Foundations of the self-force problem in arbitrary dimensions
FFoundations of the self-force problem in arbitrary dimensions
Abraham I. Harte , Peter Taylor and Éanna É. Flanagan , Centre for Astrophysics and Relativity, School of Mathematical SciencesDublin City University, Glasnevin, Dublin 9, Ireland Cornell Center for Astrophysics and Planetary Science, Cornell University, Ithaca, NY 14853 and Department of Physics, Cornell University, Ithaca, NY 14853
The self-force problem—which asks how self-interaction affects a body’s motion—has been poorlystudied for spacetime dimensions d (cid:54) = 4 . We remedy this for all d ≥ by nonperturbatively con-structing momenta such that forces and torques acting on extended, self-interacting electromagneticcharges have the same functional forms as their test body counterparts. The electromagnetic fieldwhich appears in the resulting laws of motion is not however the physical one, but a certain ef-fective surrogate which we derive. For even d ≥ , explicit momenta are identified such that thissurrogate field satisfies the source-free Maxwell equations; laws of motion in these cases can beobtained similarly to those in the well-known four-dimensional Detweiler-Whiting prescription. Forodd d , no analog of the Detweiler-Whiting prescription exists. Nevertheless, we derive its replace-ment. These general results are used to obtain explicit point-particle self-forces and self-torquesin Minkowski spacetimes with various dimensions. Among various characteristics of the resultingequations, perhaps the most arresting is that an initially-stationary charge which is briefly kickedin dimensions asymptotically returns to rest. I. INTRODUCTION
The motion of small bodies is central to some of themost enduring problems in physics. If such a body is cou-pled to an electromagnetic, gravitational, or other long-range field, it may be subject to net forces exerted by itsown contributions to that field. This “self-force” stronglyinfluences, for example, charged particles circulating inparticle accelerators and the shrinking orbits of blackhole binaries due to the emission of gravitational radi-ation. However, the apparent simplicity of the statementof the self-force problem belies a number of physical andmathematical subtleties. This has led to more than acentury of literature on the subject; see [1–10] for someelectromagnetic examples.While motivations for working on the self-force prob-lem have varied considerably over the years, the pasttwo decades have seen a concerted effort—motivatedlargely by gravitational wave astronomy—to understandthe gravitational self-force problem in general relativity.This has led to a number of theoretical and computa-tional advances which considerably improve our under-standing of classical self-interaction, in both the gravita-tional and electromagnetic contexts [11–14]. Separately,new aspects of the electromagnetic self-force are begin-ning to be accessible to investigation via high-power laserexperiments [15, 16].In this paper, we move beyond the existing literature,which almost exclusively focuses on four spacetime di-mensions, to rigorously study self-interaction in all di-mensions d ≥ . There are several reasons for this:First, considerations in different numbers of dimensionsrefine our understanding of precisely what is importantand what is not; lessons learned in this way may signif-icantly inform future considerations even in four dimen-sions, particularly in more complicated theories whichhave not yet been understood. Second, considerations of theories in non-physical numbers of dimensions can,via holographic dualities, be related to ordinary four-dimensional systems; for example, the five-dimensionalself-force might be used to understand jet quenching infour-dimensional quark-gluon plasmas [17]. Third, self-forces in odd numbers of spacetime dimensions are qual-itatively very different from those in even dimensions. Inparticular, one crucial ingredient of the self-force frame-work is the derivation of an appropriate map from thephysical field to an effective surrogate in which, e.g., theLorentz force law is preserved despite the possible pres-ence of radiation reaction and similar effects; this maphad not been previously understood in odd dimensionsand differs considerably from its even-dimensional coun-terpart. Despite these differences, a significant portion ofthis paper is devoted to developing a unified formalismwhich applies for any parity of dimension. Our resultssubsume the Detweiler-Whiting scheme which was origi-nally given in four dimensions [11, 12, 18].Our final reason for considering different numbers ofdimensions is that the self-force in three spacetime di-mensions may be proportionally stronger than in four di-mensions, both in terms of instantaneous magnitude [19]and—as argued below—in the particularly slow decay offields in this context. The latter property implies thatself-interaction encodes a strong “memory” of a system’spast. Moreover, systems in which this is relevant maybe accessible to experiment. For example, “pilot wavehydrodynamics” involves a number of striking phenom-ena observed to be associated with oil droplets bouncingon a vibrating bath [20]. Each bounce generates surfacewaves on the bath, but these waves also affect the hori-zontal motion of the droplet. This type of feedback witha long-range field (the surface waves) is reminiscent of aself-force problem in two spatial dimensions. There arealso a variety of condensed matter systems which act asthough they are confined to one or two spatial dimensions a r X i v : . [ g r- q c ] A p r [21, 22], and some of our considerations may be relevantthere as well.We do not consider any particular fluid or condensedmatter system in this paper, but instead explore a stan-dard electromagnetic self-force problem in different num-bers of dimensions. To the best of our knowledge, theliterature does not contain any rigorous derivations ofthe self-force other than in four dimensions, except forrecent work restricted to static bodies [19] (see howeverwork on the radiation reaction component of the self-forcein effective field theory [23, 24]). The fully-dynamicalself-force considered here is considerably different andmore rich than its static counterpart. We choose to fo-cus on the standard electromagnetic self-force problempurely for clarity of exposition; our results generalizeto extended bodies coupled to Klein-Gordon fields andto linearized gravitational fields satisfying the Einsteinequations. There are also no conceptual obstacles to con-sidering non-trivial boundary conditions such as thoseappropriate for describing analogs of the pilot-wave hy-drodynamics experiments mentioned above.A layout of the paper is as follows: In Section II,we briefly review, and then apply and extend, a non-perturbative formalism [10, 12, 25, 26] which provides ageneral framework for the problem of motion of stronglyself-interacting extended bodies. Not all of the intricatedetails of this formalism are required to absorb the essen-tial points of this paper, although some of the most rele-vant aspects are recounted here. We apply them to deriveequations of motion for an extended body coupled to anelectromagnetic field in arbitrary dimensions, includingall self-interaction effects. The resultant equations arestructurally identical to extended test-body equations,except that the physical field in the test-body equationsis mapped to an effective field which encapsulates all self-force and self-torque effects. The cost of this map is thatthe stress-energy tensor of the body is renormalized asit appears in the laws of motion. This renormalizationis well-controlled, however, being both finite and quasi-local (in a sense we make precise later). Moreover, ourlaws of motion admit well-defined point-particle limits.In Section III, we discuss these point-particle limits withretarded boundary conditions in Minkowski spacetimeswith various dimensions, obtaining explicit point-particleself-forces and self-torques in these cases. In Section IV,we discuss some interesting phenomenology that occursin odd numbers of spacetime dimensions. Some of theissues that arise here remain open problems which wehope will inspire further interest. We particularly fo-cus on self-force phenomenology in d = 3 , where self-interaction effects can be relatively large and where thereis a very strong history dependence. One particularlystriking example of the latter shows up in the case of acharge which is initially stationary and is then given akick by an externally-imposed force. Our analysis showsthat the slowly-decaying fields in this case cause such aparticle to return to rest at late times, a phenomenonreminiscent of Aristotelian physics. Throughout this paper, units are chosen in which G = c = 1 , the metric signature is positive, abstract indicesare denoted by a , b , . . . , spacetime coordinate indices by µ , ν , . . . , and spatial coordinate indices by i , j , . . . . II. EXTENDED BODIES ANDNON-PERTURBATIVE LAWS OF MOTION
Our strategy is not to obtain a “point particle self-force” as any kind of fundamental concept, but insteadto derive laws of motion first for extended charge distri-butions and then to evaluate point-particle limits of thoselaws. Although we focus for concreteness on the electro-magnetic self-force problem, analogous results are easilyobtained for the scalar and (at least the first order) grav-itational self-force. The approach adopted here is basedon a nonperturbative formalism developed by one of us[10, 12, 25, 26], which provides a rigorous framework withwhich to analyze problems of motion in a wide variety ofcontexts. Crucially, most of this framework is agnosticto the number of spacetime dimensions.
A. Preliminaries
In the electromagnetic context, a finite extended bodyin a d -dimensional spacetime ( M, g ab ) is associated with anonsingular conserved current density J a and a nonsingu-lar stress-energy tensor T ab B = T ( ab ) B . The support of T ab B may be identified with the body’s worldtube W ⊂ M ,and that of J a is assumed not to extend beyond W .Furthermore, we suppose that the body’s worldtube isspatially compact and that the electromagnetic field F ab satisfies Maxwell’s equations ∇ [ a F bc ] = 0 , ∇ b F ab = ω d − J a , (1)in a neighborhood of W , where ω d − ≡ π ( d − Γ( ( d − (2)is equal to the area of a unit sphere in R d − . The dy-namical evolution of such an extended body may be un-derstood, at least in part, via energy and momentumexchanges between that body and the electromagneticfield. Such exchanges are more precisely described bythe conservation of the system’s total stress-energy ten-sor T ab = T ( ab ) , in the sense that ∇ b T ab = 0 . (3)Although various arguments can be made for how to“most naturally” split T ab into electromagnetic and ma-terial components inside of a body, particularly if thatbody possesses nontrivial dielectric or related properties[27], we pragmatically extend the vacuum expression T ab EM ≡ ω d − (cid:18) F ac F bc − g ab F cd F cd (cid:19) (4)for the electromagnetic stress-energy tensor into the in-terior of W : T ab = T ab B + T ab EM everywhere of interest.This sum may in fact be viewed as a definition for T ab B in terms of T ab and F ab . Adopting it, (1) and (3) implythat ∇ b T ab B = F ab J b . (5)Every portion of an extended charge is thus acted uponby the Lorentz force density F ab J b .The question we now ask is how this force density“integrates up” to affect a body’s overall motion. Onedifficulty is that Maxwell’s equations imply that F ab J b depends nonlinearly and nonlocally on J a , and can bealmost arbitrarily complicated. Despite this, experiencesuggests that there are physically-interesting regimes inwhich the (appropriately-defined) net force is not com-plicated at all: Laws of motion arise in which net forcesinvolve a body’s internal structure only via its first non-vanishing multipole moments. That these laws do notdepend on finer details of J a lends them a certain degreeof “universality.” Deriving this universality and makingit precise is the essence of the self-force problem.We emphasize that it is only certain “bulk” featuresof an extended body which can be described as univer-sal. Individual objects may develop internal oscillations,turbulent flows, or other fine details which strongly de-pend on material composition, thermodynamic effects,and other characteristics which are difficult both to spec-ify and to model. As in Newtonian celestial mechanics,our goal here is to ignore as many of these details as possi-ble, and to instead identify certain features of T ab B and J a which i) describe a body’s behavior “as a whole,” and ii)whose evolution is only weakly coupled to a body’s inter-nal details. In Newtonian celestial mechanics, these twoproperties are well-known to hold for the linear and angu-lar momenta associated with each celestial body. We nowgeneralize the concept of momentum to describe relativis-tic extended bodies coupled to electromagnetic fields in d -dimensional spacetimes. B. Momentum
Although there is a strong physical expectation thatsome momentum-like quantity can be defined for ex-tended, relativistic charge distributions, non-pathologicaldefinitions are not so easily written down. The problemis particularly acute for bodies with significant self-fields,essentially because, i) those self-fields carry energy andmomentum, ii) they rearrange themselves to adjust toany motions associated with W , and iii) self-fields mayextend far outside a body’s material boundaries. Thefirst and second of these points suggest that a body’s self-field contributes, e.g., to its inertia—a fact already rec-ognized by the end of the 19th century [1]. Indeed, four-dimensional electromagnetic self-fields are now known tocontribute to (or “renormalize”) not only a body’s appar-ent mass, but all of its stress-energy tensor [10, 12, 28]. While there must be some sense in which these ef-fects generalize to any number of dimensions, they arenontrivial to compute even when d = 4 . Fundamen-tally, this is because it is difficult to guess a suitabledefinition for the momentum which treats a body’s ma-terial and self-field components as one. Consider, forexample, a “renormalized body momentum” defined byappropriately integrating T ab B + T ab self , where T ab self is anelectromagnetic stress-energy tensor which is quadraticin a suitably-defined self-field. Any “mass” associatedwith such a definition would clearly depend on prop-erties of the self-field at arbitrarily-large distances, andhence on the state of system over arbitarily-long times.It would not describe a body’s instantaneous resistanceto applied forces, and would thus be a poor definition ofmass. Indeed, the degree of nonlocality associated withnaive momentum definitions such as this renders themphysically unacceptable, at least in a non-perturbativecontext. While the problem is considerably alleviated atlow orders in the perturbative expansions commonly as-sociated with point-particle limits [9, 29], it is difficult toextend the methods employed in those contexts to higherorders in perturbation theory, or even to apply them atlow orders when Huygens’ principle is violated.The nonperturbative formalism [12] employed here af-fords a different approach, allowing us to derive—ratherthan postulate—physically-acceptable definitions for thelinear and angular momenta of a charged extended body.Our results do not depend on any limiting process, andare nonlocal only in the sense that they depend on thestate of the system over spatial and temporal scales com-parable to a suitably-defined diameter for W . No smallerdegree of nonlocality could reasonably be expected, evenfor a body with no self-field whatsoever. Schematically,we obtain this definition by first proposing a “bare mo-mentum,” essentially a guess which need not take intoaccount any self-field effects, and then deriving a cor-rection to that definition which maintains locality whilealso decoupling forces and torques from a body’s internaldetails.Our momenta depend on a choice of origin, which wetake to be a timelike worldline parametrized by γ ( τ ) .Such origins are required even in Newtonian mechanicsat least to define the angular momentum, and it is onlythe global parallelism of Euclidean space which preventsthem from also being necessary to define the Newtonianlinear momentum. Global parallelism cannot be assumedhere, so an origin is needed for both our linear and angu-lar momenta. We additionally need to specify a family ofspacelike hypersurfaces B τ (cid:51) γ ( τ ) which foliate W , thusfixing a notion of time inside a body’s worldtube. Specificworldlines and specific hypersurfaces may be fixed usingspin supplementary and similar conditions; see SectionII E. At this stage, however, we assume only that someprescription has been given. Its details do not matter.Next, we define the bare “generalized momentum” P τ [ · ] at time τ via P τ [ ξ a ] ≡ (cid:90) B τ dS a (cid:34) T ab B ( x ) ξ b ( x ) + J a ( x ) (cid:90) duu − × ∇ b (cid:48) σ ( y (cid:48) ( u ) , γ ( τ )) F b (cid:48) c (cid:48) ( y (cid:48) ( u )) ξ c (cid:48) ( y (cid:48) ( u )) (cid:35) , (6)where σ ( x, x (cid:48) ) denotes Synge’s world function, defined toequal one half of the squared geodesic distance betweenits arguments, y (cid:48) ( u ) is an affinely-parametrized geodesicfor which y (cid:48) (0) = γ ( τ ) and y (cid:48) (1) = x , and ξ a ( x ) is anyvector field drawn from a certain finite-dimensional vec-tor space referred to as the space of generalized Killingfields (GKFs). The GKFs are defined more preciselyin [12, 30], and coincide with the space of all ordinaryKilling vector fields in maximally-symmetric spacetimes.More generally, the GKFs always form a vector spacewith dimension d ( d + 1) . The generalized momentumat fixed time is a linear operator on this space, and maytherefore be viewed as a d ( d + 1) -dimensional vector inthe vector space dual to the space of GKFs. It simulta-neously encodes both the linear and angular momentumof an extended body. Just as electric and magnetic fieldsare best thought of as two aspects of a single electro-magnetic field, a body’s linear and angular momenta aretwo aspects of a single more-fundamental structure: thegeneralized momentum.More precisely, a bare linear momentum p a ( τ ) and abare angular momentum S ab = S [ ab ] ( τ ) may be implicitlydefined by combining (6) with P τ [ ξ a ] ≡ p a ( τ ) ξ a ( γ ( τ )) + 12 S ab ( τ ) ∇ a ξ b ( γ ( τ )) . (7) P τ [ ξ a ] thus returns a linear combination of the linear andangular momenta. The particular choice of ξ a controlswhich linear combination is obtained. If ψ a is, e.g., atranslational Killing field in flat spacetime, P τ [ ψ a ] re-turns the component of linear momentum associated withthat translation. Regardless, varying over all possibleGKFs results in integral formulae for p a and S ab whichinvolve T ab B , J a , and F ab . These formulae coincide withdefinitions originally proposed by Dixon [31–33], whosought multipole moments for T ab B in which stress-energyconservation implies differential constraints only on themonopole and dipole moments. That goal was achieved,meaning that there is a sense in which ordinary differen-tial equations with the form ˙ p a = ( . . . ) and ˙ S ab = ( . . . ) are fully equivalent to the partial differential equation(5). Regardless, these definitions for the momenta reduceto textbook ones [34] in flat spacetime and with vanishingelectromagnetic fields. They also give rise, more gener-ally, to simple conservation laws whenever there existsa Killing vector field which is also a symmetry of F ab [12, 32]. We note that this last property would fail tohold if the (less familiar) electromagnetic terms in (6)were omitted.Now that a bare momentum has been proposed, itsevolution may be understood by differentiating (6) with respect to τ while applying (5). The resulting rate ofchange may be interpreted as encoding bare forces andbare torques, which can again be written as integrals overa body’s interior: ddτ P τ [ ξ a ] = F G τ [ T ab B ; ξ c ] + F EM τ [ F ab , J c ; ξ d ] , (8)where the gravitational generalized force is given by thebilinear functional F G τ [ T ab B ; ξ c ] = 12 (cid:90) B τ dST ab B L ξ g ab , (9)and the electromagnetic generalized force by the trilinearfunctional F EM τ [ F ab , J c ; ξ d ] = (cid:90) B τ dSJ b (cid:20) ξ a F ab + ∇ b (cid:90) duu − × ∇ b (cid:48) σ ( y (cid:48) ( u ) , γ ( τ )) F b (cid:48) c (cid:48) ( y (cid:48) ( u )) ξ c (cid:48) ( y (cid:48) ( u )) (cid:21) . (10)The ξ a here again denote GKFs and the L ξ are Lie deriva-tives. Any particular GKF which is substituted into theseequations may be viewed as selecting the correspondingcomponent of the generalized gravitational and electro-magnetic force vectors.If one has full knowledge of T ab B and F ab , and hence J a = ω − d − ∇ b F ab , electromagnetic forces and torquescan be computed by directly evaluating the integral F EM τ [ F ab , J c ; ξ d ] for all possible GKFs. However, thereis little point to such descriptions if the system is alreadyunderstood in full detail. Momenta and related conceptsare most valuable precisely when a system’s details areonly partially specified, or alternatively, if one would liketo understand a class of distinct systems which neverthe-less share some bulk features.With this context in mind, forces and torques associ-ated with the definition (6) may be shown to have cer-tain undesirable characteristics. First, unless self-fieldsare negligible, the functionals F EM τ [ F ab , J c ; ξ d ] are diffi-cult to approximate as-is. In particular, it is not clearhow to evaluate them if, e.g., only the first few multipolemoments of J a are known. More precisely, it is the ac-tual self-field which must be negligible in order to makereasonable approximations, and not only some suitableintegral of that self-field. There are very few realisticsettings in which such an approximation can be justified,particularly if smallness is also demanded for the deriva-tives of a body’s self-field. In some cases, this difficulty ismerely a question of calculational practicality. In others,it is essential: There are important examples in whichbare forces and torques may be directly shown to dependon the detailed nature of J a . When this occurs, the lawsof motion associated with p a and S ab cannot be deemeduniversal. Nevertheless, we find closely-related momentawhich do obey universal laws of motion.The formalism we employ accomplishes this by pro-viding a set of tools which allow one to easily establishidentities with the form F EM τ [ F ab , J c ; ξ d ] = F EM τ [ ˆ F ab , J c ; ξ d ] − ddτ δP τ [ J a ; ξ b ]+ F G τ [ δT ab B [ J c ]; ξ d ] , (11)for a wide variety of nonlocal field transformations F ab (cid:55)→ ˆ F ab . The nature of this field transformation determinesspecific forms for the functionals δT ab B [ J a ] and δP τ [ J a ; ξ b ] ,both of which are nonlinear in J a . The point of (11) isto describe how electromagnetic forces and torques ex-erted on a charge distribution J a by a field F ab can becomputed in terms of forces and torques exerted by apotentially-simpler “effective field” ˆ F ab [ F cd ] . Substitut-ing (11) into (8) suggests the definition ˆ P τ [ ξ a ] ≡ P τ [ ξ a ] + δP τ [ J a ; ξ b ] (12)for the renormalized momentum, which is seen to satisfy ddτ ˆ P τ [ ξ a ] = F G τ [ ˆ T ab B ; ξ c ] + F EM τ [ ˆ F ab , J c ; ξ d ] . (13)Here, ˆ T ab B ≡ T ab B + δT ab B and the dependence of ˆ P τ on T ab B and J a has been suppressed. It follows that if themap F ab (cid:55)→ ˆ F ab satisfies appropriate conditions—to bediscussed in more detail below—electromagnetic forcesand torques appear to be exerted not by the physicalfield F ab , but by a particular surrogate ˆ F ab . The costfor this replacement is effectively a renormalization ofthe body’s apparent stress-energy tensor, which affectsits apparent momenta and the couplings which appearin gravitational forces and torques. Both of these effectscan be physically interpreted as contributions due to thebody’s self-field. Note that although T ab B is renormalizedhere, J a is not, essentially because Maxwell’s equationsare linear.Eq. (13) has an advantage over (8) when ˆ F ab be-haves more simply inside a body than F ab . More specif-ically, there should be a wider variety of circumstancesin which a well-chosen ˆ F ab varies slowly throughout each B τ . Whenever this occurs, useful multipole expansionsmay be found for F EM τ [ ˆ F ab , J c ; ξ d ] , resulting in electro-magnetic forces and torques which are identical in formto standard test body expressions, but with all fields inthose expressions equal to ˆ F ab . Similarly, a multipoleexpansion for F G τ [ ˆ T ab B ; ξ d ] results in standard expressionsfor the gravitational forces and torques acting on an ex-tended test body, but with stress-energy multipole mo-ments which are somewhat different from those whichmight have been computed using T ab B alone. These gravi-tational effects involve only quadrupole and higher ordermoments, and vanish entirely in maximally-symmetricspacetimes. C. Effective test bodies and effective fields
Laws of motion which are structurally identical tothose satisfied by test bodies moving in an effective ficti- tious field are used today to organize essentially all known d = 4 self-force results—whether in electromagnetism,scalar field theory, or general relativity [11, 12, 18]. Suchprinciples have also been employed to understand static scalar and electromagnetic self-interaction for more gen-eral values of d [19]. Indeed, we may view the motion ofa self-interacting body moving in a certain physical fieldas “equivalent” to the motion of an “effective test body”in an appropriate effective field.It is instructive to note that a particularly simple prin-ciple of this sort holds even in Newtonian gravity [12, 25],where it provides the foundation for celestial mechanics.Using standard definitions for the linear and angular mo-menta of a Newtonian extended body with mass density ρ coupled to a gravitational potential φ , the Newtoniangeneralized force at time τ may be shown to be ddτ P N τ [ ξ a ] = F N τ [ φ, ρ ; ξ a ]= − (cid:90) B τ ρ ( x , τ ) L ξ φ ( x , τ ) d x , (14)where B τ ⊂ R denotes the body’s location at time τ , ξ a is any Euclidean Killing vector field, and L ξ again de-notes the Lie derivative with respect to ξ a . As in electro-magnetism, this generalized force is a trilinear functionalof the potential, its source, and a vector field. Now, it isstraightforward to show that if G ( x , x (cid:48) ) = G ( x (cid:48) , x ) and L ξ G ( x , x (cid:48) ) = 0 for all Killing vector fields ξ a , and if ˆ φ ( x , τ ) ≡ φ ( x , τ ) − (cid:90) B τ ρ ( x (cid:48) , τ ) G ( x , x (cid:48) ) d x (cid:48) , (15)forces and torques computed using φ must be identicalto those computed using ˆ φ : F N τ [ φ, ρ ; ξ a ] = F N τ [ ˆ φ, ρ ; ξ a ] . (16)This result is directly analogous to (11). Although thelanguage used here is unconventional, the conclusion isnot; it is standard to apply a result equivalent to (16) spe-cialized so that G ( x , x (cid:48) ) = −| x − x (cid:48) | − . In that context, G ( x , x (cid:48) ) is a Green function for the Laplace equationand ˆ φ satisfies the source-free field equation ∇ ˆ φ = 0 .Indeed, the effective field here is just the external po-tential. Given (15) and the Newtonian field equation ρ = ∇ φ/ π , the external potential may be viewed as anonlocal linear functional of the physical potential φ .The freedom to compute forces using ˆ φ instead of φ isessential for the development of useful approximations.For example, if F N τ [ ˆ φ, ρ ; ξ a ] is evaluated for a body inwhich ˆ φ does not vary too much throughout B τ , it isstraightforward to recover the usual gravitational force − m ∇ a ˆ φ known to act on a massive body in Newtonianmechanics. The superficially-similar expression − m ∇ a φ would arise if φ varied slowly as well, although this israrely the case. Indeed, the former approximation forthe force is valid much more generally than the latter.This difference is most striking in a point-particle limit,wherein − m ∇ a ˆ φ remains a valid approximation while − m ∇ a φ is not even computable. This type of improve-ment persists also at higher multipole orders, and theusual laws of motion in Newtonian celestial mechanicsare written as laws of motion associated with (possiblyextended) test bodies moving in the external potential,not the physical one; even in Newtonian gravity, the ef-fects of self-fields must be properly understood beforeobtaining useful laws of motion.Our goal here is to extend these ideas to fully-dynamical electromagnetically-interacting systems in alldimensions d ≥ . Much of the content of the generalprinciple that self-interacting bodies move like effectivetest bodies is embedded in the precise specifications forthe renormalized momenta and the effective fields forwhich the statement is true, so this perspective suggeststhat the problem of motion can be solved by finding “ap-propriate” momenta and effective fields. Doing so turnsout to be possible much more generally and simply thanthe older, more-explicit approach to the self-force prob-lem, where forces were directly computed using some spe-cific definition for the momentum, some specific approxi-mation scheme, specific boundary conditions, and so on.We now search for a nonlocal transformation F ab (cid:55)→ ˆ F ab [ F cd ] such that i) Eq. (11) holds, ii) all renormaliza-tions implicit in that equation are physically acceptable,and iii) the transformed field has an “external character”similar to that of the Newtonian external field. We do soby first using the Newtonian field transformation (15) asa model and defining the effective electromagnetic fieldvia ˆ F ab ( x ) ≡ F ab ( x ) − (cid:90) ∇ [ a G a ] a (cid:48) ( x, x (cid:48) ) J a (cid:48) ( x (cid:48) ) dV (cid:48) , (17)in terms of some as-yet unspecified two-point “propaga-tor” G aa (cid:48) ( x, x (cid:48) ) . This ansatz reduces our problem to thesearch for a propagator whose properties imply our re-quirements. We find that very different propagators arisedepending on the parity of d .For later convenience, it will be convenient to denotethe integral portion of (17) as being generated, via F S ab ≡ ∇ [ a A S b ] , by the vector potential A S a ≡ (cid:90) G aa (cid:48) ( x, x (cid:48) ) J a (cid:48) ( x (cid:48) ) dV (cid:48) . (18)The “ S ” here has historically been short for “singular”[18], as A S a is indeed singular for pointlike sources, atleast when using the Detweiler-Whiting propagator de-scribed below. Here, we are not considering point par-ticle sources, so A S a is not typically singular. It is moreappropriate to instead interpret this as a (propagator-dependent) definition for the “bound portion” of a body’sself-field. It generalizes what is sometimes referred to asthe “Coulomb portion” of the field. D. Choosing a propagator
When d = 4 , well-behaved effective fields are known[10, 12] to be generated by a certain propagator G DW aa (cid:48) ,referred to as the Detweiler-Whiting Green function [18].Setting G aa (cid:48) = G DW aa (cid:48) in (17) fixes a precise, physically-reasonable definition for the momentum and its cor-responding laws of motion—laws which admit a well-defined point particle limit, well-controlled multipole ex-pansions to all orders for the force and torque, and otherdesirable properties. Although we discuss Detweiler-Whiting Green functions more explicitly in Section II D 2below, it is convenient at this stage to characterize themimplicitly, via three of their properties:1. G DW aa (cid:48) ( x, x (cid:48) ) = 0 for all timelike-separated x , x (cid:48) ,2. G DW aa (cid:48) ( x, x (cid:48) ) = G DW a (cid:48) a ( x (cid:48) , x ) ,3. G DW aa (cid:48) ( x, x (cid:48) ) is a Green function for the Lorenz-gaugevector potential A a ( x ) .Although these are sometimes referred to as theDetweiler-Whiting axioms, they were originally found byPoisson [11]. Any propagator which satisfies them in-duces a field transformation F ab (cid:55)→ ˆ F ab that can be shown[10] to imply the identity (11). There is a precise sense inwhich they imply laws of motion derivable from (13), im-plying that self-interacting charges act like effective testcharges in the field ˆ F ab . Moreover, since G DW ab is a Greenfunction, the associated effective field is source-free ina neighborhood of W , just like the external Newtonianpotential ˆ φ .It was noted in [19] that the arguments used to estab-lish these results in four dimensions trivially generalizeto any number of dimensions, at least if a propagatorsatisfying the above axioms does indeed exist. Such apropagator does exist, at least in finite regions, for all even d ≥ . However, existence appears to fail when d isodd.We resolve this by finding an appropriate generaliza-tion of the above axioms—valid for all d ≥ , botheven and odd—and then constructing explicit propaga-tors which satisfy those axioms. Note that throughout,although we refer to certain statements as axioms, theseare to be understood merely as vehicles with which toorganize and interpret our results. They are not axiomsin the sense of being unproven assumptions. All of ourresults are derived from first principles.
1. Generalizing the axioms
Of the three axioms stated above, it is the third whichis most easily modified. To be more precise, that axiomrequires that G DW aa (cid:48) satisfy ∇ b ∇ b G DW aa (cid:48) − R ab G DW ba (cid:48) = − ω d − g aa (cid:48) δ ( x, x (cid:48) ) , (19)where R ab ( x ) denotes the Ricci tensor and g aa (cid:48) ( x, x (cid:48) ) theparallel propagator. The differential operator on the left-hand of this equation is motivated by the Maxwell equa-tion ∇ b ∇ b A a − R ab A b = − ω d − J a (20)for a Lorenz-gauge vector potential. Demanding that G DW aa (cid:48) be a Green function in this sense is useful becauseit may be shown to guarantee that under very generalconditions, ˆ F ab varies slowly inside each cross-section B τ of a body’s worldtube. It therefore ensures that the effec-tive field generated by G DW aa (cid:48) is not only associated withthe law of motion (13), but also that generalized forces inthat equation admit the well-controlled multipole expan-sions which are so essential to practical computations.Multipole expansions such as these can be maintainedby supposing that G aa (cid:48) is not necessarily a Green func-tion, but rather a more general type of parametrix [19].The right-hand side of (19) would then be replaced by − ω d − [ g aa (cid:48) δ ( x, x (cid:48) ) + S aa (cid:48) ( x, x (cid:48) )] , (21)where S aa (cid:48) ( x, x (cid:48) ) is sufficiently smooth and satisfies cer-tain other constraints required to maintain the validityof (11). Such generalizations can be useful because i)parametrices are more easily computed than Green func-tions, and ii) there may be topological obstructions toconstructing Green functions, even when d = 4 . Never-theless, allowing for a nonzero S aa (cid:48) is still not sufficientto solve the odd-dimensional self-force problem; a furthergeneralization is needed.The generalization we choose is motivated by a desireto demand only what is directly needed, namely that ˆ F ab “vary slowly” throughout each B τ . Although this state-ment is imprecise as it stands, we note that in the limitthat a body’s size becomes arbitrarily small, a continuousfield cannot vary significantly in any single cross-section.Smoothness in a point-particle limit may thus be used asa proxy for slow variation in more general contexts.We now replace the three axioms described above bydemanding the existence of a propagator G aa (cid:48) ( x, x (cid:48) ) withthe four properties:1. G aa (cid:48) ( x, x (cid:48) ) = 0 for all timelike-separated x , x (cid:48) .2. G aa (cid:48) ( x, x (cid:48) ) = G a (cid:48) a ( x (cid:48) , x ) .3. G aa (cid:48) ( x, x (cid:48) ) is constructed only from the geometryand depends only quasilocally on the metric, in asense defined below.4. For any point charge moving on a smooth timelikeworldline, the source ω − d − ∇ b ˆ F ab for the effectivefield defined by (17) is itself smooth, at least in aneighborhood of that worldline.The first two of these axioms are unchanged from thosegiven by Poisson [11]. Axiom 3 is similar to one employedin [19], while Axiom 4 is new. Axiom 3 demands moreprecisely that for any vector field ψ a , the Lie derivative L ψ G aa (cid:48) ( x, x (cid:48) ) can be written as a functional which de-pends only on the Lie derivative of the metric, and onlyin a compact region determined by x and x (cid:48) . If consid-erations are restricted to a single flat spacetime, Axiom3 may be simplified by demanding simply that G aa (cid:48) bePoincaré-invariant.Physically, Axiom 2 describes a type of reciprocity inthe self-field definition associated with G aa (cid:48) [12]. It isessential to the establishment of (11), and thus to therenormalized laws of motion encoded in (13). Axioms1 and 3 guarantee that the renormalizations inherent inthose laws of motion involve only physically-acceptabledegrees of nonlocality.As suggested above, our fourth axiom provides a sensein which the renormalized laws of motion can admitwell-behaved multipole expansions. It suggests that the F EM τ [ ˆ F ab , J c ; ξ d ] appearing in (13) is generally simplerto evaluate than its bare counterpart F EM τ [ F ab , J c ; ξ d ] .Although Axiom 4 refers only to point particles, theseshould be interpreted as “elementary currents” whoseeffects can be summed over—as is common in kinetictheory—to yield an overall field for a nonsingular ex-tended charge distribution J a . If the effective field associ-ated with each elementary current is sufficiently smooth,the short-distance behavior associated with any given J a is considerably suppressed by the appropriate convolu-tion integral. Indeed, there is no obstruction to replacingAxiom 4 by a statement which demands somewhat lessregularity. We note as well that there is a sense in whichAxiom 4 is “gauge-agnostic,” unlike the statement thatthe Detweiler-Whiting Green function must satisfy thegauge-fixed equation (19).Now, any G aa (cid:48) which satisfies our four axioms pro-vides a useful definition for the generalized momentum ˆ P τ associated with an extended body. Moreover, thelaws of motion associated with this momentum admitwell-behaved multipole expansions. Our next task is toshow that such propagators actually exist. It is easilyestablished that any Detweiler-Whiting Green function G DW aa (cid:48) satisfies our axioms, so that choice can be madewhenever such a Green function exists—i.e., when d iseven. The freedom to choose other propagators can nev-ertheless be useful even in those cases. This freedom ishowever essential when d is odd.
2. Even-dimensional propagators If d ≥ is even, the four axioms given in Sec-tion II D 1 are satisfied by a Green function G aa (cid:48) = G DW aa (cid:48) which directly generalizes the four-dimensionalDetweiler-Whiting Green function known from [11, 18].These generalizations have the more-explicit form G DW aa (cid:48) = 12 (cid:104) U aa (cid:48) δ ( d/ − ( σ ) + V aa (cid:48) Θ( σ ) (cid:105) , (22)where U aa (cid:48) and V aa (cid:48) are smooth symmetric bitensorswhich depend only quasilocally on the metric. Essen-tially the same bitensors also appear in the retarded andadvanced Green functions, although there they are to beevaluated only when their arguments are timelike or null-separated. A more direct specification for the bitensorsappearing in the Detweiler-Whiting Green function maybe found by substituting (22) into (19), which results inthe equations collected in Appendix A 1.It is easily shown that if the spacetime is Minkowski, ∇ a ∇ a σ = d . Substituting this into (A1), one finds thatthe van Vleck determinant is everywhere constant: ∆ =1 . Moreover, ∇ b ∇ b (∆ / g aa (cid:48) ) = 0 , implying that theunique nonsingular solutions to the Hadamard transportequations (A9) are U { n } aa (cid:48) = 0 for all n ≥ . Moreover,(A10) implies that V aa (cid:48) = 0 when its arguments are null-separated. Combining this with (A4), (A6), and (A8), itfollows that U aa (cid:48) = α d g aa (cid:48) , V aa (cid:48) = 0 (23)everywhere in even-dimensional Minkowski spacetimes,where the dimension-dependent constant α d is explicitly α d ≡ ( − d/ λ d √ π Γ(1 / − λ d ) (24)in terms of λ d ≡ − d/ . (25)Substitution of these results into (22) fully specifiesthe flat-spacetime, even-dimensional Detweiler-WhitingGreen functions. They can also be characterized some-what differently in this special case, in terms of the ad-vanced and retarded solutions to (19): G DW aa (cid:48) = ( G ret aa (cid:48) + G adv aa (cid:48) ) . If F ab is taken to equal the body’s retarded field F ret ab , it follows from (17) that the effective field ˆ F ab whichdetermines how bodies move coincides with the so-calledradiative field ( F ret ab − F adv ab ) .Similar relations between Detweiler-Whiting and ad-vanced and retarded Green functions do not generalize tocurved spacetimes, essentially because Huygens’ principleis violated; the “tail” V aa (cid:48) is typically nonzero. Althoughfew closed-form results for U aa (cid:48) and V aa (cid:48) are known incurved spacetimes, U aa (cid:48) = ∆ / g aa (cid:48) whenever d = 4 .The bitensor V aa (cid:48) is also known for d = 4 plane wavespacetimes [35], although it is “pure gauge” in the sensethat ∇ [ a V b ] b (cid:48) = 0 . Expressions in maximally-symmetricspacetimes with arbitrary d may also be extracted fromthe results of [36]. More generally, numerical or pertur-bative methods can be used to solve the equations inAppendix A 1.We have already alluded to our four axioms being moregeneral than the original Detweiler-Whiting axioms. Thisgenerality is associated with a lack of uniqueness, mean-ing that other propagators besides (22) are possible when d is even. For example, it is acceptable to choose anypropagator with the form G aa (cid:48) = G DW aa (cid:48) + U aa (cid:48) K (2 σ/(cid:96) )Θ( σ ) , (26) where K is some smooth function which vanishes in aneighborhood of zero and (cid:96) > is a constant lengthscale.If K is fixed, each choice for (cid:96) defines a different propa-gator, a different ˆ P τ , a different effective field ˆ F ab , and adifferent generalized force F τ . These differences do not,however, signal any kind of contradiction. Physical con-sistency is maintained by the fact that all of these quan-tities vary simultaneously, and in very particular ways.Differing forces arise, for example, because they describerates of change associated with slightly different aspectsof the same physical system. One might experimentallyassociate a particular value of a coupling parameter—such as a mass—with measurements which assume onevalue of (cid:96) , although the same experiments performed onthe same system would generically yield a different valuewhen inferred using a different choice of (cid:96) ; a particularpropagator must be fixed before even attempting to inter-pret experimental data. Nevertheless, there is a sense inwhich “true” observables do not depend on these choices.Further discussion may be found in [19].
3. Odd-dimensional propagators If d ≥ is odd, no Detweiler-Whiting Green functionappears to exist. It is thus essential to exploit the free-dom afforded by the four axioms listed above. Beforeconstructing an odd-dimensional propagator which sat-isfies those axioms, note that the retarded Lorenz-gaugeGreen function in this context has the form G ret aa (cid:48) = [( − σ ) λ d U aa (cid:48) Θ( − σ )] ret , (27)where λ d is again given by (25). The retarded Greenfunction here involves a bitensor U aa (cid:48) which may beshown to be symmetric and to depend only quasilocallyon the metric. Also note that the “ ret ” on the whole ex-pression denotes that it has support only for x (cid:48) in the pastof x . As in the even-dimensional context, U aa (cid:48) = α d g aa (cid:48) in Minkowski spacetime, although the odd-dimensionalconstants here are given by α d ≡ ( − / λ d Γ( − λ d ) √ π Γ(1 / − λ d ) (28)instead of (24). In more general spacetimes, a prescrip-tion to compute U aa (cid:48) is described in Appendix A 2.Whether in Minkowski spacetime or not, it is evi-dent from (27) that Huygens’ principle is violated when d is odd. Signals travel not only on null cones, butalso inside of them. Although Huygens’ principle issimilarly violated for Maxwell fields in curved even-dimensional spacetimes, the odd-dimensional case is dif-ferent in that G ret aa (cid:48) is unbounded even when its argu-ments are timelike-separated. Indeed, the tail here isnot even locally integrable in general. Eq. (27) is thuscloser to a schematic than a precise description for theretarded Green function. The correct distributional so-lution can more precisely be constructed by considering [( − σ ) λ U aa (cid:48) Θ( − σ )] ret for values of λ in which the sin-gularity is integrable and then analytically continuingthe result to λ → λ d . Another mathematical detail isthat [( − σ ) λ Θ( − σ )] ret should be regarded as a singlesymbol, not a product of singular distributions. Theseand other details associated with the odd-dimensional re-tarded Green functions are made precise in, e.g., [37, 38].The propagators which allow us to solve the self-forceproblem in odd numbers of dimensions can also definedusing analytic continuation. They are G odd aa (cid:48) ≡ ( − − λ d π U aa (cid:48) lim λ → λ d (cid:96) λ ∂∂λ (cid:2) (2 σ/(cid:96) ) λ Θ( σ ) (cid:3) , (29)where the overall prefactor has been chosen in order toenforce Axiom 4. The U aa (cid:48) appearing here is constructedin the same way as for the retarded and advanced Greenfunctions. Performing the differentiation in (29) whileleaving the limit λ → λ d implicit, our propagator canalternatively be written as G odd aa (cid:48) = ( − ( d − π (2 σ ) λ d U aa (cid:48) ln(2 σ/(cid:96) )Θ( σ ) . (30)In either form, these expressions fix a 1-parameter fam-ily of propagators which depend on an arbitrary length-scale (cid:96) > . This lengthscale is introduced in order toensure that the quantity differentiated with respect to λ is dimensionless in (29). Choosing different valuesfor (cid:96) would result in propagators which differ by mul-tiples of U aa (cid:48) (2 σ ) λ d Θ( σ ) , a propagator which generatessource-free solution to Maxwell’s equations. Althoughvariations in (cid:96) generically change effective fields and thusforces, such shifts have no observable consequences. Theymerely parametrize different ways to describe the samephysical system. The situation here is fully analogous tothat associated with the (cid:96) -dependence of (26) and alsowith the non-uniqueness of the static propagators dis-cussed in [19].We now verify that the propagator G odd aa (cid:48) satisfies thefour axioms described in Section II D 1. That the firstof these holds is immediately clear from the presence ofthe Θ -function in (30). The second and third axiomsare verified by noting that σ and U aa (cid:48) are symmetric intheir arguments and depend quasilocally on the metric,as elaborated in Appendix A. Considerably more effort is required to show that ourpropagator also satisfies Axiom 4. We do so using thedirect calculations summarized in Appendix D: Considera point particle with timelike worldline Γ and let F ab beidentified with that particle’s retarded field. Then the ˆ F ab generated by G odd aa (cid:48) is given by combining (D18) and(D19). In those equations, the only position dependenceis via smooth functions, at least if the metric and theworldline are themselves smooth. We thus conclude that ˆ F ab , and hence its source ω − d − ∇ b ˆ F ab , must be smooth inthe presence of retarded boundary conditions. Repeatingthe problem with more general boundary or initial con-ditions would merely change ˆ F ab by a homogeneous solu-tion to Maxwell’s equations. The source is thus smooth ingeneral, verifying Axiom 4. As claimed, all axioms givenin Section II D 1 are satisfied by the propagator (29).Although G odd aa (cid:48) is not a Green function or more gen-eral parametrix for Lorenz-gauge vector potentials, someintuition for it may nevertheless be gained by notingthat the derivative with respect to λ which appears inits definition (30) evinces a procedure which “infinitesi-mally varies d .” This suggests that our map F ab (cid:55)→ ˆ F ab may reduce to dimensional regularization in a point par-ticle limit, and may provide an underlying physical andmathematical origin for that procedure at least in thepresent context. We are not aware of any other examplesin which dimensional regularization arises as the natu-ral limit of a more-general nonsingular operation whichfollows from first principles. E. Laws of motion
To summarize our development at this point, we haveshown that for all d ≥ , two-point propagators G aa (cid:48) maybe found which satisfy the four axioms given in SectionII D 1. If d is even, one possibility is to set G aa (cid:48) = G DW aa (cid:48) ,where G DW aa (cid:48) is given by (22). If d is odd, one may insteaduse G aa (cid:48) = G odd aa (cid:48) , where G odd aa (cid:48) satisfies (29). Regard-less, any specific choice for G aa (cid:48) which satisfies the givenaxioms may be associated with a particular renormaliza-tion ˆ P τ of the bare generalized momentum defined by(6). More specifically, the methods reviewed in [12] maybe used to show that the appropriate relation betweenthese momenta is ˆ P τ = P τ + 12 (cid:18)(cid:90) B + τ dV J a L ξ (cid:90) B − τ dV (cid:48) G aa (cid:48) J a (cid:48) − (cid:90) B − τ dV J a L ξ (cid:90) B + τ dV (cid:48) G aa (cid:48) J a (cid:48) (cid:19) + (cid:90) B τ dS a J a × (cid:18)(cid:90) B τ dV (cid:48) ξ b G bb (cid:48) J a (cid:48) − (cid:90) duu − ∇ b (cid:48) σF b (cid:48) c (cid:48) S ξ c (cid:48) (cid:19) , (31)where B ± τ denotes the portion of the body’s worldtube which lies to the future ( + ) or past ( − ) of B τ , and the0primes in the u -integral are associated with points onthe same geodesic y (cid:48) ( u ) which appeared in (6). The im-portant point here is that the renormalizing terms areappropriately-local: The first of our axioms for G aa (cid:48) im-plies that the momentum at time τ can depend on T ab B , J a , and F ab only in the body’s worldtube, and only onthose portions of the worldtube which are spacelike ornull-separated from B τ . This is in strong contrast toany attempt which might be made to directly compute a“self-momentum” associated with T ab EM . Nevertheless, thetwo procedures do coincide in simple cases where nonlo-cality is not an issue; see [10] for the d = 4 discussion.Continuing our summary, fixing an appropriate G aa (cid:48) fixes a particular definition for ˆ P τ , and we have shownthat this momentum must satisfy the laws of motion (13).These laws are instantaneously identical to those whichhold for an extended test body with stress-energy tensor ˆ T ab B and current density J a , coupled to a spacetime met-ric g ab and an electromagnetic field ˆ F ab . The effectiveelectromagnetic field here depends on G aa (cid:48) and is givenmore precisely by (17). The renormalized stress-energy ˆ T ab B also depends on G aa (cid:48) , and at least in static contexts,it can be written in terms of functional derivatives of theappropriate propagator [19].Regardless, once a propagator has been fixed, the lawsof motion are fixed as well. The force on a body maybe found by computing ˆ F ab from F ab and substitutingthe result into an appropriate test body equation. Forexample, the lowest-order electromagnetic force actingon a body with charge q is given by the usual Lorentzexpression ˆ f a = q ˆ F ab ˙ γ b . (32)Similarly, the lowest-order electromagnetic torque on abody with electromagnetic dipole moment q ab = q [ ab ] is ˆ n ab = 2 q c [ a ˆ F b ] c . (33)Although these expressions might appear superficiallysimilar to test-body expressions, they encode all leading-order self-force and self-torque effects in general space-times.More generally, the full multipole expansion for theelectromagnetic generalized force can be shown to be F EM τ [ ˆ F ab , J c ; ξ d ] = q ˆ F ab ξ a ˙ γ b + ∞ (cid:88) n =1 n ( n + 1)! q b ··· b n a L ξ ˆ F ab ,b ··· b n , (34)where q b ··· b n a denotes the n -pole moment of J a and ˆ F ab,c ··· c n the n th tensor extension of F ab . Letting ˆ I c ··· c n ab denote the n -pole moment of ˆ T ab B and g ab,c ··· c n the n th tensor extension of g ab , the gravitational gener-alized force may be similarly expanded as F G τ [ ˆ T ab B ; ξ c ] = 12 ∞ (cid:88) n =2 n ! ˆ I c ··· c n ab L ξ g ab,c ··· c n . (35) Tensor extensions are discussed in more detail in [12, 33];the first nontrivial ones are g ab,cd = 23 R a ( cd ) b , F ab,c = ∇ c F ab . (36)Regardless, the gravitational expression here involvesonly quadrupole and higher moments, and the the tan-gent ˙ γ a to the reference worldline appears explicitly onlyin the Lorentz force (and not in the higher-order electro-magnetic terms or in any gravitational terms).Eqs. (34) and (35) may now be combined with (13) toyield the full laws of motion. It is however more conven-tional to split ˆ P τ into its linear and angular componentsvia a “hatted” analog of (7). Doing so, it is convenient todefine a renormalized force ˆ f a and a renormalized torque ˆ n ab = ˆ n [ ab ] using the similar implicit equation ddτ ˆ P τ [ ξ a ] = ˆ f a ξ a + 12 ˆ n ab ∇ a ξ b . (37)This definition provides forces and torques which mea-sure the degree by which the Mathisson-Papapetrouequations are violated: Ddτ ˆ p a = 12 R bcda ˆ S bc ˙ γ d + ˆ f a , (38) Ddτ ˆ S ab = 2ˆ p [ a ˙ γ b ] + ˆ n ab . (39)That the first term on the right-hand side of the sec-ond equation is not considered a torque is natural inthe sense that an analogous term exists even for theangular momentum of an isolated system in Newtonianphysics. This is so essentially because a Euclidean rota-tion about one origin can be decomposed into a rotationabout another origin plus a translation. If the originabout which the angular momentum is defined is mov-ing, it must “mix” over time with the linear momentumconjugate to the translations generated by that motion.The R bcda ˆ S bc ˙ γ d term on the right-hand side of (38) issimilarly interpreted as arising from the fact that in acurved spacetime, pure translations at one point are notnecessarily pure translations at another point. Both thisterm and the p [ a ˙ γ b ] in (39) are thus kinematic in origin,an interpretation which persists even in the absence ofany true symmetries.Now, explicit multipole expansions for our force andtorque may be derived by combining (13), (34), (35),(38), and (39) while varying over all GKFs ξ a . The resultis no different than it is when d = 4 , and is given by Eqs.(193) and (194) of [12]. The monopole truncation for theresulting force is simply (32), while the dipole truncationfor the torque is (33). Gravitational effects do not enteruntil quadrupole order. To all multipole orders, our ex-pansions for ˆ f a and ˆ n ab are structurally identical to themultipole expansions derived by Dixon for an extendedtest body [33]. All differences are implicit in our hat no-tation, which alters the definitions for the momenta, thestress-energy moments, and the electromagnetic field in1such a way that multipole expansions remain useful evenin the presence of strong self-interaction.Thus far, all of our discussion has allowed foressentially-arbitrary reference worldlines γ ( τ ) and foliat-ing hypersurfaces B τ . It is however conventional to iden-tify the worldline with some kind of mass center and thefoliation with the instantaneous rest frames associatedwith that center. The first of these demands is typicallyaccomplished by imposing a “spin supplementary condi-tion” which asks that the mass dipole moment associatedwith the body vanish in an appropriate reference frame.There are different ways to make this precise. Althoughit is not essential, here we do so by choosing γ ( τ ) suchthat ˆ S ab ˆ p b = 0 . (40)We can also fix the foliation by demanding that each B τ is constructed from the hyperplane formed by allgeodesics which pass through γ ( τ ) and are orthogonalto ˆ p a ( τ ) at that point. These conditions may now beused to relate ˆ p a to ˙ γ a ; they are not necessarily paral-lel. Differentiating (40) while using (38) and (39), themomentum-velocity relation is found to be ˆ m ˙ γ a = 1ˆ m ( I − ) ab [( − ˆ p · ˙ γ )ˆ p b − ˆ S bc ˆ f ◦ c − ˆ n bc ˆ p c ] , (41)where we have defined the renormalized mass by ˆ m ≡ − ˆ p a ˆ p a , (42)used the inverse of I ab ≡ δ ab + 1ˆ m ˆ S ac (cid:18) R bcdf ˆ S df − q ˆ F bc (cid:19) , (43)and let ˆ f ◦ a ≡ ˆ f a − q ˆ F ab ˙ γ b be the non-Lorentz portionof the force (which is relevant because the Lorentz forceis the only component which depends explicitly on ˙ γ a ).A more explicit momentum-velocity relation can be ob-tained if the matrix rank of ˆ S ab is no greater than two[12, 39], although such a condition can be guaranteedonly when d < . Eq. (41) may instead be applied when-ever I ab is invertible. Components of ˆ p a which fail tobe parallel to ˙ γ a are referred to as hidden momentum[40, 41]. Although the equations presented here are com-plicated, they differ from their test-body counterpartsonly via physically-ignorable renormalizations and thenonlocal map F ab (cid:55)→ ˆ F ab . No simpler result could reason-ably be expected, at least in the absence of a particularapproximation scheme. III. POINT PARTICLES IN FLAT SPACETIMES
One useful class of approximations may be interpretedas point particle limits. Certain limits of this type havebeen discussed in detail in [9] when d = 4 , while oth-ers, valid for all d , were considered in [19]. Regardless of details, one considers a 1-parameter family of extendedbodies whose sizes scale linearly with a control parameter δ > which is eventually taken to zero. Various physicalconstraints require that other properties of the bodies—such as their net charges—scale at rates which dependon particular powers of δ , powers which depend both on d and on the specific property being considered. Rea-sonable motivations can be found for different approxi-mations, although a general feature is that self-force ef-fects can be “more important” in lower numbers of dimen-sions; they generically compete in magnitude with test-body effects associated with lower multipole orders. Con-versely, leading-order self-force effects are strongly sup-pressed relative to leading-order test-body effects when d is large. Self-interaction should thus be understoodnot in isolation, but in combination with test-body ef-fects up to an appropriate multipole order. Nevertheless,our discussion below focuses for simplicity mainly on thecomputation of leading-order self-forces and self-torques.We now apply the results derived in Section II toperform these computations for “point particles” inMinkowski spacetimes of various dimensions. Althoughwe have in mind a point particle limit, we do not discussdetails of the associated family of extended charges. In-stead, we suppose that in this limit, the family of world-tubes associated with the extended bodies used to con-struct the point particle limit shrink to a timelike world-line Γ = { γ ( τ ) : τ ∈ R } , where the parametrization hasbeen chosen such that ˙ γ a ˙ γ a = − . We take Γ to be thereference worldline for the constructions of the previoussection, and assume that it satisfies the spin supplemen-tary condition (40). In the limit, the bodies’ net chargestypically tend to zero along with their diameters; a bodywith too much charge for its size and mass cannot holditself together without exerting stresses which violate en-ergy conditions. Regardless, it is convenient to considera point particle limit in which the current densities as-sociated with members of the given family of extendedcharge distributions approach an appropriate function of δ multiplied by the point-particle current density J a pp ( x ) = q (cid:90) ˙ γ a ( τ ) δ ( x, γ ( τ )) dτ. (44)The q appearing here is a fixed parameter which rep-resents a δ -dependent rescaling of the charges associatedwith different members of the family in the limit δ → + .Despite this, we refer to it as “the” charge for simplic-ity. Leading-order self-forces and self-torques may nowbe computed by evaluating the effective field ˆ F ab associ-ated with J a pp and then inserting the result into (32) and(33). No regularization is required. A. Even dimensions
In even-dimensional Minkowski spacetimes, the pre-scription described in Section II implies that it is use-ful to define a body’s renormalized momentum using the2propagator G aa (cid:48) = G DW aa (cid:48) , where G DW aa (cid:48) is given by (22).The S -field generated by this propagator and associatedwith a current of the form (44) is found by substituting(23) into (C2), which yields A S a = qα d (cid:88) τ ∈{ τ ± } | ˙ σ | (cid:18) − ∂∂τ σ (cid:19) d/ − g aa (cid:48) ˙ γ a (cid:48) , (45)where the advanced and retarded times τ ± ( x ) are definedin Appendix B and α d depends on the dimension via (24).In the special case where the physical field F ab coincideswith the particle’s retarded field, a vector potential forthe effective field ˆ F ab can be written as in (C4). Special-izing that equation to flat spacetime, ˆ A a = qα d | ˙ σ | (cid:18) − ∂∂τ σ (cid:19) d/ − g aa (cid:48) ˙ γ a (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ = τ − τ = τ + . (46)Leading-order self-forces and self-torques may now becomputed by evaluating ˆ F ab = 2 ∇ [ a ˆ A b ] on the particle’sworldline and substituting the result into (32) and (33).If d = 4 , this procedure is reasonably straightforwardusing the expansion techniques and limits collected inAppendix B; carrying out the relevant calculations re-sults in ˆ F ab = 43 q ˙ γ [ a ... γ b ] (47)on the particle’s worldline. It follows that the leading-order self-force with retarded boundary conditions infour-dimensional Minkowski spacetime is ˆ f a = 23 q h ab ... γ b , (48)where h ab ≡ g ab + ˙ γ a ˙ γ b denotes a projection opera-tor associated with the particle’s rest frame. This maybe recognized as the standard Abraham-Lorentz-Diracradiation-reaction force; see, e.g., [4, 11]. The leading-order four-dimensional self-torque follows immediately aswell: ˆ n ab = 43 qq c [ a ( ˙ γ b ] ... γ c − ... γ b ] ˙ γ c ) . (49)Although the fields for our particle have been obtainedwithout a dipole moment q ab , including one would stillresult in this self-torque at leading order. Also note that ˆ n ab ˙ γ b need not vanish in (49). Such components may beseen from (41) to induce a hidden momentum in whichthe direction of ˆ p a differs from that of ˙ γ a .Deriving analogous results in higher numbers of evendimensions is straightforward but tedious. For d = 6 , we find by expanding (46) that the effective field on aparticle’s worldline is ˆ F ab = 29 q (cid:18) γ (5)[ a ˙ γ b ] + γ (4)[ a ¨ γ b ] − | ¨ γ | ... γ [ a ˙ γ b ] − d | ¨ γ | dτ ¨ γ [ a ˙ γ b ] (cid:19) , (50)implying that the leading-order flat-spacetime self-forcewith retarded boundary conditions is ˆ f a = − q h ab (cid:18) γ (5) b − | ¨ γ | ... γ b − d | ¨ γ | dτ ¨ γ b (cid:19) . (51)This force agrees with expressions which have been ob-tained elsewhere using different methods [23, 42–45]. Ourapproach trivially allows a self-torque to be obtained aswell, by substituting (50) into (33), although we omit thisfor brevity.Continuing, the d = 8 effective field with retardedboundary conditions may be computed by again expand-ing (46) using the techniques of Appendix B. We omit thefull result, noting only that the leading-order self-force is ˆ f a = 2 q h ab (cid:20) γ (7) b − | ¨ γ | γ (5) b − d | ¨ γ | dτ γ (4) b + 79 (cid:18) | ¨ γ | + 7 | ... γ | − d | ¨ γ | dτ (cid:19) ... γ b + 76 ddτ (cid:18) | ¨ γ | + 7 | ... γ | − d | ¨ γ | dτ (cid:19) ¨ γ b (cid:21) . (52)Taking into account differing sign conventions and a ty-pographical error in which ˙ u ¨ u should really be ¨ u ¨ u , thisagrees with an expression found in [45].Although our flat-spacetime self-forces agree with ex-isting expressions in Minkowski spacetimes with evennumbers of dimensions, our odd-dimensional predictionsdo not. B. Odd dimensions
Self-forces and self-torques acting on point charges inodd-dimensional Minkowski spacetimes may now be ob-tained by fixing the definition for the renormalized mo-mentum by identifying the propagator G aa (cid:48) with the G odd aa (cid:48) given by (30). The constant lengthscale (cid:96) whichappears in the definition for G odd aa (cid:48) is assumed to havebeen fixed as well, although its precise value is irrele-vant. With these choices, it is shown in Appendix Dthat the S -field near the worldline of a point charge withcurrent density (44) may be expanded in powers of theradar distance r associated with Γ :3 A S a = Γ( d − π / Γ( d − ) (cid:40) ∞ (cid:88) n = ( d − Γ( n + )Γ(2 − d )(2 n )!Γ( n + (5 − d )) (cid:104) (cid:16) H n − ( d − − H − d − r/(cid:96) ) (cid:17) W { n } a − ∂ λ W { n } a (cid:105) × r n − ( d − + ( − ( d − ( d − (cid:88) n =0 ( − n Γ( n + )Γ(2 − d )Γ( ( d − − n )(2 n )! W { n } a r ( d − − n (cid:41) . (53)Here, the sum in the second line is understood to existonly for d ≥ , r ( x ) is defined more precisely by (B3), H µ denotes the µ th harmonic number, and the coeffi-cients W { n } a ( x ; λ ) are defined by (D7) in terms of theflat-spacetime specialization W a ( x, τ ; λ ) = qg aa (cid:48) ( x, γ ( τ )) ˙ γ a (cid:48) ( τ )Σ λ ( x, τ ) (54)of (D5) and the “factorized world function” Σ( x, τ ) de-fined by (B1). All implicit instances of λ in (53) are tobe evaluated at λ = λ d = 1 − d/ . Some results for thefirst few W { n } a and their exterior derivatives on the par-ticle’s worldline are collected in (B15) and (B16). Alsonote that although we are focusing here on flat space-times, the derivation in Appendix D 2 shows that (53) isactually valid in all odd-dimensional spacetimes, as long as (54) is replaced by the more-general (D5).Regardless, (53) is the odd-dimensional analog of (45).It generically involves non-negative even powers of r ,non-negative even powers multiplied by ln r , and nega-tive even powers down to r − ( d − . The self-force andself-torque may be evaluated by subtracting this from aphysical vector potential and then using (17) to compute ˆ F ab . The result is automatically finite, at least in theabsence of impulsive incoming waves or other singularphenomena external to the body itself.As in the even-dimensional context, it is interesting tosuppose that the true electromagnetic field F ab is equal tothe retarded field F ret ab . Assuming this, the relevant sub-traction with F S ab is performed in Appendix D 3, whichculminates in the effective field (D21). That result isvalid for general odd-dimensional spacetimes. Specializ-ing it to the flat case by introducing Minkowski coordi-nates x µ while using (27) and (28), we find that ˆ F µν = 2( − ( d − Γ( d/ − √ π Γ( ( d − (cid:34) ( d − q (cid:90) τ − (cid:15) −∞ X [ µ ( τ, τ (cid:48) ) ˙ γ ν ] ( τ (cid:48) )[ − X ( τ, τ (cid:48) )] d/ dτ (cid:48) − d − (cid:88) n =0 ( − n n ! (cid:18) ∇ [ µ W { n } ν ] d − − n + 1 (cid:15) ˙ γ [ µ W { n } ν ] (cid:19) × (cid:15) d − − n − d − (cid:18) (cid:15) ˙ γ [ µ W { d − } ν ] − ∇ [ µ W { d − } ν ] ln( (cid:15)/(cid:96) ) − ∂ λ ∇ [ µ W { d − } ν ] − d −
2) ˙ γ [ µ W { d − } ν ] (cid:19) (cid:35) (55)on Γ , where we have omitted an implicit limit (cid:15) → + and defined X µ ( τ, τ (cid:48) ) ≡ γ µ ( τ ) − γ µ ( τ (cid:48) ) . Although individ-ual terms here involve negative powers of (cid:15) and also ln (cid:15) ,these cancel similarly-divergent terms in the integral; theoverall limit here is well-behaved. Also nmote that eventhough G odd aa (cid:48) is not a Green function and the effectivefield here is not in general a solution to the source-freeMaxwell equations, it is source-free for inertially-movingparticles. Indeed, it vanishes in those cases.Two qualitative differences may now be observed be-tween our flat-spacetime effective fields in even andodd numbers of dimensions. First, the odd-dimensional ˆ F ab depends on the particle’s past history. Its even-dimensional counterpart does not. Second, our odd-dimensional field depends on the arbitrary parameter (cid:96) > which appears in the definition for G odd aa (cid:48) . Vary-ing (cid:96) results in different propagators, different definitions for a body’s momentum, and different forces. In prac-tice, one can choose a convenient value and then use itto infer masses and other parameters from available ex-perimental data. Although those inferences would differsomewhat with different choices for (cid:96) , they would do soin predictable ways which could be computed from theexpressions found in Section II.Having now noted that the even and odd-dimensionaleffective fields discussed here differ both in their historydependence and their parameter dependence, we empha-size that neither of these differences are essential. Pa-rameter dependence can appear for even d if, e.g., oneconstructs momenta using a family of propagators withthe form (26). Furthermore, history dependence generi-cally occurs in even-dimensional effective fields wheneverthe spacetime is curved. Indeed, it arises even in flateven-dimensional spacetimes if a body is coupled to a4massive field (as opposed to the massless Maxwell cou-plings considered here).
1. Special cases
In the absence of closed-form expressions for the coef-ficients W { n } µ and ∇ [ µ W { n } ν ] which appear in (55), it isnot possible to provide fully-explicit formulae for all odd-dimensional self-forces. However, those coefficients canbe computed, for each n , using the methods of AppendixB. Explicit self-forces may thus be obtained for any spe-cific odd d . We now discuss three and five-dimensionalMinkowski spacetimes as special cases.Assuming retarded boundary conditions, substitutionof (B15) and (B16) into (55) results in the d = 3 effectivefield ˆ F µν = 2 q (cid:34)(cid:90) τ − (cid:15) −∞ (cid:32) X [ µ ˙ γ (cid:48) ν ] ( − X ) / (cid:33) dτ (cid:48) + 12 ¨ γ [ µ ˙ γ ν ] ln( (cid:15)/e(cid:96) ) (cid:35) (56)on Γ , where e denotes the base of the natural logarithmand the limit (cid:15) → + has again been left implicit. Com-bining this with (32) immediately yields the leading-orderthree-dimensional self-force ˆ f µ = 2 q (cid:34)(cid:90) τ − (cid:15) −∞ (cid:32) X [ µ ˙ γ (cid:48) ν ] ˙ γ ν ( − X ) / (cid:33) dτ (cid:48) −
14 ln( (cid:15)/e(cid:96) )¨ γ µ (cid:35) . (57)Similarly, substituting (56) into (33) yields the leading-order three-dimensional self-torque ˆ n µν = 2 qq ρ [ µ (cid:34) (cid:90) τ − (cid:15) −∞ (cid:32) X ν ] ˙ γ (cid:48) ρ − ˙ γ (cid:48) ν ] X ρ ( − X ) / (cid:33) dτ (cid:48) + 12 (¨ γ ν ] ˙ γ ρ − ˙ γ ν ] ¨ γ ρ ) ln( (cid:15)/e(cid:96) ) (cid:35) , (58)which depends both on a particle’s charge q and on itselectromagnetic dipole moment q µν . It is clear in thiscontext that varying (cid:96) changes the force only by con-stant multiples of ¨ γ µ . Different values for (cid:96) thus providedifferent renormalizations of a particle’s apparent mass,at least to leading nontrivial order.Additional insight into our d = 3 forces and torquesmay be gained by evaluating them in a slow-motion ap-proximation. Applying such an approximation while in-tegrating (57) once by parts shows that the spatial 2-vector components of the self-force are explicitly ˆ f ( τ ) = − q (cid:20) (cid:90) τ − (cid:15) −∞ dτ (cid:48) (cid:18) ¨ γ ( τ (cid:48) ) τ − τ (cid:48) (cid:19) + ¨ γ ( τ ) ln (cid:18) (cid:15)e (cid:96) (cid:19) (cid:21) , (59)where it has been assumed that the acceleration falls offaccording to lim τ (cid:48) →−∞ ( τ − τ (cid:48) ) ˙ γ ( τ (cid:48) ) − [ γ ( τ ) − γ ( τ (cid:48) )]( τ − τ (cid:48) ) = 0 (60) in the distant past. If this falloff condition is indeed sat-isfied, the three-dimensional self-force thus depends ona weighted history of the charge’s past acceleration. Be-yond noting that the relevant weighting factor decays like /τ , the (cid:15) → + limit makes it difficult to interpret (59)directly. A manifestly-finite form for the self-force can beobtained by integrating by parts once more. Assumingthat lim τ (cid:48) →−∞ ¨ γ ( τ (cid:48) ) ln( τ − τ (cid:48) ) = 0 , (61)the d = 3 self-force may be seen to reduce to ˆ f ( τ ) = − q (cid:90) τ −∞ ... γ ( τ (cid:48) ) ln (cid:18) τ − τ (cid:48) e (cid:96) (cid:19) dτ (cid:48) . (62)This depends on a past history of the particle’s jerk, witha weighting factor which increases logarithmically in theincreasingly-distant past.Similar expressions may be obtained for the slow-motion limit of the d = 3 self-torque (58). If the falloffconditions (60) and (61) are assumed to hold and theelectric and magnetic components of the particle’s dipolemoment may be considered comparable, the time-spacecomponents of the self-torque reduce to ˆ n i ( τ ) = q ij ( τ ) ˆ f j ( τ ) /q, (63)where i, j ∈ { , } , the ˆ f j appearing here is given by (62),and we have assumed that q (cid:54) = 0 . Differences between ˆ p and ˆ m ˙ γ are thus controlled, in part, by the coupling of aparticle’s magnetic dipole moment to a logarithmically-weighted history of its jerk.The remaining space-space components of the nonrel-ativistic d = 3 self-torque, which directly affect a body’sspin evolution, are determined by ˆ n ij ( τ ) = 2 q i ( τ ) ˆ f j ] ( τ ) /q. (64)The spin, which has only one component in this case, isthus affected by misalignments between a body’s electricdipole moment and the same logarithmically-weightedhistory of its jerk.Analogous expressions are more complicated when d =5 . We give only the leading-order self-force, which isagain found by substituting (B15) and (B16) into (55),and then substituting the effective field which results into(32). The fully-relativistic force is thus ˆ f µ = − q (cid:34) (cid:90) τ − (cid:15) −∞ X [ µ ˙ γ (cid:48) ν ] ˙ γ ν ( − X ) / dτ (cid:48) + h µν (cid:18) γ ν (cid:15) − ... γ ν (cid:15) −
316 ( γ (4) ν − | ¨ γ | ¨ γ ν ) ln( (cid:15)/e (cid:96) ) − | ¨ γ | ¨ γ ν (cid:19)(cid:35) . (65)Changing (cid:96) is this context may be seen to shift more thanjust the apparent mass; noting that h µν (cid:18) γ (4) ν − | ¨ γ | ¨ γ ν (cid:19) = ddτ (cid:18) ... γ µ − | ¨ γ | ˙ γ µ (cid:19) , (66)5it affects both the direction and magnitude of the renor-malized 5-momentum.We note also that in a slow-motion limit, the spatialcomponents of (65) reduce to ˆ f ( τ ) = 3 q (cid:90) τ −∞ γ (5) ( τ (cid:48) ) ln (cid:18) τ − τ (cid:48) e − (cid:96) (cid:19) dτ (cid:48) , (67)at least if derivatives of the particle’s position fall offsufficiently rapidly in the distant past. This differs fromits d = 3 counterpart (62) mainly by an overall sign andby the replacement of ... γ with γ (5) in the integral.
2. Comparisons
We close this section by comparing with other odd-dimensional self-force results which have appeared in theliterature. First, a five-dimensional force similar to (67)has recently been obtained using the methods of effectivefield theory [46]. In that context, (cid:96) appears as a free pa-rameter in a dimensional regularization procedure. Thisis not so different from our usage of (cid:96) as a free parameterin the choice of propagator used to define a body’s mo-mentum: Our propagator induces an (cid:96) -dependent map F ab (cid:55)→ ˆ F ab , and this turns into an (cid:96) -dependent regu-larization in the point particle limit. Nevertheless, wenote that our results differ conceptually in that we haveprovided precise “microscopic” (or “UV-complete”) defini-tions for the mass, mass center, other quantities appear-ing in the laws of motion; we do not merely assert thatquantities satisfying such laws exist and that they havephysical interpretations consistent with their names.Other odd-dimensional self-forces which have appearedin the literature differ much more significantly from ours.These have been obtained by the use of heuristic ar-guments to directly regularize point-particle self-fields[43, 47], expressions for the momenta associated withthose fields [48], or similar quantities. In at least onecase, the claimed force law is IR-divergent; see Eq. (4.4)in [47]. Other proposals use counterterms which de-pend on a particle’s entire past history [43, 48], implyingthat a body’s momentum could not be computed withoutknowledge of that history—a physically-unacceptable op-tion. Another result predicts a time-varying mass even atleading order [48]. While mass variations are normal andexpected when including effects due to a body’s highermultipole moments [32], they should not arise when con-sidering only monopole interactions with an electromag-netic field. Indeed, it is clear from (32), (38), and (42)that mass variations do not arise in our leading-orderexpressions. IV. PHENOMENOLOGY OF THEODD-DIMENSIONAL SELF-FORCE
Although the results of Section III B may be used toevaluate odd-dimensional point-particle self-forces and self-torques, the physical implications of those resultsare not immediately apparent. We now elucidate someof those implications, with a particular emphasis onnon-relativistic systems in flat, three-dimensional space-times. This setting i) possesses features which are par-ticularly distinct from the d = 4 case, and ii) may findexperimentally-accessible analogs in certain condensed-matter or fluid systems. Nevertheless no attempt is madehere to provide a comprehensive discussion of d = 3 self-force effects. Rather, we seek mainly to highlight someof the subtleties and unusual features of these effects. A. Approximations
We begin our discussion of odd-dimensional self-forcephenomenology by remarking on some of the relevant ap-proximations. Although we have already noted that theresults of the previous section assume a type of point par-ticle limit, the details of that limit were not fixed. Indeed,a number of different point particle limits can be consis-tently discussed, and without a specific physical systemin mind, it is difficult to settle on a particular approxi-mation. Despite this, one generic constraint which canbe used is that physically-realisable bodies cannot existwith arbitrary combinations of physical size, charge, andmass. Energy conditions may be violated if the stressesrequired to counteract a body’s internal electrostatic re-pulsion become larger than its mass density. Letting L characterize a charge’s linear dimension, those stressesmight be estimated to be order ( q/L d − ) . Noting thatthe mass density is approximately m/L d − , energy con-ditions thus demand that q (cid:46) mL d − , (68)where we have used the bare mass m associated with thebare momentum P τ , which is defined by (6). The renor-malized mass ˆ m is however derived from ˆ P τ , which isdistinguished from P τ via (31). The bare and renormal-ized masses can differ from one another by terms of order q /L d − and q ln( L/(cid:96) ) /L d − . If (cid:96) is held fixed, saturat-ing the bound in (68) might then result in an “imaginary ˆ m ,” i.e., a spacelike ˆ p a . Other pathologies could arise aswell. Our formalism breaks down in such cases, whichwe avoid by additionally requiring that q (cid:46) mL d − | ln( L/(cid:96) ) | . (69)This is sufficient to imply that m and ˆ m have similarmagnitudes.If a charge moves in an externally-imposed electricfield, the self and external forces acting on it may nowbe estimated to scale like f self ∼ ( q / ˆ m ) f ext τ d − ∗ (cid:46) ( L/τ ∗ ) d − | ln( L/(cid:96) ) | f ext , (70)6where τ ∗ is a characteristic timescale associated with theexternal field. If L is sufficiently small and τ ∗ is indepen-dent of L , self-forces thus remain at least logarithmically-suppressed in comparison with external forces, even forobjects which are “maximally charged” according to (69).We note however, that this statement is not precise.What meaning it does have is global, in the sense thatnontrivial tails imply that self-forces can be instanta-neously significant even when external forces vanish.It would be interesting to now write down and sys-tematically analyze the consequences of a complete, self-consistent approximation scheme which saturates thegiven bounds. We do not do so, however. Instead,we consider a simpler model problem in which only themass and charge monopoles are significant. In this case,the momentum-velocity relation (41) reduces simply to ˆ p a = ˆ m ˙ γ a and the force is given entirely by the Lorentzterm (32). With these assumptions, (62) implies that thenon-relativistic d = 3 equation of motion is given by theintegral equation ˆ m ¨ γ ( τ ) = q E ext ( γ ( τ )) − q (cid:90) τ −∞ ... γ ( τ (cid:48) ) ln (cid:18) τ − τ (cid:48) e (cid:96) (cid:19) dτ (cid:48) , (71)where E ext denotes the external electric field. Similarly,(67) implies that with the same assumptions, the d = 5 equation of motion is ˆ m ¨ γ ( τ ) = q E ext ( γ ( τ ))+ 3 q (cid:90) τ −∞ γ (5) ( τ (cid:48) ) ln (cid:18) τ − τ (cid:48) e − (cid:96) (cid:19) dτ (cid:48) . (72)More systematic approximations would also include var-ious test body effects involving the spin and higher-orderelectromagnetic multipole moments. B. Runaway solutions
The simplest applications for the equations of motion(71) and (72) concern the behavior of free particles. Un-accelerated trajectories are of course valid solutions when E ext = 0 , although they are not the only solutions. Thespace of possible initial data for these integral equationsis infinite dimensional, and nontrivial choices for this datagenerically lead to nontrivial trajectories. Physically, thisis as expected. However, there also exist solutions whichare not physically reasonable. These “runaway solutions”accelerate exponentially and without bound: Letting a denote an arbitrary constant vector, suppose that ¨ γ ( τ ) = a exp( τ /τ run ) . (73)If d = 3 , substitution of this expression into (71) showsthat it is a solution when τ run = (cid:96) exp( γ E + 1 / − m/q ) , (74) where γ E denotes the Euler-Mascheroni constant. Notethat although τ run may appear to depend on thearbitrarily-chosen lengthscale (cid:96) , the implicit dependenceof ˆ m on ln (cid:96) ensures that it does not.More importantly, the existence of runaway solutionssuggests that a particle upon which no force has been ap-plied might spontaneously and violently accelerate with-out any apparent cause. One may hope that the runawaysolutions are artifacts of the initial data (or lack thereof),in that solutions for which ¨ γ ( τ ) = 0 for all τ < τ mightbehave more sensibly. Unfortunately, this is not so. Thethree-dimensional equation of motion (71) may be solvedusing Laplace transforms, and doing so shows that withtrivial initial data, almost any applied force excites a run-away mode with growth timescale τ run .The situation is somewhat better when d = 5 . Substi-tuting the ansatz (73) into (72), the runaway timescalemay be seen to satisfy ˆ mq = 164 τ {
17 + 12[ln( τ run /(cid:96) ) − γ E ] } . (75)However, the right-hand side here has a maximum whenvarying over all τ run > , implying that runaway solu-tions can exist (with the given form) only when q / ˆ m ≥ (cid:96) exp(2 γ E − / . (76)If this bound holds but is not saturated, there are in facttwo solutions to (75), and thus two runaway timescales. Ifthe bound is violated, solutions to our equation of motionappear not to be unstable in five dimensions.Although we are not aware of runaway solutions havingpreviously been discussed in odd-dimensional spacetimes,they are well-known features of the d = 4 Abraham-Lorentz-Dirac equation. One objection to them (be-sides their manifest disagreement with observation) isthat their associated timescale is extremely short—of or-der q / ˆ m when d = 4 . However, (68) implies that awell-defined four-dimensional point particle limit requiresthat q / ˆ m be of order L or smaller. Additionally, stan-dard derivations assume that all dynamical timescalesare much longer than L . Runaway solutions in four di-mensions are thus solutions to an equation whose prop-erties violate the conditions under which that equationhas been derived. In this sense, they are not genuinepredictions.Similar conclusions may be reached also when d = 3 or d = 5 ; the runaway solutions discussed above cannotbe considered genuine predictions of the theory. This ismost easily seen in the five-dimensional case, for which(69) implies that q / ˆ m → + in a point particle limit.The bound (76) is thus violated for sufficiently-small bod-ies, which means that runaway solutions do not exist inthe relevant portion of parameter space. If d = 3 , run-away solutions do exist formally, although they violatethe conditions under which the equation of motion maybe expected to hold. A body which is maximally chargedaccording to (69) has a runaway timescale (74) which is7short compared to its light-crossing time L , and parti-cles with less charge have runaway timescales which areeven shorter. However, our derivation breaks down fortimescales of order L ; runaway solutions are thus unphys-ical also in three dimensions. C. Reduction of order
Although runaway solutions are not true predictions ofour equations, it would be desirable to be able to system-atically extract solutions which are physically and math-ematically justified—well-behaved trajectories which aresufficiently close to satisfying, e.g., (71) and for which allsignificant timescales are much larger than L . By analogywith the d = 4 case, we accomplish by “reducing order,”which corresponds to supposing that the external forcealone generates a “zeroth order” trajectory determined by ¨ γ ≈ q E ext / ˆ m , and that it is this trajectory which shouldbe substituted into the self-force integrals. If d = 3 , sucha procedure results in ˆ m ¨ γ ( τ ) = q E ext ( γ ( τ )) − q m (cid:90) τ −∞ ˙ E ext ( γ ( τ (cid:48) )) ln (cid:18) τ − τ (cid:48) e (cid:96) (cid:19) dτ (cid:48) . (77)When d = 5 , one finds instead that ˆ m ¨ γ ( τ ) = q E ext ( γ ( τ ))+ 3 q
16 ˆ m (cid:90) τ −∞ ... E ext ( γ ( τ (cid:48) )) ln (cid:18) τ − τ (cid:48) e − (cid:96) (cid:19) dτ (cid:48) . (78)These replacements do not change the order of the ap-proximation as long as q / ˆ m is sufficiently small. Moreto the point, they mollify the high-frequency characterof the Fourier transforms associated with the unmodi-fied accelerations (as is made more clear in Section IV Dbelow). Regardless of justification, these equations nolonger admit runaways and there is a sense in which theirsolutions nearly satisfy their parent equations as long asthe self-force is sufficiently small. However, as we shallsee below, the reduced-order equations can still be prob-lematic when applied over very long times.To briefly remark on our terminology, the reduction-of-order procedure applied to the d = 4 Abraham-Lorentz-Dirac equation yields what is sometimes referred to asthe Landau-Lifshitz equation. In that case, it has themathematical effect of reducing the order of the rele-vant differential equation from three to two. Here, thereduced-order terminology is retained even though we arenot changing the order of a differential equation.We also note that the reduction-of-order procedure isnot as ad hoc as it might appear. It arises naturally whenconstructing more careful point particle limits; see [9] atleast for the d = 4 case. D. Exact and approximate solutions withoutrunaways
We next discuss how physically-acceptable exact andapproximate solutions—i.e., solutions which do not runaway—of the integro-differential equation of motion (71)can be obtained when d = 3 , how the reduced-order equa-tion (77) arises in a certain limit, and how reduction oforder breaks down over very long timescales.First note that our original equation (71), which as-sumes that the acceleration vanishes in the distant past,can be recast as q E ext ( γ ( τ )) = (cid:90) τ −∞ ... γ ( τ (cid:48) ) ln (cid:32) τ − τ (cid:48) (cid:96) exp( − mq ) (cid:33) dτ (cid:48) . (79)This may be viewed as a linear integral equation forthe particle’s jerk ... γ in terms of the prescribed exter-nal force q E ext . In particular, it is a Volterra equationof the first kind. Such equations are often solved usingLaplace transforms. If the initial data is trivial, solutionsobtained in this way generically display the runaway be-havior mentioned above. However, there does exist non-trivial initial data for which no such problems arise. Thisdata is selected automatically by using Fourier trans-forms instead of Laplace transforms, as the former cannotbe used to represent exponentially-growing solutions. In-deed, we view the solution obtained by Fourier transformto be “the” physical one in a wide range of scenarios.It is first convenient to define the body’s accelerationas it would be in the absence of self-interaction: a ext ≡ q ˆ m E ext . (80)Also defining the dimensionless time variable s ≡ ( τ /τ run ) e γ E (81)and its primed equivalent in terms of the runaway time(74) and the Euler-Mascheroni constant γ E , the body’strue acceleration a = ¨ γ is found from Eq. (79) to satisfy a ext ( s ) = q m (cid:90) s −∞ d a ds (cid:48) ( s (cid:48) ) ln( s − s (cid:48) ) ds (cid:48) . (82)Since this equation is linear, a general solution can bewritten as a ( s ) = 2 ˆ mq (cid:90) ∞−∞ K ( s − s (cid:48) ) a ext ( s (cid:48) ) ds (cid:48) (83)for some kernel K , where the factor m/q has been in-cluded for later convenience.To solve for K , we now assume that the Fourier trans-form of the solution exists. As mentioned above, this as-sumption excludes runaway solutions, and so yields onlya certain class of solutions of the original equation. Defin-ing the Fourier transform of the kernel by ˜ K ( ω ) = 1 √ π (cid:90) ds e iωs K ( s ) , (84)8and substituting into (82) and (83), we find that ˜ K ( ω ) = i π ω ˜ G ( ω ) , (85)where G ( s ) ≡ Θ( s ) ln( s ) . Evaluating the Fourier trans-form of G ( s ) now yields ˜ K ( ω ) = − √ π + ( ωe γ E ) − iπ/ , (86)where ln + ( ω ) is the function obtained by analyticallycontinuing ln( ω ) from the positive real axis into the up-per half ω plane. In particular, for real ω , we have ln + ( ω ) = ln | ω | + iπ Θ( − ω ) . (87)One consequence is that (cid:90) ∞−∞ K ( s ) ds = √ π ˜ K (0) = 0 . (88)In combination with (83), it follows that with appropriatefalloff conditions on a ext , ∆ v ≡ (cid:90) ∞−∞ a ( τ ) dτ = 0 . (89)Initially-stationary particles thus return to rest at latetimes, an effect which is discussed further in SectionIV E 4 below.A particle’s motion at finite times can be understoodby obtaining an expression for the kernel in the time do-main, which of course follows from the inverse Fouriertransform of (86): K ( s ) = − π (cid:90) e − iωs dω ln + ( ωe γ E ) − iπ/ . (90)We note that the Fourier transform ˜ G ( ω ) ∝ / ˜ K ( ω ) isanalytic in the upper half ω plane, which reflects thecausal nature of G ( s ) : G ( s ) = 0 , s < . (91)By contrast, taking the reciprocal of ˜ G ( ω ) to find ˜ K ( ω ) results in a simple pole at ω = ie − γ E , (92)indicating that the kernel K ( s ) does not vanish for s < .The motion given by the solution (83) thus exhibitsa degree of “preacceleration,” just as for solutions ofthe Abraham-Lorentz-Dirac equation in four dimensions.Preacceleration arises in both of these cases when one im-poses that the solution does not diverge at late times.Although the three and four-dimensional equations ofmotion are mathematically quite different, such an im-position necessarily requires knowledge of the future—violating causality. We now show that this violation is confined to very small timescales which are effectivelynegligible.For s < , the inverse Fourier transform (90) can beevaluated by completing the contour into a semicircle inthe upper half plane and evaluating the residue at thepole (92), yielding K ( s ) = e − γ E exp( − e − γ E | s | ) , s < . (93)Although the kernel is acausal, its acausality is thus lim-ited to a specific timescale over which s varies of or-der e γ E . Recalling (81), this corresponds to a physicaltimescale equal to the runaway time τ run , given by (74).As argued in Section IV B, this timescale is short com-pared to the body’s size L ; it is negligible.If s > , one can instead complete the contour in (90)into a semicircle in the lower half ω plane, with a detouraround branch cut at Arg( ω ) = − π/ . This yields an ex-pression for the kernel in the form of a Laplace transform K ( s ) = − (cid:90) ∞ e − sσ dσ ln( σe γ E ) + π , s > . (94)While we have been unable to find an explicit analyticexpression for K ( s ) for s positive, it follows that an upperbound is | K ( s ) | ≤ π s (95)for all s > . This indicates that the memory of anexternal force on a body’s acceleration decays at least asfast as /τ .To summarize up to this point, we have found, forgeneric external fields, exact, physically-acceptable solu-tions to the d = 3 equation of motion (71). The accelera-tions corresponding to these solutions are given by (83),where a ext is defined by (80), s is defined by (81), andwhere K ( s ) satisfies (93) and (94).The Laplace transform expression (94) for the kernel K ( s ) for s > is not very transparent. We now developa useful approximation to this kernel. We have in mindtwo small quantities. First, the limiting process discussedin Section IV A above requires that q (cid:28) ˆ m . Second, wedefine τ ∗ to be a timescale over which the external electricfield varies, and define the dimensionless quantity ν by ν ≡ τ run /τ ∗ . (96)We assume ν to be small and throw away terms that aresuppressed by one or more powers of it.An approximate expression for the kernel (94) at large s can now be obtained as follows: Changing the variableof integration from σ to u = sσ we first obtain K ( s ) = − s (cid:90) ∞ e − u du [ln( ue γ E ) − (ln s ) ] + π . (97)9Expanding the integrand here at large ln s gives K ( s ) = − s (ln s ) (cid:90) ∞ due − u (cid:34) ue γ E )ln s + O (cid:32)(cid:18) ln u ln s (cid:19) (cid:33) (cid:35) , (98)which is an approximation that breaks down both at large u and at small u . At large u , the errors in the inte-grand become of order unity when u (cid:38) s , but because ofthe exponential suppression factor in the integrand, theoverall fractional corrections to the integral scale as e − s ,which we neglect. At small u , the errors in the integrandare of order unity or larger for u (cid:46) /s , and the cor-responding overall fractional corrections to the integralscale as the size of this region compared with the value u = u peak ∼ / ln s at which the integrand in (97) takesits maximum value; they are of order /su peak ∼ ln ss . Terms with this relative magnitude are also neglectedhere. Evaluating the integral (98) thus gives K ( s ) = − s (ln s ) (cid:20) O (cid:18) s ) (cid:19)(cid:21) (99)for s > .We now argue that the asymptotic form (99) of thekernel is sufficient for deriving a useful explicit approxi-mation for the acceleration (83). We start by writing thelatter expression in the form a ( s ) = 2 ˆ mq (cid:18)(cid:90) ¯ s −∞ + (cid:90) ∞ ¯ s (cid:19) ds (cid:48) K ( s (cid:48) ) a ext ( s − s (cid:48) ) , (100)for some parameter ¯ s . Although this parameter is clearlyarbitrary, we find it convenient to set ¯ s = 1 /ν = (cid:112) τ ∗ /τ run (cid:29) . (101)This accomplishes two goals. First, it allows us to use a ext ( s − s (cid:48) ) = a ext ( s )[1 + O ( ν )] (102)for | s (cid:48) | (cid:46) ¯ s , at least if we are not too close to the bound-ary of the support of a ext . Second, if (99) is used toapproximate the kernel in the second integral in (100),the relative error in doing so is bounded by ε , where ε − ≡ ln s = ˆ mq + ln (cid:114) τ ∗ (cid:96) exp( + γ E ) ≈ ˆ mq . (103)The first integral in (100) can now be approximatedby substituting (102) when | s (cid:48) | (cid:46) ¯ s and noting that con-tributions from larger negative values of s (cid:48) are exponen-tially suppressed due to (93). Combining this with (88) and (99), it follows that a ( s ) = 2 ˆ mq (cid:90) ∞ ¯ s ds (cid:48) s (cid:48) (ln s (cid:48) ) [ a ext ( s − s (cid:48) ) − a ext ( s )] × (cid:2) O ( ν, ε ) (cid:3) . (104)A somewhat simpler expression arises when integratingby parts, which yields a ( s ) = 2 ˆ mq (cid:90) ∞ ¯ s ds (cid:48) ln s (cid:48) d a ext ( s − s (cid:48) ) ds (cid:2) O ( ν, ε ) (cid:3) (105)if it is assumed that a ext ( s ) → as s → ∞ . Note that theomission of the s (cid:48) = ¯ s boundary term in this expression,which is equal to − mq (cid:18) [ a ext ( s − ¯ s ) − a ext ( s )]ln ¯ s (cid:19) (106)up to terms of relative order ν or ε , results in errors oforder d a ext ds ε − ¯ s ln ¯ s ∼ a ext (cid:18) τ run τ ∗ (cid:19) ¯ s ∼ a ext ν. (107)This is absorbed into the overall O ( ν, ε ) relative errorin (105). Using (81), our approximation (105) can finallybe rewritten in terms of the physical time τ : Letting ¯ τ ≡ e − γ E ¯ sτ run = e − γ E √ τ run τ ∗ [which is not to be confusedwith the ¯ τ defined by (B2)], a ( τ ) = q ˆ m (cid:90) ∞ ¯ τ dτ (cid:48) (cid:32) ˙ E ext ( γ ( τ − τ (cid:48) ))1 + ( q / m ) ln( τ (cid:48) /e (cid:96) ) (cid:33) × (cid:2) O ( ν, ε ) (cid:3) . (108)This is our approximate solution to the d = 3 equationof motion (71).The reduced-order equation (77) can now be ob-tained directly from (108) by assuming that E ext ( γ ( τ )) is nonzero only for a finite time, which we assume to beshort compared to the timescale (cid:96) exp(2 ˆ m/q ) (cid:29) (cid:96) (cid:29) L. (109)If, furthermore, we evaluate a ( τ ) at times τ which aresmall compared to this timescale, we can expand the de-nominator in (108) in a Taylor series in q / m . Thisyields a ( τ ) = (cid:40) a ext ( τ ) − q m (cid:90) τ −∞ dτ (cid:48) ˙ a ext ( τ (cid:48) ) ln (cid:18) τ − τ (cid:48) e (cid:96) (cid:19) × (cid:20) O (cid:18) q m ln (cid:16) τ(cid:96) (cid:17)(cid:19)(cid:21) (cid:41) (cid:2) O ( ν, ε ) (cid:3) , (110)where we have used (102) and also the fact that the lowerlimit of ¯ τ in (108) can be replaced by a lower limit of while incurring relative errors only of order ν . Thisresult coincides with the expression (77) obtained ear-lier by reduction of order. At times large compared tothe timescale (109), the approximation (110) is no longervalid, and one must instead use the original expression(108).0 E. Special types of motion
Our equations of motion may now be used to answerat least two types of questions:1. How does a small charge move in response to agiven external field?2. Which external field is required in order for a chargeto move on a given trajectory?The first of these questions cannot generally be answeredusing the exact equations of motion (71) and (72), sincetheir solutions generically involve unphysical instabilitiesas discussed in Section IV B above. However, at least if d = 3 , one can instead use the reduced-order equation(77) over short timescales, or more generally (108) overall timescales. Either of these possibilities yield approxi-mate solutions with no unphysical instabilities.The second potential question we can address, con-cerning the force required to hold a particle on a giventrajectory, can be computed using either (71) or (77)when d = 3 , although it is the former unmodified equa-tion which is typically simpler for this purpose. Answerswill in any case be similar using either method, at leastif all timescales associated with the given trajectory aresufficiently long and q / ˆ m is sufficiently small. We nowdiscuss some simple examples.
1. Exponential growth
Our first case is that of exponential motion: Considertrajectories with the form (73), where τ run is now re-placed by a generic positive constant τ ∗ . At least for-mally, (71) predicts that if τ ∗ = τ run , no external force isrequired to effect such a trajectory when d = 3 . A ratherdifferent prediction follows, however, from the reduced-order equation (77). If τ ∗ (cid:29) τ run , both equations pre-dict similar results; the unmodified one gives the exactly-exponential external force q E ext = (cid:20) q m (cid:18) ln( τ ∗ /(cid:96) ) − − γ E (cid:19)(cid:21) ˆ m a e τ/τ ∗ (111)in three dimensions, while the reduced-order equation im-plies that if this force is applied, the particle’s accelera-tion will be ¨ γ = (cid:40) − (cid:20) q m (cid:18) ln( τ ∗ /(cid:96) ) − − γ E (cid:19)(cid:21) (cid:41) a e τ/τ ∗ . (112)The relative difference between this and our startingansatz (73) is of order ( q / ˆ m ) , as expected when com-paring equations in which order reduction has and hasnot been applied.
2. Harmonic motion
A more interesting example which can be understoodanalytically (and is mathematically similar) is that ofharmonic motion. Suppose that the trajectory is givenby γ ( τ ) = (cid:60) [ γ exp( iωτ )] , (113)where ω is real and the constant vector γ may be com-plex. Such an acceleration violates the falloff condition(61) but not the weaker condition (60). We therefore sub-stitute into (59) to find that the leading-order externalforce required to maintain harmonic motion is q E ext = (cid:20) − q m (cid:18) ln | ω | (cid:96) + 12 + γ E (cid:19)(cid:21) ˆ m ¨ γ + π q | ω | ˙ γ (114)when d = 3 . The analogous d = 5 expression is verysimilar except for an additional overall factor of ω inthe self-interaction terms. Regardless, if the motion isconfined to one spatial dimension, the self-force acts toprovide i) a damping force, and ii) a ln | ω | (cid:96) shift to acharge’s apparent inertia. If the motion is instead circu-lar, similar interpretations apply, except that it is onlythe component of the self-force which is proportional tothe velocity that performs work.
3. Power laws and analytic trajectories
Another example which is easily understood is one inwhich the acceleration vanishes for all τ < τ , while ¨ γ ( τ ) = a n [( τ − τ ) /τ ∗ ] n (115)thereafter, where a n , τ , τ ∗ , and n are constants (the lat-ter two of which are assumed to be positive). Substitut-ing this into (71) shows that the external force requiredto produce such an acceleration has a somewhat-differenttime dependence than the acceleration itself: In terms ofthe harmonic number H n , q E ext = (cid:26) q m (cid:20) ln (cid:18) τ − τ e (cid:96) (cid:19) − H n (cid:21)(cid:27) ˆ m ¨ γ (116)when d = 3 and τ > τ . The logarithm here implies thateven at late times, there remains a strong “memory” ofthe “turn-on event” at τ = τ .This result allows us to understand which externalforces are needed to hold a charge on a more-general tra-jectory which is analytic for all all τ > τ . Suppose that ¨ γ ( τ ) = 0 for τ < τ and ¨ γ ( τ ) = ∞ (cid:88) n =1 a n [( τ − τ ) /τ ∗ ] n (117)1when τ ≥ τ , where the a n are constants. Combining(115) and (116), the required external force is seen to be q E ext ( γ ( τ )) = (cid:20) q m ln (cid:18) τ − τ e (cid:96) (cid:19)(cid:21) ˆ m ¨ γ ( τ ) − q ∞ (cid:88) n =1 a n H n [( τ − τ ) /τ ∗ ] n . (118)
4. Kicks
Our last—and most interesting—example is concernedwith a charge which is briefly “kicked” by some externalforce. Focusing again on three dimensions, we initiallyask which external field must be imposed in order fora particle to be only momentarily accelerated: Supposethat a charge is initially stationary, is subjected to a briefacceleration near τ = τ , and moves inertially thereafterwith velocity ˙ γ ( τ ) = ∆ v . Substituting this into (62)shows that the self-force at late times must be balancedby an external force satisfying q E ext ( γ ( τ )) = q (cid:18) ∆ v τ − τ (cid:19) , (119)where we have neglected terms of order / ( τ − τ ) . Theself-force thus acts to push the particle back towards rest.This effect persists indefinitely, suggesting that the par-ticle’s initially-stationary state creates a “preferred restframe” to which it always attempts to return.We now change perspective, asking not for the exter-nal force required to maintain a briefly-accelerated tra-jectory, but instead for the trajectory of a particle inwhich the external force is only briefly nonzero. Thereare potential physical issues associated with this sce-nario, essentially because it is not clear if the strongtails present in three dimensions preclude any possibilityof setting up a prescribed, confined electric field; theremay be unavoidable and significant remnants of the pro-cess by which any experiment might be assembled. See,e.g., [49] for some recent remarks—in a somewhat dif-ferent context—on persistent memory effects in odd di-mensions. Regardless, there is no mathematical difficultywith assuming a prescribed external field and we proceedwithout further comment.The net force acting on a charge for which the ex-ternal field has a Gaussian profile is plotted in Figure1, assuming the reduced-order equation of motion (77).Self-interaction is seen to slightly increase the peak mag-nitude of the force in this case, and also to shift that peakearlier in time. That the peak of the net force appearsto anticipate the peak of the applied force might initiallyappear to violate causality, and to be reminiscent of thepreacceleration seen in the Abraham-Lorentz-Dirac equa-tion (and in the d = 3 results discussed in Section IV Dabove). Causality is not violated here, however. The re-sult arises from the explicitly-causal integral in (77), and - - ττ * - FIG. 1. Net force as a function of τ /τ ∗ for a Gaussian externalfield proportional to exp( − ( τ /τ ∗ ) ) , as computed using thereduced-order d = 3 equation of motion (77). Here, (cid:96) = τ ∗ > and all results are normalized so that the maximumexternal force is equal to unity. The solid line correspondsto the external force, the dashed line to the net force when q / m = 1 / , and the dotted line to the net force when thisparameter is equal to / . appears because the self-force is sensitive to ˙ E ext , whichdecreases near the peak of the external force.One can also see in the figure that the self-force even-tually switches sign and only slowly returns to zero. Acharge thus continues to decelerate long after the externalfield decays away. The late-time behavior of this processdoes not depend on whether or not the external field isGaussian, and we now analyze more generally the asymp-totic motion of a kicked charge.Long after a briefly-nonzero external force has beenapplied, the reduced-order equation (77) would suggestthat the acceleration decays like / ( τ − τ ) . However, anacceleration which decays this slowly implies a velocitywhich grows logarithmically at late times. Such growth isunphysical. It may be traced back to a failure of the orderreduction procedure at late times; cf. the derivation of(110) from (108).A more careful analysis using the methods of SectionIV D shows that in fact, a particle asymptotically returnsto its initial “pre-kick” velocity; see (88). In essence,this recovers the Aristotelian idea that perturbed masseseventually return to rest when all perturbations are re-moved. More precisely, (108) shows that the asymptoticvelocity of a particle which is initially at rest decays like ˙ γ ( τ ) = ∆ v q / m ) ln[( τ − τ ) / ( e (cid:96) )] (120)at late times, where ∆ v is the time integral of a ext ( τ ) =( q/ ˆ m ) E ext ( γ ( τ )) . V. DISCUSSION
We have developed a general formalism with whichto understand the motion of extended, self-interacting2charges in all spacetime dimensions d ≥ . Before under-standing how such objects move, it is first necessary tofix precisely what should be meant by the concept of mo-tion. We do so by giving precise definitions for a body’slinear and angular momenta. One of the central proper-ties of the momenta introduced here is that their laws ofmotion are structurally identical to the laws of motionsatisfied by extended test bodies. This statement holdsto all multipole orders, and for both an object’s transla-tional and rotational degrees of freedom. For example,the lowest-order force is given by the usual Lorentz ex-pression (32), and the lowest-order torque by (33). Theonly difference between these results and their test bodycounterparts is that the field ˆ F ab which appears in themis a certain nonlocal linear transformation of the physi-cal electromagnetic field F ab . It is in the details of thisfield that the most visible effects of self-interaction maybe found. Note as well that it is the same effective fieldwhich appears in expressions for both forces and torques,and that the prescription for this field remains the sameat all multipole orders.To be somewhat more precise, we do not find onlya single momentum definition which obeys laws of mo-tion structurally identical to test-body laws, but rathera class of such definitions. Different elements of thisclass become distinct only when self-interaction is signifi-cant, and they may be characterized by a certain 2-point“propagator” G aa (cid:48) ( x, x (cid:48) ) ; see (6), (7), and (31). Phys-ically, this propagator fixes a sense in which a chargeelement at x (cid:48) can source a field at x whose net effecton the body’s motion may be removed by finite renor-malization of its multipole moments. We show from firstprinciples that any propagator which satisfies the four“axioms” given in Section II D 1 has this interpretation,and that such propagators may be used to define mo-menta with physically-desirable properties. Our axiomsgeneralize the three originally proposed by Poisson [11](in a somewhat different context) in order to characterizethe d = 4 propagators originally constructed by Detweilerand Whiting [18].The axioms we introduce are essential to understand-ing the odd-dimensional self-force, and can be useful alsoin certain even-dimensional scenarios. However, theydo not single out a unique propagator. Consequently,we do not have a unique momentum, a unique effectivefield, or even unique multipole moments associated witha body’s stress-energy tensor. All of these quantities maydepend on the choice of propagator. Nevertheless, suchdifferences do not signal any kind of physical inconsis-tency. They merely reflect that one can choose to focuson slightly different aspects of the same physical system,and there is no reason to expect that all such aspectsbehave identically. A somewhat simpler “gauge freedom”of this kind arises even in d = 4 discussions of extendedtest bodies, wherein different spin supplementary condi-tions may be applied to yield distinct centroids whichnevertheless describe different aspects of the same phys-ical system [41]. Having established an appropriate class of propaga-tors with which to construct physically-useful momenta,it is essential to be able to find explicit examples in thatclass. In even numbers of dimensions, a straightforwardgeneralization of the Detweiler-Whiting “ S -type” Greenfunction satisfies our constraints, and may therefore beused to generate suitable momenta for extended chargedistributions. Adopting such definitions, the laws of mo-tion involve effective electromagnetic fields which locallysatisfy the source-free Maxwell equations. Extended self-interacting charges in even numbers of dimensions maythus be viewed as obeying laws of motion which are struc-turally identical to those of extended test bodies, andwhere the effective field appearing in those laws is source-free. This is a relatively straightforward generalizationof existing d = 4 results on relativistic motion in genericspacetimes [10]. It may also be viewed as a generalizationof the well-known statement that massive bodies inter-acting via Newtonian gravity or electrostatics satisfy lawsof motion which involve only source-free external fields.The odd-dimensional case is different. One of our mainresults is the identification of an odd-dimensional prop-agator, namely (29), which satisfies the four constraintsgiven in Section II D 1. This propagator is quite differ-ent from its even-dimensional Detweiler-Whiting coun-terpart; it is not a Green function or even a more gen-eral parametrix for Maxwell’s equations. The effectivefield which appears in the laws of motion may thus failto satisfy the source-free Maxwell equations when d isodd. This difference is reasonably subtle at lower multi-pole orders. However, it may be qualitatively importantwhen higher-order extended-body effects become signif-icant: In that context, all components of a body’s mul-tipole moments may affect its motion, rather than onlytheir (more familiar) trace-free components.Another interesting feature of the odd-dimensional ef-fective fields identified here is that in a point-particlelimit, the map which translates the physical field into theeffective field appears to turn into a kind of dimensionalregularization procedure. This procedure arises as thelimit of a map which is generically non-singular, makesno symmetry assumptions, and applies in a single space-time with fixed integer dimension. A better understand-ing of this link may provide an improved understandingof dimensional regularization more generally.Regardless, whether in even numbers of dimensions orodd, our formalism can be applied together with pointparticle limits to generate explicit laws of motion. We doso in Section III, restricting to flat spacetimes for simplic-ity. Assuming retarded boundary conditions, we providethe general prescription for all dimensions d ≥ , andapply it in full to find leading-order point-particle self-forces for d = 3 , , , , , and leading-order self-torquesfor d = 3 , , . Our explicit self-forces agree with existingresults in the literature when d = 4 , , , although for d (cid:54) = 4 , our approach is more systematic and includes mi-croscopic definitions which were previously lacking. Theodd-dimensional cases are different, and we identify sig-3nificant problems with most other proposals which havebeen suggested in that context.Finally, Section IV analyzes solutions to the nonrel-ativistic limits of our d = 3 and d = 5 results. Theparticularly-slow decay of odd- d fields—particularly inthree dimensions—results in a very strong dependence ona charge’s past history: It follows from (62) that the self-force acting on a particle in dimensional Minkowskispacetime depends on the past history of its jerk ... γ ( τ ) ,with a weighting factor which grows logarithmically inthe increasingly-distant past.Some physical consequences of this can be illustratedby considering a charge which is briefly kicked by anexternally-imposed electric field in a d = 3 Minkowskispacetime. If this external field is Gaussian, one seesfrom Figure 1 that self-interaction causes the peak of thenet force to arrive before the peak of the applied force.Despite appearances, this effect is causal. Moreover, forany external force—whether Gaussian or not—we showthat if a charge is stationary for all time before an ex-ternal field is applied, the slowly-decaying remnant ofits self-field causes that charge to asymptotically returnto rest at late times. The slow decay of the self-field abody produces while it is initially at rest in three space-time dimensions thus provides a preferred, dynamically-produced rest frame which persists and remains signifi-cant even in the distant future.We note that although this paper has focused on themotion of bodies coupled to electromagnetic fields, ouranalysis extends straightforwardly for other types of in-teractions. For example, our odd-dimensional electro-magnetic propagator (29) is replaced by G odd = ( − − λ d U π lim λ → λ d (cid:96) λ ∂∂λ (cid:2) (2 σ/(cid:96) ) λ Θ( σ ) (cid:3) , (121)for a body coupled to a Klein-Gordon field in an odd-dimensional spacetime, where U is a smooth biscalarwhich also appears in the retarded Green function. Fur-thermore, point-particle scalar fields can be obtained di-rectly from our electromagnetic vector potentials by re-placing the W a ( x, τ ; λ ) given by (D5) with W ( x, τ ; λ ) = q ( τ ) α d U ( x, γ ( τ ))Σ λ ( x, τ ) . (122)We note as well that our methods generalize almost aseasily for bodies coupled to (at least the linearized) d -dimensional Einstein equation.As a simple application in the scalar setting, we notethat masses can vary here even at monopole order, andthat charges which source Klein-Gordon fields are notnecessarily conserved. If an initially-uncharged bodyrapidly acquires a net charge q ∞ around τ = τ , ourequations show that the mass “evaporates” according to ˆ m ( τ ) − ˆ m ( τ (cid:48) ) = q ∞ ln (cid:18) τ (cid:48) − τ τ − τ (cid:19) (123) for a stationary particle in a d = 3 Minkowski spacetime,where τ, τ (cid:48) (cid:29) τ . Accounting for differences in numericalconventions, this matches an earlier result [50] obtainedusing different methods. It is also conceptually similarto the scalar charge evaporation found for freely-fallingcharges in d = 4 de Sitter spacetimes [51].Whether in electromagnetic or other contexts, thereare various directions in which the results presented inthis paper may be extended or applied. One possibilitywould be to relax our assumptions regarding retardedboundary conditions and trivial topology. Some discus-sion of motion in topologically-nontrivial spacetimes hasalready been given [52], although mainly in cases wherethe formally-divergent portion of the point-particle self-field could be clearly seen not to contribute to the self-force. The formalism developed here lays the ground-work for extending these kinds of results for generic typesof motion: All of the formalism developed in SectionII holds regardless of boundary or initial conditions, ortopology, and the S -fields given by (C2) and (53) aresimilarly-agnostic to these features. If a physical field F ab can be computed in some physical system—whether bynumerical, perturbative, or other methods—the S -fieldsgiven here can be used to straightforwardly determine theforce. One motivation for such generalizations is the po-tential for connecting this work with the behavior of cer-tain lower-dimensional condensed matter systems, sys-tems which are often characterized by nontrivial bound-ary conditions or topology. Moreover, experimental workin pilot-wave hydrodynamics [20] suggests—although themathematics applicable there is not precisely analogousto ours—that self-interaction problems in two spatial di-mensions can have very rich and surprising behavior inthe presence of nontrivial boundary conditions. Appendix A: Propagators and Hadamard series
This appendix explains how to determine the bitensorswhich appear in the even-dimensional Detweiler-WhitingGreen functions G DW aa (cid:48) with the form (22), and also in theodd-dimensional propagators G odd aa (cid:48) given by (29). In bothcases, it is convenient to introduce the van Vleck deter-minant ∆( x, x (cid:48) ) , which is a symmetric biscalar satisfying[11, 19] σ a ∇ a ln ∆ = d − ∇ a ∇ a σ, (A1)and also ∆( x, x ) = 1 . In general, σ a ≡ ∇ a σ ( x, x (cid:48) ) liestangent to the geodesic which passes through x and x (cid:48) ,so (A1) may be viewed as a first-order ordinary differen-tial equation for ∆( x, x (cid:48) ) along that geodesic. If Synge’sfunction is known, d −∇ a ∇ a σ is easily computed and thesolution can be written as an explicit integral along thatgeodesic. Integral solutions for this and similar “trans-port equations” may be found in, e.g., Appendix B of[19].4
1. Even-dimensional propagators
The even-dimensional Detweiler-Whiting Green func-tion G DW aa (cid:48) involves two bitensors, U aa (cid:48) and V aa (cid:48) . Sub-stituting its form (22) into (19) shows that these mustsatisfy σ b ∇ b U aa (cid:48) + ( ∇ b ∇ b σ − d ) U aa (cid:48) ] δ ( d/ − ( σ )+ [ ∇ b ∇ b U aa (cid:48) − R ab U ba (cid:48) ] δ ( d/ − ( σ )+ [2 σ b ∇ b V aa (cid:48) + ( ∇ b ∇ b σ − V aa (cid:48) ] δ ( σ )+ [ ∇ b ∇ b V aa (cid:48) − R ab V ba (cid:48) ]Θ( σ ) (A2)when x (cid:54) = x (cid:48) , and also lim x (cid:48) → x U ab (cid:48) ( x, x (cid:48) ) = α d g ab , (A3)where α d is given by (24). The first three lines restrict U aa (cid:48) and V aa (cid:48) on the σ = 0 light cones, while the last re-quires that V aa (cid:48) satisfy the homogeneous Maxwell equa-tion ∇ b ∇ b V aa (cid:48) − R ab V ba (cid:48) = 0 , (A4)at least when σ > . We note that the bitensors deter-mined by these equations also arise in the retarded andadvanced Green functions, via G ret , adv aa (cid:48) = [ U aa (cid:48) δ ( d/ − ( σ ) − V aa (cid:48) Θ( − σ )] ret , adv , (A5)although they may be evaluated at different locationshere than in G DW aa (cid:48) .In order to complete the solution to (A2), it is firstconvenient to factor out the square root of the van Vleckdeterminant and to expand in the Hadamard series U aa (cid:48) = ∆ / d/ − (cid:88) n =0 σ n n ! U { n } aa (cid:48) . (A6)Note that this is not a Taylor expansion; the “coefficients” U { n } aa (cid:48) may be nontrivial functions of x and x (cid:48) . Regardless,using (A1) and the identity σ n δ ( p ) ( σ ) = ( − n p !( p − n )! δ ( p − n ) ( σ ) , (A7)while setting to zero explicitly-equal numbers of deriva-tives of δ ( σ ) , we find that U { } aa (cid:48) = α d g aa (cid:48) , (A8)and that for all n ∈ { , . . . , d/ − } , ( σ b ∇ b + n ) U { n } aa (cid:48) = nd − − n × [∆ − / ∇ b ∇ b (∆ / U { n − } aa (cid:48) ) − R ab U { n − } ba (cid:48) ] . (A9)These constitute a tower of transport equations for each U { n } aa (cid:48) in terms of U { n − } aa (cid:48) . The lone nonsingular solutions to these differential equations are the physical ones. Theyguarantee that the first two lines of (A2) vanish.The last line of that equation vanishes by (A4), whilethe third can be eliminated by imposing the “boundarycondition” [ σ b ∇ b + ( d/ − − / V aa (cid:48) ) = ( − d/ − × [∆ − / ∇ b ∇ b (∆ / U { d/ − } aa (cid:48) ) − R ab U { d/ − } ba (cid:48) ] (A10)on V aa (cid:48) when its arguments are null-separated. Eqs.(A4), (A8), (A9), and (A10) together provide a completesolution to (A2), and thus a complete determination of U aa (cid:48) and V aa (cid:48) .Note that unlike when finding these bitensors for theretarded or advanced Green functions, solving (A4) withboundary data (A10) constitutes a peculiar type of “ex-terior” characteristic problem: Data is specified on thepast and future light cones and we seek a solution to thewave equation outside of those light cones. Although thegeneral mathematical status of such problems is not par-ticularly clear, a Hadamard-like series analogous to (A6)can be developed for V aa (cid:48) , resulting in an infinite tower oftransport equations for the Hadamard coefficients V { n } aa (cid:48) .We assume V aa (cid:48) to be specified in this sense, and thatthe resulting series is well-behaved. In fact, it can be ac-ceptable to use Detweiler-Whiting propagator in whichthe Hadamard series for V aa (cid:48) is truncated at some finiteorder. Forces and torques due to the associated effec-tive field would be slightly altered by this truncation,although that would be due to them describing rates ofchange of slightly different quantities; such propagatorsstill generate correct and useful laws of motion.It is clear from this discussion that since eachHadamard coefficient U { n } aa (cid:48) or V { n } aa (cid:48) can be written as aline integral along the geodesic segment which connectsits arguments, it can depend on the geometry only onthat geodesic. This establishes that each coefficient isquasilocal in the sense of Axiom 3 in Section II D 1. Theworld function and the van Vleck determinant are sim-ilarly quasilocal, so this description holds for G DW aa (cid:48) as awhole.Note as well that each of the Hadamard coefficients issymmetric, so U aa (cid:48) ( x, x (cid:48) ) = U a (cid:48) a ( x (cid:48) , x ) , V aa (cid:48) ( x, x (cid:48) ) = V a (cid:48) a ( x (cid:48) , x ) . (A11)This may be argued in various ways. Most simply, theself-adjointness of the differential operator δ ba ∇ c ∇ c − R ba ,Stokes’ theorem, and the causal properties of the ad-vanced and retarded Green functions imply that G ret aa (cid:48) = G adv a (cid:48) a . Eq. (A5) thus implies (A11), at least for null andtimelike-separated points. Symmetry of the Detweiler-Whiting Green function merely requires that this prop-erty extend also to spacelike-separated points. Such anextension is argued to be valid in Section 6.4 of [37]; seealso [53].5
2. Odd-dimensional propagators
The bitensor U aa (cid:48) which appears in our odd-dimensional propagator G odd aa (cid:48) is the same as the onewhich appears in the retarded and advanced Green func-tions associated with (19). It may be found by factor-ing out the van Vleck determinant and expanding in theHadamard series U aa (cid:48) = ∆ / ∞ (cid:88) n =0 σ n n ! U { n } aa (cid:48) . (A12)Unlike its even-dimensional analog (A6), the sum heredoes not necessarily terminate at finite n . Nevertheless,the zeroth term in the series is again given by (A8),although the odd-dimensional α d is now computed us-ing (28) instead of (24). Substituting this and (27)into (19), the higher-order Hadamard coefficients U { n } aa (cid:48) may be shown to be the nonsingular solutions to thesame transport equations (A9) which determine the even-dimensional Hadamard coefficients. The U aa (cid:48) appear-ing here is again symmetric and quasilocally dependenton the metric, by the same arguments as in the even-dimensional case.Also note that again, there is no obstacle to work-ing instead with a somewhat-different propagator whoseHadamard series is truncated at finite n . Appendix B: Expansion methods and coincidencelimits
We now collect various expansion methods and resultsrelevant to the point-particle fields computed in Appen-dices C and D.Many of these expansions involve σ ( x, γ ( τ )) , Synge’sworld function specialized to cases in which one argumentis evaluated at a specific proper time on a given timelikeworldline Γ . This is assumed to be a smooth function of x and τ , at least if g ab and Γ are themselves smooth and x and γ ( τ ) are sufficiently close. More precisely, we sup-pose that these points always lie within a convex normalneighborhood. Then, if τ is varied while x is held fixednear (but not on) Γ , there exist exactly two “nearby” ze-ros. We call the larger of these the advanced time τ + ( x ) and the smaller the retarded time τ − ( x ) . While it is pos-sible to approximate these times in terms of some givencoordinate system, we have no need to do so. Instead,we note that Synge’s function must factorize via σ ( x, γ ( τ )) = [ τ + ( x ) − τ ] [ τ − τ − ( x )] Σ( x, τ ) , (B1)where Σ( x, τ ) is assumed to be positive and smooth in allregions of interest. For an inertial worldline in flat space-time, Σ( x, τ ) = 1 . More generally, everything we need isencoded in the various derivatives of Σ( x, τ ) evaluatedusing coincidence limits in which x → γ ( τ ) . It is convenient to also use the retarded and advancedtimes to introduce the “radar time” ¯ τ ( x ) ≡
12 [ τ + ( x ) + τ − ( x )] , (B2)and the “radar distance” r ( x ) ≡
12 [ τ + ( x ) − τ − ( x )] , (B3)associated with points x near Γ . In terms of these func-tions, (B1) may be rearranged to read σ/ Σ = r − ( τ − ¯ τ ) , from which it follows that ¯ τ = τ + ∂∂τ (cid:16) σ Σ (cid:17) , r = 2 σ Σ + (cid:20) ∂∂τ (cid:16) σ Σ (cid:17)(cid:21) . (B4)These expressions imply that if σ/ Σ is smooth, so are ¯ τ and r .Now solve (B1) for Σ( x, τ ) and consider the substitu-tion x = γ ( τ (cid:48) ) , in which case τ + = τ − = τ (cid:48) : Σ( γ ( τ (cid:48) ) , τ ) = − σ ( γ ( τ (cid:48) ) , γ ( τ ))( τ (cid:48) − τ ) . (B5)Coincidence limits for the left-hand side or its derivativesfollow by evaluating the right-hand side or its derivativesas τ (cid:48) → τ , using well-known coincidence limits for thederivatives of Synge’s function. For example, applyingL’Hôpital’s rule twice gives Σ( γ ( τ ) , τ ) = − lim τ (cid:48) → τ ˙ γ a (cid:48) ˙ γ b (cid:48) ∇ a (cid:48) ∇ b (cid:48) σ. (B6)where we have used the vanishing coincidence limits of σ and ∇ a (cid:48) σ . Further applying lim x (cid:48) → x ∇ a (cid:48) ∇ b (cid:48) σ ( x, x (cid:48) ) = g ab , (B7)it follows that Σ( γ ( τ ) , τ ) = 1 . (B8)Supplementing (B7) by, e.g., lim x (cid:48) → x ∇ a (cid:48) ∇ b σ ( x, x (cid:48) ) = − g ab , (B9)coincidence limits of τ -derivatives of Σ( x, τ ) may be de-rived similarly. If we specialize to flat spacetime, in whichthird and higher derivatives of σ vanish, it may be shownthat ˙Σ = 0 , ¨Σ = 16 | ¨ γ | , ... Σ = 12 (¨ γ · ... γ ) , Σ (4) = 115 (cid:16) | ... γ | + 9¨ γ · γ (4) (cid:17) , Σ (5) = 13 (cid:16) γ · γ (5) + 5 ... γ · γ (4) (cid:17) , Σ (6) = 128 (cid:16) γ · γ (6) + 64 ... γ · γ (5) + 45 | γ (4) | (cid:17) , (B10)6when x = γ ( τ ) .We also need coincidence limits for ∇ a Σ( x, τ ) and its τ derivatives. First note that differentiating (B1) withrespect to x and rearranging using (B2) and (B3) impliesthat ∇ a Σ( x, τ (cid:48) ) = 2 ∇ a σ + ∇ a [(¯ τ − τ (cid:48) ) − r ]Σ( τ + − τ (cid:48) )( τ (cid:48) − τ − ) , (B11)where we have added a prime to the second argument forlater convenience. Noting that ∇ a ¯ τ ( γ ( τ )) = − ˙ γ a ( τ ) , ∇ a r ( γ ( τ )) = 0 , (B12)substituting x = γ ( τ ) into (B11) gives ∇ a Σ( γ ( τ ) , τ (cid:48) ) = − τ (cid:48) − τ ) (cid:104) ∇ a σ ( γ ( τ ) , γ ( τ (cid:48) ))+ ( τ (cid:48) − τ ) ˙ γ a ( τ )Σ( γ ( τ ) , τ (cid:48) ) (cid:105) . (B13)Repeatedly applying L’Hôpital’s rule to this expressionagain allows us to evaluate coincidence limits τ (cid:48) → τ for ∇ a Σ( γ ( τ ) , τ (cid:48) ) and its τ -derivatives. Specializing to flatspacetime while using (B10), the first few such limits are ∇ a Σ = ¨ γ a , ∇ a ˙Σ = 16 (2 ... γ a − | ¨ γ | ˙ γ a ) , ∇ a ¨Σ = 16 (cid:104) γ (4) a − γ · ... γ ) ˙ γ a (cid:105) , ∇ a ... Σ = 130 (cid:104) γ (5) a − (cid:16) | ... γ | + 9¨ γ · γ (4) (cid:17) ˙ γ a (cid:105) , ∇ a Σ (4) = 115 (cid:104) γ (6) a − (cid:16) γ · γ (5) + 5 ... γ · γ (4) (cid:17) ˙ γ a (cid:105) , ∇ a Σ (5) = 184 (cid:104) γ (7) a − (cid:16) γ · γ (6) + 64 ... γ · γ (5) + 45 | γ (4) | (cid:17) ˙ γ a (cid:105) . (B14)Point-particle electromagnetic fields in odd numbers ofdimensions are expressed below in terms of coincidencelimits of W { n } a ( x ; λ ) and its derivatives, functions definedby (D5) and (D7). However, we specialize here to flatspacetime, in which case the first of these equations isreplaced by (54). Recalling that ∇ b g aa (cid:48) = ∇ b (cid:48) g aa (cid:48) = 0 inMinkowski spacetimes, the W { n } a ( x ; λ ) can depend onlyon the particle’s worldline and on Σ( x, τ ) . Using (B10)and (B14), the first coincidence limits in flat spacetimemay be shown to be W { } a = q ˙ γ a , W { } a = q ¨ γ a ,W { } a = q (cid:16) ... γ a + λ | ¨ γ | ˙ γ a (cid:17) ,W { } a = q (cid:104) γ (4) a + λ (cid:0) | ¨ γ | ¨ γ a + (¨ γ · ... γ ) ˙ γ a (cid:1) (cid:105) , (B15) and ∇ [ a W { } b ] = − q (1 + λ ) ˙ γ [ a ¨ γ b ] , ∇ [ a W { } b ] = − q (1 + 13 λ ) ˙ γ [ a ... γ b ] , ∇ [ a W { } b ] = 16 q (cid:104) λ ¨ γ [ a ... γ b ] − λ (4 + λ ) | ¨ γ | ˙ γ [ a ¨ γ b ] − (6 + λ ) ˙ γ [ a γ (4) b ] (cid:105) . (B16) Appendix C: Point-particle fields in even dimensions
This appendix computes various electromagnetic fieldsassociated with monopole point charges in potentially-curved spacetimes for which d ≥ is even. We startby evaluating the vector potential (18) for the S -fieldassociated with the point-particle current (44). Identify-ing G aa (cid:48) with the Detweiler-Whiting Green function (22),this is more explicitly A S a ( x ) = q (cid:90) (cid:2) U aa (cid:48) ( x, γ ( τ )) δ ( d/ − ( σ ( x, γ ( τ )))+ V aa (cid:48) ( x, γ ( τ ))Θ( − σ ( x, γ ( τ ))) (cid:3) ˙ γ a (cid:48) ( τ ) dτ. (C1)The range of τ values over which this integration is toperformed are to be understood as restricted to a normalneighborhood of x , in which case the only relevant zerosof σ ( x, γ ( τ )) are, for fixed x , at τ = τ ± ( x ) [cf. (B1)].Hence, A S a = q (cid:34) (cid:88) τ ∈{ τ ± } | ˙ σ | (cid:18) − ∂∂τ σ (cid:19) d/ − U aa (cid:48) ˙ γ a (cid:48) + (cid:90) τ + τ − V aa (cid:48) ˙ γ a (cid:48) dτ (cid:35) , (C2)where ˙ σ = ∂σ ( x, γ ( τ )) /∂τ and the sum denotes that oneis to substitute τ = τ + and then add to that the same ex-pression evaluated at τ = τ − . While other null geodesicsmay exist between the particle’s worldline and x , it isonly the “closest two” which are included in this expres-sion.The retarded Green function may be shown to have tohave the form (A5) at least within a normal neighbor-hood, so the retarded vector potential is A ret a = q (cid:34) | ˙ σ | (cid:18) − ∂∂τ σ (cid:19) d/ − U aa (cid:48) ˙ γ a (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ = τ − + lim (cid:15) → + (cid:90) τ − − (cid:15) −∞ G ret aa (cid:48) ˙ γ a (cid:48) dτ (cid:35) . (C3)The Green function in the second line here is left as-is toallow for integrations beyond the normal neighborhood,in which case the Hadamard form (A5) can fail to remainvalid.7If the full electromagnetic field F ab is identified withthe retarded field F ret ab = 2 ∇ [ a A ret b ] , it follows from (17)that ˆ F ab = 2 ∇ [ a ( A ret b ] − A S b ] ) . In Minkowski spacetime,this is equivalent to what is often called the radiativefield, one-half of the retarded minus advanced fields. Inmore general spacetimes, a vector potential for ˆ F ab withretarded boundary conditions may be written as the dif-ference between (C3) and (C2): ˆ A a = q (cid:34) | ˙ σ | (cid:18) − ∂∂τ σ (cid:19) d/ − U aa (cid:48) ˙ γ a (cid:48) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ = τ − τ = τ + − (cid:90) τ + τ − V aa (cid:48) ˙ γ a (cid:48) dτ + lim (cid:15) → + (cid:90) τ − − (cid:15) −∞ G ret aa (cid:48) ˙ γ a (cid:48) dτ (cid:35) . (C4)Although it is not obvious from this expression, the ef-fective field is finite, and indeed smooth, even on theworldline, essentially because it satisfies the source-freeMaxwell equations. Appendix D: Point-particle fields in odd dimensions
We now compute point-particle fields in odd numbersof dimensions. Section D 1 starts by obtaining a vectorpotential A S a for the S -field F S ab associated with the point-particle current (44), identifying the propagator G aa (cid:48) bywhich these fields are defined with the G odd aa (cid:48) given in (30).The final result, summarized by (53), is a series involvingthe radar distance r away from the particle’s worldline Γ [as defined by (B3)]. This series involves positive andnegative powers of r , ln r , and coefficients which can de-pend smoothly on x .Next, the point-particle retarded field A ret a ( x ) is com-puted in Section D 2, again as a series involving r . Boththe retarded and S fields diverge on Γ , although we showin Section D 3 that their difference is smooth. This lastresult essentially constitutes our verification that Axiom4 of Section II D 1 is satisfied by G odd aa (cid:48) .
1. The S -field The point-particle S -field vector potential may befound by evaluating (18) with G aa (cid:48) = G odd aa (cid:48) and J a = J a pp .Given the form (29) for G odd aa (cid:48) , it is useful to introduce theauxiliary family of propagators ˜ G aa (cid:48) ( x, x (cid:48) ; λ ) ≡ U aa (cid:48) ( x, x (cid:48) )[2 σ ( x, x (cid:48) )] λ Θ( σ ( x, x (cid:48) )) , (D1)and the associated point-particle fields ˜ A a ( x ; λ ) = q (cid:90) τ + ( x ) τ − ( x ) [2 σ ( x, γ ( τ ))] λ U aa (cid:48) ( x, γ ( τ )) ˙ γ a (cid:48) ( τ ) dτ. (D2) Once this potential is known, the point-particle S -fieldfollows from A S a = ( − − λ d π lim λ → λ d (cid:96) λ ∂∂λ ( (cid:96) − λ ˜ A a ) , (D3)where (cid:96) > is the arbitrary lengthscale used in the con-struction of G odd aa (cid:48) , the dimension-dependent number λ d is given by (25), and the limit implies an analytic con-tinuation in λ .A series expansion for ˜ A a may now be found by sub-stituting the factorization (B1) for σ into (D2). Doing soresults in ˜ A a ( x ; λ ) = α d (cid:90) τ + ( x ) τ − ( x ) dτ [ τ + ( x ) − τ ] λ [ τ − τ − ( x )] λ × W a ( x, τ ; λ ) , (D4)where α d is given by (28) and it convenient to define W a ( x, τ ; λ ) ≡ qα d U aa (cid:48) ( x, γ ( τ )) ˙ γ a (cid:48) ( τ )Σ λ ( x, τ ) . (D5)Expanding W a ( x, τ ; λ ) about τ = ¯ τ ( x ) as defined in (B2),we find that W a ( x, τ ; λ ) = ∞ (cid:88) n =0 n ! [ τ − ¯ τ ( x )] n W { n } a ( x ; λ ) , (D6)in terms of the coefficients W { n } a ( x ; λ ) ≡ ∂ n ∂τ n W a ( x, τ ; λ ) (cid:12)(cid:12)(cid:12)(cid:12) τ =¯ τ ( x ) . (D7)Substituting these expressions into (D4) now yields ˜ A a = 2 α d ∞ (cid:88) n =0 W { n } a (2 n )! (cid:90) τ + ¯ τ dτ ( τ − ¯ τ ) n + λ ) × (cid:34)(cid:18) rτ − ¯ τ (cid:19) − (cid:35) λ . (D8)If λ > − , the integral on the right-hand side is well-defined and ˜ A a = α d ∞ (cid:88) n =0 Γ( n + )Γ( λ + 1)(2 n )!Γ( n + λ + ) W { n } a r n + λ ) . (D9)However, it follows from (25) that λ d < − in five ormore dimensions. The analytic continuation associatedwith the limit in (D3) nevertheless implies that the right-hand side of (D9) remains valid as long as it may beanalytically continued to λ → λ d .Carrying out this continuation, Γ( n + λ + ) divergesfor all n ≤ ( d − . Such terms therefore go to zero inthe sum (D9) and lim λ → λ d ˜ A a ( x ; λ ) = α d ∞ (cid:88) n = ( d − Γ( n + )Γ(2 − d )(2 n )!Γ( n + (5 − d )) × W { n } a ( x ; λ d ) r n − ( d − . (D10)8This depends only on non-negative even powers of r ( x ) and on the smooth functions W { n } a ( x ; λ d ) ; the overallresult is smooth near the particle’s worldline.Computing A S a requires not only ˜ A a continued to theappropriate value of λ , but also a continuation for the λ -derivative of that field. Differentiating (D9), one finds that that is ∂ λ ˜ A a = α d ∞ (cid:88) n =0 Γ( n + )Γ( λ + 1)(2 n )!Γ( n + λ + ) r n + λ ) (cid:104) ∂ λ W { n } a + (cid:16) H λ − H n + λ + + 2 ln r (cid:17) W { n } a (cid:105) (D11)for general values of λ , where H µ denotes the µ th har-monic number. Taking the λ → λ d limit here requiressome care since the factor of Γ( n + λ + ) in the denomina-tor and the harmonic number H n + λ + in the numeratorboth diverge in that limit, for all n ≤ ( d − . What isimportant however is the limit of their ratio, which maybe shown to be lim λ → λ d H n + λ + Γ( n + λ + ) = ( − ( d − − n Γ( ( d − − n ) (D12)for all n ≤ ( d − . Hence, lim λ → λ d ∂ λ ˜ A a = α d (cid:40) ∞ (cid:88) n = ( d − Γ( n + )Γ(2 − d )(2 n )!Γ( n + (5 − d )) (cid:104) (cid:16) H − d − H n − ( d − + 2 ln r (cid:17) W { n } a + ∂ λ W { n } a (cid:105) r n − ( d − − ( − ( d − ( d − (cid:88) n =0 ( − n Γ( n + )Γ(2 − d )Γ( ( d − − n )(2 n )! W { n } a r n − ( d − (cid:41) . (D13)The full S -field is found by substituting this equation and(D10) into (D3). The result is (53) in the main text.
2. The retarded field
We now derive the retarded point-particle field, an ex-pansion of which may be found using an integral analo-gous to (D8). Unfortunately, the relevant integration isno longer performed over the interval τ ∈ ( τ − , τ + ) , butinstead runs over all τ < τ − . There are various reasonsfor which it is undesirable to attempt expansions over thisinfinite domain, so we initially consider integrals for theretarded vector potential which are truncated at somefinite time T < τ − . We eventually find it convenient tolet T be only slightly less than τ − , although it may beviewed more generally for now.Convolving the odd-dimensional retarded Green func-tion (27) with the point-particle current density (44)while using the expansion coefficients defined by (D7), the appropriate truncated field can be shown to be A Ta = α d ∞ (cid:88) n =0 ( − n n ! W { n } a (cid:90) τ − T dτ (¯ τ − τ ) n +2 λ × (cid:34) − (cid:18) r ¯ τ − τ (cid:19) (cid:35) λ , (D14)where we have omitted the implicit limit λ → λ d . Fromthis, the full retarded field strength follows via F ret ab = 2 ∇ [ a A Tb ] + 2 q (cid:90) T −∞ ∇ [ a G ret b ] b (cid:48) ˙ γ b (cid:48) dτ. (D15)We choose to consider F ret ab here instead of A ret a in orderto avoid convergence problems when d = 3 .Now, the truncated vector potential may be evaluatedby applying the binomial theorem to expand the term insquare brackets in (D14), giving A Ta = α d ∞ (cid:88) n =0 ∞ (cid:88) k =0 ( − n + k Γ(1 + λ ) n ! k !Γ(1 + λ − k ) W { n } a r k × (cid:90) ¯ τ − rT dτ (¯ τ − τ ) n +2( λ − k ) . (D16)9Evaluating the λ → λ d limit of this expression requires some care. Omitting details, the result is that A Ta = α d (cid:40) ∞ (cid:88) n =0 ∞ (cid:88) k (cid:54) = k n ( − k + n Γ(2 − d )(¯ τ − T ) n +3 − d − k W { n } a r k n ! k !( n + 3 − d − k )Γ(2 − d − k ) + ( d − (cid:88) n =0 Γ(2 − d )Γ( ( d − − n ) W { n } a n )!Γ( − n ) r ( d − − n − ∞ (cid:88) n = ( d − ( − n + ( d − Γ(2 − d ) r n +3 − d n )!Γ( − n )Γ( n − ( d − (cid:20) H − − n − H n − ( d − + 2 ln (cid:18) r ¯ τ − T (cid:19)(cid:21) W { p } a (cid:41) , (D17)where k n ≡ [ n − ( d − . This can be substituted into (D15) to obtain the full (non-truncated) retarded field for apoint particle in an odd-dimensional spacetime.
3. The effective field
Our final task in this appendix is to compute the point-particle effective field with retarded boundary conditionsin odd dimensions. Defining the effective cut-off potential by ˆ A Ta ≡ A Ta − A Sa and comparing (D17) with (53), alllogarithms and negative powers of r exactly cancel, leaving ˆ A Ta = α d (cid:40) ∞ (cid:88) n =0 ∞ (cid:88) k (cid:54) = k n ( − k + n Γ(2 − d )(¯ τ − T ) n +3 − d − k W { n } a r k n ! k !( n + 3 − d − k )Γ(2 − d − k ) + ∞ (cid:88) n = ( d − ( − n + ( d − Γ(2 − d ) r n − ( d − n )!Γ( − n )Γ( n − ( d − × (cid:104)(cid:16) H − − n − H − d − τ − T ) /(cid:96) ) (cid:17) W { n } a − ∂ λ W { n } a (cid:105) (cid:41) . (D18)This depends on x only via non-negative even powers of r ( x ) , the smooth coefficients W { n } a ( x ; λ d ) , and ¯ τ ( x ) . Moreover,it follows from (17) and (D15) that the full effective field strength with retarded boundary conditions is ˆ F ab = 2 ∇ [ a ˆ A Tb ] + 2 q (cid:90) T −∞ ∇ [ a G ret b ] b (cid:48) ˙ γ b (cid:48) dτ. (D19)As long as the series in (D18) converge, any number of derivatives of the effective field exist, even on Γ , because ¯ τ ( x ) − T (cid:54) = 0 everywhere of interest and ¯ τ ( x ) and r ( x ) are smooth; see Appendix B.Evaluating the leading-order self-force and self-torque acting on a point particle requires that we evaluate ˆ F ab ( x ) on the particle’s worldline, where r → . Discarding terms in the truncated potential (D18) which are O ( r ) , we findthat ˆ A Ta = α d ∞ (cid:88) n (cid:54) = d − ( T − ¯ τ ) n − ( d − n !( n + 3 − d ) W { n } a + 1( d − (cid:20) W { d − } a ln ((¯ τ − T ) /(cid:96) ) + 12 ∂ λ W { d − } a (cid:21) . (D20)Using this in (D19) and letting r → + , individual terms in the resulting expression for ˆ F ab ( γ ( τ )) depend on thearbitrarily-chosen cutoff time T . Nevertheless, all such terms taken together cannot depend on T . We are thereforefree to choose T = τ − (cid:15) for some (cid:15) > , and then to take the limit (cid:15) → + . Doing so eliminates the infinite sum in n , leaving only ˆ F ab ( γ ( τ )) = 2 lim (cid:15) → + (cid:40) q (cid:90) τ − (cid:15) −∞ ∇ [ a G ret b ] b (cid:48) ˙ γ b (cid:48) dτ (cid:48) − α d (cid:34) d − (cid:88) n =0 ( − n n ! ∇ [ a W { n } b ] d − − n + 1 (cid:15) ˙ γ [ a W { n } b ] (cid:15) d − − n + 1( d − (cid:18) (cid:15) ˙ γ [ a W { d − } b ] − ∇ [ a W { d − } b ] ln( (cid:15)/(cid:96) ) − ∂ λ ∇ [ a W { d − } b ] − d −
2) ˙ γ [ a W { d − } b ] (cid:19) (cid:35)(cid:41) . (D21)A version of this expression specialized to flat spacetime is given by (55) in the main text. In either form, the0coefficients W { n } a ( γ ( τ ); λ ) and ∇ [ a W { n } b ] ( γ ( τ ); λ ) whichappear here are to be evaluated in their coincidence lim-its, and it is implicit that λ = λ d = 1 − d/ . The first fourundifferentiated and the first three differentiated coeffi-cients of this kind are given explicitly in flat spacetime by (B15) and (B16). These are sufficient to determine ˆ F ab in full for d = 3 and d = 5 . Higher-dimensional re-sults follow by extending the limit calculations describedin Appendix B. [1] J. J. Thomson, Phil. Mag. , 229 (1881).[2] M. Abraham, Ann. Phys. (Leipzig) , 105 (1902).[3] H. A. Lorentz, The Theory of Electrons and its Applica-tions to the Phenomena of Light and Radiant Heat , 2nded. (Teubner, Leipzig, 1915).[4] P. A. M. Dirac, Proc. Roy. Soc. A , 148 (1938).[5] B. S. DeWitt and R. W. Brehme, Ann. Phys. (N.Y.) ,220 (1960).[6] R. J. Crowley and J. S. Nodvik, Ann. Phys. (N.Y.) ,98 (1978).[7] H. Spohn, Dynamics of charged particles and their radi-ation field (Cambridge University Press, 2004).[8] A. D. Yaghjian,
Relativistic dynamics of a charged sphere (Springer, 2006).[9] S. E. Gralla, A. I. Harte, and R. M. Wald, Phys. Rev.
D80 , 024031 (2009).[10] A. I. Harte, Class. Quantum Grav. , 155015 (2009).[11] E. Poisson, Living Rev. Relativ. , 6 (2004).[12] A. I. Harte, in Equations of Motion in Relativistic Grav-ity , Fundamental Theories of Physics, Vol. 179, edited byD. Puetzfeld, C. Lämmerzahl, and B. Schutz (Springer,2015) p. 327.[13] A. Pound, in
Equations of Motion in Relativistic Grav-ity , Fundamental Theories of Physics, Vol. 179, edited byD. Puetzfeld, C. Lämmerzahl, and B. Schutz (Springer,2015) p. 399.[14] B. Wardell, in
Equations of Motion in Relativistic Grav-ity , Fundamental Theories of Physics, Vol. 179, edited byD. Puetzfeld, C. Lämmerzahl, and B. Schutz (Springer,2015) p. 487.[15] D. A. Burton and A. Noble, Contemp. Phys. , 110(2014).[16] A. di Piazza, T. N. Wistisen, and U. I. Uggerhøj, Phys.Lett. B , 1 (2017).[17] E. Shuryak, H.-U. Yee, and I. Zahed, Phys. Rev. D ,104006 (2012).[18] S. Detweiler and B. F. Whiting, Phys. Rev. D , 024025(2003).[19] A. I. Harte, É. É. Flanagan, and P. Taylor, Phys. Rev.D , 124054 (2016).[20] J. W. M. Bush, Annu. Rev. Fluid Mech. , 269 (2015).[21] V. Ambegaokar et al. , Phys. Rev. B , 1806 (1980).[22] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010).[23] O. Birnholtz and S. Hadar, Phys. Rev. D , 045003(2014). [24] O. Birnholtz and S. Hadar, Phys. Rev. D , 124065(2015).[25] A. I. Harte, Class. Quantum Grav. , 235020 (2008).[26] A. I. Harte, Class. Quantum Grav. , 055012 (2012).[27] S. M. Barnett, Phys. Rev. Lett. , 070401 (2010).[28] A. I. Harte, Class. Quantum Grav. , 135002 (2010).[29] J. Moxon and É. É. Flanagan, arXiv:1711.05212 .[30] A. I. Harte, Class. Quantum Grav. , 205008 (2008).[31] W. G. Dixon, J. Math. Phys. , 1591 (1967).[32] W. G. Dixon, Proc. Roy. Soc. A , 499 (1970).[33] W. G. Dixon, Phil. Trans. Roy. Soc. A , 59 (1974).[34] C. Misner, K. Thorne, and J. Wheeler, Gravitation (W.Freeman, 1973).[35] A. I. Harte, Phys. Rev. D (2013).[36] B. Allen and T. Jacobson, Comm. Math. Phys. , 669(1986).[37] F. G. Friedlander, The Wave Equation on a CurvedSpace-Time (Cambridge University Press, 2010).[38] I. M. Gel’fand and G. E. Shilov,
Generalized Functions,Vol. I: Properties and Operations (Academic Press, NewYork, 1964).[39] J. Ehlers and E. Rudolph, Gen. Rel. Grav. , 197 (1977).[40] S. E. Gralla, A. I. Harte, and R. M. Wald, Phys. Rev.D , 104012 (2010).[41] L. F. O. Costa and J. Natário, in Equations of Motionin Relativistic Gravity , Fundamental Theories of Physics,Vol. 179, edited by D. Puetzfeld, C. Lämmerzahl, andB. Schutz (Springer, 2015) p. 215.[42] D. V. Gal’tsov and P. A. Spirin, Gravitation Cosmol. ,241 (2007).[43] P. O. Kazinski, S. L. Lyakhovich, and A. A. Sharapov,Phys. Rev. D , 025017 (2002).[44] B. P. Kosyakov, Th. Math. Phys. , 493 (1999).[45] D. Galakhov, Sov. J. Exp. Theor. Phys. Lett. , 452(2008).[46] C. R. Galley, Private communication.[47] D. V. Gal’tsov, Phys. Rev. D , 025016 (2002).[48] Y. Yaremko, J. Math. Phys. , 092901 (2007).[49] G. Satishchandran and R. M. Wald, Phys. Rev. D (2018).[50] L. M. Burko, Class. Quantum Grav. , 3745 (2002).[51] L. M. Burko, A. I. Harte, and E. Poisson, Phys. Rev. D (2002).[52] A. Matas, D. Müller, and G. Starkman, Phys. Rev. D (2015).[53] V. Moretti, Comm. Math. Phys.212