Self-forces on extended bodies in electrodynamics Abraham I. Harte Institute for Gravitational Physics and Geometry, Center for Gravitational Wave Physics, Department of Physics, The Pennsylvania State University, University Park, PA 16802 (Dated: January 16, 2006) In this paper, we study the bulk motion of a classical extended charge in flat spacetime. A formalism developed by W. G. Dixon is used to determine how the details of such a particle’s internal structure influence its equations of motion. We place essentially no restrictions (other than boundedness) on the shape of the charge, and allow for inhomogeneity, internal currents, elasticity, and spin. Even if the angular momentum remains small, many such systems are found to be affected by large self-interaction effects beyond the standard Lorentz-Dirac force. These are particularly significant if the particle’s charge density fails to be much greater than its 3-current density (or vice versa) in the center-of-mass frame. Additional terms also arise in the equations of 6 motion ifthedipolemomentistoolarge, andwhenthe‘center-of-electromagnetic mass’isfar from 0 the ‘center-of-bare mass’ (roughly speaking). These conditions are often quite restrictive. General 0 equationsofmotionwerealsoderivedundertheassumptionthattheparticlecanonlyinteractwith 2 the radiative component of its self-field. These are much simpler than the equations derived using n thefull retarded self-field; as are theconditions required to recoverthe Lorentz-Dirac equation. a J PACSnumbers: 03.50.De,04.25.-g 0 2 2 v I. INTRODUCTION 3 2 Originally motivated by the discovery of the electron, the behavior of small (classical) electric charges has been 1 8 studied in various contexts for over a century. The first results were obtained by Abraham [1] for a non-relativistic 0 non-spinning rigid sphere. This calculation was later repeated by Schott [2] and Lorentz [3] within special relativity, 5 and by Crowley and Nodvik [4] in general (background) spacetimes. Similar results have since been obtained by a 0 number of authors for more general charge distributions [5, 6, 7, 8, 9, 10, 11] in flat spacetime. These results have / c very recently been extended to curved backgrounds as well [39]. Still, these derivations did not allow for significant q elasticity, charge-currentcoupling, and/or rotation. - Detailedreviewsofdifferentaspectsoftheself-forceproblem(inelectromagnetismaswellasscalarfieldtheoryand r g general relativity) have been given by Poisson [17], Havas [18], and Spohn [19]. To summarize, though, a common : theme throughoutallofthese workshasbeen thatthe equationsofmotiondescribingsufficiently smallparticleswere v i found to be independent of the details of their internal structure to a considerable degree of precision. Only a few X parameters such as the rest mass and total charge entered into these equations. Ignoring the effects of spin and r internal currents, the apparently universal correction to the Lorentz force law is given by the well-known Lorentz- a Dirac self-force. Ifa particle withchargeq has acenter-of-masspositionza(s¯)(with s¯being a propertime), then this self-force is equal to (using a metric with signature 2, and units in which c=1) − 2 ... q2 za(s¯)+z˙a(s¯)z¨b(s¯)z¨ (s¯) . (1.1) b 3 (cid:16) (cid:17) Although this is only an approximation to the full self-force for any realistic (extended) charge, it is natural to introduce a new concept into the theory for which it is exact – that of a point particle. The immediate problem with this is of course that the self-field diverges at the locationof any point-like source,which would appear to imply that its self-force and self-energy are not well-defined. Dirac [12] removed this problem by noting that self-forces acting on a structureless particle should only arise as a reaction to emitted radiation. Letting Fab denote the advanced self(+) self-fieldandFab the retardedone,we candefine ‘radiative’and‘singular’portionsofFab =Fab +Fab : self(−) self(−) self(S) self(R) 1 Fab = Fab +Fab , (1.2) self(S) 2 self(−) self(+) 1(cid:16) (cid:17) Fab = Fab Fab . (1.3) self(R) 2 self(−)− self(+) (cid:16) (cid:17) As the names imply, the singular field contains the entire divergent part of the retarded field. It is also derived from a time-symmetric Green function, so it would not be expected to contain any radiation. The self-force on a point 2 particle should therefore be determined entirely by the radiative self-field. Combining it with the Lorentz force law immediately recovers (1.1) [12]. This prescription is much simpler than any direct derivation of the equations of motion for a finite charge, and for this reason, it has been generalized to work in curved spacetime, as well as for scalar and gravitational self-forces [17,34](althoughthisisnottheonlywayof‘renormalizing’pointparticleself-fields[26,27]). Asbefore,therelevance oftheseextensionstorealisticextendedbodieshasalsobeenestablishedincertainspecialcases. Inlinearizedgravity, for example, a small nearly-Schwarzchildblack hole has been found to obey the same equations of motion as a point particle [17, 25]. More generally, it has been shown that a nonspinning body’s internal structure is irrelevant to very highorderwithin Post-Newtoniantheory[13]. Still, thereremainquestionsofexactlyhowuniversaltheseresultsare. Will a spherical neutron star in near-equilibrium fall into a supermassive black hole in the same way as another one that is spinning rapidly and experiencing internal oscillations [38]; or one having a mountainous (solid) surface [37]? Such systems could be important sources for the upcoming Laser Interferometer Space Antenna (LISA). Rather than addressing such questions directly, we have chosen to study the motions of charged bodies in flat spacetime (in part) as a model problem. The methods used here were specifically chosen so that very few conceptual changes would be required to consider particles (charged or not) moving in a fully dynamic spacetime. This has led to some additional complexity not strictly necessary to solve the problem at hand, although the majority of the complicating issues have been placed in the the appendix. Our derivation is based on W. G. Dixon’s multipole formalism [20, 21, 22, 23, 24]. This gives a relatively simple, unified, and rigorous way of understanding the motions of arbitrarily structured bodies in both electrodynamics and general relativity. Despite the fact that Dixon’s theory decomposes the source functions into multipole moments, we never ignore any of them. By using generating functions, the entire infinite set is retained throughout all of our calculations. This onlyappearstobe possible inthis formalism. The lawsofmotionthatareusedarethereforeexact (despite being ordinarydifferentialequations). All ofthe approximationsthatwe makeare only used to compute the self-field. Takingahybridpointofviewwhereanextendedbodycanonlyinteractwithitsradiativeself-field,itisfoundthat the Lorentz-Dirac equation follows for a wide range of nonspinning charges with small dipole moments. If the body instead interacts with its full retarded self-field, this result no longer holds. In this case, the equations of motion are drastically different if the charge and (3-) current densities have remotely similar magnitudes in the center-of-mass frame. Although it is impractical to define exactly what this means at this point, it will be shown that one of these quantities must be at least ‘second order’ compared to the other in order for these extra terms to vanish. This is becausethese casesusuallyallowthe self-fieldsto dosignificantamountsofinternalwork(e.g. Ohmicheating). Even whenthecharge-currentcouplinginnegligible,therearestillextracomplicationsintheretardedcase. Theconditions required to generically exclude these and other complicating effects are derived, and turn out to be surprisingly restrictive. Sec. II reviews the various steps involved in calculating the motion of matter interacting with an electromagnetic field. Sec. III then summarizes the appropriate definitions of the center-of-mass and its laws of motion as obtained from Dixon’s formalism. It also decomposes the 4-current in a particular way that happens to be convenient in this framework. Although this reduction is not strictly required for the current problem, it is adopted throughout on the groundsthat it wouldbe essentialin curvedspacetime. It is derivedin detail inthe appendix, which alsocontainsan in-depth review of of Dixon’s ideas. With these basic ideas in place, Sec. IV goes on to derive expansions for the advanced and retarded self-fields of an arbitrarily-structured charge using a slow motion approximation. Sec. V then combines these results with those from Sec. III to find general expressions for the self-force and self-torque. Finally, Sec. VI examines the equations of motion for certain simpler classes of charges, and derives some conditions under which the Lorentz-Dirac equation is applicable. We use units in which c = 1 throughout. In order to facilitate a more direct comparison to Dixon’s papers, the metric is chosen to have signature 2 (although the rest of the notation used here frequently differs from Dixon’s). − Wealsoassumethatthespacetimeisflat,andadoptMinkowskicoordinatesforsimplicity. Latinindicesrefertothese coordinates, while Greek ones are triad labels running from 1 to 3. II. THE PROBLEM OF MOTION In studying the dynamics of any system in a classical field theory, one has to specify ‘laws of motion’ for both the field and matter variables. In our case,the only field is ofcourse the electromagneticone, Fab =F[ab]. As usual, this 3 is governed by Maxwell’s equations ∂ Fab =Ja , (2.1) b ∂[aFbc] =0 . (2.2) We assume that the matter in our problem is completely described by its stress-energy tensor Tab and 4-current vector Ja. Taking the divergence of (2.1) immediately gives our first constraint on these quantities: ∂ Ja =0 . (2.3) a This equation acts as one of the laws of motion for the matter fields. The other is derived from the requirement that ∂ Tab+Tab =0, where Tab is the stress-energy tensor of the electromagnetic field. Combining the standard form a em em of Tab [8] with Maxwell’s equations then shows that the matter moves according to (cid:0)em (cid:1) ∂ Tab = FabJ . (2.4) b b − (2.1)-(2.4) are essentially the entire content of classical continuum mechanics in flat spacetime. Of course, different types ofmatter do not allmoveinthe same way,so these equations by themselvesare notsufficientto determine Tab and Ja for all time (even if the Fab were given). One also needs to specify something analogous to a (generalized) equation of state, which can take on a rather unwieldy form. This sort of procedure is the standard one in continuum mechanics. Unfortunately, the resulting nonlinear partial differential equations are notoriously difficult to solve. Such a detailed description of the system should not really be required,however,forproblemswhereweareonlyinterestedinthebody’sbulkmotion. Inthesecases,arepresentative worldlinecouldbedefinedinsidethe(convexhullofthe)spacelike-compactsupportofTab. Giventhatthisworldline canalwaysbeparametrizedbyasinglequantity,itstangentvectormightbeexpectedtosatisfyanordinarydifferential equation – at least when using certain approximations. Solving such an equation would clearly be much more straightforwardthan the partial differential equations that we started with. This sort of simplification is one of the main motivations behind the many (source) multipole formalisms in the literature [8, 30]. In these, one first fixes some particular reference frame which has, among other properties, a preferred time parameter s. The quantity being expanded – say Ja(x) – can then be written in terms of an infinite set of tensors depending only on s: Qa(s), Qab(s),... The reverse is also true. Given Ja, there is a well-defined way to compute any moment. The set Q... is therefore completely equivalent to Ja. This implies that the conservation { } equation (2.3) may be used to find restrictions on the individual moments. Such restrictions depend on the precise definitions that are being adopted, but can usually be divided into two general classes. The first of these consists of purely algebraic equations imposed at a fixed value of s. We call these the constraints. There are also a number of evolution equations which usually take the form of ordinary differential equations. Multipole expansions can therefore be used to convert (2.3) and (2.4) into a number of algebraic and ordinary differential equations (without any approximation). This does not actually simplify things as muchas it at first might appear. The reasonis that almostall definitions for the source multipoles will lead to an infinite number of (coupled!) evolution equations. This is often dealt with in practice by assuming that all moments above a particular order are irrelevant, which leaves one with only a finite number of evolution equations. There are, however, interesting questions that require knowing the higher moments. Ja (orTab)forexample,cannotbereconstructedwithoutthem. Thismeansthattheself-fieldcannotbecalculatedin thenearzonefromonlythefirstfewmoments. Althoughtheself-forceandself-torquecanbefoundincertaincasesby examiningenergyandmomentumfluxes inthe farzone [12,15, 17, 25](whichcanbe adequatelyapproximatedusing only a finite number ofmoments), this is considerablyless accuratethanintegratingthe forcedensity throughoutthe charge’s interior. Forthesereasonsandothers,itisdesirabletodefineasetofmultipolesthatdonotrequireanycutoff. Remarkably, suchasetexists[20],andisessentiallyunique[23]. Withoutanyapproximation,momentsforbothJa andTab canbe defined which satisfy a finite number of evolutionequations. There remain(uncoupled) constraintequations for each moment, although these are easily solved. We adopt this formalismdue to Dixon for the remainder of this paper. Its neteffectistoallowustorelatethemotiontothe fieldsinamorerigorouswaythanisusuallydone(shortofdirectly solving (2.3) and (2.4)). It does not, however,have anything to say about the fields themselves. We therefore obtain the self-field in a standard way, and then use Dixon’s equations to find how the matter moves in response to it. III. LAWS OF MOTION As noted, we use Dixon’s method [20, 21, 22, 23, 24] to decompose Ja and Tab into multipole moments. Each of these sets is designed to describe as simply as possible all possible forms of Ja and Tab satisfying their respective 4 conservationequations. Wefirstassumethatthesematterfieldsareatleastpiecewisecontinuous,andhave(identical) supports W. Any intersection of a spacelike hypersurface with this worldtube is assumed to be compact. Now choose a timelike worldline Z W, and a timelike unit vector field na(s) defined on Z. It is assumed that this isalwayspossibleinanyphysically⊂interestingsystem. Z is thenparameterizedby the coordinatefunctionza(s), and the tangent vector to it is denoted by va(s):=dza/ds=:z˙a(s). va need not be equal to na, although it will be convenient to normalize s such that nav =1 (so vav =1 in general). va is called the kinematical velocity, while na a a 6 is the dynamical velocity. The set Z,na then defines a reference frame for the definition of the multipole moments. { } At this point, it should be thought of as arbitrary, although physical conditions will later be given that pick out a unique ‘center-of-mass’frame. A collection of spacelike hyperplanes Σ can easily be constructed from Z and na. Each Σ(s) is to pass through { } za(s), and be (everywhere) orthogonal to na(s). Assume that any point in W is contained in exactly one of these planes. Unless na is a constant, it is clear that this property cannot be true throughout the entire spacetime. For x Σ(s) W, we must therefore have that max n˙a(s) x z(s) <1 (among other conditions), which gives a weak ∈ ∩ | − a| restriction on the body’s maximum size. It is not really important, though, as any reasonable type of matter would (cid:0) (cid:1) be ripped apart long before this condition was violated. The main results that we need from Dixon’s theory at this point are his definitions of the linear and angular momenta. These disagree with the usual ones when either Ja or Fab are nonzero, although it is still convenient to label them by pa(s) and Sab(s) respectively. Unless otherwise noted, the words ‘linear and angular momenta’ will always refer to the quantities [20, 21] 1 pa(s) = dΣ Tab+Jbr duFac z(s)+ur , (3.1) b c ZΣ(s) (cid:20) Z0 (cid:21) 1 (cid:0) (cid:1) Sab(s) = 2 dΣ r[aTb]c+Jcr duur[aFb]d z(s)+ur , (3.2) c d ZΣ(s) (cid:20) Z0 (cid:21) (cid:0) (cid:1) where ra :=xa za(s), and u is just a dummy parameter used to integrate along the line segment connecting za(s) − to xa =za(s)+ra. Detailed motivations for these definitions can be found in [20, 21, 23], as well as the appendix. In short, though, it canbeshownthatthegivenquantitiesareuniquelydeterminedbydemandingthatstress-energyconservationdirectly affect only the first two moments of Tab (once the concept of a moment has been defined in a reasonable way). If pa andSab areknowninsometime intervalandsatisfythe appropriateevolutionequations,the classofallstress-energy tensors with these moments can be constructed without having to solve any differential equations. Eachof these will exactly satisfy (2.4). This property implies that evolution equations for the quadrupole and higher moments of the stress-energy tensor arenearlyunconstrained. They canbe thoughtofasthe ‘equationofstate’ofthe materialunder consideration. This type of independence of the higher moments from the conservation laws also occurs in Newtonian theory [24], and there are considerable advantages in preserving as much of that structure as possible in the relativistic regime. In particular,there is no need to discard multipole moments above a certain order. The choices (3.1) and (3.2) allow us to retain many of the conveniences of a multipole formalism without its classic limitations. The same definitions for the momenta can also be motivated by considering charged particles in curved spacetime [21, 24]. There, one can study the conserved quantities associated with Killing vectors in appropriate spacetimes (where both the metric and electromagnetic field are assumed to share the same symmetries). Fixing Z and na(s) allows each such quantity to be written as a linear combination of vector and antisymmetric rank 2 tensor fields on Z. Crucially, the definitions of these quantities do not depend on the Killing vector under consideration, so we can suppose that they are meaningful even in the absence of any symmetries. If the metric is taken to be flat, these objects reduce to (3.1) and (3.2) [21]. They are therefore the natural limits of what would generally be referred to as the linear and angular momenta of symmetric spacetimes. It is now convenient to define a coordinate system on W that is more closely adapted to the system than the Minkowskicoordinatesusedsofar. Firstchooseanorthonormaltetrad na(s),ea(s) (α=1,2,3)alongZ. Requiring { α } that each of these vectors remain orthonormalto the others implies that e˙a = nan˙ eb , (3.3) α − b α whichis essentiallyFermi-Walkertransport. A spatial(rotation)termmay alsobe addedto this, althoughourcalcu- lations would then become considerably more tedious. On its own, the choice of tetrad has no physical significance, so we choose the simplest case. Since it was assumed that Σ foliates W, any point x Σ(s) W may now be uniquely written in terms of s and a ‘triad radius’ rα { } ∈ ∩ xa =ea(s)rα+za(s) . (3.4) α 5 Varying rα with a fixed value of s clearly generates Σ(s). Also, the Jacobian of the coordinate transformation (xa) (rα,s) is equal to the lapse N of the foliation, → N(x) = 1 n˙ (s)ra , a − = 1 n˙ (s)ea(s)rα . (3.5) − a α We will be extensively transforming back and forth between these two coordinate systems, so it is convenient to abuse the notation somewhat by writing f(rα,s) = f(xa) for any function f. The intended dependencies should alwaysbe clearfromthecontext. Itisalsousefultodenotequantitiessuchasn˙ ea byn˙ . Notethatinthis notation, a α α n¨ =n¨ ea =dn˙ /ds. α a α 6 α Using these conventions, it is natural to split Ja into the charge and 3-current densities seen by an observer at r =0 (the ‘center-of-mass observer’) Ja =ρna+eajα . (3.6) α It is then shown in the appendix that there exist ‘potentials’ ϕ and Hα which generate Ja through the equations ρ = ∂ (rαϕ) , (3.7) α jα = N−1 Hα+vβ∂ (rαϕ) rαϕ˙ . (3.8) β − h i Denoting the total charge by q and letting r2 := rαr 0, ϕ(rα,s) was found to be continuous, and equal to α q/4π r3 outside W. Similarly, Hα rβ,s is g|iv|en by−(A40)≥outside W, and is piecewise continuous in rβ. Hα also | | satisfies ∂ Hα = 0. These properties guarantee that Ja has support W, and is everywhere continuous. A direct α (cid:0) (cid:1) calculation also shows that ∂ ∂ Ja = N−1n vβ∂ Jb+∂ jβ , (3.9) a b β β ∂s − (cid:18) (cid:19) = 0 , (3.10) as required by (2.3). Any physically reasonable current vector can now be constructed by choosing potentials satisfying these rules. A physical interpretation of one’s choice is then given by substitution into (3.7) and (3.8). For example, a uniform spherical charge distribution with time-varying radius D(s) is described by (assuming na =va) 3 q D(s) ϕ(r,s) = Θ D(s) r + Θ r D(s) , (3.11) 4πD3(s)" −| | (cid:18) |r| (cid:19) | |− # (cid:0) (cid:1) (cid:0) (cid:1) Hα(r,s) = 0 , (3.12) where Θ() is the Heaviside step function. (3.7) and (3.8) then show that the tetrad components of Ja are · 3q ρ(r,s) = Θ D r , (3.13) 4πD3 −| | 3qr(cid:0)α (cid:1) D˙(s) jα(r,s) = Θ D(s) r . (3.14) 4πN(r,s)D3(s) D(s)! −| | (cid:0) (cid:1) It is also clearly possible to calculate ϕ and Hα from any given Ja. This requires inverting (3.7), which acts as a partial differential equation for ϕ on each time slice. The solution to this equation would usually have to be obtained numerically, which is clearly inconvenient. Largely for this reason, we shall consider ϕ and Hα to be the given quantities for the remainder of this paper. Ja can be derived from them using operations no more complicated than differentiation. Another reason for this unconventional choice is that ϕ and Hα contain all of the multipole moments of Ja in a natural way. As shown in the appendix, they are closely related to the Fourier transform of a generating function for these moments. An arbitrary current moment can be obtained essentially by differentiating the inverse Fourier transforms of the potentials a suitable number of times. For example, (A1), (A24), (A30), (A31), and (A38) can be used to show that the dipole moment has the general form q Qab = d3r 2n[aeb]rβN ϕ(r,s) +eaebH¯αβ(r,s) , (3.15) β − 4π r3 α β Z (cid:20) (cid:18) | | (cid:19) (cid:21) 6 where H¯αβ =H¯[αβ] is defined by Hα =∂ H¯αβ . (3.16) β Given (A40), we let qv[αrβ] H¯αβ = (3.17) 2π r3 | | outside W. For the example given in (3.11) and (3.12), the dipole moment is equal to 1 Qab = qD2(s)n[an˙b] . (3.18) 5 This might have been expected to vanish in spherical symmetry, although it should be noted that it is only nonzero when sphericity is being defined in an accelerated reference frame. TheseconstructionscannowbeusedtosimplifythedefinitionsofpaandSab. Lettingr¯α :=urα,theelectromagnetic term in (3.1) is equivalent to 1 d3r¯eβr¯ Fab(r¯,s) duu−4ρ(r¯/u,s) . (3.19) b β Z Z0 But (3.7) shows that 1 1 ∂ q duu−4ρ(r¯/u,s) = du u−3 ϕ(r¯/u,s) , (3.20) − ∂u − 4π r¯/u3 Z0 Z0 (cid:20) (cid:18) | | (cid:19)(cid:21) q = ϕ(r¯,s) , (3.21) − − 4π r¯3 (cid:18) | | (cid:19) so (3.1) and (3.2) can be rewritten as q pa = d3r Tabn ϕ r eγFac , (3.22) b− − 4π r3 γ c Z (cid:20) (cid:18) | | (cid:19) (cid:21) q Sab = 2 d3r rαe[aTb]cn ϕ r eγrαe[aFb]c . (3.23) α c− − 4π r3 γ c α Z (cid:20) (cid:18) | | (cid:19) (cid:21) We now need evolutionequations for these quantities, which are most easily obtainedby direct differentiation. For any function Ib(x), relating the x-coordinates of (rα,s+ds) to those of (rα,s) shows that d dΣ Ib(x)= d3r n˙ Ib+(vc ncn˙ rα)∂ n Ib . (3.24) b b α c b ds − ZΣ(s) Z h (cid:0) (cid:1)i If Ib vanishes outside of some finite radius, this expression simplifies to d dΣ Ib(x)= d3rN∂ Ib . (3.25) b b ds ZΣ(s) Z (2.4), (3.22), and (3.23) can now be used to show that ∂ q p˙a = d3r NFabJ + ϕ r eγFac , (3.26) −Z ( b ∂s(cid:20)(cid:18) − 4π|r|3(cid:19) γ c (cid:21)) and ∂ q S˙ab = 2 d3r Nrαe[aFb]cJ +v[aTb]cn + ϕ r eγrαe[aFb]c , − Z ( α c c ∂s(cid:20)(cid:18) − 4π|r|3(cid:19) γ c α (cid:21)) ∂ q = 2p[avb] 2 d3r Nrαe[aFb]cJ + ϕ r eγrαe[aFb]c − ( α c ∂s − 4π r3 γ c α Z (cid:20)(cid:18) | | (cid:19) (cid:21) q + ϕ r eγv[aFb]c . (3.27) − 4π r3 γ c ) (cid:18) | | (cid:19) 7 If the momenta had been defined in the usual way, the (ϕ q/4π r3) terms would be absent from these expressions. − | | The extra complication in the present case derives from the electromagnetic couplings in (3.1) and (3.2). Unsurprisingly, (3.26) and (3.27) can be directly related to the monopole and dipole moments of the force density FabJ . Denoting such moments by Ψa(s) and Ψab(s) respectively, it is possible to prove (A56) and (A57). It is then b natural to refer to Ψa as the net force, and 2Ψ[ab] as the net torque acting on the body. Viewing these quantities − − as force moments allows one to derive (A61) and (A62). But these results are no different than (3.26) and (3.27). If desired, the reader may therefore view Ψa and Ψ[ab] to be defined by (A56) and (A57). In cases where Fab varies slowly in the center-of-mass frame (both spatially and temporally), the expressions for the force and torque can be expandedin Taylor seriesinvolving successivelyhigher multipole moments of the current density (denoted by Q···). Deriving such equations would be awkward using the methods introduced in this section, so we simply refer to (A63) and (A64). Despite the peculiar definitions of the current moments being used, these expansions are exactly what one would expect out of a multipole formalism, and can therefore be considered a check that Dixon’s definitions are reasonable. In most cases where (A63) and (A64) are any simpler than the exact expressions for the force and torque, only the monopole and dipole moments will be significant. These moments can be computed from (A13) and (3.15) respectively. In the limit that the particle is vastly smaller than any of the field’s length scales, only the monopole momentwillenterthe equationsofmotion. Inthis case,the forcereducestothe standardLorentzexpressionandthe torque vanishes, as expected. The final ingredients required to complete this formalism are unique prescriptions for Z and na. Simply knowing the linear andangularmomentaatanypoint intime does notnecessarilydetermine the body’s locationin anyuseful way. The problem is compounded by the fact that these quantities are strongly dependent on the choice of reference frame itself. Indeed, without knowing where W is, it is essentially impossible to specify Ja in any meaningful way. TheseproblemscanberemovedbychoosingZ andna appropriately,andthenassumingthattheresultingworldline provides a reasonable representation of the body’s ‘average’position. Following [21, 23, 24, 28], we first assume that for any point z W, there exists a unique future-directed timelike unit vector na(z) such that ∈ pa(z;n)=M(z;n)na(z) , (3.28) for some positive scalar M. Here, we have temporarily changed the dependencies of pa and na for clarity. It is seen from (3.1) that pa depends nontrivially on both the base point z, and on na, which defines the surface of integration. (3.28) is therefore a highly implicit definition of na. In any case,another conditionmust be alsobe givento fixz. We wantthis to lie ona ‘center-of-massline’ in some sense, so it would be reasonable to expect the ‘mass dipole moment’ defined with respect to it to vanish: n n tabc =0 , (3.29) b c where tabc is the full dipole moment of the stress-energy tensor. This is defined by (A47), so (3.29) is equivalent to n (z)Sab(z;n)=0 . (3.30) a The integral form of (3.30) reduces to the standard center-of-mass condition when Fab = 0 and na = va. It also allows us to write the angular momentum in terms of a single 3-vector Sa Sab =ǫabcdn S . (3.31) c d Of course, both of these properties would also have been satisfied by instead requiring v Sab = 0. We reject this a choice due to the fact that it leads to nonzero accelerations even when Fab = 0 [32]. Replacing va by na avoids this peculiar behavior, which we consider to be an important requirement for anything deserving to be called a center-of-mass line. It also seems more natural to define the mass dipole moment in terms of na rather than va. For the remainder ofthis paper, we assumethat (3.28)and(3.30) arealwayssatisfied,andcallthe resultingZ and na(s) the center of mass frame [21, 24, 28]. It is not obvious that solutions to these highly implicit equations exist, although existence and uniqueness have been proven in the gravitational case [29]. There, it is also true that (given certain reasonable conditions) Z is necessarily a timelike worldline inside the convex hull of W. We assume that the same results extend to electrodynamics. The uniqueness of these definitions is actually not very critical for our purposes. The important point is that a solution with the given properties can presumably be chosen for a sufficiently large class of systems. While it is clearlyverydifficulttofindthecenter-of-massdirectlyfromTab,Ja,andFab,itisrelativelystraightforwardtosimply constructsets of moments whichautomatically incorporate(3.28) and(3.30). (3.7) and(3.8) show,for example, that these definitions do not have any effect on our ability to construct arbitrary current vectors adapted to them. There 8 is nothing preventing the moment potentials ϕ and Hα from being appropriately centered around rα =0. Although it is not obvious that this can also be done for the stress-energy tensor, we conjecture that it can. Now that (3.28) and (3.30) have been assumed to hold, we need evolution equations for M, na, and za. These are easily found from (A56) and (A57): M˙ = naΨ , (3.32) a − Mn˙a = haΨb , (3.33) − b M(va na) = Sabn˙ 2Ψ[ab]n , (3.34) b b − − where ha is the projection operator δa nan . Note that the last of these equations shows that na = va if the spin b b − b and torque both vanish. In general, (A57) and (3.32)-(3.34) may be used together to find the motion of the body’s center-of-mass in terms of Ψa and Ψ[ab]. Once the field is known, these quantities follow from (A61) and (A62). Some recipe for evolving the dipole and higher current moments in time – most conveniently expressed in terms of ϕ˙ and H˙α – is also required. Combining all of these elements together leaves us with a well-defined initial value problem that will determine za, na, M, Sab, and Ja. If the stress-energy tensor is also desired, possible forms of it could in principle be constructed from (A51) and (A54) in the same way that Ja was derived from ϕ and Hα. Combining all of these steps would completely characterize the system, although we shall omit the last one. The result is still sufficient to answer most questions that one would be interested in asking of a nearly isolated particle. IV. THE FIELD It is clear from the previous section that the equations of motion will easily follow once Ψa and Ψ[ab] are known. These depend on the field, which we now calculate. For a reasonably isolated body, it is first convenient to split Fab into two parts Fab =Fab +Fab . (4.1) ext self Theexternalfieldisassumedtobegeneratedbyoutsidesources,whiletheself-fieldisentirelyduetothechargeitself. Since (A61) and (A62) are linear in Fab, the force and torque can also be split up into ‘self’ and ‘external’ portions. In Lorenz gauge, the vector potential sourced by the particle is given by [8] Aa (x)= d4x′Ja(x′)δ σ(x,x′) , (4.2) self − Z (cid:0) (cid:1) where 1 σ(x,x′)= (x x′)a(x x′) , (4.3) a 2 − − is Synge’s world function [17, 31]. In realistic systems, this potential will couple to the external one via the outside matter fields. These may act as reflectors or dielectrics, or there may be an n-body interaction where the self-fields influence the motion of some external charged particles (obviously affecting the fields in W). Although it would be reasonable to group together all portions of Fab causally related to our particle in some way as the ‘self-field,’ this would be impossible to do with any generality. Instead, we simply define the Fab to be the field derived from (4.2) self in the usual way (this differs from the point of view taken in e.g. [35]). The interactions of the self-field with the outside world will all be categorized as ‘external’ effects that we presume can be accounted for by separate methods. IfalloftheexternalmatterissufficientlyfarawayfromW (andslowlyvarying),Fab willbeapproximatelyconstant ext within eachΣ(s) W slice (and fromone slice to the ‘next’). Ψa and Ψ[ab] can thereforebe approximatedby (A63) and(A64)inthes∩ecases. Findingtheself-forceandself-torqueisexmtorecomexptlicated. Forthis,Fab hastobecombined self withthe exactintegralexpressionsforthe forceandtorque–(A61)and(A62). The detailedstructureofthe self-field must therefore be known throughout W. This easily follows from Aa : self Fab (x) = 2∂[aAb] , self self = 2 d4x′δ′(σ)(x x′)[aJb] . (4.4) − − Z 9 Writing x in terms of (rα,s), and x′ in terms of (r′α,τ), and defining σ˙(x,x′):=∂σ/∂τ, Fab becomes self 1 d N Fab (x)=2 d3r′ (x x′)[aJb] . (4.5) self(±) Z (|σ˙|dτ (cid:20)σ˙ − (cid:21))τ=τ± τ+ (>s) represents the advanced time, and τ− the retarded one. These are found by solving σ(x,x′)=0 with x and r′ held fixed. Although we only consider the retardedfield to be real,the advanced solutionis alsoretained for now. This allows us to later construct the radiative self-field, which is considerably simpler than the full retarded field. It would be quiteconvenientiftheself-forcesgeneratedbythetwofieldswereidentical(asDiracassumedforpointparticles[12]), although we will show that this is not true in general. Returning to the explicit form for Fab , splitting up Ja according to (3.6) shows that self N σ¨ Fab (x) = 2 d3r′ ρ (x x′)[an˙b] +N−1n¨αr′ (x x′)[anb] v[anb] +ρ˙(x x′)[anb] self Z σ˙|σ˙|( (cid:20) − −(cid:18)σ˙ α(cid:19) − − (cid:21) − σ¨ jα (x x′)[aeb] +N−1n¨βr′ +n˙ (x x′)[anb]+ Nn[a+(h v)[a eb] − − α σ˙ β α − · α (cid:20) (cid:18) (cid:19) (cid:16) (cid:17) (cid:21) +j˙α(x x′)[aeb] . (4.6) − α) Here, j˙α(r′,τ):=∂jα(r′,τ)/∂τ, which differs from our usual convention (e.g. n˙α(τ):=eα(τ)dna(τ)/dτ). a Moving on, (4.3) implies that σ˙(x,x′)= N(r′,τ)na(τ)+(h v)a(τ) (x x′) , (4.7) a − · − (cid:18) (cid:19) and d(h v)a σ¨(x,x′)=N2+vαv Nn˙a nan¨βr′ + · (x x′) . (4.8) α− − β dτ − a (cid:18) (cid:19) If we now specify how ϕ and Hα vary in time (which determines ρ˙ and j˙α), we would have all of the ingredients necessary to find the body’s motion without any approximation. Unfortunately, inserting (4.6) into (A57), (A61), (A62), and (3.32)-(3.34) leads to a set of delay integro-differential equations for the object’s motion. Such a system would be extremely difficult to solve (or even interpret) in general, although it could be a useful starting point for numerical simulations. It might also be interesting analytically if one were looking for the forces required to make a body move in some pre-determined way (e.g. a circular orbit). Such questions will not be discussed here. We instead consider the simplest possible approximations that allow us to gain insight into a generic class of systems. In particular, it is assumed that all of the quantities in (4.6) which depend on τ± may be written in terms of quantities at s (via Taylor expansion). This requires that nothing very drastic happen on timescales less than the body’s light-crossing time. This is not as trivial a condition as it might seem to be (see e.g. [33]), although we will not attempt to justify it. Expressingeverythingintermsofsratherthanτ± firstrequirescalculating∆± :=s τ±. τ± wasdefinedbyσ =0, so ∆± canbe foundby Taylorexpanding this equationin ∆±. Assuming that both na−andva areat leastC3 in time for s [τ±,s], ∈ 1 1 ... eaα(τ±) = eaα(s)+∆±na(s)n˙α(s)− 2∆2± n˙a(s)n˙α(s)+na(s)n¨α(s) − 6∆3±eaα ξ±(1) , (4.9) 1 (cid:16) 1 1 ... (cid:17) (cid:16) (cid:17) za(τ±) = za(s) ∆±va(s)+ ∆2±v˙a(s) ∆3±v¨a(s)+ ∆4±va ξ±(2) , (4.10) − 2 − 6 24 (cid:16) (cid:17) where ξ±(1) and ξ±(2) are some numbers between τ± and s. Then (3.4) shows that 1 1 (x−x′)a ≃ eaα(r−r′)α+∆± Nna+(h·v)a + 2∆2± nan¨αrα′ −Nn˙a + 6∆3±n¨a. (4.11) (cid:16) (cid:17) (cid:16) (cid:17) 10 ... Everything here has been written in terms of s, and all terms involving quantities such as n˙ 3, n¨ 2, n , h v 2, and v˙ n˙ have been removed. Since ∆± r r′ , these terms can be reasonably neglecte|d|if .|n..| 3| | n¨| ·2 | n˙ | −1,|and v˙ n˙ h v n˙ | , w|h∼er|e −:=|max r, r′ . In a sense, we are expanding u|p|tRo se≪co|nd|Rord≪er | |R≪ | − |R≪| · |≪| |R R | | | | in powers of n˙ , and up to first order in v . The requir|em|Rent n˙ 1 must hold fo|r|all (r,r′) pairs(cid:0), so it(cid:1)is useful to define the largest possible value of . | |R ≪ R We call this the body’s ‘radius’ D(s):= max r . (4.12) Σ(s)∩W| | Using it, n˙ 1 implies n˙ D 1. This technically restricts the allowable size of the charge, although very few | |R ≪ | | ≪ reasonable systems would actually be excluded. Interpretingthe relationsatisfiedby(h v)a (=va na) isn’tquite assimple. Given(3.34), it is roughlyequivalent · − toassumingthatspineffectsarepresentonlytolowestnontrivialorder. Thisisnotcompletelyaccurate,though,and more precise statements will be given in the following section. In any case, the inequalities following (4.11) will be assumed to hold from now on. Using them to expand σ = 0, we find that 1 1 R2 := r r′ 2 ∆± 2vα(r r′)α+∆±N(r,s)N(r′,s)+ ∆2±n¨α(r+2r′)α+ ∆3± n˙ 2 . (4.13) | − | ≃ − 3 12 | | (cid:20) (cid:21) All but the second term here is already ‘small,’ but not quite negligible in our approximation. The zeroth order expressionfor ∆± (= R)may therefore be substituted into eachof these terms without any overalllossof accuracy. ∓ The resulting equation is easily solved: 1 1 1 1 3 2 ∆± ≃ ∓R 1+ 2n˙α(r+r′)α− 2n˙αn˙βrαrβ′ ± 6Rn¨α(r+2r′)α+ 24R2n˙αn˙α+ 8 n˙α(r+r′)α (cid:20) (cid:16) (cid:17) R−1vα(r r′) , (4.14) α ± − (cid:21) where everything is evaluated at s. Although it would still be straightforward at this stage to compute the exact error in (4.14), it is not necessary. ... Dimensionalanalysisshowsthattheneglectedtermshavemagnitudes n 4, n¨ 2 5, d(h v)/ds 2, d2(h v)/ds2 3, | |R | | R | · |R | · |R h v 2 , and so on (where each of these can be evaluated anywhere in the interval [τ±,s]). | · | R Continuing to expand quantities appearing in (4.6), a long but straightforward calculation shows that (4.7) and (4.8) can be approximated by 1 1 2 1 1 σ˙ R 1 n˙α(r+r′) n˙α(r r′) Rn¨α(r+2r′) + R2 n˙ 2 , (4.15) α α α ≃± − 2 − 8 − ∓ 3 8 | | (cid:20) (cid:16) (cid:17) (cid:21) and 1 σ¨ 1 n˙α(r+r′) Rn¨α(r+2r′) +n˙αn˙βr r′ + R2 n˙ 2 . (4.16) ≃ − α∓ α α β 2 | | Looking at (4.6), the charge and current densities are the only quantities that have not yet been expanded away fromτ±. Itis instructive to temporarilyleavethem like this, but write out allother terms infield ats. The resulting expression is quite long, so we break it up into several smaller pieces by writing Fab in the form self(±) 1 Fsaeblf(±) =2 d3r′R3 ρ(r′,τ±)f(a1b)+ρ˙(r′,τ±)f(a2b)−jβ(r′,τ±)fβa(b3)+j˙β(r′,τ±)fβa(b4) . (4.17) Z (cid:18) (cid:19) The definitions of each of these coefficients is obvious from comparison with (4.6). Before computing them, we first simplify the notation by defining a quantity T(s) > 0 such that T−1 remains (marginally) less than n˙ , n¨ 1/2, h v 1/2/D, etc. This is useful because a number of different objects were assumed | | | | | · | to be negligible in this section. Some of these may be much larger than the others, so using them to writing down error estimates would become rather awkward. Introducing T removes this difficulty.