Variational formulation of the electromagnetic radiation-reaction problem
aa r X i v : . [ phy s i c s . c l a ss - ph ] S e p Variational formulation of the electromagnetic radiation-reaction problem
M. Tessarotto a,b , C. Cremaschini c and M. Dorigo d a Department of Mathematics and Informatics, University of Trieste, Italy b Consortium of Magneto-fluid-dynamics, University of Trieste, Italy c International School for Advanced Studies (SISSA), Trieste, Italy d Department of Physics, University of Trieste, Italy (Dated: October 22, 2018)A fundamental issue in classical electrodynamics is represented by the search of the exact equationof motion for a classical charged particle under the action of its electromagnetic (EM) self-field -the so-called radiation-reaction equation of motion (RR equation).
In the past, several attemptshave been made assuming that the particle electric charge is localized point-wise (point-charge).These involve the search of possible so-called ”regularization” approaches able to deal with theintrinsic divergences characterizing point-particle descriptions in classical electrodynamics. In thispaper we intend to propose a new solution to this problem based on the adoption of a variationalapproach and the treatment of finite-size spherical-shell charges. The approach is based on threekey elements: 1) the adoption of the relativistic synchronous Hamilton variational principle recentlypointed out (Tessarotto et al, 2006); 2) the variational treatment of the EM self-field, for finite-size charges, taking into account the exact particle dynamics; 3) the adoption of the axioms ofclassical mechanics and electrodynamics. The new RR equation proposed in this paper, departingsignificantly from previous approaches, exhibits several interesting properties. In particular: a)unlike the LAD (Lorentz-Abraham-Dirac) equation, it recovers a second-order ordinary differentialequation which is fully consistent with the law of inertia, Newton principle of determinacy andEinstein causality principle and b) unlike the LL (Landau-Lifschitz) equation, it holds also in thecase of sudden forces. In addition, it is found that the new equation recovers the customary LADequation in a suitable asymptotic approximation.
PACS numbers: 47.27.Ak, 47.27.eb, 47.27.ed
The goal of this paper is to investigate a well-knowntheoretical issue of classical electrodynamics. This isconcerned with the solution of the so-called radiation-reaction problem (
RR problem ), i.e., the description ofthe dynamics of classical charges (charged particles) inthe presence of their EM self-fields. For contemporaryscience the possible solution of the RR problem repre-sents not merely an unsolved intellectual challenge, buta fundamental prerequisite for the proper formulation ofall physical theories which are based on the description ofrelativistic dynamics of classical charged particles. Theseinvolve, for example, the consistent formulation of therelativistic kinetic theory of charged particles and of therelated fluid descriptions (i.e., the relativistic magneto-hydrodynamic equations obtained by means of suitableclosure conditions), both essential in plasma physics andastrophysics.Surprisingly, until recently [1] (hereon denoted Ref.A)the problem has remained substantially unsolved, despiteefforts spent by the scientific community in more thanone century of intensive theoretical research (see relateddiscussion in Ref.[2]; for a review see Refs.[3]). In par-ticular, still missing is the exact relativistic equation ofmotion for a classical charged particle in the presence ofits electromagnetic (EM) self-field, also known as ( exact ) RR equation . For definiteness, in the following we shallconsider the RR problem in the case of a flat (Minkowski) space-time, although a similar problem can be posed, inprinciple, also for curved space-time and in the contextof a general-relativistic formulation. This requires that g ≡ det { g µν } = − , g µν denoting the Minkowski metrictensor with signature (1 , − , − , − non-asymptotic. Namely, the exact RR equation shouldnot rely on any asymptotic expansion (i.e., a truncatedperturbative expansion), in particular for the electromag-netic field generated by the charged particle, to be per-formed in terms of any possible infinitesimal parameterwhich may characterize the particle itself (assuming thatin some sense the particle has a finite ”size”, i.e., it is notpoint-like). On the other hand, by assumption, a classi-cal particle should satisfy at least two basic properties: a)to have no ”internal structure” and b) to be sphericallysymmetric (when seen with respect to the particle rest-frame). These hypotheses (which are manifestly satisfiedby point-particles), should be fulfilled also by finite-sizeparticles in which the mass and/or the electric chargehave a finite-size distribution. Hence, these parametersshould (only) be related to the radii of the mass and/orcharge distributions. Following the prescription pointedout in Ref.A, here we intend to prove that such an equa-tion can be obtained explicitly, without introducing anyso-called ”regularization” scheme, i.e., leaving unchangedthe axioms of classical electrodynamics. The result isreached by considering classical finite-size charges, and,more precisely, finite-size spherical-shell charges (in anal-ogy to the classical Lorentzian model [4]). For these par-ticles the charge is considered spatially distributed in abounded 3D domain (i.e., characterized by a finite-sizecharge distribution). In detail, the charge density - whenseen with respect to each particle rest-frame - is takenby assumption: a) spherically symmetric , b) radially lo-calized on a spherical surface δ Ω σ having a finite radius σ >
0; c) quasi-rigid , i.e., to remain constant on δ Ω σ withrespect to the same reference frame. Hypotheses a)-c) aremanifestly all consistent with the above requirements fora classical particle. Instead, as far as the mass distribu-tion is concerned, it is assumed as point-wise localizedin the center of the spherical surface δ Ω σ . This permitsus to neglect the additional degrees of freedom occurringin such a case. Thus, from this viewpoint the particle isstill treated as a point-particle. In this paper we intendto show - in particular - that, unlike the point-chargecase, for a finite-size classical charge of this type the ex-act RR equation can be explicitly constructed based onthe synchronous Hamilton variational principle.
1a - Motivations and historical background
The occurrence of self-forces, in particular the electro-magnetic (EM) one which is produced by the EM fieldsgenerated by the particles themselves, is an ubiquitousphenomenon which characterizes the dynamics of classi-cal charged particles. It is well-known that the self-forceacts on a (charged) particle when it is subject also tothe action of an arbitrary external force (Lorentz, 1892[4]; see also for example Landau and Lifschitz, 1957 [5]).This phenomenon is usually called as radiation reaction ( RR ) (Pauli [6]) or radiation damping (see [7]), althougha distinction between the two terms is actually made bysome authors [9].In classical mechanics the RR problem was first posedby Lorentz in his historical work (Lorentz, 1985 [4]; seealso Abraham, 1905 [10]). Traditional approaches arebased either on the RR equation due to Lorentz, Abra-ham and Dirac (first presented by Dirac in 1938 [11]),nowadays popularly known as the LAD equation or theequation derived from it by Landau and Lifschitz [5], viaa suitable ”reduction process”, the so-called
LL equa-tion.
As recalled elsewhere (see related discussion inRef.[2]) several aspects of the RR problem - and of theLAD and LL equations - are yet to find a satisfactoryformulation/solution. Common feature of all previousapproaches is the adoption of an asymptotic expansionfor the EM self-field (or for the corresponding EM 4-potential), rather than the exact representation of theforce-field. This, in turn, implies that such methods per-mit to determine - at most - only an asymptotic approxi-mation for the (still elusive) exact equation of motion fora charged particle subject to its own EM self-field.
1b - Difficulties with previous RR equations
Since Lorentz famous paper [4] many textbooks andresearch articles have appeared on the subject of RR.Many of them have criticized aspects of the RR theory,and in particular the LAD and LL equations (for a re-view see [3], where one can find the discussion of therelated problems). More recently, another equation hasbeen proposed by Medina [12], here denoted as
Medinaequation, which applies for spherically symmetric andfinite-size classical particles. In these approaches, thecharged particles are typically considered quasi-rigid, i.e.,their charge densities are assumed stationary, when seenwith respect to the corresponding particle rest-frame, andeventually also point-like , i.e., both the radii of the mass(if larger than zero) and of charge distributions are as-sumed much smaller (i.e., infinitesimal) with respect toany other classical scale-length characterizing the parti-cle dynamics.It is often said that current formulations of the RRproblem are unsatisfactory, because of their possible vio-lation of basic principles of classical dynamics as well asfor some of their properties. These include in particular: • for the LAD equation : 1) The violation of New-ton’s principle of determinacy (NPD), because theLAD equation requires the specification of the ini-tial acceleration , besides the initial state; 2) Theexistence of so-called runaway solutions, i.e., so-lutions which blow up in time. In fact, if a con-stant external force is applied one can show thatthe general solution of the LAD equations divergesexponentially in the future (blow-up). 3) For thesame reason, the LAD equation violates also an-other fundamental principle of classical mechanics,the Galilei principle of inertia (GPI), according towhich an isolated particle must have a constant ve-locity in any inertial Galilean frame. • for the LL equation : 1) The use of an iterative ap-proach for its derivation (from the LAD equation)does not appear justifiable for fully relativistic par-ticles. In such a case, in fact, the EM self-forcecannot generally be considered a small perturba-tion of the external EM force. 2) The LL equationbecomes invalid in the case of sudden forces, i.e.,forces which are not smooth functions of time. 3)The neglect of the EM mass: in the original deriva-tion of the LL equation, given by Landau and Lif-schitz [5], the so-called ”EM mass” was ignored,which amounts to neglect all possible EM relativis-tic corrections to the inertial mass produced by theEM self-force. In the framework of classical elec-trodynamics the latter position appears unfounded(see discussion in Ref. [2] and Ref.A). However,in the formulation [of the LL equation] given byRohrlich [13] this effect has been included. • for both equations : the derivations of both equa-tions (LAD and LL) are made under the implicitassumption that all the expansions in powers usednear the particle trajectory are valid for the wholerange of values of particle velocity, in particular, ar-bitrarily close to that of the light in vacuum. How-ever, it is easy to see that this is not the case. • for the Medina equation : the use of a perturbativeapproach, in particular to evaluate the RR force inthe rest frame. This is, however, a non-relativisticequation. Therefore, the corresponding relativisticequation is also necessarily asymptotic in character.In our view this clearly indicates that the route to thesolution of the RR problem should be based on the searchof the exact relativistic RR equation, i.e., the construc-tion of a non-perturbative equation of motion for a par-ticle in the presence of its EM self-field .
1c - The search of an exact RR equation
A critical aspect of the RR problem is, however, relatedto the search of the exact relativistic RR equation forclassical charged particles, in the sense specified above . Despite previous attempts, this equation is still missing.As far as the LAD equation is concerned this is obviousbecause to obtain it the EM self-field is usually evalu-ated by means of an asymptotic expansion. This is true,of course, also for the LL equation, which according toRohrlich should be considered as the ”exact” relativisticequation of motion for a classical point-like spherically-symmetric charge, having a charge distribution with aninfinitesimal radius σ [13] (in this case the equation isintrinsically asymptotic since it depends on the infinites-imal parameter σ ).This feature - as pointed out in Ref.A - is also re-flected by the circumstance that these equations are non-variational [2] , i.e., they do not admit a varia-tional formulation, at least in the customary sense of thestandard Hamilton principle, used in classical mechan-ics and electrodynamics [14], i.e., for the conventional8-dimensional phase-space spanned by the 4-vectors { r µ , u µ = g µν dr ν /ds } , g µν denoting the (Minkowski)metric tensor. This result is clearly in contrast to thebasic principles both of classical mechanics and electro-dynamics . In particular, it conflicts with Hamilton’s ac-tion principle, which - under such premises (i.e., the va-lidity of LAD and/or LL equations) - should actuallyhold true only in the case of inertial motion (or neglect-ing altogether the EM self-force)! A consequence whichfollows is that the dynamics of point-like charged parti-cles described by these approximate model equations isnot Hamiltonian. However, it is not clear whether thisfeature is only an accident, i.e., is only due to the ap- proximations introduced so far, or is actually an intrinsicfeature of the RR problem.Another key issue is, however, related to the treatmentof the RR problem for point-particles in a proper sense,and in particular to the conditions of validity of the rela-tivistic Hamilton variational principle [14] in such a case.Actually, difficulties with the treatment of point-particlesin classical electrodynamics and general relativity havebeen known for a long time. They are due to intrinsicdivergences produced by the EM self-field [8]. In factone can show that this problem is ill-posed since the self-fields diverge in the neighborhood of a point-particle’sworld line. For this reason in the past several authors,including Born and Infeld, Dirac, Wheeler and Feynman(see discussion in Ref.[7]), tried to modify classical elec-trodynamics in an effort to eliminate all divergent con-tributions arising due to EM self-interactions. This isthe so-called regularization problem for point-particles,based on the introduction of suitable modifications ofMaxwell’s electrodynamics.There is an extensive literature devoted to possibleways to achieve this goal. These theories either directlyintroduce ’ad hoc’ modified definitions for the EM self-force (or of the EM self 4-potential) or introduce ax-iomatic approaches involving modifications of classicalelectrodynamics. Examples of the first type is providedby Dirac [11] and Dewitt and Brehme [15] who deter-mined the RR self-force for a point particle belongingrespectively to the Minkowski and curved space-times byimposing local energy conservation on a tube surround-ing the particle’s world line and subtracting the infinitecontributions to the force through a so-called mass renor-malization scheme. More recently, Ori [16] who suggesteda regularization scheme involving averaging of multipolemoments. Another attempt is based on the adoption ofan axiomatic approach in order to produce the generalequation of motion for a point particle coupled to a scalarfield. In recent years several different methods have beenproposed for calculating the motion of a point particlecoupled to its EM self-fields (for a review and referenceson the subject see for example [17]). Finally, still anotherpossible strategy involves introducing appropriate modi-fications of the EM self 4-potential. Typically this is done(see for example Rohlich [18]) by assuming that there ex-ists a decomposition of the EM field, whereby each par-ticle ”feels” only the action of external particles and of asuitable part of the EM self-field. While this decomposi-tion becomes clearly questionable for finite-size particles,its consistency with first principles - and in particularwith standard quantum mechanics - seems dubious, tosay the least [7].Another possible approach for the search of an exactRR equation is represented by the description of clas-sical charges by means of finite-size extended particles.An example of this type is provided by Medina [12], whoinvestigated the dynamics of a point particle character-ized by an arbitrary spherically-symmetric charge. Inhis approach a formal integral representation for the RRforce in the particle rest frame is achieved. This result isused to extrapolate the same force for point-like particlesand to evaluate the general form of the RR 4-force in anarbitrary reference frame, thus yielding an approximaterepresentation of the relativistic RR equation ( MedinaRR equation ). By doing so, however, an asymptotic ap-proach is inevitably adopted again. Another interestingfeature of the Medina’s approach is that the extended-phase space variational approach is achieved [19] by intro-ducing an acceleration-dependent Lagrangian function.In this paper, we intend to follow a similar routechoosing, however, to consider: 1) spherical-shell charges and 2) the adoption, from the beginning, of a relativisticvariational approach based on the customary phase-spaceHamilton variational principle . As we intend to prove inthe following, this permits us to obtain an exact, i.e.,non-asymptotic, RR equation .
1d - Main results
In this paper we intend to pose, for classical finite-sizecharged particles represented by shell-charges, the prob-lem of the construction of the exact RR equation, in thesense indicated above. We want to show that its explicitconstruction can be achieved in the framework of classi-cal electrodynamics, based on a straightforward general-ization of Hamilton variational principle.
The approachis based on the adoption of the relativistic (hybrid) syn-chronous Hamilton variational principle recently pointedout [21]. Its basic feature is that it can be expressedvirtually in terms of arbitrary ”hybrid” variables (i.e.,generally non-Lagrangian and non-canonical variables).The traditional approach, valid for point-particles, is ex-tended to finite-size spherical-shell charges, by takinginto account the contribution of the retarded EM self-potential generated by the particles themselves. Thus,based on the construction of the Euler-Lagrange equa-tions stemming from the variational principle, the exact relativistic equation of motion for a charged particle ofthis type, immersed in a prescribed EM field and sub-ject to the simultaneous action of its EM self-field , canbe achieved explicitly in this way (THM.1-THM.3). Inparticular, it is found that the exact RR equation in co-variant form is (see THM’s.1 and 2): m o cdu µ ( s ) = qc F ( ext ) νµ dr ν ( s ) + dsG µ . (1)Here F ( ext ) νµ is the surface-average Faraday tensor - act-ing on a point particle located at the 4-position r ≡{ r µ , µ = 0 , } - which is generated by the external EMfield. In particular, the surface-averaging operator acting on a smooth position-dependent function A, and denotedas A, is defined according to Appendix A [see Eq.(81)].Moreover, G µ is the (surface-average) RR 4-vector pro-duced by the EM self-field and due to the action ofthe particle on itself. The rest of the notation is stan-dard. Thus, c is the speed of light in vacuum, m o and q are respectively the inertial rest-mass and charge ofthe particle, r µ ( s ) ≡ r µ ( t ( s )) denotes its position 4-vector parametrized in terms of the arc lengths s and u µ ( s ) = dr µ ( s ) ds is the corresponding 4-velocity. The ex-plicit form of G µ is found to be (see THM.2) G µ = 2 c (cid:16) qc (cid:17) R ′ α u α ( t )] (cid:20) dr µ ( t − t ret ) ds + (2) − R ′ µ u k ( t ) dr k ( t − t ret ) ds R ′ α u α ( t ) . Here t − t ret is the retarded time, with t ret denoting asuitable delay-time [see Eq.(32)], while R ′ α = r α ( t ) − r α ( t − t ret ) and r α ( t − t ret ) is the 4-position vector evalu-ated at the retarded time t ′ . As a consequence of Eq.(2)the properties of G µ can be immediately established (seeTHM.3). In particular, G µ depends, besides r µ ( s ) and u µ ( s ) evaluated at the local time t = t ( s ) , also on the4-position and 4-velocity [of the particle itself], evaluatedat the retarded time t ′ , i.e., r µ ( t − t ret ) and dr µ ( t − t ret ) ds . Itfollows that G µ is a smooth function which is generallydefined everywhere in a suitable extended phase space.Hence, the RR equation Eq.(1) is a retarded second-orderordinary differential equation . As a main consequence,the equation, together with the initial conditions r µ ( s o ) = r µo , (3) u µ ( s o ) = u µo , (4)prescribed so that u µo u oµ = 1, defines locally a well-posedproblem (THM.1). In addition, its solution results con-sistent with all basic principles of classical mechanics,including the principles of Galilei inertia, Newton deter-minacy and Einstein causality (THM.3).To gain deeper insight and to allow comparisons withprevious approaches, various asymptotic approximationsand limits are considered in the sequel. These include:1) the proof of the non-existence of the point-particlelimit for the present theory (see THM.4), i.e., that theexact RR equation is not defined in such a case; 2) the”short-time” asymptotic approximation for the RR equa-tion obtained in the so-called ”short-time” ordering (seeTHM.5). This is obtained by introducing a Taylor expan-sion in terms of the dimensionless ratio ξ ≡ ( t − t ′ ) /t > delay-time ratio ) , to be assumed infinitesimal; 3) theweakly-relativistic approximation for the RR equation,obtained by introducing a Taylor expansion in terms ofthe dimensionless ratio β ≡ v ( t ) /c, again to be consid-ered infinitesimal (as appropriate for the description ofnon-relativistic particle dynamics; see THM.6).The analysis is useful to assess the accuracy and limitsof validity of the customary LAD equation, either in therelativistic or weakly-relativistic descriptions. In bothcases it is found that the LAD equation (as well as therelated LL equation) provided, at most, only an asymp-totic approximation to the exact RR equation (1). Thisconclusion can typically be reached, however, only pro-vided the external EM field, defined in terms of the Fara-day tensor F ( ext ) νµ , is a suitably smooth function of theparticle proper time τ ≡ s/c. In particular, both LADand LL equations may not be valid for ”sudden forces”,i.e., external fields which are locally discontinuous withrespect to τ .
1e - Scheme of presentation
The scheme of the presentation is as follows. In Sec-tions 1 and 2 a brief overview of previous treatments isgiven in order to analyze the intrinsic difficulties met byprevious point-charges descriptions for the RR problem.In the subsequent sections (Sec. 3 to 7) the new treat-ment which applies for finite-size charges is presented. Inparticular: • In Sec.3 the exact EM 4-potential generated by afinite-size spherical shell is evaluated. • In Sec.4 the Hamilton synchronous variational prin-ciple for a finite-size charge is developed and anexplicit form of the RR equation is obtained (seeTHM.1). In particular, it is proven that the re-sulting relativistic RR equation is a second-orderordinary differential equation which defines a well-posed problem, i.e., that the solution of the corre-sponding initial-value problem locally exists and isunique. • As a consequence (Sec.5), the 4-vector ( G µ ) is in-troduced which describes the generalized Lorentzforce acting on the particle generated by its EMself-field (see THM.2). • In Sec.6 the main properties of G µ are investigated.As a result it is proven that the RR equation isconsistent with all basic principles of classical me-chanics (THM.3). • In Sec.7 the non-existence of the point-particle limit[for the RR equation] is proven (THM.4).In Sec.8 the short-time approximation for G µ is ob-tained. The resulting asymptotic RR equation is foundconsistent with the customary relativistic LAD equation(THM.5). Finally, in Sec.9 possible weakly-relativisticapproximations of the RR equation are discussed. Also inthis case, the resulting RR equation can be realized, un-like the customary weakly-relativistic LAD equation, bymeans of a second-order differential equation (THM.6). A corner-stone of classical mechanics is represented bythe Hamilton variational principle, which permits to de-termine the coupled set of equations formed by the parti-cle dynamical equations and Maxwell’s equations [5, 14].As a consequence, both the particle state and the EMfield in which the particle is immersed are uniquely de-termined by means of this variational principle. Thechoice of the dynamical variables which define the par-ticle state remains in principle arbitrary. Thus, theycan always be represented by so-called ”hybrid” vari-ables, i.e., superabundant variables which generally donot define a Lagrangian state. This implies, thanks toDarboux theorem, that it should always be possible toidentify them locally with canonical variables. As a ba-sic consequence, classical systems of charged particlesare expected to define Hamiltonian systems, i.e., theircanonical states should be extrema of the correspond-ing Hamiltonian action, while the corresponding particledynamics, provided by the Euler-Lagrange equations de-termined by the same variational principle, necessarilyshould coincide with Hamilton’s equations of motion.Nevertheless, it is easy to prove that for charged point-particles the Hamilton principle fails (see Ref.A). In fact,one can show that, if the Hamilton principle is expressedvia a synchronous hybrid variational principle [21], thepoint-charge action integral can be written in the form(here the notation is given according to Ref.A) S ( r µ , u µ , χ ) = (5)= Z (cid:16) m o cu µ + qc A µ ( r ) (cid:17) dr µ ++ Z s s dsχ ( s ) [ u µ ( s ) u µ ( s ) − A µ is considered prescribed. It is immediateto prove that the functional is actually not-defined. Thereason is due to the intrinsic divergences appearing inthe point-particle self 4-potential A ( self ) µ (see AppendixB) . In fact, due to the superposition principle the EM4-potential A µ ( r ) can always be represented in terms ofthe fundamental decomposition A µ = A ( self ) µ + A ( ext ) µ , (6)where A ( self ) µ and A ( ext ) µ denote respectively the point-particle self 4-potential and the external 4-potential. Inparticular, by assumption A ( self ) µ ≡ A ( self ) µ ( r ( s )) is a so-lution of the Maxwell’s equations which in flat space-timeare given by ∂ ν F νµ ( self ) = 4 πc j µ , (7)with j µ ( r ν ) = R s s ds ′ u µ ( s ′ ) δ (4) ( r − r ( s ′ )) denoting the4-current carried by the point charge. Hence, in the func-tional S ( r µ , u µ , χ ) , the 4-vector function A µ must beconsidered a prescribed function of the varied 4-vector r µ ( s ). Invoking the causality principle, the explicit formof A ( self ) µ for a point particle immersed in the Minkowskispace-time M ≡ R can be easily recovered (see Ap-pendix B) and is provided by the well-known retardedEM 4-potential (in covariant form) A ( self ) µ ( r ) = qc u µ ( t ′ ) R α u α ( t ′ ) , (8)which can be represented in the equivalent integral formgiven by Eq.(92). Here R µ , u µ ( t ′ ) , u µ ( t ′ ) and t ′ are re-spectively the bi-vector R µ ≡ r µ − r ′ µ , with r ′ µ ≡ r µ ( t ′ ) , the 4 − velocity u µ ( t ′ ) ≡ dds ′ r µ ( t ′ ) = γ ′ ddt ′ r µ ( t ′ ) and its co-variant components u µ ( t ′ ), while t ′ is a suitable retardedtime. In particular this is defined so that t − t ′ = | r − r ′ | c , (9)where r ′ ≡ r ( t ′ ) . We now notice that for an arbitraryvaried curve r ( s ) , the inf of t − t ′ is generally not strictlypositive in the case of a point-charge. As a conse-quence, the contributions carried by A ( self ) µ in the func-tional S ( r µ , u µ , χ ) contain essential divergences. Thismeans that, when the self 4-potential A ( self ) µ is properlytaken into account in the point-charge action functional S ( r µ , u µ , χ ) , the functional cannot actually be defined. A prerequisite for the subsequent developments is thedetermination of the EM self-potential ( A ( self ) µ ) producedby a prescribed charge distribution. As indicated above,in this paper we wish to consider the case of a classicalparticle characterized by point-particle mass and - re-spectively - finite-size charge distributions. For definite-ness, here we shall determine the EM 4-potential gener-ated by a finite-size spherical-shell particle immersed inthe Minkowski space-time . In particular, we assume thatwhen observed with respect to the particle rest-frame thecharge density takes the form ρ ( r , t ) = q πσ δ ( | r − r ( t ) | − σ ) . (10)First, let us evaluate the retarded electrostatic (ES) po-tential generated by ρ ( r , t ) and measured at a position r defined in such a frame. This is manifestly defined asΦ ( self ) ( r , t ) = Z d r ′ R ρ ( r , t − Rc ) , (11) with R ≡ | R | and R = r − r ′ . It is well known thatΦ ( self ) ( r , t ) can be determined conveniently by introduc-ing an expansion in Legendre polynomials for the inte-grand R ρ ( r , t − Rc ). As a result one can readily showthat for a finite-size spherical-shell charge the retardedES potential is (see for example [20])Φ ( self ) ( r , t ) = (cid:26) qR R ≥ σ qσ R < σ, (12)where R ≡ | R | , (13) R = r − r ( t − (cid:12)(cid:12) r − r ( t − Rc ) (cid:12)(cid:12) c ) . (14)Therefore, in the internal domain ( R < σ ) the EM self-potential does not produce any self-field. Instead, in theexternal domain ( R ≥ σ ) its expression is the same asthat produced by a point-charge. In both cases the ESpotential is manifestly spherically symmetric, thereforeit follows by construction that in the rest frame:Φ ( self ) ( r ,t ) = Φ ( self ) ( r , t ) , (15)where Φ ( self ) ( r ,t ) is the surface-average (80). The cor-responding expression of the EM 4-potential in a movingframe can be easily obtained by applying a Lorentz trans-formation. In particular, since in this case the externaldomain is defined by the inequality R α R α ≥ σ the cor-responding surface-average EM self 4-potential A ( self ) µ isgiven again by Eq.(8), namely A ( self ) µ ( r ) = qc u µ ( t ′ ) R α u α ( t ′ ) . (16)Instead, in the internal domain ( R α R α < σ ) there re-sults necessarily A ( self ) µ = const., so that F µν ( self ) ≡ r α is the4-position vector of the point-particle, there results (seeAppendix C) A ( self ) µ ( r ) = 2 qc Z dr ′ µ δ ( R α R α − σ ) , (17)where R α = r α − r ′ α . In this section we wish address the key issue posed inthis paper, i.e., the problem of the explicit constructionof the relativistic RR equation for a finite-size sphericalshell charge. Here we intend to prove that, as earlierpointed out in Ref.A, this goal can be uniquely estab-lished based on a suitable formulation of the Hamiltonvariational principle. More precisely, we intend to provethat: • the exact RR equation can be obtained by mak-ing use of a suitably modified form of the syn-chronous Hamilton variational principle appropri-ate for finite-size charges (see THM.1); • the solution of the related initial-value problem ex-ists and is unique, i.e., the RR equation defines awell-posed problem (THM.1).
4a - Treatment of finite-size particles
First, let us generalize the
Hamilton action functional[Eq.(5)] to treat finite-size particles.
This is obtainedformally by introducing in S ( r µ , u µ , χ ) the replacements ds → W ( r, s ) d Ω √− g , (18) dr µ → dr µ ds W ( r, s ) d Ω √− g , (19)where W ( r, s ) is the so-called ”wire function”, generallyto be identified with a suitable distribution. For a generic W ( r, s ) the appropriate form of the variational functional(to be expressed again in synchronous form [21]) becomes S ( r µ , u µ , χ ) = (20)= Z d Ω √− g W ( r, s ) (cid:16) m o cu µ + qc A µ ( r ) (cid:17) dr µ ds ++ Z s s d Ω √− g W ( r, s ) χ ( s ) [ u µ u µ − .
4b - The wire function of a spherical-shell charge
Let us now consider, in particular, the case of aspherical-shell charge, while requiring that the mass isstill point-wise localized, i.e., it is a point-particle (seediscussion in Sec.1a) with 4-position r µ ( s ) and 4-velocity u µ ( s ). To obtain the appropriate representation of thewire function in this case, let us introduce the coordi-nate transformation r µ → (cid:0) s, ξ , ξ , ρ (cid:1) . Here s is thearc length along the particle world line, ξ and ξ aretwo curvilinear angle-like coordinates on the surface ∂ Ω σ and ρ is the 4-scalar defined so that ρ = b R α b R α , with b R α = r α − r α ( s ) and r α ( s ) denoting the 4-position ofthe point particle. It follows that the wire-function for aspherical-shell charge can be defined as W = 14 πσ δ ( ρ − σ ) , (21)while the invariant volume element is d Ω √− g = dsρ dρ d Σ( n ) √− g , with √− g = 1 for flat space-time and d Σ( n ) √− g denoting a suitable invariant surface element. It followsthat W is non-zero only if r α = r α ( s ) + σn α ( ξ , ξ ) , (22) where n α is a unit 4-vector ( n α n α = 1) depending onlyon ( ξ , ξ ). Thus, in particular, in the rest-frame of thesame particle W takes the form W = 14 πσ δ ( | r − r ( s ) | − σ ) , (23)while d Σ( n ) can be identified with the solid angle (surfaceelement of 3-sphere of unit radius centered at the particleposition r ), ρ = | r − r ( s ) | and the 4-vector n reads n =(0 , n ) , n denoting the normal unit 3-vector to the surface ∂ Ω σ . It follows that for an arbitrary 4-tensor A ( r ( s ) + σn ) evaluated at the 4-position (22) one can define anappropriate surface-average (see Appendix A).
4c - Spherical-shell charge Hamilton principle
The appropriate form of the Hamilton action func-tional for a spherical-shell charge is found to be givenby the following Lemma.
LEMMA 1 - Spherical-shell charge action inte-gral
For a finite-size spherical-shell charge the Hamiltonianaction integral defined by Eq.(20) reads: S ( r µ , u µ , χ ) = (24)= 14 π Z d Σ( n ) Z [ m o cu µ ( s ) ++ qc A ( ext ) µ ( r ( s ) + σn ) i dr µ ++∆ S ( r µ )++ 14 π Z d Σ( n ) Z s s χ ( s ) [ u µ ( s ) u µ ( s ) − ds (Hamilton action integral), where ∆ S ( r µ ) is thefunctional carrying the contribution of the EM self 4-potential ∆ S ( r µ ) ≡ π Z d Σ( n ) Z qc A ( self ) µ ( r ( s ) + σn )) dr µ . (25) In view of Eq.(17) and the surface average (81) thereresults ∆ S ( r µ ) = (26)= 2 (cid:16) qc (cid:17) Z dr µ Z dr ′ µ δ ( R α R α − σ ) . Proof - The proof of Eq.(24) follows from the wire-function functional [Eq.(21)] upon invoking Eq.(23) forthe wire function. Instead, the specific form of thefunctional ∆ S ( r µ ) [Eq.(26)], which carries the EM self4-potential, follows invoking the integral representation(17). Q.E.D.
As a basic consequence, invoking in particular thesurface-average of A ( ext ) µ ( r ( s )+ σn ) given by Eq. (81), allterms in the integrand of the action functional (24) be-come independent of the surface element d Σ( n ) . Hence,the action functional reduces simply to: S ( r µ , u µ , χ ) = (27)= Z (cid:16) m o cu µ ( s ) + qc A ( ext ) µ ( r ( s )) (cid:17) dr µ ++2 (cid:16) qc (cid:17) Z dr µ Z dr ′ µ δ ( R α R α − σ ) ++ Z s s χ ( s ) [ u µ ( s ) u µ ( s ) − ds. Let us now prove that the relativistic dynamics of a(finite-size) spherical shell particle is uniquely prescribedby the Hamilton variational principle defined in terms of S ( r µ , u µ , χ ) , specified according to Eq.(24). In partic-ular, in this case, due to the assumption that the massof the particle is point-wise localized, the extremal curvemust be necessarily of the form r ≡ r ( s ) [see Assumption3 in THM.1]. The following result then holds: THM.1 - Hamilton principle for a spherical-shell charge
Let us assume that: 1) the real varied functions f ( s ) ≡ [ r µ ( s ) , u µ ( s ) , χ ( s )] belong to a suitable functional class { f } in which end points and boundaries are kept fixed;2) the Hamilton action integral S ( r µ , u µ , χ ) defined byEq.(24) is assumed to exist for all f ( s ) ∈ { f } . Here, u µ ( s ) = g µν u ν ( s ) , while g µν = g µν ( r ( s )) denotes thecounter-variant components of the metric tensor, eachone to be considered dependent on the generic variedcurve r ( s ); furthermore, m o and q are respectively theconstant rest mass and electric charge of a point particleand ds the line element; 3) an extremal curve f ∈ { f } of S is assumed of the form f ( s ) , i.e., to be independent of n ;
4) if r ( s ) is an extremal curve of S the line element ds satisfies the constraint ds = g µν ( r ( s )) dr µ ( s ) dr ν ( s ) . Then it follows that:T1 ) if the synchronous variations δf ( s ) [see alsoAppendix D] are considered as independent, the Euler-Lagrange equations following from the synchronous vari-ational principle δS ( r µ , u µ , χ ) = 0 (28) yield identically the RR equation of motion for a finite-size spherical-shell charged particle, which reads: m o cdu µ ( s ) = qc F ( ext ) νµ ( r ( s )) dr ν ( s ) + (29)+ dr k H µk , where F ( ext ) νµ ( r ( s )) is the surface-average of the Faraday4-tensor F ( ext ) µν ≡ ∂ µ A ( ext ) ν − ∂ ν A ( ext ) µ evaluated at the 4-position r ( s ) . In addition , r µ ≡ r µ ( t ) , r ′ µ ≡ r µ ( t ′ ) , while u µ = dr µ ds is the 4-velocity, v µ ( t ) denotes v µ ( t ) = dr µ dt and H µk is the function H µk = 2 (cid:16) qc (cid:17) c (cid:12)(cid:12)(cid:12) ( t − t ′ ) − c d r ( t ′ ) dt ′ · ( r − r ′ ) (cid:12)(cid:12)(cid:12) (30) ddt ′ v µ ( t ′ ) R k − v k ( t ′ ) R µ c (cid:12)(cid:12)(cid:12) ( t − t ′ ) − c d r ( t ′ ) dt ′ · ( r − r ′ ) (cid:12)(cid:12)(cid:12) t ′ = t − t ret = 0 . Finally, r ≡ r ( t ) and r ′ ≡ r ( t ′ ) , while t ′ = t − t ret denotesthe retarded time and t ret a suitable delay-time;T1 ) the delay-time t ret is the positive root of the equa-tion R α R α = σ (31) (delay-time equation) which is t ret ( t ) ≡ t − t ′ = (32)= 1 c q [ r ( t ) − r ( t − t ret ( t ))] + σ > T1 ) let us require that the 4-vector-field A ( ext ) µ ( r ) issuitably smooth in the whole Minkowski space-time M ,i.e., is at least C (2) ( M ); then the initial-value problemset by the Euler-Lagrange equation (29), with the initialconditions x ( t o ) = x o , (33) [where x ( t o ) ≡ [ r µ ( t o ) , u µ ( t o )] and x o ≡ [ r µo , u µo ] denotesa suitable initial state], is locally well-posed.Proof - T1 ) It is immediate to construct explicitlythe Euler-Lagrange equations of the Hamilton action S ( r µ , u µ , χ ). In fact, first, since ∂∂u µ δ ( R α R α − σ ) = ∂∂u ′ µ δ ( R α R α − σ ) ≡
0, the variations with respect to χ ( s ) and u µ deliver respectively u µ ( s ) u µ ( s ) − , (34) m o cdr µ + 2 χ ( s ) u µ ( s ) ds = 0 , (35)while it must result for consistency 2 χ ( s ) = − m o c (asin the case in which A ( self ) µ is assumed to vanish identi-cally). To reach Eq.(29), instead, let us invoke Lemma2 [see Appendix D]. Then, thanks to assumption 3), thevariation with respect to r µ can easily be proven to yieldthe Euler-Lagrange equation defined by Eq.(29). To-gether with Eq.(35), this manifestly defines the RR equa-tion , i.e., the exact relativistic equation of motion for apoint charge subject to the simultaneous action of a pre-scribed external EM field and of its self-EM field.T1 ) Recalling that in the Minkowski metric theretarded-time equation [Eq.(31)] reads R α R α = c ( t ′ − t ) − ( r − r ′ ) = σ , (36)with R α = r α ( t ) − r α ( t ′ ) and r = r ( t ) , r ′ = r ( t ′ ) , theproof of Eq.(32) is straightforward . T1 ) Finally, it is immediate to show that the problemdefined by Eq.(29), together with the initial conditionsdefined by Eq.(33), admits a local existence and unique-ness theorem (fundamental theorem). In fact it is obviousthat Eq.(29) can be cast in the form of a delay-differentialequation, i.e., d x ( t ) dt = X ( x ( t ) , x ( t − t ret ) , t ) , (37)where x ( t ) and x ( t − t ret ) denote respectively the ”in-stantaneous” and ”retarded” states x ( t ) ≡ [ r µ ( t ) , u µ ( t )]and x ( t − t ret ) ≡ [ r µ ( t − t ret ) , u µ ( t − t ret )] , while X ( x ( t ) , x ( t − t ret ) , t ) is a suitable C (2) real vector fielddepending smoothly on both of them. It is manifest thatthe fundamental theorem holds for Eqs.(33)-(37). In fact,by considering [in X ] x ( t − t ret ) as a prescribed function oftime, the previous equation recovers the canonical form d x ( t ) dt = b X ( x ( t ) , t ) , (38)with b X ( x ( t ) , t ) denoting the corresponding C (2) real vec-tor field. This proves the statement. Q.E.D. G µ A basic consequence of THM.1 is that the RR equationcan be expressed in covariant form. This permits us toidentify the
RR 4-vector G µ , which represents the (gen-eralized) Lorentz force produced on a charged particle byits EM self-field . Here we intend to show, in particular,that G µ can be expressed in covariant form and uniquelyparametrized in terms of the proper length s, defined atthe point-particle 4-position vector r µ . The main resultis represented by the following theorem which providesalso an explicit representation of the 4-vector G µ . THM.2 - Covariant representation of G µ For the Minkowski metric the covariant RR equationreads m o cdu µ ( s ) = qc F ( ext ) νµ ( r ( s )) dr ν ( s ) + (39)+ G µ ds. Here the 4-vector G µ ≡ ( G o , G ) is defined as: G µ = 2 c (cid:16) qc (cid:17) u k ( s ) (cid:20) R α u α ( t ′ ) (40) dds ′ ( dds ′ r µ ( t ′ ) R k − dds ′ r k ( t ′ ) R µ R α u α ( t ′ ) ) t ′ = t = t ret (covariant representation with respect to s ′ ). Here s and s ′ are defined respectively by ds = cdt p − β ( t ) and ds ′ = cdt ′ p − β ( t ′ ) , where t ′ = t − t ret is the retardedtime and β ( t ) = c (cid:16) d r ( t ) dt (cid:17) . An equivalent representa-tion of G µ in terms of the particle arc length s is: G µ = (41)= 2 c (cid:16) qc (cid:17) R ′ α u α ( t )] (cid:20) dr µ ( t − t ret ) ds + − R ′ µ u k ( t ) dr k ( t − t ret ) ds R ′ α u α ( t ) (covariant representation with respect to s ), where R ′ α = r α ( t ) − r α ( t − t ret ) . This can be proven to yield also aparametric representation of G µ in terms of s. Proof - The proof of the first covariant representa-tion of G µ [given by Eq.(40)] follows immediately. Infact, by definition there results ddt ′ = cR ′ α u α ( t ′ ) dds ′ , where u α ( t ′ ) = γ ( t ′ ) v α ( t ′ ) , with γ ( t ′ ) = 1 / p − β ( t ′ ) and v α ( t ′ ) denoting v α ( t ′ ) = dr α ( t ′ ) dt ′ . Instead, to prove therepresentation (41) we first notice that by construction d ( R α R α ) = 0 . Hence the two differential constraints dr k ( t ) R k = dr k ( t ′ ) R k and R α u α ( t ′ ) dds ′ = R α u α ( t ) dds (seealso Lemma 3 in Appendix D) must be fulfilled too. Thisimplies that the following differential identity must hold dds ′ r µ ( t ′ ) R k − dds ′ r k ( t ′ ) R µ R α u α ( t ′ ) = (42)= dds r µ ( t ) R k − dds r k ( t ) R µ R α u α ( t ) . Substituting this expression in Eq.(40) there follows G µ = 2 c (cid:16) qc (cid:17) (cid:20) u µ ( t ′ ) R α u α ( t ′ ) R β u β ( t ) (43) − R µ R α u α ( t ′ ) [ R β u β ( t )] u m ( t ′ ) u m ( t ) t ′ = t − t ret . Invoking Lemma 3 this delivers Eq.(41). Here wenotice that the proper-time derivatives dr µ ( t − t ret ) ds and d r µ ( t − t ret ) ds are evaluated invoking the chain rule This isobtained by introducing the diffeomorphism t → s ( t ) ≡ s [and similarly t ′ → s ′ ( t ′ ) ≡ s ′ ] with its inverse transfor-mation s → t ( s ) . It follows t ′ ( s ′ ) = t ( s ) − t ret ( t ( s )) , whichproves that Eq.(41) delivers a parametric representationof G µ in terms of the local arc length s . Q.E.D.
Thus, remarkably, Eq.(41) shows that, whenparametrized in terms of the local arc length s, the 4-vector G µ depends - at most - on first-orderderivatives (with respect to s ) of the 4-position, i.e., is afunction only of the 4-bi-vector b R α and of the derivatives u k ( t ) ≡ dr k ( t ) ds and dr k ( t − t ret ) ds . G µ In this section we intend to investigate the main prop-erties of the 4-vector G µ (and hence of the RR equationgiven above). We intend to show that they are fully con-sistent with the basic principles of classical mechanics.In particular it is immediate to prove that G µ fulfills : • Galilei’s principle of inertia: in fact, in the case ofinertial motion it results identically G µ ≡ • the characteristic property of the Lorentz force, i.e.,the Lorentzian constraint G µ u µ = 0 . (44) • Newton’s principle of determinacy and Einstein’scausality principle.Finally, it can be shown that: • G µ is defined also in the case of ”sudden forces”.These results are summarized in the following theorem: THM.3 - Properties of G µ In validity of THM.1 and THM.2, the vector G µ fulfillsthe following properties : T3 ) in case of inertial motion in a given proper-timeinterval [ s , s ] , there results identically G µ ≡ ;T3 ) if F ( ext ) µν ( r ( s )) ≡ ∀ s in a given proper-time in-terval [ s , s ] and with respect to an inertial frame , thenthere results identically G µ ≡ , ∀ s ∈ [ s , s ] (Galilei’sinertia principle);T3 ) G µ satisfies the Lorentzian constraint condition G µ u µ = 0 . (45) Moreover, assuming that the RR equation (39), with(40), admits smooth solutions in the proper-time interval [ s a , s b ] , in such an interval:T3 ) G µ fulfills the Einstein’s causality principle,namely for any s ∈ [ s a , s b ] , r µ ( s ) depends only on thepast history of r µ ( s ) , i.e. { r µ ( s ∗ ) , ∀ s ∗ ≤ s } ; T3 ) G µ fulfills Newton’s determinacy principle,namely for any s o ∈ [ s a , s b ] , the knowledge of the parti-cle initial state { r µ ( s o ) , u µ ( s o ) } determines uniquely theparticle state { r µ ( s ) , u µ ( s ) } at any s ≥ s o which belongsto [ s a , s b ] ; T3 ) G µ is defined also in the case of ”sudden forces”.For example, let us require that the external EM field hasthe form F ( ext ) µν ( r ( s )) ≡ (cid:26) s ≤ s F (0) µν s > with F (0) µν a constant 4-tensor and s ∈ [ s a , s b ] . In sucha case one can prove that the solution of the RR equationexists and is unique. Proof - To prove propositions T and T let usassume that in the interval [ s , s ] the motion is in-ertial, namely that dds u µ ≡ , ∀ s ∈ [ s , s ] . This im-plies, that in [ s , s ] , u µ ≡ u µ , with u µ denoting aconstant 4-vector velocity. It follows ∀ s, s ′ ∈ [ s , s ] ,r µ ( s ) = r µ ( s ′ ) + u µ ( s ′ )( s − s ′ ) and R µ = u µ ( s )( s − s ′ ) . Hence, there results identically dr µ ( t − t ret ) ds R k − dr k ( t − t ret ) ds R µ R α u α ( s ) = (47)= u µ u k ( s )( s − s ′ ) − u µ u k ( s )( s − s ′ ) s − s ′ ≡ . Propositions T , T3 and T3 follow, similarly, by directinspection of Eqs.(39) and (40), or similarly Eq.(41). Inparticular, T3 is an immediate consequence of THM.1and the fact that the RR equation defines a well-posedinitial-value problem. Finally, the proof of proposition T3 can be obtained by explicit construction of thesolution of the RR equation (see analogous treatmentgiven in Ref.[2] for the weakly-relativistic LAD equation). Q.E.D.
An important aspect of the present formulation con-cerns the validity of the RR equation obtained letting σ → + (48)( point-charge limit ) in the definition of G µ [see Eq.(40)or (41)]. Here we intend to prove that: • the exact RR equation is not defined in the limit (48) [see following THM.4]. In other words, thepoint-charge limit [for G µ ] is not defined .To establish the result let us introduce yet anotherrepresentation of the 4-vector G µ which makes explicitits dependence in terms of the parameter σ, the radius ofthe spherical charge distribution. For definiteness let usintroduce the position w ≡ v ( t ′ ) + 1( t − t ′ ) t Z t ′ dt a ( t )( t − t ) . (49)Eq.(36) can also be written as R α R α = c ( t − t ′ ) (cid:26) − w c (cid:27) = σ , (50)so that the delay-time t ′ − t = t ret , with t ret > , reads t ret = σc q − w c . (51)1Here it is obvious that for all σ > − w /c > ,t ret = σc q − w c > σc ,R α v α ( s ) = cσ q − w c h − c d r ( t ) dt · w i > , − c d r ( t ) dt · w > . (52)Thus, introducing the 4-vector X k ≡ cσ q − w c R k = { c, w } one obtains for G µ the representation G µ = − c (cid:16) qc (cid:17) q − w c u k ( s ) σc h − c d r ( t ) dt · w i (53) dds dr µ ( t − t ret ) ds X k − dr k ( t − t ret ) ds X µ (cid:16) − c d r ( t ) dt · w (cid:17) r − v ( t ) c , which displays explicitly its dependence in terms of σ. The singular limit σ → + Let us now investigate the limit σ → + for G µ . Thefollowing (non-existence) theorem holds:
THM.4 - Non-existence of the point-chargelimit for G µ The limit lim σ → + G µ is not defined. In other words:for spherically symmetric charges the RR 4-vector is notdefined in the limit (48).Proof - Let us introduce the absurd hypothesis thatthe following limits exist:lim σ → + (cid:18) − w c (cid:19) > , (54)lim σ → + (cid:18) − c d r ( t ) dt · w (cid:19) > , (55)and moreover that for non-inertial motion there results0 < lim σ → + (cid:12)(cid:12)(cid:12)(cid:12) dds b H µk (cid:12)(cid:12)(cid:12)(cid:12) < ∞ , (56)where b H µk ≡ dr µ ( t − t ret ) ds X k − dr k ( t − t ret ) ds X µ (cid:16) − c d r ( t ) dt · w (cid:17) (57) r − v ( t ) c . In such a case, invoking Eq.(53) for G µ , it follows neces-sarily lim σ → + G µ ∝ lim σ → + c σ = ∞ . (58) Hence, in validity of (54)-(56) the limit lim σ → + G µ doesnot exist. On the other hand if one of the inequali-ties (54)-(56) is violated, the motion defined by the RRequation [Eq.(39)] is non-physical, which brings again thesame conclusion. Q.E.D.
A crucial point in the Dirac evaluation of the LADequation [11] was the power-series expansion of the re-tarded potential in terms of a suitably defined small di-mensionless parameter ξ, related to the proper-time dif-ference between emission ( t ′ ) and observation ( t ) times,0 < ξ ≡ ( t − t ′ ) t , (59)to be assumed as infinitesimal ( short-time ordering ). Thesame approach was also adopted by DeWitt and Brehme[15] in their covariant generalization of the LAD equationvalid in curved space-time.In analogy, here we introduce a power-series expansionwith respect to the dimensionless parameter ξ of the form G µ = ∞ X k =0 ξ k G ( k ) µ , (60)which is assumed to converge for ξ ≪ short-time asymptotic ordering ). The power series ex-pansion is actually obtained by introducing a Taylor ex-pansion for the 4-position vector r µ ( t − t ret ) in terms ofthe retarded time t ′ , namely letting r µ ( t − t ret ) = ∞ X k =0 ( t ′ − t ) k k ! d k r µ ( t ) dt k . (62)Manifestly, for the validity (i.e., the convergence) of theseries, a prerequisite is that r µ ( s ) ≡ r µ ( t ( s )) is a C ( ∞ ) function. In turn, this requires that also the Faradaytensor generated by the external EM field, F ( ext ) νµ , mustbe C ( ∞ ) . The use of the expansion (59) to represent the4-vector G µ reduces, formally, the RR equation to a lo-cal and infinite-order ordinary differential equation. Inview of THM.1 and the assumed convergence of the se-ries its (infinitely) smooth solution must still exist andbe uniquely defined. As a side consequence, this meansthat the initial conditions for such an equation must nec-essarily be considered as uniquely prescribed in terms ofthe initial conditions defined above [see Eqs.(3) and (4)]and the same RR-equation. Nevertheless, despite thesefeatures, the full series-representation of the RR equa-tion obtained in this way appears practically useless foractual applications.2As an alternative, however, assuming ξ as infinitesi-mal an asymptotic approximation for G µ [and the exactRR equation Eq.(29) or equivalent Eq.(39)] can in prin-ciple be achieved, subject again to suitable smoothnessassumptions to be imposed on the external field. Herewe intend to prove, in particular, that in this way: • the relativistic LAD equation is recovered as aleading-order asymptotic approximation to the ex-act RR equation . In fact, provided suitable smooth-ness conditions are met by the external field, the 4-vector G µ recovers asymptotically - in a suitableapproximation - the usual form of RR equationprovided by the LAD equation. This conclusionis achieved by introducing for the 4-vector G µ anasymptotic expansion with respect to the dimen-sionless parameter ξ ≪ , obtained by means of atruncated Taylor expansion in terms of the retardedtime t ′ , i.e., of the form r µ ( t − t ret ) = N X k =0 ( t ′ − t ) k k ! d k r µ ( t ) dt k , (63)with N >
THM.5 - First-order, short-time asymptotic ap-proximation for G µ Let us now assume that the EM-4-potential of the ex-ternal field A ( ext ) µ ( r ) is a smooth function of r. In such acase, in validity of the asymptotic ordering (61) and ne-glecting corrections of order ξ N , with N ≥ (first-orderapproximation ) , the following asymptotic approximationholds for G µ G µ ∼ = (cid:26) m oEM c dds u µ + g µ (cid:27) [1 + O ( ξ )] , (64) with g µ denoting the 4-vector g µ = 23 q c (cid:20) d ds u µ − u µ ( s ) u k ( s ) d ds u k (cid:21) , (65) and m oEM ≡ q c σ h ( t − t ′ )2 dds γ i (66) the EM mass.Proof - To reach the proof let us first evaluate asymp-totic expansions for the 4-vectors R k , dr µ ( t − t ret ) ds , the 4-scalar R α u α ( s ) and the time delay t − t ′ ≡ t ret . Neglect-ing corrections of order ξ N with N > , and denoting γ ≡ γ ( t ( s )) ≡ / p (1 − v ( t ( s )) /c and u k ≡ u k ( t ( s )) , one obtains by Taylor expansion R k ∼ = c ( t − t ′ ) γ u k − c ( t − t ′ ) γ dds (cid:18) u k γ (cid:19) + (67)+ c ( t − t ′ ) γ dds (cid:18) γ dds u k γ (cid:19) and similarly denoting r µ ≡ r µ ( t ( s )) ,dr µ ( t − t ret ) ds ∼ = (68) ∼ = dr µ ds − c ( t − t ′ ) γ d r µ ds + c ( t − t ′ ) γ d r µ ds . Thus, Eqs.(67) and (68) imply R α u α ( s ) ∼ = c ( t − t ′ ) γ − c ( t − t ′ ) γ dds γ + (69)+ c ( t − t ′ ) γ dds (cid:18) γ dds γ u α (cid:19) u α , where dds γ = dds q − v c = − γ v ( t ) · a ( t ) c . Finally, we noticethat there results t − t ′ = σc q − w c ∼ = (70) ∼ = σγc (cid:20) σγc v ( t ) · a ( t ) v ( t ) (cid:21) By substituting Eqs.(67)-(70) in Eq.(40) [or equivalentin Eq.(41)] it is immediate to recover after straightfor-ward calculations Eq.(64).
Q.E.D.
We remark that Eq.(65) for g µ coincides formally withthe usual expression of the EM self-force adopted inthe LAD equation (see related discussion in Ref.[2]).However, to recover the customary expression of the EMmass usually given [for the LAD equation] (see for ex-ample Ref.[13]), requires retaining only the leading-orderapproximation m oEM ∼ = q c σ [1 + O ( ξ )] , (71)This amounts to ignore the correction factor1 h ( t − t ′ )2 dds γ i ∼ = 1 + σc v ( t ) · a ( t ) c , (72)i.e., a term of order ξ in Eq.(66). Hence this approxima-tion is not sufficient, since the term g µ in Eq.(65) is oforder ξ too. We conclude that, for consistency, in placeof (71), the more accurate approximation (66) should beused for the EM mass m oEM .3An important issue is related to the conditions ofsmoothness - required by THM.5 for the validity ofEqs.(64)-(66) - which must be imposed on the externalEM field, i.e., on A ( ext ) µ ( r ( s )) . It is obvious, in particu-lar, that locally discontinuous (in s ) external fields mustgenerally be excluded , since the previous expansions [seeEqs.(67)-(70)] manifestly do not hold near the discon-tinuities. An example is provided by so-called ”suddenforces”. These occur when the corresponding Faradaytensor F ( ext ) µν ( r ( s )) is permitted to be locally discontinu-ous with respect to s (which may be achieved by turningon and off repeatedly the external EM field). For thevalidity of THM.5 this case must generally be excluded.In fact, it is obvious that the Taylor expansions (67)-(70)generally do not hold in the neighborhood of the discon-tinuities. This clearly prevents also the validity of the LLequation as well of analogous asymptotic approximationof Eq.(64) (see also related discussion in Ref.[2]). Although the covariant representation given by Eq.(41)is of general validity, it is worth discussing here alsoits weakly-relativistic approximation. This enables a di-rect comparison with the original Lorentz approach [4]and the known result obtained by Sommerfeld, Page,Caldirola and Yaghjian [22, 23, 24, 25] in the case ofa finite-size spherical-shell charge, a fact which is rele-vant not merely for historical reasons. Indeed, as previ-ously pointed out [2], also the weakly-relativistic LADequation exhibits the same difficulties characteristic ofthe relativistic LAD equation. In particular, it yields athird-order ordinary differential equation which exhibitsthe known physical inconsistencies (violation of NPD andGPI, existence of runaway solutions which blow up intime, etc.). Here we intend to show how, even in theweakly-relativistic approximation, the present theory isable to overcome such difficulties. For the sake of defi-niteness, let us determine the asymptotic approximationfor G µ , obtained by assuming β ≡ v ( t ) /c ≪ weakly-relativistic approximation ). For this purpose letus introduce a Taylor expansion with respect to β, whileleaving unchanged the dependence in terms of the re-tarded time t ′ . As shown in Appendix E, in such a casethe following result holds: THM.6 - Weakly-relativistic asymptotic ap-proximation for G µ In validity of the asymptotic ordering (73) and neglect-ing corrections of order β n , with n ≥ , the followingasymptotic approximation holds for G µ : G µ ∼ = ( G = 0 , G ) , (74) where: T ) first asymptotic approximation: in the case of therepresentation (40) the 3-vector G reads: G ∼ = − σ (cid:16) qc (cid:17) (cid:20) ddt v ( t − σc )+ (75)+ cσ v ( t − σc ) − c σ n r ( t ) − r ( t − σc ) o(cid:21) ; T ) second asymptotic approximation: in the case of therepresentation (41), instead, the 3-vector G becomes: G ∼ = 2 c (cid:16) qc (cid:17) σ (cid:20) d r ( t − t ret ) dt − (76) − r ( t ) − r ( t − t ret ) | ( t − t ′ ) | (cid:21) ; T ) finally, upon invoking also the short-time order-ing (59) and a suitable condition of smoothness for theexternal EM field, one recovers in both cases [Eqs.(75) or(76)] the usual weakly relativistic approximation: G ∼ = g + m EM ·· r ( t ) , (77) where g ≡ − q c ··· r , (78) m EM ≡ q c σ , (79) are respectively the well-known weakly-relativistic EMself-force 3-vector and the EM mass.Proof (see Appendix E). We notice that the ap-parent non-uniqueness of the two representations givenabove [Eqs.(75) and (76)] can be resolved by notingthat the β − expansion should be actually carried outalso in terms of the delay-time t ret (which should beconsidered itself of order β α , with α > . Indeed, if the short-time expansion is in-troduced, as found in Appendix E, Eqs.(77) both im-ply Eqs.(78) and (79). In the same sense, Eqs.(75)and (76) can also be proven to be in agreement withthe well-known Sommerfeld-Page-Caldirola-Yaghjian re-sult [22, 23, 24, 25] for weakly-relativistic spherical-shellcharges. The resulting equations, (78) and (79), are man-ifestly consistent with the customary weakly-relativisticapproximation for the LAD equation (see, for example,also related discussion in Ref.[2]).
10 - CONCLUDING REMARKS
In this paper an exact solution has been obtained forthe RR problem. The result has been achieved in thecase of a spherical-shell finite-size charge. As a main con-sequence, the exact RR equation, describing the relativis-tic dynamics of such a particle in the presence of its EM4self-field has been achieved (see THM.1 and THM.2).Although its charge has been assumed as spatially dis-tributed, we have shown that, by assuming the mass aspoint-wise localized, the dynamics is reduced to that ofa point particle. The resulting RR equation appears freefrom all the difficulties met by previously classical RRequations (THM.1-THM.3). In particular, besides be-ing fully relativistic, the new equation:1) has been achieved via a variational formulation basedon the adoption of the Hamilton variational principle .The treatment has been made transparent by adopting asynchronous form of the variational principle;2) unlike the LAD equation : results consistent withthe Newton’s principle of determinacy, Einstein principleof causality, Galilei law of inertia and does not exhibitso-called runaway solutions;3) unlike the LL equation : does not involve the adop-tion of iterative approaches for its derivation;4) unlike the LAD and LL equations : is valid also in thecase of sudden forces and does not exhibit any singularbehavior (i.e., provided the radius of the charge σ remainsstrictly positive);5) unlike all previous equations ( LAD and LL and theMedina equations): it is not asymptotic.6) unlike in the Medina approach : the variational ap-proach is based on the Hamilton variational principle inthe ordinary phase-space, which allows us to retain thecustomary formulation of classical mechanics and classi-cal electrodynamics.In addition, as a side result, we have pointed out acorrection to the LAD equation, appearing in the EMmass, which is demanded by the perturbative expansion[see Eq.(66)].The theory developed in this paper has, potentially,deep and wide-ranging implications. These are related,in particular, to the description of relativistic dynam-ics of systems of classical finite-size charged particles.The conceptual simplicity of the present approach andits general applicability to arbitrary systems of chargesof this type make the present results of extraordinaryrelevance for relativistic theories (such as kinetic theoryof charged particles and gyrokinetic theory for magneto-plasmas) and related applications in astro- and plasmaphysics.
Acknowledgments
Useful comments by A. Beklemishev (Budker Instituteof Nuclear Physics, Novosibirsk, Russia Federation), J.Miller (International School for Advanced Studies, Mi-ramare, Trieste, Italy and Oxford University, Oxford,UK) and P. Nicolini (Department of Mathematics andInformatics, Trieste University, Italy) are acknowledged.This work has been developed in cooperation with theCMFD Team, Consortium for Magnetofluid Dynamics (Trieste University, Trieste, Italy), within the frameworkof the MIUR (Italian Ministry of University and Re-search) PRIN Programme:
Modelli della teoria cinet-ica matematica nello studio dei sistemi complessi nellescienze applicate . Support is acknowledged from GNFM(National Group of Mathematical Physics) of INDAM(Italian National Institute for Advanced Mathematics).
APPENDIX A: SURFACE-AVERAGE OPERATOR
Following the notations introduced in Sec.4b and incase of flat space-time, if A ( r + σn ) is a smooth (tensor)function of the 4-position vector r + σn, we define itssurface-average as A ( r ) = 14 π Z d Σ( n ) A ( r + σn ) . (80)In particular, identifying A with the Faraday tensor F νµ ( r + σn ) , its surface-average is F νµ ( r ) = 14 π Z d Σ( n ) F νµ ( r + σn ) . (81) APPENDIX B: INTEGRAL REPRESENTATIONFOR A ( self ) µ (CASE OF A POINT CHARGE) The integral representation (17) for A selfµ can also beobtained directly from Maxwell’s equations . Let us con-sider first the case of a point-charge. By assumption A selfµ satisfies Maxwell’s equations (in flat space-time) ∂ µ F µν ( self ) = 4 πc j ν , (82)where for a point particle: j µ ( r ν ) = q Z ds ′ u µ ( s ′ ) (83) δ (4) ( r − r ( s ′ )) (84)(with δ (4) ( r − r ( τ )) denoting the 4-dimensional Diracdelta). There results therefore A µ ( self ) = 4 πc Z d r ′ G ( r − r ′ ) j µ ( r ′ ) , (85)where G ( r − r ) is the retarded Green function which sat-isfies the equation ⊡ G ( r − r ′ ) = δ (4) ( r − r ′ ) (86)and is such that G ( r − r ′ ) = 0 (87)5for r < r ′ . It follows G ( r − r ′ ) = 12 π δ ( R µ R µ )Θ( r − r ′ ) , (88)and hence A µ ( self ) ( r ) = 4 πc Z d r ′ π δ ( R µ R µ )Θ( r − r ′ ) (89) q Z ds ′ u µ ( s ′ ) δ (4) ( r ′ − r ( s ′ )) , namely A ( self ) µ ( r ) = 2 c Z d r ′ δ ( R µ R µ )Θ( r − r ′ ) q (90) Z ds ′ u µ ( s ′ ) δ (4) ( r ′ − r ( s ′ )) . This implies also A ( self ) µ ( r ) = 2 qc Z ds ′ u µ ( s ′ ) δ ( R µ ( s ′ ) R µ ( s ′ )) , (91)where R µ ( s ′ ) = r µ − r µ ( s ′ ) . The last integral can also bewritten as A ( self ) µ ( r ) = 2 qc Z dr ′ µ δ ( R µ R µ ) . (92)This is an integral representation for A ( self ) µ ( r ) , by con-struction equivalent to Eq.(85). APPENDIX C: INTEGRAL REPRESENTATIONFOR A ( self ) µ (CASE OF A SPHERICAL-SHELLCHARGE) To prove that the differential and integral representa-tions for A selfµ (16) and (17) are equivalent it is sufficientto notice that the following identity holds: δ ( R α R α − σ ) = (93)= δ ( t − t ′ − t ret )12 c (cid:12)(cid:12)(cid:12) ( t − t ′ ) − c d r ( t ′ ) dt ′ · ( r − r ′ ) (cid:12)(cid:12)(cid:12) . In fact there follows A ( self ) µ ( r ) = 2 qc Z dr ′ µ δ ( R α R α − σ ) = (94)= 2 qc c (cid:12)(cid:12)(cid:12) ( t − t ′ ) − c d r ( t ′ ) dt ′ · ( r − r ′ ) (cid:12)(cid:12)(cid:12) dr ′ µ dt ′ t ′ = t − t ret == qc (cid:20) u µ ( t ′ ) R α u α ( t ′ ) (cid:21) t ′ = t − t ret , which recovers immediately Eq.(16). APPENDIX D - OTHER LEMMAS
LEMMA 2 - Synchronous variation of ∆ S The synchronous variation of ∆ S ( r µ ) reads δ ∆ S = δA + δB, (95) where δA ≡ − (cid:0) qc (cid:1) g µν π R d Σ( n ) R δr µ d hR dr ′ ν δ ( R α R α − σ ) i ,δB ≡ (cid:0) qc (cid:1) g αβ π R d Σ( n ) R dr ′ β R dr α δr µ ∂∂r µ δ ( R k R k − σ ) . (96) There results respectively: δA ≡ (cid:16) qc (cid:17) g µν c π Z d Σ( n ) (97) Z δr µ dr k [ A νk ] t ′ = t − t ret ,δB ≡ − (cid:16) qc (cid:17) g αβ c π Z d Σ( n ) (98) Z dr α δr µ (cid:2) A βµ (cid:3) t ′ = t − t ret , where A νk ≡ c (cid:12)(cid:12)(cid:12) ( t ′ − t ) − c d r ( t ′ ) dt ′ · ( r ′ − r ) (cid:12)(cid:12)(cid:12) (99) ddt ′ v ′ ν ( t ′ ) R k c (cid:12)(cid:12)(cid:12) ( t ′ − t ) − c d r ( t ′ ) dt ′ · ( r − r ′ ) (cid:12)(cid:12)(cid:12) . Proof - In fact let us assume that the metric tensor g µν is constant and symmetric (Minkowski space-time).In this case the synchronous variation of ∆ S is given byEqs.(95) and (96) where d (cid:20)Z dr ′ ν δ ( R α R α ) (cid:21) = dr k Z t t cdt ′ (100) s − c (cid:12)(cid:12)(cid:12)(cid:12) d r ′ dt ′ (cid:12)(cid:12)(cid:12)(cid:12) u ′ ν ( t ′ ) ∂∂r k (cid:2) δ ( R α R α − σ ) (cid:3) . Hence it follows, δA = − (cid:16) qc (cid:17) g µν π Z d Σ( n ) Z δr µ dr k Z t t cdt ′ s − c (cid:12)(cid:12)(cid:12)(cid:12) d r ′ dt ′ (cid:12)(cid:12)(cid:12)(cid:12) u ′ ν ( t ′ ) (101) ∂∂r k (cid:2) δ ( R α R α − σ ) (cid:3) . δB ≡ (cid:16) qc (cid:17) g αβ π Z d Σ( n ) Z dr α δr µ Z t t cdt ′ s − c (cid:12)(cid:12)(cid:12)(cid:12) d r ′ dt ′ (cid:12)(cid:12)(cid:12)(cid:12) u ′ β ( t ′ ) (102) ∂∂r µ δ ( R k R k − σ ) . Let us now evaluate the partial derivative ∂∂r k δ ( R α R α − σ ) . There results, thanks to the chain rule ∂∂r k δ ( R α R α − σ ) = (103)= ∂ ( R α R α ) ∂r k dδ ( R α R α − σ ) d ( R α R α ) ≡≡ R k dδ ( R α R α − σ ) dt ′ d ( R α R α ) dt ′ . Hence, there follows the identity ∂∂r k δ ( R α R α − σ ) = (104)= R k c (cid:12)(cid:12) ( t − t ′ ) − c d r ′ dt ′ · ( r − r ′ ) (cid:12)(cid:12) ddt ′ { δ ( t − t ′ − t ret )12 c (cid:12)(cid:12)(cid:12) ( t − t ′ ) − c d r ( t ′ ) dt ′ · ( r − r ′ ) (cid:12)(cid:12)(cid:12) , where r ′ ≡ r ( t ′ ) . Integrating by parts one obtains man-ifestly Eq.(97). In a similar manner, thanks again toEq.(103), the term δB can be cast in the form (98). Q.E.D.LEMMA 3 - Differential identity 1
The following identity holds dt ′ dt = p σ + ( r − r ′ ) − d r dt · ( r − r ′ ) p σ + ( r − r ′ ) − d r ′ dt ′ · ( r − r ′ ) . (105) Proof - In fact there results dt ′ dt = dtdt − ddt p σ + ( r − r ′ ) = 1 − (106) − p σ + ( r − r ′ ) (cid:20) d r dt − dt ′ dt d r ′ dt ′ (cid:21) · ( r − r ′ )namely dt ′ dt " − p σ + ( r − r ′ ) d r ′ dt ′ · ( r − r ′ ) = (107)= 1 − p σ + ( r − r ′ ) d r dt · ( r − r ′ ) . Q.E.D.
APPENDIX E - WEAKLY RELATIVISTICAPPROXIMATION
In validity of the asymptotic ordering (73) there results[from Eq.(40)] by Taylor expansion in β, while retainingexactly all dependencies in terms of the retarded time t ′ , G µ ∼ = (108) ∼ = 2 c (cid:16) qc (cid:17) (cid:20) c | ( t − t ′ ) | ddt ′ (cid:26) v ′ µ ( t ′ ) c ( t − t ′ ) − cR µ c | ( t − t ′ ) | (cid:27)(cid:21) t ′ = t − t ret . Instead, in the same approximation Eq.(41) yields G µ ∼ = (109) ∼ = 2 c (cid:16) qc (cid:17) c | ( t − t ′ ) | (cid:20) dr µ ( t − t ret ) dt + − R ′ µ c c | ( t − t ′ ) | (cid:21) where R ′ α ≡ { ct ret , r ( t ) − r ( t − t ret ) } . Here the delaytime t ret ≡ t − t ′ , evaluated in a similar way from Eq.(51)neglecting corrections of order β , reads: t ret ∼ = σc . (110)It follows G µ ∼ = (0 , G ) . (111)In particular, the spatial 3-vector G reads in case ofEq.(108) : G ∼ = (112) ∼ = − c (cid:16) qc (cid:17) (cid:20) c | ( t − t ′ ) | ddt ′ (cid:26) v ( t ′ ) − r ( t ) − r ( t ′ )( t − t ′ ) (cid:27)(cid:21) t ′ = t − t ret . This equations, with (110), implies Eq.(75). Finally, letus evaluate also the corresponding short-time approxima-tion, obtained invoking also the ordering (59). By Taylorexpansion in ξ ≡ ( t − t ′ ) /t there results to leading order (cid:20) ddt ′ v ( t ′ ) + v ( t ′ )( t − t ′ ) − r ( t ) − r ( t ′ )( t − t ′ ) (cid:21) t ′ = t − t ret ∼ = (113) ∼ = 12 ddt v ( t ) − R c d dt v ( t ) . Therefore, one obtains finally the weakly-relativistic (andshort-time) approximation G ∼ = (cid:16) qc (cid:17) (cid:20) − σ ddt v ( t ) + 23 c d dt v ( t ) (cid:21) , (114)7which similarly recovers Eq.(77). Instead, in the case ofEq.(109) in an analogous way there results: G ∼ = (115) ∼ = 2 c (cid:16) qc (cid:17) c | ( t − t ′ ) | (cid:20) d r ( t − t ret ) dt −− r ( t ) − r ( t − t ret ) | ( t − t ′ ) | (cid:21) , which implies Eq.(76). Hence it follows d r ( t − t ret ) dt − r ( t ) − r ( t − t ret ) | ( t − t ′ ) | ∼ = (116) ∼ = − R c d r ( t ) dt + 13 R c d r ( t ) dt , which implies again Eq.(114) and therefore recoversthe same weakly-relativistic approximation given by Eq.(77). [1] M. Tessarotto, C. Cremaschini, M. Dorigo, P. Nicoliniand A. Beklemishev, The exact radiation-reaction equa-tion for a classical charged particle , contributed paperat RGD26 (Kyoto, Japan, July 2008); arXiv:0807.1819(2008).[2] M. Dorigo, M. Tessarotto, P. Nicolini and A. Bek-lemishev,
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