i , k , , , . and which satisfies the allowance conditions g ; g dx dx ; , , , isallowed. Theorem
The curvature tensor becomes zero identically: R iklm in anyFR of the Minkowski space, including non-inertial (accelerated) one. Thetransformations connecting FR’s of the real moving bodies (particles of matter) withthe initial inertial Galilee FR (1.5) can be either linear or non-linear ones.The firstones (which are always non-orthogonal ones for the real bodies and, therefore, notcoinciding with the Lorentz transformations!) form a so-called generalized inertialFR (with non-orthogonal axes t , x ) the metrics of which has the off-diagonal part g , the second ones give in turn the non-inertial (accelerated) FR’s (also with anondiagonal metrics). The particular case of the generalized inertial FR is the FRconnected with the inertial one (1.5) by the classic Galilee transformation: x x v t ; t t , and corresponding to rotation of the axis t with fixed orientation ofthe axis x .Let us pass from Galilee’s coordinates t , x , y , z with the metric (1.5) tooordinates x , y , z , t by arbitrary linear transformation. This transformation isequivalent up to a space axis rotation to a transformation in plane t , x : x ax bt ; t qx pt ; y y ; z z . . Substituting (1.7) in (1.5), in coordinates x , y , z , t the metric gets the form: ds c g dt dtdx g dx dy dz . where g p b / c ; g c pq ab / c ; g c q a . The transformation (1.7)describes the rotation of the axes x , t in the plane x , t , with after the rotation theaxis x can be not orthogonal to the axis t , i.e. x and t rotate on angles, which maydiffer. The metric (1.8) gives a generalized inertial frame of reference in the SRT.The Lorentz transformations are a particular case of the general lineartransformations 3, corresponding to the choice g , g , g in (1.8).Hence, the metric (1.8), in contrast to (1.5), is not forminvariant with respect to theLorentz transformations. Let us consider a rotation of axis t without changing of the x orientation as particular case of the transformation (1.7). It is the classic Galileetransformation: x x v t ; t t , v const . . corresponding to the choice of parameters in (1.7) as p a ; q ; b v . Thus, the metric (1.8) get the form: ds c v c dt v dtdx dx dy dz . . Theorem L mi . The metricsof the generalized inertial FR is that with respect to the so-called generalizedinertial Lorentz-Poincare group L mi connected with the classic one by the relation: L kn x k in L mi km x k , . where ki is the matrix of the linear (non-orthogonal) transformations forming thegeneralized inertial FR x i ki x k . . The transformations (1.11) are orthogonal but connect the number ofnon-orthogonal FR’s with the same nondiagonal metrics.In the particular case ofthe Galilee transformation with metric (1.10) the group of transformations,keepingthe metric (1.10) forminvariant, takes the form: x v v v c x v c v t , t v v c x v v c t .
1. 13 At u transformations (1.13) coincides with the Lorentz transformation,naturally.Almost all physical frames of reference in the universe are, strictly speaking,non-inertial because of the long range action of the gravitational force. Thus it isdesirable that the laws of physics and the universal and fundamental constants arenderstood or known not only in inertial frames but also in non-inertial frames.So itis natural and necessary to require that the laws of physics in non-inertial framesmust display the 4-dimensional symmetry of the generalized Lorentz-Poincaregroup in the limit of zero acceleration.Such a requirement is postulated as theprinciple of Generalized limiting 4-dimensional symmetry Axiom . ( Generalized Principle of limiting 4-dimensional symmetry )The laws of physics in non-inertial frames must display the 4-dimensionalsymmetry of the Generalized Lorentz-Poincare group in the limit of zeroacceleration.In particular from Axiom 4 we obtain
Hsu’s Principle of limiting 4-dimensional symmetry [3]:The laws of physics in non-inertial orthogonal frames must display the4-dimensional symmetry of the Lorentz and Poincar ґ e groups in the limit of zeroacceleration.Let us consider transformations for the constant-linear-acceleration frame F w x and an inertial frame F I x I x t x w a , x x w b y y , z z , w t , , , a w , b w .
1. 14 where the velocity t is a linear function time t . The result (1.14) is called theWu transformation. It reduces to the Möller transformation when provided achange of time variable t w tanh w t is made [3].Furthermore, in the limit of zero acceleration, w the Wu transformation(1.14) reduces to the 4-dimensional transformations (which form the Lorentzgroup), t t x , x x t y y , z z .
1. 15 Thus, limiting 4-dimensional symmetry of the Lorentz and Poincar ґ e invariance issatisfied. The differential form of the Wu transformation (1.14) for constant-linear-acceleration is t W c dt dx , dx dx W c dt dy dy , dz dz , W c w x .
1. 16 Based on the differential from in (1.16) , we can consider the generalization ofthe Wu transformation (1.15) to a more general non-inertial frame F x ; w t movingwith an arbitrary velocity t or arbitrary acceleration w t along the x -axis t t , w t d t dt d t dt . , w w .
1. 17 A simple and general spacetime transformation for GLA frames is [3]: t x w t w , x x w t w , y y , z z .
1. 18 where the two constants of integration a and b are determined by the limiting4-dimensional symmetry as w w From the transformation (1.18), we can obtain a simple transformations for thedifferentials dx and dx : dt t W t dt t dx , dx t dx t W t dt dy dy , dz dz . W t t w t x t J e t w t W t t w t x J e t w t t
1. 19 The invariant infinitesimal interval ds in GLA frames can be obtained from (1.19). ds dt dx dy dz g dx dx W t , x dt U t dtdx dx dy dz , U t J e t w t , J e t dw t dt .
1. 20 When the jerk J e t vanishes, we have w t w and one can see that thetransformation (1.20) reduces to the Wu transformation (1.14) for aconstant-linear-acceleration frame F c x , in which the time axis is everywhereorthogonal to the spatial coordinate curves. I.Generalized Principle of limiting 4-dimensionalsymmetry
Let us considered general (acceleration) transformations between the tworelativistic frames one of which inertial frame K K t , x , y , z will be considered tobe at "rest", while another one accelerated frame K K t , x , y , z will move withrespect to the first one by the law: x t , x , t t , x , y y , z z .
2. 1 or in equivalent infinitesimal form: dx D t , x x dx t dt , dt D t , x x dx t dt , dy dy , dz dz .
2. 2 Thus dt x dx t dt x dx x t dtdx t dt , dx x dx t dt x dx x t dtdx t dt
2. 3 By substituting Eqs.(2.3) into ds c dt dx dy dz we obtain ds c x dx c x t dtdx c t dt x dx x t dtdx t dt dy dz c t t c dt c x t x t dtdx c x x dx dy dz .
2. 4 In the limit of zero acceleration, transformations Eqs.(2.2) becomes to the nextform dx a x dx b x dt ; dt q x dx p x dt ; y y ; z z .
2. 5 and metric (2.4) gets the form: ds c g x dt cg x dtdx g x dx dy dz
2. 6 where p x b x / c ; g c p x q x a x b x / c ; g c q x a x .
2. 7 Theorem
The metrics (2.4) of the general noninertial FR in the limit of zeroaccelerationis is forminvariant that with respect to the so-called generalized inertialLorentz-Poincare group L mi x connected with the classic Lorentz-Poincare group L mi one by the relation: L kn x x k in x L mi x km x k , . where ki x is the matrix of the linear (non-orthogonal) transformations x i ki x x k . . forming the generalized inertial FR (2.6). III.Generalized Principle of limiting 4-dimensionalsymmetry for the case of the uniformly acceleratedframes of reference
Let us considered (acceleration) transformations between the two relativisticframes one of which inertial frame K K t , x , y , z will be considered to be at"rest", while another one uniformly accelerated frame K K t , x , y , z will move withrespect to the first one by the law: x x t v d , t t ,
3. 1 or in equivalent infinitesimal forms dx dx v t dt , v t c ; t t ,
3. 2 Thus general metric (2.4) gets the form: ds c v t c dt v t dtdx dx dy dz .
3. 3 In the limit of zero acceleration t T lim a t we have v t v T v T , t T andtransformations (3.1) becomes to the form (1.9) and metric (3.3) gets the form: (seeEq.1.10) s c v T c dt v T dtdx dx dy dz .
3. 4 Theorem L mi connected with the classicLorentz-Poincare group L mi one by the relation: L kn x k in L mi km x k , . where ki is the matrix of the linear (non-orthogonal) transformations x i ki x k . . forming the generalized inertial FR (3.4). Remark
Relativistic motion with uniformly acceleration is a motion underthe influence of a uniformly force f t , that is uniformly in value and direction r .According to the equations of relativistic motion we have ddt v t v t c f t m a t . . Integrating equation (3.6) over time, we obtain v t v t c t v . a t a d . b . Setting the constant v to zero, which corresponds to zero initial velocity, we findafter squaring v t c t c . . Taking into account this expression in (3.7), we obtain v t dr dt t t c . . Integrating this equation, we find t r d c . . In the limit of zero acceleration t T lim a t from Eq.(3.8.b) we obtain t T lim t T a d T .
3. 12 Taking into account Eqs.(3.10),(3.12) we obtain v T t T lim v t v T T T c .
3. 13 Thus in the limit of zero acceleration generalized Lorentz transformations (3.5) getsthe form: x V v T V c x v T c V t , t V V c x v T V c t .
3. 14 IV.Physical and coordinate values in SRT. Relativisticlength expansion in accelerated system revisited.
As known [4] constructing the covariant SRT on a general non-inertial frame(1.6) one should exactly distinguish a coordinates (in some senseformal-mathematical) of a particles and physical distance (experimentallymeasurable) one.The latter is defined as the ratio of the physical distance dl ph and physical time d ph : s c d ph dl ph , dl ph g g g g dx dx , d ph g dt g dx c g .
4. 1 Let us consider measurement, in a general non-inertial reference frame (1.6),of the physical length l ph of a rod. We first determine the method for measuringthe length of a moving infinite small rod . Consider an observer in the non-inertialreference system, who records the ends of the rod, X ph t and X ph t , atthe same moment of physical time ph t , i.e. d ph t g dt g dx t c g
4. 2 Hence using Eq.(4.2) one obtain g g c g dx dt
4. 3 Suppose that on the time interval t t , t admissible solution x t of the Eq.(4.3) exist and the corresponding boundary conditions: x t x t , x t x t is satisfied. Under conditions (4.2)-(4.3) interval ds c d ph dl ph , changes from the interval with spatial part only: ds dl ph , dl ph g x t , t g x t , t g x t , t g x t , t dx dt dx dt dt , l ph sgn t t t t g x t , t g x t , t g x t , t g x t , t dx dt dx dt dt .
4. 4 xample. J.S. Bell’s problem [7] (see also [8]). Its gist consists in thefollowing.Two rockets B and C are set in motion (say, to the velocity v t t c )so that the distance between them may remains constant and equal to the startingone L from the viewpoint of an external observer A. One can simpler imaging herethat the observer A operates the flight of the moving-away rockets and has a radar,by means of which he controls the constancy of the distance between them. Fromthe viewpoint of an external observer A rockets C and B moving by laws x C t and x B t : x C t c a a t c x B t L c a a t c
4. 5 Pic.1.In the case of motion with constant proper acceleration a the interval (1.6)of the FR comuving both rockets takes the form (see [9] Eq.12.12): ds c dt a t c atdtdx a t c dx dy dz .
4. 6 From Eqs.(4.3),(4.5) one obtain g c g dx dt dx dt cg g c a t c at a t c catc a t c dx dt c t t , t atc .
4. 7 Hence x t c a ln 1 t t A .
4. 8 Setting t t and take into account corresponding boundarycondition x t x t x C one obtain: c a ln 1 t t A t
4. 9 Thus A t c a ln 1 t t .
4. 10 Setting t t and take into account corresponding boundaryondition x t x B L one obtain: c a ln 1 t t A t L .
4. 11 Substitution Eq.(4.10) into Eq.(4.11) gives c a ln 1 t t c a ln 1 t t L . Hence c a ln 1 t t t t L .
4. 12 From Eq. (4.12) by simple calculation (see appendix C Eqs.(C.4-C.6)) oneobtain t t cosh aL c sinh aL c t
4. 13 Substitution Eqs.(4.13), into Eq.(4.4) gives l ph t sgn t t c a t tt t dtt c a ln cosh aL c sinh aL c t .
4. 14 Suppose that aL c From Eq. (4.12) by simple calculation(see appendix C Eqs.(C.1)-(C.3)) one obtain t t t exp aL c
4. 13 Substitution Eqs.(4.13 ), into Eq.(4.4) gives l ph t sgn t t c a t tt t dtt c a ln 12 exp aL c
12 exp aL c t L c a ln atc , t , t atc .
4. 14 Eq.(4.14) originally was obtained S.A.Podosenov by the next cleargeometrical consideration: Let us consider nonlinear transformations fromMinkowski frame (1.5) to an noninertial FR. x I x y k ,
0, 1, 2, 3, 4 k
1, 2, 3
4. 15 From Eq.(4.15) one obtain dx I dx d y k , y k dy k d .
4. 16 Substitution Eq.(1.16) into Eq.(1.5) gives s dy g y n y k dy n dy k dy dl , g g V V , V dx d , dy d V y n dy n V dx .
4. 17 Using Eqs.(4.17) one obtain V k dy k d V k V y k g k V g k V g k g .
4. 18 Hence d l g n g k g g n k dy n dy k .
4. 19 Using orthogonality condition (Pfaff equation) dx
0, 1, 2, 3.
4. 20 by substitution Eq.(4.5) into Eq.(4.20) we obtain: dx dx dx dt c t t , t atc .
4. 21 Hence x t c a t ln 1 t t A .
4. 22 Setting t t and take into account boundary condition x t x y , t y and corresponding with J.S. Bell’s problem Eq.(4.23) (see Eq.(4.5)) x C y , t x y , t y C c a t x B y , t x y , t y B c a t y C y B L ,
4. 23 one obtain a t ln 1 t t A t c a t hence c a ln 1 t t A t c a .
4. 24 Hence A t c a ln 1 t t c a .
4. 26 Setting t t and take into account boundary condition x t x B y L , t and corresponding (with J.S. Bell’s problem)Eq.(4.23) one obtain c a t ln 1 t t A t L c a t hence c a ln 1 t t A t L c a .
4. 27 Substitution Eq.(4.26) into Eq.(4.27) gives L c a ln 1 t t t t
4. 28 By using Eqs.(4.28),(4.13) one obtain t sgn t t c a t tt t dtt c a ln cosh aL c sinh aL c t .
4. 29 Eq.(4.29) coincide with Eq.(4.14).
V. J.S. Bell’s problem in canonical parametrization byusing proper time . Let us consider Bell’s problem in canonical parametrization by using propertime such that ( c ) [8]: x B x B , x B a cosh a x B , x A x A , x A a cosh a x A , x B x A L , t a ln a t a t , t a sinh a , x a cosh a t .
5. 1 By the canonical way the interval ds of the FR comuving both rockets takes theform [8]: s d d d d sinh a dy dz . g g g sinh a , g yy g zz
5. 2 By using Eqs.(4.1) we obtain ds d d sinh a d cosh a dy dz .
5. 3 By using Eq.(5.3) and Eq.(4.3) we obtain: dx dx d d a .
5. 4 By integration Eq.(5.4) we obtain a ln tanh a A .
5. 5 Setting and take into account boundary condition x A one obtain: a ln tanh a A x A . .
5. 6 Setting and take into account corresponding boundarycondition | x B L one obtain: a ln tanh a A L .
5. 7 By subtracting Eq.(5.7) from Eq.(5.6) one obtain: a ln tanh a a L .
5. 8 From Eq.(5.8) we obtain tanh a a exp a L ,tanh a tanh a exp a L .
5. 9 Hence a Arth tanh a exp a L
5. 10 By using Eqs.(4.4),(5.2),(5.4) we obtain: l ph sgn cosh a d sgn cosh a d d d sgn coth a d sgn a ln sinh a sinh a .
5. 11 By substitution Eq.(5.10) into Eq.(5.11) we obtain: ph sgn a ln sinh a sinh a sgn a ln sinh 2 Arth tanh a exp a L sinh a .
5. 12 By using equality: sinh Arth z z z , z one obtain sinh 2 Arth tanh a exp a L a exp a L tanh a exp a L .
5. 13 By substitution Eq.(5.13) into Eq.(5.12) finally we obtain L ph a ln 2 tanh a exp a L tanh a exp a L sinh a L a ln sinh a a ln 2 tanh a a ln 1 tanh a exp a L .
5. 14 Suppose that: a L a Thus from Eq.(5.14) we obtain L ph L a ln sinh a
5. 15 ppendix A. We denote space-time indices
0, 1, 2, 3 in Greek (where corresponds to thetime dimension), while spatial indices
1, 2, 3 are denoted in Roman. We assumethat summation takes a place on two same indices met in the same term. Weassume that ds g dx dx , x ct . A . 1 Targeting the A.Zelmanov’s chronometrically invariantformulae for the elementary "length" [4] dl , the it is a purespatial metric tensors h ik and h ik , and the fundamentaldeterminant h | h ik |, one obtain dl h ik dx i dx j , h ik g ik g ik g k g , h ik g ik , h gg . A . 2 where g | g |. The spatial metric determined in such a way coincideswith " radar metric" that assumed by Landau and Lifshitz, see (84.6) and (84.7) in[5], and that assumed by Fock, see (55.20) in [6]. For the elementarychronometrically invariant interval of time d and the elementary worldinterval ds , one obtain cd g dx g , ds c d dl . A . 3 ppendix B. Arth z ln 1 z z , z B . 1 sinh Arth z sinh ln 1 z z
12 exp ln 1 z z exp ln 1 z z
12 exp ln 1 z z exp ln 1 z z
12 1 z z z z z z z z
12 4 z z z z . B . 2 Appendix C. c a ln 1 t t t t C . 1 By using Eq.(C.1) one obtain t t t t exp aL c d C . 2 Eq.(C.2) gives t t d t t d ,1 t d t ,1 t d t t d t d t d t d t t d d d , aL c t t t exp aL c C . 3 t t d , d d t t t d t t t t d t t d t d d d t t t d t t d d t t t t d t t d d t t d t t t d t d t t d t t d t d t d d t d t d d t d d t d d t t d d d d t . C . 4 ence t d d t d d d d t t d d d d t t
12 exp aL c exp aL c