Motion in classical field theories and the foundations of the self-force problem
MMotion in classical field theories and thefoundations of the self-force problem
Abraham I. Harte ∗ Albert-Einstein-InstitutMax-Planck-Institut f¨ur GravitationsphysikAm M¨uhlenberg 1, 14476 Golm, GermanyAugust 7, 2015
Abstract
This article serves as a pedagogical introduction to the problem ofmotion in classical field theories. The primary focus is on self-interaction:How does an object’s own field affect its motion? General laws governingthe self-force and self-torque are derived using simple, non-perturbativearguments. The relevant concepts are developed gradually by consideringmotion in a series of increasingly complicated theories. Newtonian gravityis discussed first, then Klein-Gordon theory, electromagnetism, and finallygeneral relativity. Linear and angular momenta as well as centers of massare defined in each of these cases. Multipole expansions for the force andtorque are derived to all orders for arbitrarily self-interacting extendedobjects. These expansions are found to be structurally identical to thelaws of motion satisfied by extended test bodies, except that all relevantfields are replaced by effective versions which exclude the self-fields ina particular sense. Regularization methods traditionally associated withself-interacting point particles arise as straightforward perturbative limitsof these (more fundamental) results. Additionally, generic mechanismsare discussed which dynamically shift — i.e., renormalize — the appar-ent multipole moments associated with self-interacting extended bodies.Although this is primarily a synthesis of earlier work, several new resultsand interpretations are included as well.
How are charges accelerated by electromagnetic fields? How do masses fall incurved spacetimes? Such questions can be answered in many different ways.Consider, for example, the Newtonian n -body problem. This is typically posedas a system of ordinary differential equations which describe the motions of ∗ Email: [email protected] a r X i v : . [ g r- q c ] A ug points in R . Besides its position, each point is characterized only by itsmass. This is a considerable abstraction from the extended stars or planetswhose motions the n -body problem is physically intended to describe, and theinternal density distributions, velocity fields, and temperatures of each bodymight be governed by complicated sets of nonlinear partial differential equations.From one point of view, it is the solutions to these continuum equations whichrepresent “the motion” of each mass.This is not, however, the approach which is typically adopted in celestialmechanics. In that context, one instead focuses only on a body’s center of mass(and perhaps its spin angular momentum): observables which describe motion“in the large.” It is a central result of Newtonian gravity that much of thedynamics of these observables can be understood without detailed knowledgeof each body’s internal structure. This is why the extended stars and associ-ated partial differential equations of the “physical n -body problem” can oftenbe modeled as discrete points satisfying a simple set of ordinary differentialequations — an enormous simplification.This work is intended as an introduction to techniques which have recentlybeen developed [1, 2, 3, 4, 5] to similarly simplify problems of motion in a widevariety of contexts. Is it possible, for example, to describe extended masses ingeneral relativity using appropriately-defined centers of mass? Do these masscenters obey simple laws of motion? Of course, the same questions may alsobe asked for charged matter coupled to electromagnetic fields. In simple cases,appropriate laws of motion are well-known in both electromagnetism and gen-eral relativity; sufficiently small test charges accelerate via the Lorentz forcelaw and sufficiently small test masses fall on geodesics. Test body motion isalso understood in cases where a body’s higher multipole moments cannot beneglected [6].Although it has historically been difficult to relax the test body assumptionin relativistic theories, several important cases have nevertheless been under-stood [7, 8, 9, 10, 11, 12, 13, 14]. The majority of this work has been intrin-sically perturbative. It makes detailed assumptions about the systems to bestudied and uses these assumptions at all stages in the analysis. Concepts likethe mass and momentum of individual objects typically arise as purely pertur-bative structures with no clear connection to the full theory. This review takesa different approach. Although approximations may be needed to understandspecific applications, we adopt the point of view that approximating exact con-cepts is preferable to considering structures which emerge only as artifacts of aparticular approximation scheme. We therefore focus on non-perturbative de-scriptions of motion. Somewhat surprisingly, considerable progress can be madefrom this perspective. Indeed, applying perturbation theory “too early” servesmainly to increase the computational burden and to obscure the underlyingphysics.The first step in our program is to define exact linear and angular momentafor arbitrary extended objects . It is these momenta which are used to charac- Classical point particles are sometimes discussed as though they were the fundamental . Nevertheless, many of the conceptsused here are likely to be unfamiliar. Considerable effort has therefore beendevoted to explaining these concepts slowly and carefully by applying them ina series of increasingly complicated contexts.The prototype for all of our discussion is Newtonian celestial mechanics. Thelaws of motion for this theory are therefore reviewed in Section 2. This servestwo purposes. First, Section 2.1 uses standard techniques to remind the readerwhich ideas are important and why they are true. What, for example, is a self-field? Why do the self-force and self-torque vanish in Newtonian gravity? Themethods used to discuss these questions make essential use of the vector spacestructure of Euclidean space, and cannot be generalized to curved spacetimes.Section 2.2 introduces techniques which remove this objection. The Newto-nian problem is reformulated such that all references to the detailed propertiesof Euclidean space are eliminated. Indeed, all that is required at this stage isa Riemannian space which admits a maximal set of Killing vector fields. Us-ing this, a “generalized momentum” is introduced which serves to describe abody’s large-scale behavior. The ordinary linear and angular momenta arise astwo aspects of this more fundamental structure, and are therefore equivalent.Many computations simplify, however, when expressed in terms of the general-ized momentum rather than its linear and angular components. Employing it,Newtonian self-forces and self-torques are seen to vanish using a one-line com-putation which employs only the symmetry of an appropriate Green function.That symmetry is physically related to Newton’s third law.From this perspective, certain generalizations of Newtonian gravity may beconsidered with almost no additional effort. The standard Euclidean back-ground space can, for example, be replaced by one which is spherical or hyper-bolic. The usual laws of motion still hold in these cases, except for the additionof Mathisson-Papapetrou spin-curvature couplings. Such terms arise kinemat-ically even in these non-relativistic problems, and are shown to have a simplegeometrical interpretation.Fully relativistic motion is first discussed in Section 3, although only in flator otherwise maximally-symmetric backgrounds. We first consider the motionof matter coupled to a linear scalar field. It is shown that a non-perturbativeself-field can be defined in this context which is only a slight generalization building blocks of all classical matter. This viewpoint is severely problematic on both mathe-matical and physical grounds, and is rejected here. That said, appropriately-regularized pointparticles do arise as mathematical structures obtained from certain well-defined limits involv-ing families of extended bodies. All mention of point particles here is to be understood in this(effective) sense. As is typical throughout physics, simple underlying principles do not imply simple appli-cations to explicit problems. Applications typically do require significant computations.
3f its Newtonian analog. Unlike in the non-relativistic case, however, forcesand torques exerted by relativistic self-fields do not necessarily vanish. Theyinstead act to renormalize an object’s linear and angular momenta. This effectis finite and non-perturbative. Physically, it represents the inertia associatedwith an object’s self-field. Mathematically, it is related to the hyperbolicity ofthe underlying field equations. Similar effects apply generically for all mattercoupled to long-range hyperbolic fields.Next, Section 4 considers motion in fully generic curved spacetimes. TheKilling vectors used to define momenta in simpler cases must then be replacedby an appropriate set of “generalized Killing fields.” This is accomplished inSection 4.1. The scalar problem of motion is analyzed first in this more generalcontext, where a new type of renormalization is found to occur which affects thequadrupole and higher multipole moments of a body’s stress-energy tensor, andmay be viewed as a consequence of the “passive gravitational mass distribution”of an object’s self-field. Matter coupled to electromagnetic fields in genericbackground spacetimes may be understood similarly, and is discussed in Section4.3. Finally, Section 4.4 considers motion in general relativity. Notation
The sign conventions used here are those of Wald [15]. Metrics have positivesignature. The Riemann tensor satisfies 2 ∇ [ a ∇ b ] ω c = R abcd ω d for any 1-form ω a , and the Ricci tensor is defined by R ab = R acbc . Abstract indices are denotedusing letters a, b, . . . from beginning of the Latin alphabet, while i, j, . . . representcoordinate components. Boldface symbols are used to denote Euclidean vectorsand tensors. Units are chosen such that c = G = 1. Consider a Newtonian test body immersed in a gravitational potential φ ( x , t ). Ifsuch a body is sufficiently small, it is well-known that its center of mass location γ t at time t evolves via ¨ γ t = − ∇ φ ( γ t , t ) . (1)This is not the correct equation of motion for (non-spherical) objects with sig-nificant self-gravity. Relaxing the test body assumption while still imposing anappropriate smallness condition instead results in¨ γ t = − ∇ ˆ φ ( γ t , t ) , (2) It is common in the literature to use the words renormalization and regularization in-terchangeably, both implying the removal of unwanted infinities. This is not the usage here.Renormalization is intended in this review essentially as a synonym for “dynamical shift.”These shifts need not be infinite. Regularizations, by contrast, always refer to rules forhandling singular behavior. Almost all discussion here focuses on finite renormalizations.Regularizations arise only in certain limiting cases. φ denotes that part of the potential which is determined only by massesexternal to the body of interest. Comparison of these two equations shows thatthe field ∇ ˆ φ which accelerates a large mass can differ from the field ∇ φ whichwould be inferred by measuring the accelerations of nearby test particles. Al-though it is not often emphasized, this is a standard result in Newtonian gravity.The well-known laws of motion which describe “Newtonian point masses” are,for example, equivalent to (2), not (1).A central goal of this review is to explain how similar results hold in morecomplicated relativistic theories. In all cases, the laws of motion are structurallyidentical to those associated with test bodies. The fields which appear in thoselaws of motion are not, however, the physical ones. Effective fields appearinstead, their details depending on appropriate notions of self-interaction. Oncethe precise nature of the effective field has been determined in a particulartheory, “point particle limits” and related approximations follow very easily.Many of the difficulties encountered in the relativistic theory motion alreadyappear in the Newtonian problem (where they can be so simple as to easily passby unnoticed). It is therefore instructive to open this review by carefully dis-cussing the Newtonian theory of motion. Section 2.1 accomplishes this usingessentially standard arguments. Concepts such as the self-field and self-forceare emphasized, as well as their connections to physical principles like Newton’sthird law. Similar discussions may be found in, e.g., [14, 16, 17]. Unfortunately,the familiar techniques used in Section 2.1 cannot be readily applied to morecomplicated theories. Section 2.2 therefore uses Newtonian gravity as a familiarsetting with which to introduce a different, more geometrical, approach. The re-sulting formulation, some of which originally appeared in [2, 5], does generalize.It is used throughout the remainder of this review. Consider an extended body residing inside a finite (and possibly time-dependent)region of space B t ⊂ R which contains no other matter. The mass density ρ and momentum density ρ v of this body are both assumed to be smooth. Localmass and momentum conservation then imply that [17, 18, 19] ∂ρ∂t + ∇ · ( ρ v ) = 0 (3)and ∂∂t ( ρ v ) + ∇ · ( ρ v ⊗ v − τ ) = f . (4) Although the gradients of φ and ˆ φ coincide at the center of a spherically-symmetric mass,they can be quite different in general. Consider, for example, a barbell constructed by joiningtwo unequal spheres with a massless strut. In simple cases, v represents a velocity field in the standard sense. More generally, itmight be only an effective construction. This occurs, for example, if a body is composed ofmultiple interpenetrating fluids. τ describes how matter interacts via contact forces,while the force density f describes longer-range interactions (also known as bodyforces). If the only long-range forces are gravitational, there exists a potential φ such that f = − ρ ∇ φ. (5)Inside B t , the potential must satisfy Poisson’s equation ∇ φ = 4 πρ. (6)The influence of masses external to B t may be encoded using, e.g., φ or itsnormal derivative on the boundary ∂ B t .Equations (3)-(6) are very general. They are not, however, complete. Im-posing appropriate boundary conditions, Poisson’s equation determines φ interms of ρ , mass conservation evolves ρ using v , and momentum conservationevolves v using τ . The stress tensor cannot, however, be determined withoutadditional assumptions. Its evolution is not universal. Stresses depend on anobject’s detailed composition, reflecting the trivial fact that different types ofmaterials move differently. This is but a minor obstacle in celestial mechanics,and no particular form for τ is assumed here.Observables which describe a body’s “large-scale” motion may be obtainedby integrating the conservation laws (3) and (4). This results in the total mass m , linear momentum p ( t ), and angular momentum S ( z t , t ): m := (cid:90) B t ρ d x , p := (cid:90) B t ρ v d x , S := (cid:90) B t ρ ( x − z t ) × v d x . (7)The general philosophy of celestial mechanics is to focus on p and S while ignor-ing ρ and v as much as possible. The vast majority of information concerningan object’s internal structure is set aside; only its momenta matter. Evolutionequations for these momenta are easily obtained from (3) and (4), which showthat the mass remains constant and that˙ p = − (cid:90) B t ρ ∇ φ d x , ˙ S = − (cid:90) B t ρ ( x − z t ) × ∇ φ d x − ˙ z t × p . (8)The gravitational force and torque acting on an extended body therefore dependon its mass distribution, its internal gravitational potential, and a “choice oforigin” parametrized by z t .One reason for considering p is its close relation to the center of mass position γ t . This can be defined by γ t := 1 m (cid:90) B t x ρ ( x , t )d x , (9)or equivalently by demanding that a body’s mass dipole moment vanish whenevaluated about γ t : (cid:90) B t ( x − γ t ) ρ ( x , t )d x = 0 . (10) This follows from noting that any sufficiently small piece of matter with finite densityresponds to gravitational forces as though it were a test body. m ˙ γ t = p . (11)Note that this is a derived result, not a definition; m , γ t and p are defined interms of ρ and v via (7) and (9). In more complicated theories, the center ofmass velocity need not be parallel to the momentum.Once γ t has been defined, its time evolution is easily found by combining(8) and (11) to yield m ¨ γ t = − (cid:90) B t ρ ∇ φ d x. (12)Evaluating S ( z t , t ) with z t = γ t isolates the spin component of the angular mo-mentum from the orbital component which can appear more generally, resultingin ˙ S = − (cid:90) B t ρ ( x − γ t ) × ∇ φ d x . (13)In astrophysical applications, ρ and ∇ φ are typically only very coarsely con-strained by observations. Integral expressions like (12) and (13) are thereforeunsuitable for applications. They must first be simplified.Such simplifications are immediate if ∇ φ varies negligibly throughout B t ,as can occur if the mass in question is a test body whose dimensions are smallcompared with the distances to all other masses in the universe. In these cases,it follows from (12) and (13) that the center of mass acceleration satisfies (1) andthat ˙ S = 0. Simplifying the laws of motion in more general contexts requiresunderstanding the influence of an object’s own gravitational field.A precise definition for the self-field may be obtained via a two-point function(or propagator) G ( x , x (cid:48) ) describing “the gravitational potential at x per unitmass at x (cid:48) .” Any potential constructed from such a propagator can reasonablybe called a self-field only if it is a Green function for the Poisson equation: ∇ G ( x , x (cid:48) ) = 4 πδ ( x − x (cid:48) ) . (14)There are, of course, many possible Green functions. A particular one may besingled out by demanding that self-fields described by G be compatible withNewton’s third law. Consider two distinct points x , x (cid:48) ∈ B t . It is then naturalto interpret “the force on mass at x due to mass at x (cid:48) ” to mean − ρ ( x , t ) ∇ (cid:2) G ( x , x (cid:48) ) ρ ( x (cid:48) , t )d x (cid:48) (cid:3) d x . (15)The weak form of Newton’s third law states that the force at x due to x (cid:48) mustbe equal and opposite to the force at x (cid:48) due to x , implying that ∇ G ( x , x (cid:48) ) = − ∇ (cid:48) G ( x (cid:48) , x ) . (16)The strong form of Newton’s third law instead requires that the force at x dueto x (cid:48) point along the line which connects these two points. Imposing this, ∇ G ( x , x (cid:48) ) ∝ x − x (cid:48) . (17)7ny G ( x , x (cid:48) ) which is compatible with the strong form of Newton’s third lawcan therefore depend only on the distance | x − x (cid:48) | between its arguments. Upto an irrelevant additive constant, it follows from (14) that G ( x , x (cid:48) ) = G ( x (cid:48) , x ) = − | x − x (cid:48) | , (18)and the total self-field is φ S ( x , t ) := (cid:90) B t ρ ( x (cid:48) , t ) G ( x , x (cid:48) )d x (cid:48) = − (cid:90) B t ρ ( x (cid:48) , t ) | x − x (cid:48) | d x (cid:48) . (19)The physical field φ may be viewed as the sum of the self-field φ S and anappropriate remainder ˆ φ : ˆ φ := φ − φ S . (20)It follows from (6), (14), and (19) that the effective potential ˆ φ satisfies thevacuum field equation ∇ ˆ φ = 0 throughout B t .Now consider the total force exerted by φ S , the “self-force.” Noting (8), itis natural to let this refer to F S := − (cid:90) B t ρ ∇ φ S d x . (21)Substituting (19) into this expression results in an integral over the productspace B t × B t : F S = − (cid:90) B t × B t ρ ( x , t ) ρ ( x (cid:48) , t ) ∇ G ( x , x (cid:48) )d x d x (cid:48) . (22)Recalling (16) or (18), the integrand is antisymmetric under interchange of x and x (cid:48) . The Newtonian self-force therefore vanishes. This is an exact result.It holds for all compact mass distributions. Whatever the shape a particularbody happens to be in B t , the self-force vanishes because that shape is triviallysymmetric when copied into B t × B t . A similar argument may be used to showthat the self-torque, the net torque exerted by φ S , vanishes as well.The main point of this discussion is that the net gravitational force exertedon any isolated extended mass satisfies˙ p = − (cid:90) B t ρ ∇ ( ˆ φ + φ S )d x = − (cid:90) B t ρ ∇ ˆ φ d x . (23)Not necessarily choosing z t to be the center of mass, the equivalent evolutionequation for S ( z t , t ) is˙ S = − (cid:90) B t ρ ( x − z t ) × ∇ ˆ φ d x − ˙ z t × p . (24)The vanishing self-force and self-torque therefore allows φ to be replaced by ˆ φ in the evolution equations for both the linear and angular momenta. Although8orces and torques may be computed using either the physical field φ or thefictitious effective field ˆ φ , the latter computation is often simpler. In most casesof practical interest, ∇ ˆ φ varies far more slowly in B t than does ∇ φ . The integralinvolving ∇ ˆ φ can therefore be amenable to approximation when the (otherwiseequivalent) integral involving ∇ φ is not.Recalling that ˆ φ is harmonic inside the body region, it must be analytic there.This means that its Taylor series about an arbitrary point z t ∈ B t converges atleast in some neighborhood of z t . If that series converges throughout the body,it may be substituted into (23) and integrated term by term. The integralexpression for the force is then equivalent to˙ p i ( t ) = − m∂ i ˆ φ ( z t , t ) − ∞ (cid:88) n =1 n ! m j ··· j n ( z t , t ) ∂ i ∂ j · · · ∂ j n ˆ φ ( z t , t ) , (25)where m j ··· j n ( z t , t ) denotes the body’s 2 n -pole mass moment about z t : m j ··· j n ( z t , t ) := (cid:90) B t ( x − z t ) j · · · ( x − z t ) j n ρ ( x , t )d x . (26)The series (25) is referred to as a multipole expansion for the force. A similarseries also exists for the angular momentum. If z t is chosen to coincide withthe center of mass γ t , the dipole moment m i ( γ t , t ) vanishes by (10). The n = 1term in (25) therefore vanishes as well, so¨ γ it = − ∂ i ˆ φ ( γ t , t ) − m ∞ (cid:88) n =2 n ! m j ··· j n ( γ t , t ) ∂ i ∂ j · · · ∂ j n ˆ φ ( γ t , t ) . (27)The utility of this expression is that there are many cases of interest wherethe multipole series can be truncated at low order without significant loss ofaccuracy. The simplest such truncation recovers (2). More generally, thereare correction to this equation which involve a body’s quadrupole and highermultipole moments.Lastly, note that the moments (26) are somewhat different from the oneswhich are found in textbooks. The harmonicity of ˆ φ implies that arbitrary tracesmay be added to the m j ··· j n without affecting the force. The m j ··· j n appearingin (25) may therefore be replaced by different moments ˜ m j ··· j n which are trace-free in all pairs of indices. It is these trace-free moments which are typicallyused in practical calculations. Besides the elimination of irrelevant components,the trace-free moments are also useful in that they may be determined purelyusing external measurements of an object’s gravitational field. The discussion which has just been presented relies heavily on the geometricpeculiarities of Euclidean space. This is not essential, however. The only char-acteristic of (three-dimensional) Euclidean space which is truly important is9hat it is maximally symmetric: There exist a total of six linearly independentKilling vector fields. The Newtonian laws of motion are now rederived usingmethods which make this manifest.As a consequence, certain aspects of the Newtonian problem are significantlyclarified. The geometrical nature of the linear and angular momenta is madeprecise, for example. These are shown to be two aspects of a more fundamentalvector which lives not in the physical space, but in a space which is dual tothe space of Killing vector fields. The approach introduced in this section alsoemphasizes the importance of symmetries, and is fundamental to understandingmotion in the more complicated theories discussed below.Another advantage of the reformulation discussed in this section is thatcertain generalizations of Newtonian gravity may be understood essentially“for free.” Noting that spherical and hyperbolic spaces are both maximally-symmetric, there are no new complications if the usual Euclidean backgroundof Newtonian gravity is replaced by a space of constant curvature. It is alsotrivial to change the number of spatial dimensions, or to add, e.g., a mass termto the field equation. For concreteness, we restrict to three spatial dimensionsand keep the gravitational field equation as-is. We do, however, allow the back-ground space to be curved. This has interesting consequences which reappearin the more complicated relativistic theories considered in later sections.
The locations of Newtonian events may be viewed as points in a four-dimensionalmanifold M . While a relativistic spacetime is defined using only a manifold anda non-degenerate metric, Newtonian spacetimes require more structure [18, 19,20]. One such structure is a preferred notion of time. This takes the form ofan equivalence class of functions which associate each event in spacetime with“the time” at which it occurs. Associated with this is a preferred foliation of M into a one-parameter family of hypersurfaces {S t } , the spaces of constant time.Newtonian spacetimes are difficult to work with directly. They simplifyconsiderably in the presence of a frame, a structure that identifies events atdifferent times as being at “the same” spatial point. It is assumed here thata frame has been fixed in such a way that all spaces S t are mapped into asingle space consisting of a three-dimensional manifold S together with a (fixed)Riemannian metric g ab . This process also fixes a particular time function. Itpermits all physical quantities in spacetime to be viewed as time-dependentquantities on S . We allow the spatial metric to be curved, but assume that itscurvature is everywhere constant. Letting ∇ a and R abcd denote the covariantderivative and Riemann tensor associated with g ab , it follows that ∇ a R bcdf = 0,and that ( S , g ab ) is maximally symmetric.Consider the motion of a material object instantaneously confined to a sub-manifold B t ⊂ S which contains no other matter and has finite volume. Denotethis body’s mass density at time t by ρ ( · , t ) and its velocity field at time t by For any single time function T : M → R and any c, d ∈ R such that c >
0, the map cT + d is also an acceptable time function. a ( · , t ). Local conservation of mass and momentum continue to hold in thiscontext, so (3) and (4) carry over essentially without change: ∂∂t ρ + ∇ a ( ρv a ) = 0 , (28) ∂∂t ( ρv a ) + ∇ b ( ρv a v b − τ ab ) = − ρ ∇ a φ. (29)The gravitational potential φ which appears here satisfies the obvious general-ization of Poisson’s equation: ∇ a ∇ a φ = g ab ∇ a ∇ b φ = 4 πρ. (30) Our first significant departure from the elementary discussion of Newtonian mo-tion found in Section 2.1 arises in the definitions for a body’s linear and angularmomenta. The usual integrals (7) make sense only when evaluated in a Carte-sian coordinate system. Alternatively, they require a canonical identification oftangent spaces associated with different points in the spatial manifold. Whilethis is easily accomplished in Euclidean space, it is not obvious what to do moregenerally. Our first task is therefore to define momenta which do not make ref-erence to a specific coordinate system. Accomplishing this provides a notion ofmomentum which is easily generalized to curved Newtonian backgrounds, andeven to completely generic relativistic spacetimes. It is a basic building blockfor all results discussed in this review.One problem with elementary definitions of mechanical momentum is thatthey attempt to represent this concept as a spatial vector or covector. This isphysically unnatural, however (except for point particles or momentum densi-ties); momenta are generically associated with extended regions, not individualpoints. There is no natural tangent or cotangent space in which to place, forexample, the momentum of an extended region R ⊂ B t . The simplest mathe-matical structure with which to represent a quasi-local quantity like a momen-tum must itself be quasi-local, and spatial tensors are not examples of suchstructures.Besides being quasi-local, momenta must also be extensive. For any twodisjoint regions R , R ⊂ B t which are “physically independent,” there mustbe a sense in which(momentum in R ) + (momentum in R ) = (momentum in R ∪ R ) (31)for some binary operation “+” which is both associative and commutative. If R and R are identically prepared, it is also natural to suppose that(momentum in R ) + (momentum in R ) = 2(momentum in R ) , (32)thus motivating a notion of scalar multiplication.11ogether, these considerations and others suggest that momenta should beelements of a vector space. The most natural vector space is not, however, thespace of tensors at any particular spatial point. A better choice may be mo-tivated by recalling that conserved linear momenta arise naturally in theorieswhich are derived from translation-invariant Lagrangians. Similarly, conservedangular momenta arise from Lagrangians which are invariant with respect torotations. This suggests that both types of momenta can be associated explic-itly with a collection of continuous symmetries. Consider, in particular, thosesymmetries — the continuous isometries — which preserve the spatial metric.While these are not necessarily symmetries for all physically-interesting quan-tities, they are extremely useful.The continuous isometries of a Riemannian space ( S , g ab ) are generated byits Killing vector fields. By definition, L ξ g ab = 0 (33)for every Killing vector ξ a , where L ξ denotes the Lie derivative with respect to ξ a . We use K to denote the vector space consisting of all Killing vector fieldstogether with obvious notions of addition and scalar multiplication. It is well-known that the dimension of this vector space is finite. More specifically, if thedimension of the physical space is dim S = dim B t = N , it may be shown that(see, e.g., Appendix C of [15])dim K ≤ N ( N + 1) . (34)This section restricts attention to maximally-symmetric spaces where dim K = N ( N + 1). When N = 3, Euclidean, spherical, and hyperbolic spaces are allmaximally-symmetric, admitting six linearly-independent Killing fields. Givena preferred point, three Killing fields may be interpreted as translations andthree as rotations. This split makes sense only near the given point, however,and is best avoided at this stage. Doing so implies that the linear and angularmomenta should be treated as elements of a single object “conjugate to” thespace of all Killing vector fields.More specifically, consider a representation for a body’s momentum as avector in the space K ∗ which is dual to K . An element of K ∗ is, by definition,a linear map from K to R . The specific linear map which has the desiredproperties is P t [ R ]( ξ ) := (cid:90) R ρ ( x, t ) v a ( x, t ) ξ a ( x )d V, (35)where ξ a ∈ K and the volume element is the natural one associated with g ab . Wecall this the generalized momentum contained in R ⊆ B t at time t . It is oftenconvenient to omit the dependence on R , in which case it is to be understoodthat P t = P t [ B t ].The dimension of K ∗ is equal to the dimension of K , so this momentumhas six components in three spatial dimensions. These components correspondto the usual three components of linear momentum and three components of12ngular momentum. Such a split can be made explicit by introducing additionalstructure, namely a preferred point z t ∈ B t , and for any such point, P t [ R ] canbe re-expressed in terms of spatial tensors p a , S a at z t . This decomposition isexplained in Section 2.2.6. For now, it suffices to consider P t [ R ] on its own.While the introduction of a preferred point allows this map to be replacedby spatial tensors, avoiding such representations whenever possible providesconsiderable calculational and conceptual simplifications.In relativistic contexts where there exists a maximally-symmetric backgroundgeometry, the generalized momentum remains essentially unchanged. The in-finitesimal momentum ρv a d V is merely replaced by T ab d S b , where T ab is anappropriate stress-energy tensor and d S b is the natural volume element on athree-dimensional hypersurface. If a spacetime is not maximally-symmetric,one also replaces K by another vector space which has the correct dimension-ality. The “generalized Killing fields” used for this purpose are discussed inSection 4.1. How does Newtonian gravity affect the time evolution of the generalized momen-tum? Using local momentum conservation (29) and assuming that the boundary ∂ R is independent of time (or that there is no matter there),dd t P t [ R ]( ξ ) = (cid:90) R (cid:2) − ρ L ξ φ + 12 ( ρv a v b − τ ab ) L ξ g ab (cid:3) d V = − (cid:90) R ρ L ξ φ d V. (36)The second equality here follows from Killing’s equation (33). If L ψ φ = 0for some specific Killing field ψ a , it is clear that the associated momentum P t [ R ]( ψ ) is conserved. This means that if φ is constant along a translationalKilling field, there can be no force in that direction. Similarly, a field whichis invariant about rotations around a given axis exerts no torque about thataxis. Both of these statements are physically obvious. They are also of limitedvalue. Once the field equation (30) is taken into account, L ψ φ = 0 implies that L ψ ρ ∝ L ψ ∇ a ∇ a φ = ∇ a ∇ a L ψ φ = 0 as well. This is clearly impossible for anycompact body if ψ a is a pure translation. Rotational symmetries fare somewhatbetter, although they are still a rather special case.Transforming (36) into a surface integral results in a more interesting con-servation law. Using the field equation and integrating by parts shows thatdd t P t [ R ]( ξ ) = − (cid:73) ∂ R T ab ξ b d S a , (37)where T ab := 14 π ( ∇ a φ ∇ b φ − g ab ∇ c φ ∇ c φ ) (38)is the stress tensor associated with φ . At least in flat space, one might imagineextending ∂ R (and perhaps ∂ B t ) far outside of all matter of interest. If φ fallsoff sufficiently fast in this region, the surface integral can be seen to vanish.13he generalized momentum is therefore conserved in such cases. Of course, themomentum associated with a single object in a larger system is not conserved.Understanding its dynamics requires a different argument. The generalized force (36) involves the physical field φ . As discussed in Section2.1, this is too complicated to work with directly. We therefore isolate itsmost complicated part — the self-field — and compute what it does directly.Once this is accomplished, the remaining undetermined portion of the force isrelatively simple to understand.The self-field in this context is defined in Section 2.1 in terms of a certaintwo-point function G . More specifically, G is a Green function. If G ( x, x (cid:48) ) = G ( x (cid:48) , x ), the two constraints (16), (17) which implied a notion of Newton’s thirdlaw in Euclidean space generalize to the statement that L ξ G ( x, x (cid:48) ) = (cid:2) ξ a ( x ) ∇ a + ξ a (cid:48) ( x (cid:48) ) ∇ a (cid:48) (cid:3) G ( x, x (cid:48) ) = 0 (39)for all ξ a ∈ K . In the Euclidean case, translational invariance alone impliesthe weak form of Newton’s third law. Further imposing rotational invariancerecovers the strong form of Newton’s third law. In general, though, symmetriesof G imply only “portions of” Newton’s third law.It is always possible to find Green functions which satisfy (39) in maximally-symmetric backgrounds. Indeed, these Green functions depend only on thegeodesic distance between their arguments. Introducing Synge’s function (alsoknown as the world function) [10, 21, 22] σ ( x, x (cid:48) ) := 12 (squared geodesic distance between x and x (cid:48) ) , (40)the Euclidean Green function (18) can be written as G = − / √ σ . Greenfunctions associated with spherical and hyperbolic spaces are merely more com-plicated functions of σ [23]. In any of these cases, L ξ G ∝ L ξ σ = 0.Using the symmetric Green function which satisfies (39) to define the self-field, let φ S ( x, t ) := (cid:90) B t ρ ( x (cid:48) , t ) G ( x, x (cid:48) )d V (cid:48) . (41)Substituting this into (36) then shows thatdd t P t = − (cid:90) B t ρ ( x, t ) L ξ ˆ φ ( x, t )d V − (cid:90) B t d V (cid:90) B t d V (cid:48) ρ ( x, t ) ρ ( x (cid:48) , t ) L ξ G ( x, x (cid:48) )= − (cid:90) B t ρ ( x, t ) L ξ ˆ φ ( x, t )d V, (42)14here ˆ φ = φ − φ S and R has been replaced by the entire body region B t . It isclear from this that the self-force and self-torque both vanish as an immediateconsequence of (39). All forces and torques may therefore be computed using ˆ φ instead of φ . Furthermore, the effective field satisfies the vacuum equation ∇ a ∇ a ˆ φ = 0 , (43)and can clearly be computed by subtracting the self-field from the physical field.Alternatively, Stokes’ theorem may be used together with (43) to write ˆ φ as akind of average of φ over a closed surface which surrounds the body of interest.It has already been mentioned that P t ( ψ ) is conserved if L ψ φ = 0. Equation(42) shows that this also true if L ψ ˆ φ = 0, a much weaker condition. For aclosed system, one typically has φ = φ S and hence ˆ φ = 0. All components ofthe generalized momentum are therefore conserved in such cases.Equation (42) has been established by showing that the generalized forceexerted by φ S always vanishes. This force involves an integral over B t × B t ,and may therefore be interpreted as a two-point interaction. It can sometimesbe interesting to also consider interactions between three or more points. Let˜ φ S ( x, t ) := n max (cid:88) n =1 c n (cid:90) B t d V · · · (cid:90) B t d V n ρ ( y , t ) · · · ρ ( y n , t ) G n ( x, y , . . . , y n ) , (44)where the c n are arbitrary constants and the ( n + 1)-point propagators G n aresymmetric in their arguments and satisfy L ξ G n for all ξ a ∈ K . It is straight-forward to show that the generalized force exerted by any such field vanishes.Given the two-point G used to define φ S , an appropriate three-point interactionmay be chosen using, e.g., G ( x, y, z ) = G ( x, y ) G ( y, z ) G ( z, x ) . (45)Other choices are also possible, of course. Higher-order propagators typicallylead to fields ˜ φ S which are not really Newtonian self-fields in the sense that ∇ a ∇ a ˜ φ S (cid:54) = 4 πρ . Series like (44) can nevertheless be useful for understandingdifferent theories where matter couples to nonlinear fields. In those cases, thesum in ˜ φ S might be compared to a kind of Dyson series for an object’s self-field.Regardless of the field equation, however, the existence of a Killing field ψ a which satisfies L ψ ( φ − ˜ φ S ) = 0 for some ˜ φ S always implies that P t ( ψ ) is con-served. Although this conservation law might be manifest only for a particularchoice of ˜ φ S , the value of P t ( ψ ) does not depend on that choice. Returning to the main development, note that (36) and (42) differ only by thereplacement φ → ˆ φ . Although both of these integrals are numerically equivalent,the latter is often simpler to evaluate. This is because L ξ ˆ φ can be readily ap-proximated throughout B t in many more physically-interesting situations thancan L ξ φ . Such approximations are based on a Taylor expansion of ˆ φ . While15his has an obvious meaning in Euclidean space, a technical diversion is neededto explain what is meant by Taylor expansions more generally.Given an origin z t ∈ B t about which a particular Taylor expansion is to beperformed, the most natural Cartesian-like coordinate systems are the Riemannnormal coordinates with origin z t . These are unique up to rotations, and maybe used to perform Taylor expansions in the usual way.To be more precise, recall that the exponential map exp x X a = x (cid:48) takes asinput a point x and a vector X a at that point. The point x (cid:48) which is returnedis found by considering an affinely-parametrized geodesic y u satisfying y = x and ˙ y a = X a . The point x (cid:48) is then equal to y . An equivalent statement maybe expressed using Synge’s function (40). Letting σ a ( x (cid:48) , x ) denote ∇ a σ ( x (cid:48) , x ),exp x [ − σ a ( x (cid:48) , x )] = x (cid:48) . (46)First derivatives of Synge’s function therefore generalize the concept of a “sep-aration vector.” The x (cid:48) − x of a conventional Taylor series in Cartesian coordi-nates naturally turns into − σ a ( x (cid:48) , x ) in more general contexts. If a scalar field λ ( x ) is to be expanded in a Taylor series about some x , it is convenient to firstrewrite this as a function on the tangent bundle by definingΛ( x, X a ) := λ (exp x X a ) . (47)Now let the n th tensor extension of λ at x be λ ,a ··· a n ( x ) := (cid:20) ∂ n Λ( x, X b ) ∂X a · · · ∂X a n (cid:21) X b =0 . (48)This is the unique tensor field which reduces to n partial derivatives of λ ina Riemann normal normal coordinate system with origin x . In flat space, λ ,a ··· a n = ∇ a · · · ∇ a n λ . More generally, the curvature can appear. Furtherdiscussion of tensor extensions may be found in [4, 6].Combining all of these concepts, a natural Taylor series for ˆ φ which appliesregardless of the background geometry isˆ φ ( x (cid:48) , t ) = ∞ (cid:88) n =0 ( − n n ! σ a ( x (cid:48) , z t ) · · · σ a n ( x (cid:48) , z t ) ˆ φ ,a ··· a n ( z t , t ) . (49)All distances are assumed to be sufficiently small that σ remains single-valuedand its derivative well-defined. Furthermore, a Taylor series like this is — evenif it does not converge everywhere of interest — assumed to be at least a usefulasymptotic approximation throughout B s . Substituting (49) into (42) and inte-grating term-by-term then results in a multipole expansion for the generalizedforce. Noting that L ξ σ a = L ξ ( g ab ∇ b σ ) = g ab ∇ b L ξ σ = 0 (50)for any Killing field ξ a , the multipole expansion for the generalized force isdd t P t ( ξ ) = − ∞ (cid:88) n =0 n ! m a ··· a n ( z t , t ) L ξ ˆ φ ,a ··· a n ( z t , t ) , (51)16here the mass moments depend on ρ via m a ··· a n ( z t , t ) := ( − n (cid:90) B t σ a ( x (cid:48) , z t ) · · · σ a n ( x (cid:48) , z t ) ρ ( x (cid:48) , t )d V (cid:48) . (52)It follows from (28) that the zeroth moment, the mass, is independent of time.Conservation laws do not, however, fix the evolution of the higher moments.These depend on the type of matter under consideration.If L ψ ˆ φ = 0 for some Killing field ψ a , it follows that L ψ ˆ φ ,a ··· a n = 0 forany n . The conservation of P t ( ψ ) in such a case is therefore preserved byany approximation which truncates the multipole series at finite n . This is animportant property which contributes to the accuracy of these approximationsover long times. Thus far, P t = P t [ B t ] has been loosely described as being equivalent to a body’slinear and angular momenta at time t . Similarly, time derivatives of the gener-alized momentum have been interpreted as “forces and torques.” These identi-fications are now made precise.Recall that the generalized momentum is a vector in K ∗ , the vector spacedual to the Killing fields K . While it is productive to view P t simply as a linearmap from K to R , it can also be useful to find its components with respect to aparticular basis. It is in this context that the linear and angular momenta arisein their more familiar form.A basis for K may be found by recalling that knowledge of a Killing fieldand its first derivative at any one point fixes it everywhere [15]. Choosing anarbitrary point x , the space of Killing vectors is in one-to-one correspondencewith the space of all 1- and 2-forms at x . There exist two-point tensor fieldsΞ a (cid:48) a ( x (cid:48) , x ), Ξ a (cid:48) ab ( x (cid:48) , x ) such that ξ a (cid:48) ( x (cid:48) ) = Ξ a (cid:48) a ( x (cid:48) , x ) A a + Ξ a (cid:48) ab ( x (cid:48) , x ) B ab (53)is an element of K for any A a and any B ab = B [ ab ] , and also A a = ξ a ( x ) , B ab = B [ ab ] = ∇ a ξ b ( x ) . (54)In a physical space of dimension N , there exist N linearly independent 1-formsand N ( N − / N ( N + 1) / i (cid:48) i ( x (cid:48) , x ) = δ ii (cid:48) , Ξ i (cid:48) ij ( x (cid:48) , x ) = ( x (cid:48) − x ) [ i δ j ] i (cid:48) . (55)More generally, Ξ a (cid:48) a and Ξ a (cid:48) ab are related to the geodesic deviation equationand form a basis for K . They can be computed using the first two derivativesof Synge’s function [24]. Defining σ ab := ∇ b σ a = ∇ b ∇ a σ , σ aa (cid:48) := ∇ a (cid:48) σ a , and H a (cid:48) a := [ − σ aa (cid:48) ] − , Ξ a (cid:48) a = H a (cid:48) b σ ba , Ξ a (cid:48) ab = H a (cid:48) [ a σ b ] . (56)17ubstituting (53) into (35) shows that P t ( ξ ) can be written as a linear com-bination of ξ a ( x ) and ∇ a ξ b ( x ). The coefficients in this combination are iden-tified with the linear momentum p a ( x, t ) and the angular momentum bivector S ab = S [ ab ] ( x, t ): P t ( ξ ) = p a ( x, t ) ξ a ( x ) + 12 S ab ( x, t ) ∇ a ξ b ( x ) . (57)This is an implicit definition. Varying amongst all possible ξ a and ∇ a ξ b recoversthe explicit formulae p a ( x, t ) = (cid:90) B t ρ ( x (cid:48) , t ) v a (cid:48) ( x (cid:48) , t ) H a (cid:48) b ( x (cid:48) , x ) σ ba ( x (cid:48) , x )d V (cid:48) , (58) S ab ( x, t ) = 2 (cid:90) B t ρ ( x (cid:48) , t ) v a (cid:48) ( x (cid:48) , t ) H a (cid:48) [ a ( x (cid:48) , x ) σ b ] ( x (cid:48) , x )d V (cid:48) . (59)In three spatial dimensions, the angular momentum bivector is dual to an an-gular momentum 1-form S a via S a = 12 (cid:15) abc S bc . (60)Introducing Cartesian coordinates in a flat background, it is easily verified thatthe p i and S i derived from P t in this way reproduce the elementary defini-tions (7). Explicit coordinate expressions are more difficult to obtain in curvedbackgrounds, but these are rarely necessary.Thus far, the spatial curvature has played no explicit role in any of ourdiscussion. It does appear, however, in the evolution equations for p a and S ab .First note the general identity [15] ∇ a ∇ b ξ c = − R bcad ξ d , (61)which holds for any Killing field ξ a . Time derivatives of the Killing data( A a , B ab ) along a path z t therefore satisfyDd t A a = ˙ z bt B ba , Dd t B ab = − R abcd ˙ z ct A d . (62)These are known as the Killing transport equations [15, 25], and are ordinarydifferential equations which can be used to relate Killing data at one point toKilling data at another point.Consider linear and angular momenta defined about some z t , so, e.g., p a = p a ( z t , t ). Substituting (54) and (62) into (57) then shows that (cid:18) D p a d t − R bcda S bc ˙ z dt (cid:19) ξ a + 12 (cid:18) D S ab d t − p [ a ˙ z b ] t (cid:19) ∇ a ξ b = dd t P t ( ξ ) (63)for all ξ a ∈ K . Varying over all Killing vector fields finally recovers the individ-ual evolution equations˙ p a = 12 R bcda S bc ˙ z dt + F a , ˙ S ab = 2 p [ a ˙ z b ] t + N ab , (64)18here the force F a and torque N ab = N [ ab ] are determined by matching appro-priate coefficients in d P t / d t . Exact expressions for these quantities follow from(42), (53), and (56): F a = − (cid:90) B t ρH a (cid:48) b σ ab ∇ a (cid:48) ˆ φ d V (cid:48) , (65) N ab = − (cid:90) B t ρH a (cid:48) [ a σ b ] ∇ a (cid:48) ˆ φ d V (cid:48) . (66)Multipole expansions for the force and torque could be obtained directly fromthese integrals, although it is simpler to instead start from (51). Regardless, F a = − ∞ (cid:88) n =0 n ! m b ··· b n ∇ a ˆ φ ,b ··· b n , (67) N ab = ∞ (cid:88) n =0 n ! g c [ a m b ] d ··· d n ˆ φ ,cd ··· d n . (68)Note that the velocity ˙ z at of the (arbitrarily-chosen) origin does not appearin F a or N ab . Those portions of (64) which do involve the velocity are spatialanalogs of the Mathisson-Papapetrou terms typically used to describe the mo-tion of spinning particles in general relativity. It is apparent here that similarterms arise even in non-relativistic theories. Their origin is essentially kinematic,being related to the decomposition of K into pure translations and pure rota-tions. It follows from (57) that p a ( z t , t ) is associated with Killing vectors whichappear translational at z t in the sense that ∇ a ξ b ( z t ) = 0. Similarly, S ab ( z t , t )is associated with Killing fields which are purely rotational in the sense that ξ a ( z t ) = 0. The Mathisson-Papapetrou terms arise in the laws of motion be-cause, e.g., a Killing vector which is purely translational at z t is not necessarilypurely translational at a neighboring point z t + dt . A given Killing field may havedifferent proportions of “translation” and “rotation” at different points, and thisinevitably mixes the evolution equations for p a and S ab . A simple version ofthis effect occurs even in flat space, where a pure rotation about one origin isnot necessarily a pure rotation about another origin. This explains the p [ a ˙ z b ] t term in (64) and the − ˙ z t × p term in (24).Essentially the same explanation for the Mathisson-Papapetrou terms ap-plies in general relativity. In that case, the spacetime may not admit any Killingvectors at all. Regardless, there still exists a ten-dimensional space of “general-ized Killing fields” as described in Section 4.1. Given a particular event, theseare naturally decomposed into a four-dimensional space of translations and asix-dimensional space of rotations and boosts. Whether or not a particular gen-eralized Killing field is, e.g., purely translational varies from point to point justas it does for ordinary Killing fields. The evolution equations for relativistic mo-menta therefore acquire velocity-dependent terms which are closely analogousto those which appear in the generalized Newtonian theory discussed here.Confining attention only to the generalized momentum whenever possibleavoids the complications associated with the Mathisson-Papapetrou terms. It19lso simplifies the discussion of conservation laws. Recall that the presence ofa particular spatial Killing field ψ a which satisfies L ψ ˆ φ = 0 implies that P t ( ψ )must be conserved. It follows from (57) that a particular linear combination of p a and S ab must be conserved as well: p a ( z t , t ) ψ a ( z t ) + 12 S ab ( z t , t ) ∇ a ψ b ( z t ) = (constant) . (69)This constant is independent of z t . Its existence implies that a particular com-bination of forces and torques must vanish. Specifically, comparison with (63)shows that F a ψ a + 12 N ab ∇ a ψ b = 0 . (70)Although these results could be deduced directly from (64)-(66), they are con-siderably more clear from the perspective of the generalized momentum and itsevolution equation (42). The laws of motion for p a and S ab have left z t undetermined. One convenientchoice is to set z t = γ t , where γ t denotes the body’s center of mass at time t .This is straightforward when the background space is flat, in which case itis standard to define the center of mass to be the origin about which the massdipole moment vanishes: m a ( γ t , t ) = 0. Enforcing this while differentiating (52)recovers the standard relation p a = m ˙ γ at between an object’s velocity and itslinear momentum. Using (64) and (67) then shows that¨ γ at = −∇ a ˆ φ ( γ t , t ) − m ∞ (cid:88) n =2 n ! m b ··· b n ( γ t , t ) ∇ a ∇ b · · · ∇ b n ˆ φ ( γ t , t ) , (71)which is equivalent to (27).Similar results do not appear to hold when the background space is curved.It is still possible to demand that the dipole moment vanish [which, among otherbenefits, eliminates the n = 1 term in (67) and the n = 0 term in (68)], butthen the velocity of such a trajectory may be shown to satisfy˙ γ bt (cid:90) B t ρ ( x (cid:48) , t ) σ ab ( x (cid:48) , γ t )d V (cid:48) = − (cid:90) B t ρ ( x (cid:48) , t ) v a (cid:48) ( x (cid:48) , t ) σ aa (cid:48) ( x (cid:48) , γ t )d V (cid:48) . (72)The integral on the left-hand side of this equation can (typically) be inverted toyield an explicit expression for ˙ γ at . Unfortunately, the result does not dependon p a in any simple way. Simplifications are possible when a body’s dimensionsare much smaller than the curvature scale, in which case σ ab and σ aa (cid:48) can beexpanded in Taylor series about γ t , yielding the ordinary momentum-velocityrelation at lowest order. More generally, though, (72) is problematic. Higher-order corrections require more information about the body than is required forthe evolution equations of the momenta alone. Moments of a body’s momentum20istribution — its “current moments” — appear to be needed together with itsmass moments.It is only in this very last step where a celestial mechanics of “curved New-tonian gravity” appears to be problematic. Similar complications do not arisein relativistic systems. Among other differences, the presence of boost-typeKilling fields (or their generalizations) on spacetime provide additional con-straints which imply well-behaved momentum-velocity relations. Results such as (64) are properly described as laws of motion, not equations ofmotion [26, 27]. They are incomplete in the sense that even if z t is chosen asa body’s center of mass, we still have not described how to compute ˆ φ or thehigher multipole moments.The traditional approach is to introduce smallness assumptions. Consider,for simplicity, the n -body problem in flat space. If a particular body in sucha system has characteristic size (cid:96) and mass m , its 2 n -pole moments must besmaller than approximately m(cid:96) n . Letting r denote a minimum distance be-tween bodies and assuming that all masses are comparable, the n th term in(71) is at most of order n ! ( m/r ) ( (cid:96)/r ) n . Considerable simplifications thereforeresult if (cid:96) (cid:28) r . At lowest order, only the monopole term is retained in the law ofmotion. Each ˆ φ in such an approximation may also be computed by assumingthat all other masses are pure monopoles, thus recovering the typical Newtonian n -body equations of motion. More details may be found in, e.g., [16, 17]. Despite being considerably more abstract than the traditional presentation ofNewtonian gravity, the formalism which has just been described is very pow-erful. It does not rely on any particular coordinate systems, and the majorityof the discussion doesn’t even require that the metric be flat. Indeed, most ofthe results well-known in ordinary Newtonian gravity continue to hold in gen-eralizations of this theory which employ spherical or hyperboloidal geometries.It is also trivial to change the number of spatial dimensions, or even to amendthe field equation in certain ways. It is physically more interesting, however,to consider motion in relativistic theories such as electromagnetism or generalrelativity.This section describes how the formalism of Section 2.2 generalizes for ob-jects coupled to relativistic fields. For simplicity, we consider the motion ofan extended mass coupled to a scalar field φ which satisfies the Klein-Gordon Large astrophysically-relevant objects like planets tend to be very nearly spherical due tothe limited shear stresses which can be supported. The trace-free components of the moments,which are all that couple to the motion, are then much smaller than m(cid:96) n . These tend to beinduced mainly by rotation and external tidal fields, and are typically modeled using Lovenumbers. ∇ a ∇ a − µ ) φ = 4 πρ. (73) µ represents a (constant) field mass and ρ the body’s charge density. Followingthe Newtonian problem as closely as possible, the four-dimensional backgroundspacetime ( M , g ab ) is assumed to be maximally-symmetric. Understanding mo-tion in more general curved spacetimes requires eliminating our reliance on amaximal set of Killing vector fields. This is indeed possible, but is somewhattechnical. Its discussion is therefore delayed to Section 4 below. Motion inelectromagnetic fields is discussed there as well.Scalar charges in maximally-symmetric spacetimes provide a simple examplewith which to introduce the relativistic theory of motion. They differ from theirNewtonian counterparts in only one important respect: Self-forces no longervanish. Still, self-forces are “almost ignorable” in the sense that they effectivelyrenormalize a body’s momentum, but do nothing else . This is a finite renor-malization, meaning that self-forces conspire to, e.g., make the mass computedby integrating over a body’s stress-energy tensor differ from the mass inferredby watching how that body accelerates in response to external fields.Physically, renormalization arises because as a charge accelerates, its fieldmust be accelerated as well. Although portions of that field may break away asradiation or otherwise change, there is a sense in which charges and their fieldsremain inseparably coupled. The energy contained in a body’s self-field impliesthat it must resist acceleration and contribute to that body’s inertia.Now, self-forces vanish in Newtonian theory because of Newton’s third law.The self-field is sourced by a body’s instantaneous mass distribution and exertsforces on that same mass distribution. Interactions are no longer instantaneous,however, in theories which involve hyperbolic field equations. Fields are sourcedby charge in a four-dimensional region of spacetime, but act only on configura-tions in three-dimensional hypersurfaces. It is impossible to maintain an exactconcept of “action-reaction pairs” in this context, and the imbalance which re-sults turns out to exert forces and torques which precisely mimic changes to abody’s linear and angular momenta. This type of effect is generic for any cou-pling to long-range fields which satisfies hyperbolic field equations (or otherwisedepends on a system’s history). The simplest relativistic modification of Newtonian gravity involves objectswith scalar charge density ρ interacting via a field φ which satisfies the waveequation (73). Suppose that the body of interest resides inside a worldtube The notion of self-force used here is consistent with the usual Newtonian definition, butis unconventional in relativistic contexts. Its precise meaning is made clear below. This is to be considered as a model problem. If interpreted as a theory of gravity, thetype of scalar field theory described here is not compatible with observations. Of course, it isnot necessary to interpret φ as a gravitational potential (so ρ needn’t be a “mass density” inany sense). ⊂ M containing no other matter, and also that W is spatially bounded inthe sense that its spacelike slices have finite volume.Our description for the motion of a compact object is based on its stress-energy tensor T ab body . This encodes many of a body’s mechanical properties,and is analogous to the ( ρ, v a , τ ab ) triplet used to analyze Newtonian objects inSection 2. Like those variables, the stress-energy satisfies differential equationswhich are independent of the specific type of material under consideration. Al-though these laws do not determine T ab body completely, they do provide significantconstraints.If φ vanishes everywhere and there are no other long-range fields, ∇ b T ab body =0. More generally, scalar fields contribute to a system’s total stress-energy. Itis only this total T ab tot = T ( ab )tot which is necessarily conserved: ∇ b T ab tot = 0 . (74)Consider splitting this total into “body” and “field” components: T ab tot = T ab field + T ab body . (75)Away from any matter, it is clear that T ab body = 0 and T ab field = 14 π (cid:20) ∇ a φ ∇ b φ − g ab ( ∇ c φ ∇ c φ + µ φ ) (cid:21) . (76)Elsewhere, local interactions between the matter and the field make it physicallydifficult to motivate any particular split.One possible way forward is to work only with T ab tot . Unfortunately, themomentum obtained from this stress-energy tensor might be very different ifcomputed first in a slice of W , and then in a slightly larger hypersurface. Thereis no natural boundary where integrations can be stopped. Although momentumintegrals might settle down when performed over very large volumes, it is unclearhow useful this is. The relevant distance scale could be so large that the only“total momenta” which are interesting encompass the entire (modeled) universe,thus precluding any ability to learn about the dynamics of individual masses.Results based on T ab tot alone are known to be useful in certain approximationsinvolving the motion of very small bodies [28], but this is insufficiently generalfor our purposes.The approach adopted here is mathematically the simplest. Let T ab field begiven by (76) throughout W . The remaining stress-energy tensor is then definedto be the body’s: T ab body = T ( ab )body := T ab tot − T ab field . Equations (73)-(76) imply thatthis satisfies ∇ b T ab body = − ρ ∇ a φ, (77)which generalizes the Newtonian conservation laws (28) and (29). In a Lagrangian formalism, the total stress-energy tensor considered here is derived froma functional derivative of the action with respect to the metric. It is conserved whenever theaction is diffeomorphism-invariant [15]. .2 Generalized momentum Recall the generalized momentum (35) defined for Newtonian mass distribu-tions. This requires very little modification for use in relativistic systems. Theone complication which does arise is that there is no longer any preferred no-tion of time. A time parameter must be supplied as an additional ingredient,which is accomplished by foliating W with a 1-parameter family of hypersurfaces { B s } . Each B s may be viewed as the body region at time s , and is assumed tohave finite volume. The precise nature of these body regions may be consideredarbitrary for now, and can be spacelike or even null .Supposing that a particular foliation has been fixed, the generalized mo-mentum contained in any three-dimensional region R ⊆ B s is most obviouslydefined as P s [ R ]( ξ ) = (cid:90) R T ab body ξ a d S b , (78)where ξ a is any Killing vector field. P s [ R ]( · ) defines a linear map on the space of K of Killing vector fields. It is therefore a vector in the dual space K ∗ . For themaximally-symmetric four-dimensional spacetimes considered here, dim K =dim K ∗ = 10. Given a particular event, four of these dimensions correspond totranslations and six to rotations and boosts. As in the Newtonian case, suchdecompositions allow the generalized momentum to be expressed in a basiswhich recovers linear and angular momenta associated with a preferred event.The details of this correspondence are described more precisely in Section 3.7. Forces and torques are determined by s -derivatives of the generalized momen-tum. Considering only the momenta in B s , it is convenient to simplify thenotation by defining P s = P s [ B s ]. Using (77) together with Killing’s equationthen shows that dd s P s ( ξ ) = − (cid:90) B s ρ L ξ φ d S (79)for all ξ a ∈ K , where d S := t a d S a and t a denotes a time evolution vectorfield for the foliation { B s } . The relativistic generalized force (79) is essentiallyidentical to its Newtonian analog (36). As in that context, the force can beimmediately put into a practical form only if the object of interest does not sig-nificantly contribute to φ . More generally, the self-field introduces considerablecomplications. Progress is made by finding a precise definition for the self-field,computing its effects analytically, and then subtracting it out. The “effectivefield” which remains after this process is typically much simpler to analyze thanthe physical one. Consider, e.g., the past-directed light cones associated with a timelike worldline. .4 The self-field At least in part, the Newtonian self-field (19) can be generalized essentially as-is. Let the relativistic self-field φ S be obtained by convolving an object’s chargedensity with a particular two-point scalar G . This must be a Green function,so ( ∇ a ∇ a − µ ) G ( x, x (cid:48) ) = 4 πδ ( x, x (cid:48) ) . (80)Still more constraints are necessary to fix G uniquely. One of these follows fromrequiring that the self-field depend only quasi-locally on a body’s “instan-taneous” configuration. It should not, for example, involve distantly-imposedboundary conditions, the behavior of other objects, or a body’s history in thedistant past. Such conditions can be ensured by demanding that G ( x, x (cid:48) ) = 0whenever x and x (cid:48) are timelike-separated. Lastly, suppose that G ( x, x (cid:48) ) = G ( x (cid:48) , x ). Such objects exist (at least in finite regions), are unique, and are re-ferred to as S-type or “singular” Detweiler-Whiting Green functions [10, 29]. Inthe maximally-symmetric backgrounds considered here, G satisfies L ξ G = 0 forall ξ a ∈ K , implying a relativistic form of Newton’s third law. For masslessfields in Minkowski spacetime, G = ( G + + G − ) where G ± are the advancedand retarded Green functions. More generally, G = 12 ( G + + G − − V ) (81)for a certain symmetric biscalar V ( x, x (cid:48) ) which satisfies the homogeneous fieldequation. G can also be expressed in terms of Synge’s function σ . Using ∆ todenote the van Vleck determinant [10] (which depends on second derivatives of σ ), δ the Dirac distribution, and Θ the Heaviside distribution, G = 12 [∆ / δ ( σ ) − V Θ( σ )] . (82)This shows that G ( x, · ) can have support on and outside the light cones of x .The self-field “due to” charge contained in a given spacetime volume R ⊆ W can now be expressed in terms of the S-type Detweiler-Whiting Green function: φ S [ R ]( x ) = (cid:90) R G ( x, x (cid:48) ) ρ ( x (cid:48) )d V (cid:48) . (83)If the argument R is omitted, the integral is understood to be carried out overan object’s entire worldtube W .We now compute the self-force. It simpler not to consider this directly, butrather its integral over a finite interval of time. Letting s f > s i , it is clear from(79) that P s f ( ξ ) − P s i ( ξ ) = (cid:90) s f s i dd s P s ( ξ )d s = − (cid:90) I ρ L ξ φ d V (84) It is also possible to introduce ( n +1)-point self-fields similar to (44). This is not consideredany further here. The term self-field is used in several different ways in the literature. The definitionadopted here is uncommon, and is sometimes described as the “Coulomb-like” component ofthe self-field. ξ a ∈ K . The 4-volume I = I ( s i , s f ) ⊂ W which appears here representsthat part of an object’s worldtube which lies in between the initial and finalhypersurfaces B s i , B s f . See Figure 1. Substitution of the self-field definition(83) into (84) shows that the total change in momentum due to φ S alone is (cid:90) I d V (cid:90) W d V (cid:48) f ( x, x (cid:48) ) , (85)where f ( x, x (cid:48) ) = − ρ ( x ) ρ ( x (cid:48) ) ξ a ( x ) ∇ a G ( x, x (cid:48) ) (86)may be interpreted as the density of generalized force exerted at x by x (cid:48) . Inde-pendently of any specific form for f , double integrals with the form (85) can berewritten as12 (cid:90) I d V (cid:32)(cid:90) W d V (cid:48) [ f ( x, x (cid:48) ) + f ( x (cid:48) , x )] + (cid:90) W \ I d V (cid:48) [ f ( x, x (cid:48) ) − f ( x (cid:48) , x )] (cid:33) (87)whenever the relevant integrals commute. This identity is very general, and iscentral to understanding motion in every relativistic theory we discuss. It istherefore worthwhile to examine it in detail.The first term in (87) can be interpreted as an average of “action-reactionpairs” in the sense of Newton’s third law. It is very similar to the types ofidentities used to simplify the motion of objects coupled to elliptic fields inSection 2. Recalling that discussion, the reciprocity relation G ( x, x (cid:48) ) = G ( x (cid:48) , x )implies that (cid:90) I d V (cid:90) W d V (cid:48) [ f ( x, x (cid:48) ) + f ( x (cid:48) , x )] = − (cid:90) I d V (cid:90) W d V (cid:48) ρ ( x ) ρ ( x (cid:48) ) L ξ G ( x, x (cid:48) ) . (88)Lie derivatives of G are again associated with sums over action-reaction pairs,and as in the Newtonian case, these sums vanish. The relativistic scalar self-force is therefore determined only by the second group of terms in (87). Thoseterms do not vanish in general, but are instead connected to the finite speed ofpropagation associated with φ . They are responsible for renormalizing a body’smomentum. The generalized force exerted by φ S is entirely determined by the final part of(87). To understand this, first let B + s (resp. B − s ) denote the four-dimensionalfuture (past) of B s inside the body’s worldtube: B ± s ( s ) := (cid:91) ± ( τ − s ) > B τ . (89)Also define E s ( ξ ) := 12 (cid:18)(cid:90) B + s ρ L ξ φ S [ B − s ]d V − (cid:90) B − s ρ L ξ φ S [ B + s ]d V (cid:19) . (90)26igure 1: Schematic illustrations of a body’s worldtube W together with hyper-surfaces B s i and B s f (drawn spacelike). The region I ( s i , s f ) ⊂ W bounded bythese hypersurfaces and appearing in (84) is indicated. The shaded 4-volumesˆ B s i and ˆ B s f [see (97)] denote the domains of dependence associated with theself-momenta E s i and E s f defined by (90). Although P s f ( ξ ) − P s i ( ξ ) dependson ρ L ξ ˆ φ throughout I ( s i , s f ), it depends on more complicated aspects of abody’s internal structure only in the shaded regions. These contributions arealways confined to within approximately one light-crossing time of the boundinghypersurfaces, and are therefore “quasi-local.”Like P s , this represents an s -dependent vector in K ∗ . Using it, the second termin the expansion (87) for the self-force is simply12 (cid:90) I d V (cid:90) W \ I d V (cid:48) [ f ( x, x (cid:48) ) − f ( x (cid:48) , x )] = − [ E s f ( ξ ) − E s i ( ξ )] . (91)Taking the limit s f → s i while combining (84), (87), (88), and (91) finally showsthat the generalized force can be written asdd s P s ( ξ ) = − (cid:90) B s ρ L ξ ˆ φ d S − dd s E s ( ξ ) . (92)Replacing the physical field φ with ˆ φ = φ − φ S can therefore be accomplishedonly at the cost of the counterterm − d E s / d s . That this is a total derivativesuggests the introduction of an “effective generalized momentum” ˆ P s satisfyingˆ P s := P s + E s . (93)For any finite scalar charge in a maximally-symmetric spacetime,dd s ˆ P s ( ξ ) = − (cid:90) B s ρ L ξ ˆ φ d S. (94)There is a sense in which E s renormalizes the (bare) momentum P s . The sumof P s and E s behaves instantaneously as though it were the momentum of a testcharge placed in the effective field ˆ φ . Furthermore,( ∇ a ∇ a − µ ) ˆ φ = 0 . (95)27hysically, it is not sufficient to motivate the renormalization P s → ˆ P s merely by fact that the self-force is a total derivative. Essentially any functionof one variable can be written as the total derivative of its integral. Indeed, onemight introduce a constant s and define˜ P s ( ξ ) := P s ( ξ ) + E s ( ξ ) + (cid:90) ss d τ (cid:90) B τ d Sρ L ξ ˆ φ. (96)This does not vary at all with s . While it may be useful for some purposes,˜ P s is not a physically acceptable momentum. This is because it depends in anessential way on the configuration of the system for all times between s and s .While ˜ P s would be approximately local for s ≈ s , it otherwise depends on asystem’s history in a complicated way.The renormalized momentum ˆ P s defined by (93) does not share this defi-ciency. Like P s , it depends only on the body’s configuration in regions “near” B s . The relevant region is, however, somewhat larger for ˆ P s than it is for P s .Definitions (83) and (90) imply that E s ( ξ ) depends on a neighborhood ˆ B s ⊃ B s defined to be the set of all points in W which are null- or spacelike-separated toat least one point in B s . In terms of Synge’s function,ˆ B s = { x ∈ W | σ ( x, y ) ≥ y ∈ B s } . (97)ˆ B s is a (finite) four-dimensional region of spacetime. It extends into the pastand future of B s by roughly the body’s light-crossing time. See Figure 1.One might have guessed that a self-momentum at time s could be definedby integrating the stress-energy tensor associated with φ S over a large hyper-surface which contains B s . Unfortunately, such integrals depend on gradientsof φ S far outside of B s . These, in turn, depend on the body’s state in thedistant past and future. Such a definition is physically unacceptable in general.Nevertheless, it does make sense in the stationary limit, and may be shown tocoincide with E s in that case [2]. In more dynamical cases, the E s defined hereappears to be the only well-motivated possibility. Forces and torques exerted on relativistic scalar charges may be expanded ex-actly as in the Newtonian theory. Assuming that ˆ φ can be accurately approxi-mated using a Taylor series about some origin z s ∈ B s , the techniques of Section2.2.5 may be used to show that (94) admits the multipole expansiondd s ˆ P s ( ξ ) = − ∞ (cid:88) n =0 n ! q a ··· a n ( z s , s ) L ξ ˆ φ ,a ··· a n ( z s ) . (98) Recalling (76), T ab field is quadratic in φ . The stress-energy tensor “associated with φ S ” istaken to mean that portion of T ab field which is quadratic in φ S . Terms linear in φ S are notincluded. n -pole moment of ρ which appears here is defined by q a ··· a n ( s ) := ( − n (cid:90) B s σ a ( x, z s ) · · · σ a n ( x, z s ) ρ ( x )d S, (99)and ˆ φ ,a ··· a n denotes the n th tensor extension of ˆ φ . Equation (98) may becompared with the Newtonian generalized force (51). Unlike its Newtoniancounterpart, however, the relativistic scalar monopole moment q may dependon time; the total charge is not necessarily conserved. Also note that the rela-tivistic multipole expansion is intended only to be asymptotic. It may requiretruncation at large n (see, e.g., [30]). Like P s , the effective generalized momentum ˆ P s is an element of K ∗ . Expandingthis in an appropriate basis recovers objects which may be interpreted as abody’s linear and angular momenta. The appropriate arguments are almostidentical to those described in Section 2.2.6.Choosing a point z s ∈ B s , every Killing field may be written as a linearcombination of 1- and 2-forms at z s [cf. (53)]. P s ( ξ ) and ˆ P s ( ξ ) are clearly linearin ξ a , so they too may be expanded in linear combinations of 1- and 2-forms at z s . Recalling (57), the coefficients in this combination may be identified as abody’s linear and angular momentum:ˆ P s ( ξ ) = ˆ p a ( z s , s ) ξ a ( z s ) + 12 ˆ S ab ( z s , s ) ∇ a ξ b ( z s ) . (100)Hats have been placed on ˆ p a and ˆ S ab to emphasize that these are renormalizedmomenta. Bare quantities defined in terms of P s may be introduced as welland shown to coincide with the momenta introduced by Dixon [6, 17, 24] forobjects without electromagnetic charge-currents. The bare momenta obey morecomplicated laws of motion, and are not considered any further.Differentiating (100) while using (61) leads to an implicit evolution equationfor ˆ p a and ˆ S ab : (cid:18) Dˆ p a d s − R bcda ˆ S bc ˙ z ds (cid:19) ξ a + 12 (cid:18) D ˆ S ab d s − p [ a ˙ z b ] s (cid:19) ∇ a ξ b = dd s ˆ P s ( ξ ) . (101)Varying over all ξ a and all ∇ a ξ b = ∇ [ a ξ b ] recovers the explicit equationsDˆ p a d s = 12 R bcda ˆ S bc ˙ z d + ˆ F a , D ˆ S ab d s = 2ˆ p [ a ˙ z b ] + ˆ N ab . (102)The force ˆ F a and torque ˆ N ab = ˆ N [ ab ] appearing here may be found in integral29orm by comparing (94) and (101). Using the multipole expansion (98) instead,ˆ F a = − ∞ (cid:88) n =0 n ! q b ··· b n ∇ a ˆ φ ,b ··· b n , (103)ˆ N ab = ∞ (cid:88) n =0 n ! g c [ a q b ] d ··· d n ˆ φ ,cd ··· d n . (104)The laws of motion (102)-(104) describe bulk features of essentially arbitraryself-interacting scalar charge distributions in maximally-symmetric backgrounds.As in the Newtonian case, all explicit dependence on ˙ z as is contained in theMathisson-Papapetrou terms. These terms are kinematic, and may again betraced to the changing character of Killing fields evaluated about different ori-gins.All conservation laws discussed in the Newtonian context generalize imme-diately. If L ψ ˆ φ = 0 for some particular Killing field ψ a ,ˆ P s ( ψ ) = ˆ p a ψ a + 12 ˆ S ab ∇ a ψ b = (constant) . (105)Similarly, ˆ F a ψ a + 12 ˆ N ab ∇ a ψ b = 0 . (106)These results are exact. They hold independently of any choices made for z s ,and also apply to approximate momenta evolved via any consistent truncationof the multipole series.Although the relativistic multipole expansions are structurally almost iden-tical to their Newtonian counterparts, it is important to emphasize that theeffective field is far more difficult to compute in relativistic contexts. In New-tonian gravity, ˆ φ is simply the external potential and is easily computed giventhe instantaneous external mass distribution of the universe. The relativisticeffective potential can, however, depend in complicated ways on boundary con-ditions, initial data, and past history. Using retarded boundary conditions, therelativistic ˆ φ typically depends on ρ , and is therefore not interpretable as apurely external field.Another property of the relativistic theory is that the angular momentumhas six independent components rather than three. This can be interpreted byintroducing a local frame at z s in the form of a unit timelike vector u a . Such avector allows ˆ S ab to be decomposed into two components ˆ S a and ˆ m a satisfyingˆ S ab = (cid:15) abcd u c ˆ S d − u [ a ˆ m b ] (107)and u a ˆ S a = u a ˆ m a = 0. Writing out ˆ S ab explicitly in flat spacetime in the limitof negligible self-interaction suggests that ˆ S a represents a body’s “ordinary”angular momentum about z s . Similarly, ˆ m a may be interpreted as the dipolemoment of a body’s energy distribution. Relativistically, these are two aspectsof the same physical structure. The split ˆ S ab → ( ˆ S a , ˆ m a ) is closely analogousto the decomposition F ab → ( E a , B a ) of an electromagnetic field into its electricand magnetic components. 30 .8 Center of mass Thus far, the foliation { B s } of W used to define the generalized momentum hasbeen left unspecified. The collection of events { z s } used to perform the multi-pole expansion (98) has been arbitrary as well. This constitutes a considerableamount of freedom.One simplifying strategy is to first associate a hypersurface with each possi-ble point in W . This could be accomplished by, e.g, defining B s to be the past-(or future-) null cone with vertex z s . A timelike worldline parametrized by { z s } then results in a null foliation of W . Alternatively, a spacelike foliation may bechosen as described in [17, 24, 31].Regardless, defining each B s in terms of z s reduces all freedom in the law ofmotion to the choice of a single worldline (and its parametrization). Recall thatin Newtonian gravity, a body’s center of mass is the location about which itsmass dipole moment vanishes. Relativistically, the dipole moment of a body’sstress-energy tensor is proportional to ˆ S ab ( z s , s ). In general, there is no choice of z s which can be used to make this vanish entirely. It is, however, possible to usetranslations to set ˆ m a = 0 as defined in (107). This requires the introductionof a frame with which to choose an appropriate dipole moment. Consider thezero-momentum frame u a where ˆ p a = ˆ mu a . (108) u a is defined to be a unit vector, so the rest mass ˆ m must satisfyˆ m := (cid:112) − ˆ p a ˆ p a . (109)A center of mass γ s may now be defined by demanding thatˆ S ab ( γ s , s )ˆ p a ( γ s , s ) = 0 . (110)This can be interpreted as requiring that the dipole moment of a body’s energydistribution vanish as seen by a zero-momentum observer at γ s . It is a highlyimplicit definition. Good existence and uniqueness results are known for theclosely-related Dixon momenta [32, 33], but not more generally. We neverthelessassume that a unique worldline (and associated foliation) can be found foundin this way. Other choices are also possible, however.A general relation between the center of mass 4-velocity and the linear mo-mentum may be found by differentiating (110). The result of this differentiationis solved explicitly for ˙ γ as in [17, 31], resulting in ˆ m ˙ γ as = ˆ p a − ˆ N ab u b − ˆ S ab [ ˆ m ˆ F b − (ˆ p c − ˆ N ch u h ) ˆ S df R bcdf ]ˆ m + ˆ S pq ˆ S rs R pqrs . (111) References [17, 31] derive the momentum-velocity relation using Dixon’s momenta withouta scalar field, but in a spacetime which is not maximally-symmetric. Here, Dixon’s momentaare modified by E s , there is a scalar field, and the spacetime is maximally-symmetric. Despitethese differences, the relevant tensor manipulations are identical. s has been chosen such that ˙ γ as ˆ p a = − ˆ m ,and also that all instances of z s have been replaced with γ s . In principle, it ispossible for the denominator ˆ m + ˆ S pq ˆ S rs R pqrs here to vanish, indicating abreakdown of the center of mass condition. This can occur only if the curvaturescale is comparable to a body’s own size, in which case it is unlikely that anysimple description of an extended body in terms of its center of mass is likelyto be useful. In more typical cases, it is straightforward to obtain multipole ap-proximations of (111) by substituting appropriately-truncated versions of (103)and (104).Also note that ˙ γ as is not necessarily collinear with ˆ p a . The difference ˆ p a − ˆ m ˙ γ as may be interpreted as a “hidden mechanical momentum.” Simple examples ofhidden momentum are commonly discussed in electromagnetic problems (see,e.g., [8, 34, 35]), but occur much more generally. Some consequences of thehidden momentum are discussed in [36, 37].The center of mass condition provides more than just a relation betweenthe momentum and the velocity. It also implies that ˆ S ab ( γ s , s ) can be writtenentirely in terms of the spin vector ˆ S a ( γ s , s ). Inverting (107) while using (110),ˆ S a ( γ s , s ) = − (cid:15) abcd u b ˆ S cd . (112)Differentiating this and applying (102), the spin vector evaluated about thecenter of mass is seen to satisfyD ˆ S a d s = − (cid:15) abcd u b ˆ N cd + u a (cid:18) ˆ S b D u b d s (cid:19) . (113)The first term here represents a torque in the ordinary sense. The second termis responsible for the Thomas precession, and may be made more explicit bysubstituting (102) and (108).An evolution equation may also be be derived for the mass ˆ m , which is notnecessarily constant. Variations in ˆ m are not an exotic effect; masses changeeven for monopole test bodies coupled to relativistic scalar fields. In general,use of (102) and (109) shows that the mass evaluated using z s = γ s satisfies [24]d ˆ m d s = − ˙ γ as ˆ F a + ˆ N ab u a D u b d s . (114)The final term here may be made more explicit by using (102) and (108) toeliminate D u b / d s . The equations derived here are quite complicated in general. Some intuitionfor them may be gained by truncating the multipole series at monopole order.Inspection of (103) and (104) then shows thatˆ F a = − q ∇ a ˆ φ, ˆ N ab = 0 . (115)32urther restricting to cases where q = (constant), it follows from (114) thatˆ m − q ˆ φ = (constant) . (116)ˆ m − q ˆ φ may therefore be viewed as a conserved energy for the system. Notethat it is the effective field ˆ φ which occurs here, not the physical field φ .Contracting (113) with ˆ S a also shows that ˆ S := ˆ S a ˆ S a = (constant), mean-ing that the spin vector can only precess in the monopole approximation. Therate at which this occurs may be simplified by first recalling that in any maximally-symmetric spacetime, there exists a constant κ such that R abcd = κg a [ c g d ] b . (117)Of course, κ = 0 in the flat background of special relativity. Regardless, thespin evolution is independent of κ :D ˆ S a d s = − ( q/ ˆ m ) u a ˆ S b ∇ b ˆ φ. (118)It experiences a purely Thomas-like precession. The momentum-velocity rela-tion (111) does, by contrast, retain explicit evidence of the curvature, reducingto ˆ m ˙ γ as = ˆ p a + (cid:32) q/ ˆ m κ ( ˆ S/ ˆ m ) (cid:33) (cid:15) abcd u b ˆ S c ∇ d ˆ φ. (119)The linear momentum is also affected by κ :Dˆ p a d s = − q ∇ a ˆ φ + q (cid:32) κ ( ˆ S/ ˆ m ) κ ( ˆ S/ ˆ m ) (cid:33) (cid:16) δ ba + u a u b − ˆ S a ˆ S b / ˆ S (cid:17) ∇ b ˆ φ. (120)The overall force is therefore a particular linear transformation of − q ∇ a ˆ φ . Upto an overall factor, the second term here extracts that component of q ∇ a ˆ φ which is orthogonal to both ˆ p a and ˆ S a .Together, (116) and (118)-(120) determine the evolution of a scalar charge inthe monopole approximation. Despite the approximations which have alreadybeen made, these equations remain rather formidable. They may be simplifiedfurther by demanding that ˆ S = 0 at some initial time. The monopole approxi-mation implies that an initially non-spinning particle remains non-spinning, soit is consistent to set ˆ S = 0 for all time. It also follows that ˆ p a = ˆ m ˙ γ as and¨ γ as = − ( q/ ˆ m )( g ab + ˙ γ as ˙ γ bs ) ∇ b ˆ φ. (121)In the test body limit where ˆ φ ≈ φ , this is the typical equation adopted forthe motion of a point particle with scalar charge q . A point particle limitof this equation which still allows for self-interaction is equivalent to what isknown as the Detweiler-Whiting regularization [10, 29]. This regularization —which originally arose via heuristic arguments associated with the singularity33tructure of point particle fields — is a special case of the much more generalresults derived here (and which first appeared in [2]). Its origin is unrelated toany point particle limits or to the singularities associated with them.It is shown in Section 4.2 below that standard self-force results follow easilyfrom (121). That section also generalizes all laws of motion derived here toapply to scalar charges moving in arbitrarily-curved spacetimes. The discussion up to this point has made extensive use of Killing vector fields.This is familiar and simple, but not necessary. The first step towards under-standing motion in generic spacetimes is to find a suitable replacement for thespace of Killing vector fields. Once this is accomplished, the problem of mo-tion for extended charges coupled to scalar fields is considered once again. Thisexample is used to illustrate a new type of renormalization which occurs inspacetimes without symmetries. The techniques used to solve the scalar prob-lem are then adapted to discuss motion in electromagnetic fields. Lastly, weconsider motion in general relativity, where the objects of interest dynamicallymodify the geometry itself.
Recalling (35) or its relativistic analog (78), the generalized momenta used inmaximally-symmetric spacetimes are defined as linear operators over the space K of Killing vector fields. There is, however, no obstacle to replacing K bysome other vector space K G . This is the approach we take to defining momentain generic spacetimes. Although several notions of generalized or approximateKilling fields exist in the literature [38, 39, 40], only one of these [1] appears tobe suitable for our purposes. We describe it now.The space of generalized Killing fields used here can be motivated axiomati-cally. First note that avoiding significant modifications to the formalism devel-oped thus far requires that all elements of K G be vector fields on spacetime (orspace in non-relativistic problems). It is also reasonable to require that:1. All genuine Killing vectors which might exist are also generalized Killingvectors: K ⊆ K G .2. K = K G in maximally-symmetric spacetimes.3. The dimensionality of K G can depend only on the number of spacetimedimensions.The first of these conditions is clearly necessary for any K G which may be saidto generalize K . The second condition ensures that there are not “too many”generalized Killing fields in simple cases. The final condition is more subtle.It guarantees that the generalized momentum contains the “same amount” ofinformation regardless of the metric. More specifically, a generalized momentum34efined using K G must be decomposable in terms of linear and angular momentawith the correct number of components. Recalling (34),dim K G = 12 N ( N + 1) ≥ dim K (122)in any N -dimensional space.The given conditions restrict K G , but do not define it. An additional con-straint is needed which describes how those elements of K G which are not alsoelements of K preserve an appropriate geometric structure. It is not, of course,possible to demand that they preserve the metric. Symmetries of the connec-tion or curvature are unsuitable as well. The only reasonable possibilities arenonlocal. First consider the Riemannian case . Recalling (50), any Killing vector fieldused to drag two points x and x (cid:48) preserves the separation vector − σ a ( x, x (cid:48) )between those points. A slightly weaker condition can be used to define gener-alized Killing vectors even when no genuine Killing vectors exist. Suppose thata particular point x has been fixed and demand that A G ( x ) be defined as theset of all vector fields ξ a satisfying L ξ σ a ( x (cid:48) , x ) = 0 . (123)The result clearly forms a vector space. Unfortunately A G ( x ) is too large.In flat space, it becomes independent of x and coincides with the space ofaffine collineations: vector fields satisfying ∇ a L ξ g bc = 0. Geometrically, affinecollineations represent symmetries which preserve the connection. In genericspaces, A G ( x ) may be described as a space of generalized affine collineationswith respect to x .The space of Killing vector fields is known to be a vector subspace of theaffine collineations. Similarly, generalized Killing fields K G ( x ) may be obtainedas an appropriate subspace of A G ( x ). It is sufficient to demand only that theappropriate vector fields be exactly Killing at x : L ξ g ab ( x ) = 0 . (124)The set of all vector fields satisfying this and (123) are denoted by K G ( x ). Theyare the generalized Killing fields with respect to x . Any genuine Killing fieldswhich may exist are also elements of K G ( x ).Many geometric structures are preserved by generalized Killing fields. Equa-tion (124), can, for example, be shown to generalize to L ξ g ab ( x ) = ∇ a L ξ g bc ( x ) = 0 , (125) The same geometric conditions can also be imposed in Lorentzian geometries. Physically,however, the vector fields discussed here are most useful in non-relativistic contexts. A morecomplicated structure described in Section 4.1.2 is better-suited to Lorentzian physics. x . More generally, the “projected Killing equations” σ a (cid:48) ( x, x (cid:48) ) L ξ g a (cid:48) b (cid:48) ( x (cid:48) ) = σ a (cid:48) ( x, x (cid:48) ) σ b (cid:48) ( x, x (cid:48) ) ∇ a (cid:48) L ξ g b (cid:48) c (cid:48) ( x (cid:48) ) = 0 (126)are valid wherever the generalized Killing fields are defined [1]. Elements of K G ( x ) also preserve all distances from x in the sense that L ξ σ ( x, x (cid:48) ) = 0. Ingeneral, ξ a (cid:48) ( x (cid:48) ) is a solution to the geodesic deviation (or Jacobi) equation alongthe geodesic connecting x and x (cid:48) .The generalized Killing fields may be shown to admit an expansion in termsof 1- and 2-forms at x . Expanding (123) in terms of covariant derivatives showsthat for every ξ a ∈ K G ( x ), ξ a (cid:48) = Ξ a (cid:48) a ξ a + Ξ a (cid:48) ab ∇ a ξ b . (127)The bitensors Ξ a (cid:48) a and Ξ a (cid:48) ab which appear here — known as Jacobi propagators— are explicitly given by (56). The expansion (53) of Killing vector fields istherefore identical to the expansion (127) of generalized Killing vector fields.Varying ξ a and ∇ a ξ b arbitrarily, it is clear that dim K G ( x ) = N + N ( N −
1) = N ( N + 1) in N dimensions.The generalized Killing fields defined by (123) and (124) provide a notion ofsymmetry with respect to a point. They may be used to analyze non-relativisticmotion in geometries which do not admit any exact symmetries. This is notpursued here. We instead focus on relativistic motion, in which case it is moreappropriate to consider a different kind of generalized Killing field which pro-vides a notion of symmetry near a worldline instead of a point. In relativistic contexts, it is useful to define a K G which takes as arguments atimelike worldline and a foliation instead than a single point. Given a worldtube W = { B s | s ∈ R } in a Lorentzian spacetime ( M , g ab ) of dimension N , considera foliation { B s } . Also consider a timelike worldline Z parametrized by z s := Z ∩ B s . The definition of K G ( Z , { B s } ) in this context is as follows: First,(125) is enforced for all z s . This provides a sense in which the elements of K G ( Z , { B s } ) generalize Poincar´e symmetries “near” Z . It implies that thegeneralized Killing fields and their first derivatives satisfy the Killing transportequations on Z . Moreover, ∇ a ∇ b ξ c | Z = − R bcad ξ d , (128)which generalizes (61). This describes, e.g., how generalized Killing fields whichmight appear purely rotational or boost-like at one point acquire translationalcomponents at nearby points. When applied to problems of motion, it leads tothe Mathisson-Papapetrou spin-curvature force.Demanding that (125) hold on Z describes the generalized Killing fields onlyon that worldline. They may be extended outwards into W by demanding that L ξ σ a (cid:48) ( x (cid:48) , z s ) = 0 (129)36or each x (cid:48) ∈ B s . This is merely a restriction of (123). It implies that (127)holds whenever there exists some s such that x = z s and x (cid:48) ∈ B s . Elementsof K G ( Z , { B s } ) may therefore be specified using an arbitrary pair of 1- and2-forms at any point on Z . As required, the generalized Killing fields form avector space with dimension N ( N + 1). Additionally, L ξ σ ( x (cid:48) , z s ) = 0 and σ a (cid:48) ( x (cid:48) , z s ) σ b (cid:48) ( x (cid:48) , z s ) L ξ g a (cid:48) b (cid:48) ( x (cid:48) , z s ) = 0 (130)whenever x (cid:48) ∈ B s [1]. The elements of K G ( Z , { B s } ) are the generalized Killingfields used in the remainder of this work. We now return to the motion of scalar charges as discussed in Section 3, butno longer require that the background spacetime admit any symmetries. Con-sider a body coupled to a Klein-Gordon field φ in a four-dimensional spacetime( M , g ab ). This body is assumed to be contained inside a worldtube W ⊂ M with finite spatial extent and no other matter. Its stress-energy tensor T ab body isassumed to satisfy (77).A generalized momentum is easily defined by reusing (78), but with the space K employed there replaced by an appropriate space K G ( Z , { B s } ) of generalizedKilling fields as described in Section 4.1. This requires the introduction of atimelike worldline Z and a foliation { B s } of W . Supposing that these structureshave been chosen — perhaps using center of mass conditions — let P s ( ξ ) := (cid:90) B s T ab body ξ a d S b (131)for all ξ a ∈ K G ( Z , { B s } ). For each s , this is a vector in the ten-dimensionalspace K ∗ G ( Z , { B s } ). The associated linear and angular momenta p a and S ab coincide with those introduced by Dixon [6, 17, 24] for matter which does notcouple to an electromagnetic field .Using stress-energy conservation to differentiate the generalized momentumwith respect to the time parameter s ,dd s P s ( ξ ) = (cid:90) B s (cid:18) T ab body L ξ g ab − ρ L ξ φ (cid:19) d S. (132)The first term on the right-hand side of this expression is not present in itsmaximally-symmetric counterpart (79); extra forces arise when the ξ a are notKilling. These may be interpreted as gravitational effects. While sufficientlysmall test bodies fall along geodesics in curved spacetimes, the same is nottrue for more extended masses. Gravity generically exerts nonzero 4-forcesand 4-torques which are described by the L ξ g ab term in (132). If expanded in a Dixon’s papers never considered matter coupled to scalar fields. The momenta associatedwith (131) are those which arise naturally for objects falling freely in curved spacetimes. s ˆ P s ( ξ ) = (cid:90) B s (cid:20) T ab body ( x ) L ξ g ab ( x ) − ρ ( x ) L ξ ˆ φ ( x ) − (cid:90) W d V (cid:48) ρ ( x ) ρ ( x (cid:48) ) L ξ G ( x, x (cid:48) ) (cid:21) d S. (133)The effective field which appears here is defined by ˆ φ := φ − φ S , where φ S satisfies (83) and G is the Detweiler-Whiting S-type Green function describedin Section 3.4. The generalized momentum P s has also been replaced by therenormalized momentum ˆ P s := P s + E s , where the self-momentum E s is givenby (90).An absence of spacetime symmetries explicitly affects self-interaction onlyvia the second line of (133). Unlike in maximally-symmetric backgrounds, theDetweiler-Whiting Green function does not satisfy L ξ G = 0 in general. Indeed,no Green function can be constructed with this property, a result which couldbe viewed as implying that it is impossible to define an analog of Newton’sthird law in generic spacetimes. It follows that the self-force cannot be entirelyabsorbed into a redefinition of the momentum. Its remainder can, however, beunderstood as equivalent to another type of renormalization which affects thequadrupole and higher multipole moments of T ab body . The self-force which remains after renormalizing P s depends on L ξ G . It may beunderstood physically by recalling that there is a sense in which G is constructedpurely from the spacetime metric. It therefore follows that for any vector field ξ a , whether it is a generalized Killing field or not, L ξ G must depend linearly on L ξ g ab . In this sense, the first and third terms on the right-hand side of (133)are both linear in L ξ g ab . They are physically very similar, both representingdifferent aspects of the gravitational force [4].As remarked in Section 3, a body’s inertia depends on both its own stress-energy tensor and the properties of its self-field. The inertia due to T ab body isdescribed by P s and the inertia due to φ S by E s . A body’s passive gravitationalmass experiences a similar split. The gravitational force exerted on a body dueto its stress-energy tensor is 12 (cid:90) B s T ab body L ξ g ab d S, (134)while the gravitational force exerted on a body’s self-field is instead describedby − (cid:90) B s d S (cid:90) W d V (cid:48) ρρ L ξ G. (135)38lthough it is difficult to do so explicitly, this latter expression can be trans-formed into a linear operator on L ξ g ab . It effectively adds to a body’s gravita-tional mass distribution as inferred by observing responses to different spacetimecurvatures. In a multipole expansion, the force (135) may be viewed as renor-malizing the quadrupole and higher multipole moments of a body’s stress-energytensor.
The failure of Newton’s third law, or equivalently L ξ G (cid:54) = 0, therefore pro-vides a second mechanism by which self-fields lead to renormalizations. It isdistinct — both in its origin and in the quantities it affects — from the mecha-nism described in Section 3.5. The renormalization of a body’s momentum wasshown to be closely connected to the hyperbolicity of the underlying field equa-tion. The geometry-induced failure of Newton’s third law can instead ariseeven for matter coupled to elliptic field equations. It affects only the quadrupoleand higher moments of a body’s stress-energy tensor. Combined, the two typesof renormalization affect all multipole moments of T ab body . In this sense, one is ledto the concept of an effective stress-energy tensor. This is defined quasi-locally,and can be identified with T ab tot only in special cases.A fully explicit demonstration of this effect is not known. It may neverthelessbe motivated more directly in terms of a wave equation satisfied by L ξ G . Notingthat L ξ δ ( x, x (cid:48) ) = − δ ( x, x (cid:48) ) g ab ( x ) L ξ g ab ( x ) , (136)a Lie derivative of (80) yields( ∇ a ∇ a − µ ) L ξ G = ∇ a (cid:2)(cid:0) g ac g bd − g ab g cd (cid:1) ∇ b G L ξ g cd (cid:3) + µ g ab L ξ g ab ) G. (137)Viewing L ξ G on the left-hand side of this equation as “independent” of the G appearing on the right-hand side suggests that L ξ G is a solution to a waveequation sourced by L ξ g ab and its first derivative. A source which is independentof G may be found by applying the Klein-Gordon operator to both sides:( ∇ a (cid:48) ∇ a (cid:48) − µ )( ∇ a ∇ a − µ ) L ξ G = 4 π ∇ a (cid:2)(cid:0) g ac g bd − g ab g cd (cid:1) ( L ξ g cd ) ∇ b δ (cid:3) − πµ ( g ab L ξ g ab ) δ. (138)This describes L ξ G as the solution to a fourth-order distributional differentialequation sourced by L ξ g ab and ∇ a L ξ g bc . This type of renormalization fundamentally arises from the connection between L ξ G and L ξ g ab which occurs for Green functions associated with the Klein-Gordon equation. In differ-ent theories, Lie derivatives of G can depend on fields other than the metric. Self-forces thenrenormalize whichever moments are coupled to these fields. T ab field , S := 14 π (cid:20) ∇ a φ S ∇ b φ S − g ab ( ∇ c φ S ∇ c φ S + µ φ ) (cid:21) (139)be the stress-energy tensor associated with φ S and define I a := 18 π ( g ac g bd − g ab g cd ) φ S ∇ b φ S L ξ g cd + 18 π (cid:90) W ρ (cid:48) ( ∇ a φ S L ξ G − φ S ∇ a L ξ G )d V (cid:48) . (140)The law of motion (133) then reduces todd s ˆ P s = (cid:90) B s (cid:20)
12 ( T ab body + T ab field , S ) L ξ g ab − ρ L ξ ˆ φ (cid:21) d S − dd s (cid:90) B s I a d S a − (cid:73) ∂ B s I a t b d S ab . (141)The gravitational force in this expression clearly acts on the combined stress-energy tensor T ab body + T ab field , S . Unfortunately, the stress-energy tensor associatedwith the self-field does not have compact spatial support. The finite integrationvolume B s must therefore be compensated by the two boundary terms involving I a . If these can be ignored, it is evident that gravitational forces are determinedonly by T ab body + T ab field , S . This cannot, however, be expected to hold generically.In general, there does not appear to be any reason to neglect I a .Approximations may instead be introduced which allow the renormalizedmultipole moments to be computed essentially using local Taylor series [4]. An-other approach, described here for the first time, is to use Hadamard series.Recalling the Hadamard form (82) for G , L ξ G = 12 (cid:104) ∆ / δ (cid:48) ( σ ) L ξ σ + ( L ξ ∆ / − H L ξ σ ) δ ( σ ) − L ξ V Θ( σ ) (cid:105) . (142)Our task is now to convert this into an expression where all Lie derivatives acton the metric.This is easily accomplished for the Lie derivatives of σ and ∆. Differentiatingthe well-known identity σ a (cid:48) σ a (cid:48) = 2 σ shows that σ a (cid:48) ∇ a (cid:48) L ξ σ − L ξ σ = 12 σ a (cid:48) σ b (cid:48) L ξ g a (cid:48) b (cid:48) . (143)The differential operator σ a (cid:48) ( x, x (cid:48) ) ∇ a (cid:48) appearing here is a covariant derivativealong the geodesic connecting x and x (cid:48) , so (143) may be viewed as an ordinarydifferential equation for L ξ σ . Letting y τ denote a geodesic which is affinelyparametrized by τ while satisfying y = x and y = x (cid:48) , it follows that L ξ σ ( x, x (cid:48) ) = 12 (cid:90) ˙ y aτ ˙ y bτ L ξ g ab ( y τ )d τ. (144)40oreover, an argument found in [4] shows that Lie derivatives of the van Vleckdeterminant ∆ depend on L ξ σ : L ξ ln ∆ / = − (cid:104) g ab L ξ g ab + g a (cid:48) b (cid:48) L ξ g a (cid:48) b (cid:48) + 2 H aa (cid:48) ∇ a ∇ a (cid:48) L ξ σ (cid:105) . (145)Both L ξ σ and L ξ ∆ / may therefore be written as line integrals — solutions totransport equations — which are linear in L ξ g ab . Substituting these expressionsinto (142) goes much of the way towards expressing L ξ G in terms of L ξ g ab .All that remains is to consider L ξ V . This is more difficult. Even V itselfis complicated to compute. It is a solution to the homogeneous field equationwhich is symmetric in its arguments and satisfies V ( x, x (cid:48) ) = G + ( x, x (cid:48) ) + G − ( x, x (cid:48) ) (146)whenever x (cid:48) is timelike-separated from x . Alternatively, V can be computedusing a transport equation along null geodesics [10]. For each x (cid:48) , this transportequation may be used as boundary data with which to obtain V ( · , x (cid:48) ). Extend-ing this data outside of the null cones is essentially an “exterior characteristicproblem:” One seeks a solution to a hyperbolic differential equation in the ex-terior of a null cone given values of the solution on that cone. Unlike interiorcharacteristic problems, the general mathematical status of such problems isunclear.One way to proceed is to construct a Hadamard series. This is an ansatzwhich supposes that V can be expanded via [41] V ( x, x (cid:48) ) = ∞ (cid:88) n =0 V n ( x, x (cid:48) ) σ n ( x, x (cid:48) ) . (147)The V n here are determined by demanding that each explicit power of σ vanishseparately when this series is inserted into ( ∇ a ∇ a − µ ) V = 0. The result is aninfinite tower of ordinary differential equations. The n = 0 case is governed by∆ / σ a ∇ a (∆ − / V ) + V = 12 ∇ a ∇ a ∆ / . (148)For each x (cid:48) , this determines V ( · , x (cid:48) ) on the light cones of x (cid:48) . Higher-order termsare needed in the exteriors of these light cones. For all n ≥ / σ a ∇ a (∆ − / V n ) + ( n + 1) V n = − n ∇ a ∇ a V n − . (149)It should be emphasized that the Hadamard series is not a Taylor expansion.The V n are two-point scalar fields, not constants. Furthermore, the Hadamardseries is known to converge only if the metric is analytic (and even then, itmight converge only in a finite region) [41]. Although analyticity is quite astrong assumption, there may be other interesting cases where a finite Hadamardseries can be used to approximate V up to a well-controlled remainder.41ssuming that (147) is valid, it is easily used to compute Lie derivatives of V . This results in L ξ V = ∞ (cid:88) n =0 (cid:2) L ξ V n + ( n + 1) V n +1 L ξ σ (cid:3) σ n . (150)Lie derivatives of σ may already be transformed into Lie derivatives of g ab via(144). Lie derivatives of the V n may instead be found by differentiating (148)and (149). The n = 0 case satisfies∆ / σ a ∇ a (∆ − / L ξ V ) + L ξ V = 12 (cid:0) L ξ ∇ a ∇ a ∆ / − V L ξ σ aa (cid:1) + ( g ab σ c L ξ g bc − ∇ a L ξ σ ) ∇ a V , (151)for example. The left-hand side of this equation may be interpreted as an ordi-nary differential operator along the geodesic which connects the two argumentsof V . All Lie derivatives in the source on the right-hand of this equation may,with the help of (144) and (145), be rewritten in terms of L ξ g ab . It follows that L ξ V can be expressed as a line integral involving L ξ g ab . Similar results alsohold for L ξ V n with n >
0. The detailed forms of these integrals are complicatedand are not displayed here. The important point, however, is that all partsof L ξ G may be expressed as line integrals involving L ξ g ab . Changing variablesappropriately and using the spatially-compact support of ρ then shows that theself-force (135) does indeed renormalize the gravitational force (134). It is useful to expand the scalar force in a multipole series when ˆ φ varies slowlyinside the body. Similarly, the gravitational force may be expanded in its ownmultipole series when there is an appropriate sense in which g ab does not vary toorapidly inside each B s . Such expansions can be obtained using the techniquesof Sections 2.2.5 and 3.6. Recalling (94), (98), and (133), first note thatdd s ˆ P s ( ξ ) = − ∞ (cid:88) n =0 n ! q a ··· a n L ξ ˆ φ ,a ··· a n + 12 (cid:90) B s T ab body L ξ g ab d S − (cid:90) B s d S (cid:90) W d V (cid:48) ρρ (cid:48) L ξ G. (152)The two integrals which remain here are intrinsically gravitational.Given (125), it is evident that multipole expansion for the generalized forcemust begin at quadrupole order. More specifically, it may be shown that [4, 6, More precisely, the coordinate components g ij must not vary too rapidly when expressed ina Riemann normal coordinate system with origin z s . Physically, this is a significant restriction.It would be far better to perform multipole expansions only using effective metrics wheregravitational self-fields have been appropriately removed. It is not known how to do to thisfor the full Einstein-Klein-Gordon system. (cid:90) B s T ab body L ξ g ab d S = 12 ∞ (cid:88) n =2 n ! I c ··· c n ab L ξ g ab,c ··· c n , (153)where g ab,c ··· c n represents the n th tensor extension of g ab and I c ··· c n ab isDixon’s [6] 2 n -pole moment of T ab body . Tensor extensions in this context aresomewhat more complicated than for the scalar case discussed in Section 2.2.5.While they are still defined as those tensors which reduce to n partial deriva-tives when evaluated at the origin of a Riemann normal coordinate system,equations like (48) must be generalized for objects with nonzero tensorial rank(see, e.g., [4]). For this reason, explicit integrals relating I c ··· c n ab to T ab body aresignificantly more complicated than their scalar analogs. These are not neededhere, and may be found in [4, 6]. Additionally, note that the first few tensorextensions of the metric are g ab,c = 0 and g ab,c c = 23 R a ( c c ) b , g ab,c c c = ∇ ( c R | a | c c ) b , (154) g ab,c c c c = 65 ∇ ( c c R | a | c c ) b + 1615 R a ( c c d R | b | c c ) d . (155)Expanding both integrals in (152) while identifying coefficients in frontof L ξ g ab,c ··· c n results in a series structurally identical to (153), but with allbare multipole moments I c ··· c n ab replaced by their renormalized counterpartsˆ I c ··· c n ab . The final multipole expansion for the generalized force acting on aself-interacting scalar charge distribution is thereforedd s ˆ P s ( ξ ) = 12 ∞ (cid:88) n =2 n ! ˆ I c ··· c n ab L ξ g ab,c ··· c n − ∞ (cid:88) n =0 n ! q a ··· a n L ξ ˆ φ ,a ··· a n . (156)Gravitational terms first arise at quadrupole order, while scalar terms appeareven at monopole order. As in Sections 2.2.6 and 3.7, the generalized momentum can be decomposedinto linear and angular components ˆ p a , ˆ S ab . These obey the law of motion(102), where the force and torque are now supplemented by gravitational termsat quadrupole and higher orders:ˆ F a = 12 ∞ (cid:88) n =2 n ! ˆ I d ··· d n bc ∇ a g bc,d ··· d n − ∞ (cid:88) n =0 n ! q b ··· b n ∇ a ˆ φ ,b ··· b n , (157)ˆ N ab = ∞ (cid:88) n =2 n ! g f [ b (cid:0) ˆ I | c ··· c n | a ] d g df,c ··· c n + n I a ] c ··· c n − dh g dh,c ··· c n − f (cid:1) + ∞ (cid:88) n =0 n ! g c [ a q b ] d ··· d n ˆ φ ,cd ··· d n . (158)A center of mass may be defined exactly as described in Section 3.8. Apply-ing (110), the momentum-velocity relation remains (111).43 .2.4 Monopole approximation Suppose that Z is identified with the center of mass worldline { γ s } . It is theninteresting to consider the laws of motion truncated at monopole order. Asexplained in Section 3.9, the spin magnitude is conserved in such cases. It istherefore consistent to consider cases where ˆ S a is negligible. Assuming this, anobject’s mass ˆ m and center of mass γ s satisfyˆ m − q ˆ φ = (constant) , ˆ m ¨ γ as = − q ( g ab + ˙ γ as ˙ γ bs ) ∇ b ˆ φ (159)whenever q = (constant). These are formally equivalent to the standard equa-tions of motion adopted for scalar test charges, but with the physical fieldreplaced everywhere by the effective field; laws of motion applicable to self-interacting charges involve ˆ φ , not φ .Classical results on the scalar self-force are easily derived from (159). Inthe absence of any external charges (which does not necessarily imply trivialmotion in curved spacetimes), it is natural to suppose that initial data for φ has been prescribed in the distant past. If all details of that data have decayedsufficiently, the only remaining field is the retarded solution associated with thebody’s own charge distribution. Generalizing this somewhat to also allow for aprescribed “external field” φ ext , φ ( x ) = φ ext ( x ) + (cid:90) W ρ ( x (cid:48) ) G − ( x, x (cid:48) )d V (cid:48) . (160)While the retarded Green function G − can be difficult to compute in nontrivialspacetimes, we assume that this problem has been solved. Using (81) and (83)then results in the effective potential ˆ φ = φ ext + 12 (cid:90) W ρ (cid:48) ( G − − G + + V )d V (cid:48) . (161)Although the “self-field” φ S has been removed from φ , it is clear from this thatremnants of the body’s charge distribution do remain. These are responsiblefor self-forces as they are commonly defined in the literature, and give rise tophysical phenomena such as radiation reaction.The “traditional” self-force problem involves point particle limits. Such lim-its consist of appropriate one-parameter families of charge distributions λ ρ andstress-energy tensors λ T ab body whose supports collapse to a single timelike world-line as λ → λ as λ →
0, suggest-ing that λ q a ··· a n scales like λ n +1 for all n ≥ λ ˆ I c ··· c n ab scales like λ n +1 for all n ≥
2. These conditions guarantee that the multipole series can betruncated at low order [as has already been assumed in (159)]. The two-point scalar V is to be understood here as equivalent to G ± when its argumentsare timelike-separated. This defines it even in the presence of caustics and other potentialcomplications.
44t is also important to demand that the time dependence of λ ρ remain smoothas λ →
0. This guarantees that a body’s internal dynamics remain slow com-pared to its light-crossing timescale, which is required to ensure that a chargedoes not self-accelerate in the absence of any external influence. Self-accelerationcould occur if, e.g., internal oscillations conspired to generate strongly-collimatedbeams of radiation. Such cases are physically possible, but are typically excludedfrom self-force discussions.The λ → φ ext . Effects typically described as self-forces occur at the fol-lowing order. In principle, interactions between dipole moments and φ ext alsooccur at this order. These are dropped here for simplicity, leaving only (159).Without entering into technical details, the appropriate approximation for theeffective field in this limit may be shown to be given by (161) with a pointparticle charge density. Dropping all labels involving λ ,ˆ φ ( x ) = φ ext ( x ) + 12 q (cid:90) [ G − ( x, γ s (cid:48) ) − G + ( x, γ s (cid:48) ) + V ( x, γ s (cid:48) )]d s (cid:48) (162)plus terms of order λ . Unlike the λ → φ , the limit of ˆ φ is well-defined even at the particle’s location. The same is also true of its gradient. Allnecessary regularizations have automatically been taken into account by firstderiving the correct laws of motion for nonsingular extended bodies.The computations needed to evaluate ∇ a ˆ φ at γ s are rather tedious, so wemerely cite the result [10]. Defining the projection operator h ab := g ab + ˙ γ as ˙ γ bs and assuming that q = (constant), ∇ a ˆ φ = ∇ a φ ext + 13 ( q / ˆ m ) h ab ˙ γ cs (cid:2) ∇ b ∇ c φ ext − q/ ˆ m ) ∇ b φ ext ∇ c φ ext (cid:3) + 112 q ( R ˙ γ as − h ab R bc γ cs ) + q lim (cid:15) → + (cid:90) s − (cid:15) −∞ ∇ a G − ( γ s , γ s (cid:48) )d s (cid:48) . (163)Substituting this into (159), the final equations of motion are¨ γ as = − ( q/ ˆ m ) h ab (cid:18) ∇ b φ ext + 13 ( q / ˆ m ) ˙ γ cs (cid:2) ∇ b ∇ c φ ext − q/ ˆ m ) ∇ b φ ext ∇ c φ ext (cid:3) − qR bc ˙ γ cs + q lim (cid:15) → + (cid:90) s − (cid:15) −∞ ∇ b G − ( γ s , γ s (cid:48) )d s (cid:48) (cid:19) (164)and ˆ m − qφ ext + q (cid:18) R − lim (cid:15) → + (cid:90) s − (cid:15) −∞ G − ( γ s , γ s (cid:48) )d s (cid:48) (cid:19) = (constant) (165) The point particle field derived in [10] includes a derivative of the particle’s acceleration.A careful treatment of the perturbation theory shows, however, that such terms refer only toaccelerations at lower order [28]. The self-consistent discussion which is implicit here thereforerequires that accelerations be simplified using the zeroth order equation of motion. This istaken into account in (163). λ . Essentially the same results were obtained by Quinnusing an axiomatic argument [42], and later derived from first principles in [2].Note that the integrals over the tail of G − which appear here indicate that acharge’s motion can depend on its past history. This is related to the failureof Huygens’ principle, implying that the field a body sources in the past canscatter back towards it at later times. This effect disappears for matter coupledto massless fields in flat spacetime, but is almost always present otherwise. The problem of electromagnetic self-force has inspired considerable discussionover the past century. Here, we show how it can be understood using a straight-forward application of the formalism just described for scalar charges. Thebody of interest is assumed to be smooth and to be confined to a finite world-tube W ⊂ M in a fixed four-dimensional spacetime ( M , g ab ). It couples to anelectromagnetic field F ab = F [ ab ] satisfying Maxwell’s equations ∇ [ a F bc ] = 0 , ∇ b F ab = 4 πJ a , (166)from which it follows as an integrability condition that the body’s 4-current J a must be divergence-free. Its total charge q := (cid:90) B s J a d S a (167)is therefore a constant independent of B s (as long as this hypersurface com-pletely contains the body of interest). Furthermore, the stress-energy tensorassociated with F ab can be defined throughout W via T ab field = 14 π ( F ac F bc − g ab F cd F cd ) . (168)Identifying all remaining stress-energy as T ab body , stress-energy conservation im-plies that ∇ b T ab body = F ab J b . (169)The right-hand side of this equation is the Lorentz force density. A generalized momentum may again be defined using (131). This requires achoice of worldline Z and a foliation { B s } , leading to a vector which resides in K ∗ G ( Z , { B s } ). Applying (169) shows that forces and torques follow fromdd s P s ( ξ ) = (cid:90) B s (cid:18) T ab body L ξ g ab + F ab ξ a J b (cid:19) d S. (170) Some sign conventions in [2] and [42] are different from those adopted here. P s has the unfortunate property that symmetries do not necessar-ily imply conservations laws. Even if there exists a vector field ψ a satisfying L ψ F ab = L ψ g ab = 0, the associated momentum P s ( ψ ) is not necessarily con-served.Dixon [6, 24, 43] has proposed a different set of linear and angular mo-menta in this context which do form simple conservation laws in the presenceof symmetries (among other desirable properties more generally). Translatinghis definitions into a generalized momentum P D s results in P D s ( ξ ) := P s ( ξ ) + (cid:90) B s d S a J a ( x ) (cid:90) d uu − σ b (cid:48) ( y (cid:48) u , z s ) ξ c (cid:48) ( y (cid:48) u ) F b (cid:48) c (cid:48) ( y (cid:48) u ) . (171)The curve { y (cid:48) u | u ∈ [0 , } describes an affinely-parametrized geodesic satisfying y = z s and y = x . The correction to P s which is used here represents a type ofinteraction between the electromagnetic field and its source. It can be motivatedusing symmetry considerations [24], the theory of multipole moments [6], orLagrangian methods [43]. Some intuition for this interaction may be gained byintroducing a vector potential A a so that F ab = 2 ∇ [ a A b ] . Then, P D s + q ( A a ξ a ) | z s = P s + (cid:90) B s d S a J a (cid:18) A b ξ b − (cid:90) d uu − σ b (cid:48) L ξ A b (cid:48) (cid:19) . (172)Although this has been written in terms of a gauge-dependent vector potential,it is manifest from (171) that P s and P D s are both gauge-invariant.Combining (172) with (170) shows that the generalized force associated withDixon’s momentum isdd s ( P D s + qA a ξ a ) = (cid:90) B s d S (cid:18) T ab body L ξ g ab + J a L ξ A a (cid:19) − dd s (cid:90) B s d S a J a (cid:90) d uu − σ a (cid:48) L ξ A a (cid:48) . (173)This is awkward to write more explicitly without introducing additional tech-nical tools. Even so, it is simple to temporarily consider special cases wherethere exists a Killing vector field ψ a satisfying L ψ F ab = L ψ g ab = 0. It is thenpossible to find a vector potential A ( ψ ) a such that L ψ A ( ψ ) a = 0. Using this, thecomponent of momentum conjugate to ψ a is seen to be conserved in the sensethat P D s ( ψ ) + qA ( ψ ) a ξ a = (constant) , (174)where A ( ψ ) a ξ a is understood to be evaluated at z s . Although this is reminiscentof the canonical momentum associated with a pointlike test charge, it is validfor essentially arbitrary extended charge distributions.47 .3.2 The self-field and self-force Electromagnetic self-forces may be defined and removed from either P s or P D s .In both cases, it is convenient to work in the Lorenz gauge: ∇ a A a = 0 . (175)Maxwell’s equations then reduce to the single hyperbolic equation ∇ b ∇ b A a − R ab A b = − πJ a . (176)Introducing the parallel propagator g aa (cid:48) ( x, x (cid:48) ) [10], consider a Green function G aa (cid:48) satisfying1. (cid:3) G aa (cid:48) − R ab G ba (cid:48) = − πg aa (cid:48) δ ( 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) ) = 0 when x , x (cid:48) are timelike-separated.These are the closest possible analogs to the constraints used to define G inSection 3.4. They characterize the S-type Detweiler-Whiting Green function forthe reduced Maxwell equation (176). In terms of the advanced and retardedGreen functions G ± aa (cid:48) , there exists a homogeneous solution V aa (cid:48) such that G aa (cid:48) = 12 ( G + aa (cid:48) + G − aa (cid:48) − V aa (cid:48) ) = 12 [ g aa (cid:48) ∆ / δ ( σ ) − V aa (cid:48) Θ( σ )] . (177)Although it can be difficult to find G aa (cid:48) explicitly in a particular spacetime, weassume that it is known.A self-field A S a may be defined by convolving J a with G aa (cid:48) . Considering aspacetime volume R ⊆ W , let A S a [ R ] := (cid:90) R G aa (cid:48) J a (cid:48) d V (cid:48) . (178)In the scalar case, the analog of this expression represents the “self-field” sourcedin the region R . The interpretation here is somewhat more obscure, as therestriction of J a to arbitrary regions is not necessarily conserved and is thereforeunphysical. This definition is nevertheless useful. Without specification of anyparticular region, it is to be understood that R = W so A S a = A S a [ W ]. This(full) self-field is sourced by a conserved current, implying that F S ab := 2 ∇ [ a A S b ] satisfies the complete Maxwell equations ∇ [ a F S bc ] = 0 , ∇ b F S ab = 4 πJ a . (179)The same cannot necessarily be said for fields 2 ∇ [ a A S b ] [ R ] where R (cid:54) = W .Now consider changes in P s due to A S a . Applying the arguments of Sections3.4 and 3.5 to (170) shows that the self-force has the form (85) with f ( x, x (cid:48) ) = 2 ξ a J b J b (cid:48) ∇ [ a G b ] b (cid:48) . (180)48he average f ( x, x (cid:48) ) + f ( x (cid:48) , x ) of action-reaction pairs reduces in this case to J a J a (cid:48) L ξ G aa (cid:48) − ∇ a ( J a J a (cid:48) ξ b G ba (cid:48) ) − ∇ a (cid:48) ( J a J a (cid:48) ξ b (cid:48) G ab (cid:48) ) . (181)The divergences which appear here are new features of the electromagneticproblem. Only their integrals matter, however, so they are easily dealt with.Defining the homogeneous effective fieldˆ F ab := F ab − F S ab , (182)the final result is thatdd s ˆ P s = (cid:90) B s d S (cid:18) T ab body L ξ g ab + ξ a J b ˆ F ab + 12 (cid:90) W d V (cid:48) J a J a (cid:48) L ξ G aa (cid:48) (cid:19) , (183)where ˆ P s := P s + E s and E s := 12 (cid:18)(cid:90) B + s d V J a L ξ A S a [ B − s ] − (cid:90) B − s d V J a L ξ A S a [ B + s ] (cid:19) + (cid:90) B s ( ξ a A S a ) J b d S b . (184)The renormalized momentum ˆ P s therefore evolves via ˆ F ab rather than F ab . Asexplained in Section 4.2.1, the forces involving L ξ g ab and L ξ G aa (cid:48) in (183) com-bine in a natural way to form an effective gravitational force. Furthermore, E s isknown to reduce in simple cases to the expected expression involving the stress-energy tensor associated with F S ab [3]. More generally, it should be thought ofas a quasi-local functional of J a .A very similar result may be derived using the Dixon momentum P D s . Thisis most easily accomplished if a new self-momentum is introduced which satisfies E D s := E s − (cid:90) B s d S a J a (cid:90) d uu − σ b (cid:48) ξ c (cid:48) F S b (cid:48) c (cid:48) . (185)Defining ˆ P D s := P D s + E D s , it follows thatdd s ˆ P D s = (cid:90) B s d S (cid:18) T ab body L ξ g ab + ξ a J b ˆ F ab + 12 (cid:90) W d V (cid:48) J a J a (cid:48) L ξ G aa (cid:48) (cid:19) + dd s (cid:90) B s d S a J a (cid:90) d uu − σ b (cid:48) ξ c (cid:48) ˆ F b (cid:48) c (cid:48) , (186)or equivalentlydd s ( ˆ P D s + q ˆ A a ξ a ) = (cid:90) B s d S (cid:18) T ab body L ξ g ab + J a L ξ ˆ A a + 12 (cid:90) W d V (cid:48) J a J a (cid:48) L ξ G aa (cid:48) (cid:19) − dd s (cid:90) B s d S a J a (cid:90) d uu − σ a (cid:48) L ξ ˆ A a (cid:48) (187)49or any vector potential ˆ A a satisfying ˆ F ab = 2 ∇ [ a ˆ A b ] .If a Killing vector field ψ a exists which satisfies L ψ ˆ F ab = L ψ g ab = 0, it ispossible to choose a vector potential for ˆ F ab with the property that L ψ ˆ A ( ψ ) a = 0.It then follows immediately that ˆ P D s + q ˆ A ( ψ ) a ξ a is conserved. No similarly-simpleconservation law is associated with ˆ P s . Integral expressions for the generalized force are not particularly useful on theirown. It is instead more interesting to consider their multipole expansions. Un-like in the scalar theories discussed earlier, more than one “reasonable” forcemay be considered in the electromagnetic case. As a matter of computation,it is simplest to expand the force associated with ˆ P s . Dixon’s momentum isotherwise more attractive, however. Multipole series for the associated forcesand torques have already been derived in a test body approximation [6]. Themechanics of the calculation are exactly the same here, and result indd s ˆ P D s = 12 ∞ (cid:88) n =2 n ! ˆ I c ··· c n ab L ξ g ab,c ··· c n + q ˆ F ab ξ a ˙ z bs + ∞ (cid:88) n =1 n ( n + 1)! q b ··· b n a L ξ ˆ F ab ,b ··· b n . (188)The coefficients q b ··· b n a represent 2 n -pole moments of J a as defined (using thenotation m b ··· b n a ) by Dixon [6, 44]. They are not to be confused with the scalarcharge moments satisfying (99). The ˆ I c ··· c n ab appearing in (188) represent 2 n -pole moments of T ab body as renormalized by Lie derivatives of G aa (cid:48) . Again, theseare different from the renormalized stress-energy moments which appear in thescalar law of motion (156). In the limit of negligible self-interaction, however,both definitions reduce to Dixon’s stress-energy moments [6]. For reference, thefirst tensor extensions of ˆ F ab are explicitlyˆ F ab,c = ∇ c ˆ F ab , ˆ F ab,cd = ∇ ( c ∇ d ) ˆ F ab −
23 ˆ F f [ a R b ]( cd ) f . (189)If there exists a Killing vector ψ a which satisfies L ψ ˆ F ab = L ψ ˆ A ( ψ ) a = 0, it hasalready been stated that ˆ P D s ( ψ ) + q ˆ A ( ψ ) a ψ a is conserved exactly. It is evidentfrom (188) that this quantity is also conserved in any consistent truncationof the multipole series. If a particular ψ a is Killing but does not necessarilypreserve ˆ F ab , all gravitational terms vanish from (188) anddd s ˆ P D s ( ψ ) = q ˆ F ab ψ a ˙ z bs + ∞ (cid:88) n =1 n ( n + 1)! q b ··· b n a L ψ ˆ F ab ,b ··· b n . (190)This holds for all ψ a ∈ K ⊆ K G . It is all that arises for charges moving in flator de Sitter spacetimes. 50 .3.4 Linear and angular momentum Linear and angular momenta may be extracted from ˆ P D s using the methodsdescribed in Sections 2.2.6 and 3.7. Let ˆ p a and ˆ S ab be defined byˆ P D s = ˆ p a ξ a + 12 ˆ S ab ∇ a ξ b . (191)Differentiating this and varying over all generalized Killing fields shows thatDˆ p a d s = q ˆ F ab ˙ z bs + 12 R bcda ˆ S bc ˙ z ds + ˆ F a , D ˆ S ab d s = 2ˆ p [ a ˙ z b ] s + ˆ N ab , (192)whereˆ F a = 12 ∞ (cid:88) n =2 n ! ˆ I d ··· d n bc ∇ a g bc,d ··· d n + ∞ (cid:88) n =1 n ( n + 1)! q c ··· c n b ∇ a ˆ F bc ,c ··· c n , (193)ˆ N ab = ∞ (cid:88) n =2 n ! g f [ b (cid:0) ˆ I | c ··· c n | a ] d g df,c ··· c n + n I a ] c ··· c n − dh g dh,c ··· c n − f (cid:1) + ∞ (cid:88) n =0 n ! g f [ b q a ] c ··· c n d ˆ F df,c ··· c n . (194)Electromagnetic forces and torques defined in this way first arise at dipole or-der. The (monopole) Lorentz force depends — unlike any other electromagneticterms — on ˙ z as . It has therefore been separated out explicitly in (192). Thereis no similarly velocity-dependent electromagnetic torque.Defining the force to exclude the Lorentz component has the advantage thatif ψ a is Killing and preserves ˆ F ab , the associated conservation law implies thatˆ F a ψ a + 12 ˆ N ab ∇ a ψ b = 0 . (195)This does not involve q , and is directly analogous to the scalar result (106). The electromagnetic laws of motion (192)-(194) depend on both the foliation { B s } and the worldline Z . Center of mass conditions may be used to fix thesestructures as described in Section 3.8. An evolution equation for the center ofmass position γ s can then be obtained by differentiating ˆ p a ˆ S ab = 0. The resultdiffers slightly from (111) due to the additional velocity-dependence associatedwith the Lorentz force. Adapting the methods of [31], the electromagneticmomentum-velocity relation may be shown to beˆ m ˙ γ as = ˆ p a − ˆ N ab u b − ˆ S ab [ ˆ m ˆ F b + (ˆ p c − ˆ N ch u h )( q ˆ F bc − R bcdf ˆ S df )]ˆ m − ˆ S pq ( q ˆ F pq − R pqrs ˆ S rs ) (196)when s has been chosen such that ˙ γ as ˆ p a = − ˆ m and u a is the unit timelike vectorsatisfying ˆ p a = ˆ mu a . 51lthough a precise set of assumptions which imply this are not known, itis assumed here that the center of mass condition (111) admits a unique time-like solution in a broad class of physical systems. Inspection of (196) shows asufficient (but not necessary) condition for this to fail isˆ m −
12 ˆ S bc ( q ˆ F bc − R bcdf ˆ S df ) = 0 . (197)While the q = 0 case of this equation was dismissed in Section 3.8 as un-likely to be physically relevant, the charged case is potentially more interest-ing. Consider an electron in a magnetic field of order B . Setting q = e ,ˆ m = m e , R abcd = 0, and ˆ S = (cid:126) /
2, the denominator in (196) can diverge when B ∼ m e /e (cid:126) ∼ Gauss. This may be viewed as the field strength at whichan electron’s dipole energy (cid:126) B/ Gauss magnetic fields are far from directlaboratory experience, such fields are believed to exist around some neutronstars [45]. Even in somewhat smaller magnetic fields, hidden momentum effectspredicted by the classical theory can become very large. Whether or not thishas qualitative consequences for neutron star astrophysics is an open question.If the center of mass can be defined and the classical laws of physics remainvalid, (196) may be combined with (192)-(194) to yield very general laws ofmotion. As in the scalar case, ˆ S ab can also be reduced to a single spin vectorˆ S a satisfying (113). Similarly, the mass varies according to (114). Unlike in thescalar case, matter coupled to electromagnetic fields can change mass only atdipole and higher orders. Such effects are related to changes in a body’s internalenergy due to work performed by (or against) the ambient fields [24]. In simple cases, the laws of motion governing an extended charge distributionmay be truncated at monopole order. Within this approximation, ˆ F a = ˆ N ab = 0and (192) reduces toDˆ p a d s = q ˆ F ab ˙ γ bs + 12 R bcda ˆ S bc ˙ γ ds , D ˆ S ab d s = 2ˆ p [ a ˙ γ b ] s . (198)Contracting the second of these equations with ˆ S ab while using (110) againshows that ˆ S := ˆ S a ˆ S a = (constant). An object which is initially not spinningtherefore remains non-spinning in this approximation. Consider these cases forsimplicity. The momentum-velocity relation then reduces to ˆ p a = ˆ m ˙ γ as , themass remains constant, and the body accelerates via the Lorentz force lawˆ m ¨ γ as = q ˆ F ab ˙ γ bs (199) It is unclear that there is any sense in which an electron’s behavior can be modeled usingequations derived for classical extended charges. Nevertheless, the example appears to besuggestive.
52n the effective electromagnetic field ˆ F ab . The effective field always satisfies thevacuum Maxwell equations, and is generically distinct from the physical field F ab which governs the acceleration of nearby test charges.In many cases of interest, it is useful to model F ab as the sum of some externalfield F ext ab and the retarded field associated with a body’s charge distribution: F ab = F ext ab + 2 (cid:90) W ∇ [ a G − b ] b (cid:48) J b (cid:48) d V (cid:48) . (200)In these cases, it follows from (177) and (182) that the effective field must satisfyˆ F ab = F ext ab + (cid:90) W ∇ [ a ( G − b ] b (cid:48) − G + b ] b (cid:48) + V b ] b (cid:48) ) J b (cid:48) d V (cid:48) . (201)Performing a point particle limit as discussed in Section 4.2.4 and [3, 28], thelowest-order self-interaction effects follow from (201) with a pointlike current.Through first order in the expansion parameter λ ,ˆ F ab = F ext ab + q (cid:90) ∇ [ a ( G − b ] b (cid:48) − G + b ] b (cid:48) + V b ] b (cid:48) ) ˙ γ b (cid:48) s d s (cid:48) . (202)Evaluating this on γ s and recalling the projection operator h ab = g ab + ˙ γ as ˙ γ bs , itmay be shown that [10]ˆ F ab = F ext ab + 43 q ˙ γ ls g l [ a h b ] c ˙ γ ds (cid:2) ( q/ ˆ m ) ˙ γ fs ∇ f F ext cd + ( q/ ˆ m ) g fh F ext cf F ext hd + 12 R cd (cid:3) + 2 q lim (cid:15) → + (cid:90) s − (cid:15) −∞ ∇ [ a G − b ] b (cid:48) ( γ s , γ s (cid:48) ) ˙ γ b (cid:48) s (cid:48) d s (cid:48) . (203)Substitution into (199) finally yields the equation of motion for a self-interacting“pointlike” electric charge:ˆ m ¨ γ as = qg ab F ext bc ˙ γ cs + 23 ( q / ˆ m ) h ab ˙ γ cs (cid:2) ˙ γ ds ∇ d F ext bc + ( q/ ˆ m ) g df F ext bd F ext fc (cid:3) + 13 q h ab R bc ˙ γ cs + 2 q lim (cid:15) → + (cid:90) s − (cid:15) −∞ ∇ [ a G − b ] b (cid:48) ˙ γ bs ˙ γ b (cid:48) s (cid:48) d s (cid:48) . (204)In curved spacetime, this is a reduced-order version — see Footnote 22 — of aresult first obtained by DeWitt and Brehme [21] (with corrections due to Hobbs[46]). The second line of (204) vanishes in flat spacetime, leaving onlyˆ m ¨ γ as = qg ab F ext bc ˙ γ cs + 23 ( q / ˆ m ) h ab ˙ γ cs (cid:2) ˙ γ ds ∇ d F ext bc + ( q/ ˆ m ) g df F ext bd F ext fc (cid:3) . (205)This is essentially the Abraham-Lorentz-Dirac equation [8], but in a reduced-order form typically attributed to Landau and Lifshiftz [47].While instructive, the neglect of spin and electromagnetic dipole effects in(204) can be overly restrictive. This restriction is easily dropped by making53se of the general multipole expansions derived above. The resulting changesare qualitatively significant [3] (see also [28]): A hidden momentum appears,external fields may change an object’s rest mass, and the spin magnitude maychange due to torques exerted by F ext ab .For monopole point particles in flat spacetime, the use of ˆ F ab instead of F ab in the laws of motion was first suggested by Dirac [48]. The appropriate gener-alization for monopole particles in curved spacetimes was obtained much morerecently by Detweiler and Whiting [29]. Both of these proposals were essentiallyphysically-motivated axioms intended to define the dynamics of point charges.The discussion here, which follows [3], shows that these regularization schemesare actually limits of laws of motion which hold rigorously for nonsingular ex-tended charge distributions. Once the general laws of motion (192)-(194) and(196) have been derived, more explicit results such as (204) follow very easily,as do their spin-dependent generalizations. All results discussed up to this point assume that the spacetime metric is knownbeforehand and has been fixed. This assumption may be relaxed. Doing soallows the consideration of self-gravitating masses in general relativity. We takea minimal approach to this problem by adapting the techniques of the previoussections as closely as possible. Although there are some deficiencies to thisstrategy, considerable progress can still be made.Let the spacetime be described by ( M , g ab ) and the body of interest becontained inside a spatially-compact worldtube W ⊂ M . Only the purely grav-itational problem is considered here, meaning that objects can interact witheach other solely via their influence on the metric; electromagnetic and similarlong-range interactions are excluded. It follows that T ab body = T ab tot inside W .Letting g ab be a solution to Einstein’s equation R ab − g ab R = 8 πg ac g bd T cd body , (206)it follows that ∇ b T ab body = 0 . (207)These two equations replace, e.g., (73) and (77) used to understand the motionof scalar charges. Unlike in the scalar or electromagnetic cases, the gravitationallaw of motion (207) is a consequence of the field equation, not an independentassumption.Both T ab body and the metric inside W are assumed to be smooth. This pre-cludes the consideration of black holes. Although unfortunate, it appears diffi-cult to remove this restriction in a non-perturbative theory which describes themotions of individual objects (and not only characteristics of the entire space-time). While quantities such as momenta might be associated with a blackhole horizon [49, 50], adopting such definitions typically excludes the discus-sion of objects without horizons. Alternatively, one may consider an effective54ackground and compute momenta associated with, e.g., the Landau-Lifshitzpseudotensor [51, 52]. This is troublesome as well. Nevertheless, the motionof black holes can be sensibly discussed within certain approximation schemes[10, 53, 54]. These consider only the metric outside of the body of interest,and apply versions of matched asymptotic expansions in appropriate “bufferregions.” We instead consider internal metrics as well as external ones, but donot require the existence of a buffer region.As in other theories, the laws of motion derived here involve an effectivefield which is distinct from the physical one. In general relativity, the relevantfield is the metric. We therefore use two metrics, which makes index raisingand lowering ambiguous. All factors of the appropriate metric are thereforedisplayed explicitly in this section. The first step to understanding the motion of a self-gravitating mass is to writedown a generalized momentum which describes an object’s large-scale behavior.Introducing a foliation { B s } of W and a worldline Z , the linear and angularmomenta proposed by Dixon [6, 17, 24] are conjugate to generalized Killingfields constructed using the physical metric g ab . Adding an extra argument to K G to reflect this metric dependence, the appropriate generalized momentumwhich contains Dixon’s definitions is P D s ( ξ ) := (cid:90) B s T ab body g bc ξ c d S a , ξ a ∈ K G ( Z , { B s } ; g ) . (208)For each s , this is an element of the ten-dimensional vector space K ∗ G ( Z , { B s } ; g ).The associated linear and angular momenta have a number of useful properties[6, 31, 55, 56]. Using (207), their time evolution satisfiesdd s P D s ( ξ ) = 12 (cid:90) B s T ab body L ξ g ab d S. (209)If L ξ g ab varies slowly throughout B s , forces and torques can be expanded inmultipole series as described in [4, 6] and in Section 4.2.2.While such assumptions can be useful for test bodies, they are too strongfor self-gravitating masses. Moving beyond the test body regime first requiresthe introduction of an effective metric ˆ g ab such that — after appropriate renor-malizations — the L ξ g ab appearing in (209) can be replaced by L ξ ˆ g ab . If L ξ ˆ g ab varies slowly, the resulting integral for the generalized force can be expanded ina multipole series in the usual way. One additional subtlety which occurs in thegravitational problem is that even if a particular ˆ g ab can itself be adequatelyapproximated using a low-order Taylor expansion, the same can not necessarilybe said for L ξ ˆ g ab when ξ a ∈ K G ( Z , { B s } ; g ). The generalized Killing fieldsassociated with Dixon’s momenta involve the physical metric and all of its at-tendant difficulties. These difficulties are partially inherited by the generalizedKilling fields used to define P D s . 55ne way out of this problem is to choose a bare momentum P s ( ξ ) definedby an integral which is structurally identical to (209), but where all ξ a areelements of K G ( Z , { B s } ; ˆ g ) rather than K G ( Z , { B s } ; g ). Let P s ( ξ ) := (cid:90) B s T ab body g bc ξ c d S a , ξ a ∈ K G ( Z , { B s } ; ˆ g ) . (210)We take this to be the bare momentum of a self-gravitating mass. The effec-tive metric which appears here is to be regarded at this stage as an additionalparameter. The generalized force d P s / d s follows from a trivial modification of(209). Furthermore, Dixon’s momenta are recovered in a test mass limit whereˆ g ab ≈ g b . If there exists a ψ a ∈ K G ( Z , { B s } ; ˆ g ) such that L ψ g ab = 0, it isevident that P s ( ψ ) must be conserved. There are many possible ways to extract an effective metric ˆ g ab from the physicalmetric g ab . The simplest generalization of the previous discussions involves atwo-point tensor field G aba (cid:48) b (cid:48) ( x, x (cid:48) ) which satisfies G aba (cid:48) b (cid:48) = G ( ab ) a (cid:48) b (cid:48) = G ab ( a (cid:48) b (cid:48) ) (211)and G aba (cid:48) b (cid:48) ( x, x (cid:48) ) = G a (cid:48) b (cid:48) ab ( x (cid:48) , x ) . (212)For any such propagator, consider an effective metric ˆ g ab defined via g ab = ˆ g ab + g S ab , (213)where g S ab [ R ] := (cid:90) R G aba (cid:48) b (cid:48) T a (cid:48) b (cid:48) body d V (cid:48) (214)and g S ab = g S ab [ W ]. Note that the volume element in this “self-field” is the oneassociated with g ab , not ˆ g ab . Substituting (213) into the appropriately-modifiedform of (209) shows that the s -integral of the self-field’s contribution to d P s / d s has the form (85) with force density f = 12 T ab body T a (cid:48) b (cid:48) body ( ξ c ∇ c G aba (cid:48) b (cid:48) + 2 ∇ a ξ c G bca (cid:48) b (cid:48) ) . (215)Applying (87) and related results then transforms the law of motion intodd s ˆ P s = 12 (cid:90) B s T ab body L ξ ˆ g ab d S + 14 (cid:90) B s d S (cid:90) W d V (cid:48) T ab body T a (cid:48) b (cid:48) body L ξ G aba (cid:48) b (cid:48) , (216) It would be more elegant to instead demand that K G ( Z , { B s } ; ˆ g ) and K G ( Z , { B s } ; g ) beidentical or otherwise closely related. Such an assumption would restrict possible relationsbetween g ab and ˆ g ab , and is an avenue which has not been explored. P s = P s + E s and E s = 14 (cid:18)(cid:90) B + s T ab body L ξ g S ab [ B − s ]d V − (cid:90) B − s T ab body L ξ g S ab [ B + s ]d V (cid:19) . (217) E s is a functional of T ab body which effectively acts like the momentum of the self-field. If L ξ G aba (cid:48) b (cid:48) is a linear functional of L ξ ˆ g ab , the term involving L ξ G aba (cid:48) b (cid:48) in (216) renormalizes a body’s quadrupole and higher multipole moments. Inthese cases, a multipole expansion of (216) yieldsdd s ˆ P s ( ξ ) = 12 ∞ (cid:88) n =2 n ! ˆ I c ··· c n ab L ξ ˆ g ab,c ··· c n . (218)The tensor extensions appearing here are extensions of ˆ g ab in a spacetime withmetric ˆ g ab . This means, for example, that ˆ g ab,c = 0 and ˆ g ab,cd = ˆ R a ( cd ) f ˆ g bf [cf. (154)].Equation (218) represents not a particular law of motion, but a class of them.This is because many different propagators may be found which satisfy (211)and (212), and whose Lie derivatives with respect to ξ a are quasi-local in L ξ ˆ g ab .Choosing different propagators with these properties leads to different effectivemetrics, different self-momenta, and different effective multipole moments. Anyof these combinations satisfies (216), and also (218) when the appropriate ˆ g ab issufficiently well-behaved. It is of course preferable to choose a propagator suchthat the associated multipole series may “typically” be truncated at low orderwithout significant loss of accuracy. This condition is vague. In the electro-magnetic and scalar theories, a (rather imperfect) proxy was the requirementthat the effective fields be solutions to the vacuum field equation. The analo-gous condition in general relativity would be ˆ R ab = 0, where ˆ R ab denotes theRicci tensor associated with ˆ g ab . This does not appear to be possible within thecurrently-considered class of effective metrics.We take a pragmatic approach and suppose that G aba (cid:48) b (cid:48) is the Detweiler-Whiting S-type Green function associated withˆ g cd ˆ ∇ c ˆ ∇ d G aba (cid:48) b (cid:48) − (cid:2) g cf ˆ R f ( ab ) d + 12 (cid:0) ˆ g cd ˆ R ab + ˆ g ab ˆ g cf ˆ g dh ˆ R fh (cid:1)(cid:3) G cda (cid:48) b (cid:48) = − π (cid:0) ˆ g ac ˆ g bd −
12 ˆ g ab ˆ g cd (cid:1) ˆ g c ( a (cid:48) ˆ g db (cid:48) ) ˆ δ ( x, x (cid:48) ) . (219)This is essentially the prescription suggested in [5]. In terms of retarded andadvanced solutions G ± aba (cid:48) b (cid:48) to (219), the S-type Green function satisfies G aba (cid:48) b (cid:48) = 12 ( G + aba (cid:48) b (cid:48) + G − aba (cid:48) b (cid:48) − V aba (cid:48) b (cid:48) ) (220)for some bitensor V aba (cid:48) b (cid:48) which is an appropriate solution to the homogeneousversion of (219). Somewhat more explicitly, G aba (cid:48) b (cid:48) = 2 (cid:0) ˆ g ac ˆ g bd −
12 ˆ g ab ˆ g cd (cid:1) ˆ g c ( a (cid:48) ˆ g db (cid:48) ) ˆ∆ / δ (ˆ σ ) − V aba (cid:48) b (cid:48) Θ(ˆ σ ) . (221)57oupling this Green function with (213) and (214) defines ˆ g ab in terms of g ab .Unlike in scalar or electromagnetic theories, the definition of the effective fieldis highly implicit in general relativity. It reduces to a simple subtraction onlyin linear perturbation theory. More generally, the Green function itself dependson the field which one intends to find, and must be found by iteration or relatedmethods.Despite the nonlinearity of the map g ab → ˆ g ab , the effective metric doesnot necessarily satisfy the vacuum Einstein equation. The specific differentialequation (219) is nevertheless inspired by Lorenz-gauge perturbation theory,and has the following desirable properties:1. If g ab is sufficiently close to a background metric ¯ g ab satisfying the vacuumEinstein equation ¯ R ab = 0, ˆ g ab satisfies the vacuum Einstein equationlinearized about ¯ g ab .2. The differential operator is self-adjoint.3. The trace of (219) can be solved independently of the full equation.The first condition guarantees that the effective metric is reasonable at least infirst order perturbation theory. Self-adjointness is useful because it allows thereciprocity condition (212) to be enforced. Finally, it is important for technicalreasons to know the trace ˆ g ab G aba (cid:48) b (cid:48) of G aba (cid:48) b (cid:48) . The form of (219) may be usedto show that this satisfies ˆ g ab G aba (cid:48) b (cid:48) = G ˆ g a (cid:48) b (cid:48) , (222)where G is an S-type Detweiler-Whiting Green function for the nonminimally-coupled scalar equation (cid:0) ˆ g ab ˆ ∇ a ˆ ∇ b + 12 ˆ R (cid:1) G = 16 π ˆ δ ( x, x (cid:48) ) . (223) Using the Green function associated with (219) to construct ˆ g ab and ˆ P s , lin-ear and angular momenta may be extracted in the usual way. For any ξ a ∈ K G ( Z , { B s } ; ˆ g ) and any z s ∈ Z , letˆ P s ( ξ ) = ˆ p a ( z s , s ) ξ a ( z s ) + 12 ˆ S ab ˆ ∇ a ξ b ( z s ) . (224)The angular momentum defined in this way is antisymmetric in the sense theˆ S ( ac ˆ g b ) c = 0. Differentiating (224) using a covariant derivative associated withˆ g ab shows thatˆDˆ p a d s = −
12 ˆ R abcd ˙ z bs ˆ S cd + ˆ F a , ˆD ˆ S ab d s = (ˆ g ac ˆ g bd − δ ad δ cb )ˆ p c ˙ z ds + ˆ N ab , (225)58here ˆ F a = 12 ∞ (cid:88) n =2 n ! ˆ I d ··· d n bc ˆ ∇ a ˆ g bc,d ··· d n , (226)andˆ N ac ˆ g bc = ∞ (cid:88) n =2 n ! ˆ g f [ b (cid:0) ˆ I | c ··· c n | a ] d ˆ g df,c ··· c n + n I a ] c ··· c n − dh ˆ g dh,c ··· c n − f (cid:1) . (227)ˆ I c ··· c n ab represents the renormalized 2 n -pole moment of the body’s stress-energytensor. Despite the notation, these are not the same as the moments appearingin the scalar and electromagnetic multipole expansions (156) and (188), whichare renormalized differently. In the test body limit where ˆ g ab ≈ g ab , (225)-(227)reduce to the multipole expansions derived by Dixon [6]. More generally, theyshow — if the multipole expansion is valid — that with appropriate renormal-izations, a self-gravitating body moves instantaneously as though it were a testbody in the effective metric ˆ g ab . Note in particular that the derivatives of themomenta which appear in the evolution equations are derivatives associatedwith ˆ g ab , not g ab . This is a consequence of choosing the generalized Killingfields to be constructed using ˆ g ab instead of g ab . A center of mass frame may be defined by choosing an appropriate foliationtogether with the worldline { γ s } which guarantees that ˆ p a ˆ S ab = 0 when z s = γ s .A body’s linear momentum is then related to its center of mass velocity via anappropriately “hatted” version of (111). If d ˆ P s / d s is sufficiently small, a body’s quadrupole and higher multipole mo-ments might be neglected. In these cases, it follows from (225) that the motionis described by the Mathisson-Papapetrou equations in the effective metric:ˆDˆ p a d s = −
12 ˆ R abcd ˙ z bs ˆ S cd , ˆD ˆ S ab d s = (ˆ g ac ˆ g bd − δ ad δ cb )ˆ p c ˙ z ds . (228)Choosing z s = γ s , the squared spin magnitude ˆ S ab ˆ S ba is necessarily conserved.It is therefore consistent to once again consider systems with vanishing spin.Assuming that ˆ S ab = 0, ˆDd s ˙ γ as = 0 . (229)Non-spinning masses whose quadrupole and higher interactions may be ne-glected therefore fall on geodesics associated with ˆ g ab . This generalizes thewell-known result that small test bodies in general relativity fall on geodesicsassociated with the background spacetime.59oving beyond the test body limit, the difficult step is to compute ˆ g ab . Whatis typically referred to as the first order gravitational self-force may neverthelessbe derived by considering an appropriate family of successively-smaller extendedmasses. To lowest nontrivial order, the effective metric looks like the metric of apoint particle moving on an appropriate vacuum background ¯ g ab which is closeto ˆ g ab . Using overbars to denote quantities associated with ¯ g ab and assumingretarded boundary conditions,ˆ g ab = ¯ g ab + 12 ˆ m (cid:90) ( ¯ G − aba (cid:48) b (cid:48) − ¯ G + aba (cid:48) b (cid:48) + ¯ V aba (cid:48) b (cid:48) ) ˙ γ a (cid:48) s ˙ γ b (cid:48) s d s (cid:48) . (230)This is well-behaved even on the body’s worldline. So is the connection asso-ciated with it, which may be computed using, e.g., methods described in [10].Substituting the result into (229) recovers the MiSaTaQuWa equation commonlyused to describe the first order gravitational self-force [5]. Comparisons havenot yet been made with second order calculations of the gravitational self-forcewhich have recently been completed using other methods [57, 58]. The formulation of the gravitational problem of motion remains somewhat un-satisfactory. Most importantly, the effective metric which has been adoptedhere (and in [5]) is somewhat ad hoc. It is inspired by Lorenz-gauge pertur-bation theory, but this has no particular significance other than being one wayto guarantee hyperbolic field equations. More seriously, the ˆ g ab defined heredoes not satisfy the vacuum Einstein equation except in certain limiting cases.This seems unnatural. It would be preferable if the general relativistic lawsof motion were completely identical in structure to the laws satisfied by testbodies moving in vacuum backgrounds. A condition like ˆ R ab = 0 would alsosuggest, at least intuitively, that the associated ˆ g ab might vary slowly in a widevariety of physical systems. The importance of slow variation to the applicationof multipole expansions makes it extremely interesting to search for an effectivemetric which admits a multipole expansion like (218) while also being an exactsolution to the vacuum Einstein equation. Although such a metric has notyet been found, there are several promising routes by which progress might bemade.The simplest conceivable modifications of the formalism described here re-tains the bare momentum (210) while altering the effective metric ˆ g ab . It istrivial to accomplish this by, for example, modifying the differential equationsatisfied by G aba (cid:48) b (cid:48) or by introducing n -point propagators similar to those in(44). It can also be useful to alter the functional relation (213) between g ab ,ˆ g ab , and any integrals which may be present. Despite being very simple ana-lytically, relating two metrics to one another via the addition of a second-ranktensor is geometrically rather awkward. It could also be interesting to consider reformulations where an effective connection issought instead of an effective metric. g ab = Ω ˆ g ab for an appropriate Ω, it is straightforward to obtain an effective metric whichexactly satisfies the trace ˆ R = 0 of the vacuum Einstein equation. This is notenough, however. More degrees of freedom are necessary. It may be possibleto go further by combining an appropriate conformal factor with a “generalizedKerr-Schild transformation” so that g ab = Ω (ˆ g ab + (cid:96) ( a k b ) ) (231)for some 1-forms (cid:96) a and k a which are null with respect to ˆ g ab (and thereforenull with respect to g ab as well). Despite the simplicity of this expansion, thereis strong evidence that it is very general: Given any analytic g ab , (Ω , (cid:96) a , k a )triplets can always be chosen, at least locally, which guarantee that ˆ g ab is flat[59]. Although it is not known how such choices interact with the laws of motion,the possibility of a flat (or conformally flat) effective metric is intriguing. Amongother benefits, it might eliminate the need for generalized Killing vectors .Combinations of these observations can perhaps be applied to provide one(implicit) map g ab → ˆ g ab with the desired properties. It is likely simpler, how-ever, to instead use them to construct a continuous flow of metrics λ ˜ g ab whichsmoothly deforms g ab = ˜ g ab into an appropriate ˆ g ab = ∞ ˜ g ab . The λ param-eter here is not necessarily physical, but might be interpreted roughly as thereciprocal of the influence of a body’s internal scales. Flows like these havethe advantage that individual “steps” λ → λ + d λ can be viewed as (easily-controlled) linear perturbations. Indeed, it is straightforward to impose differ-ential relations on the λ -dependence of λ ˜ g ab which ensure that a flow removesany initial stress-energy as λ → ∞ . This requires using a 1-parameter familyof Green functions λ G aba (cid:48) b (cid:48) associated with Einstein’s equation linearized abouteach λ ˜ g ab . Separately, it is also straightforward to construct flows which leadto well-behaved laws of motion. What is more difficult is to find a flow whichaccomplishes both of these tasks simultaneously. If this were found, varying λ would likely vary an object’s effective metric, its effective momentum, and itseffective multipole moments. While only the λ → ∞ limit might be physical,such variations are highly reminiscent of the running couplings which arise inrenormalization group flows.Regardless, a great deal of freedom clearly exists and may be exploited tobetter understand the problem of motion in general relativity. The resultinginsights may also shed new light on nonlinear problems more generally. Analyzing the effect of a conformal factor on the laws of motion is similar to consideringobjects coupled to a particular type of nonlinear scalar field. Despite the nonlinearity, suchsystems can be understood exactly using only minimal adaptations of the formalism used toanalyze the (linear) Klein-Gordon problem. Quasi-local momenta have recently been proposed in general relativity which use isometricembeddings to lift flat Killing fields into arbitrary spacetimes [60, 61]. See also [62] fora proposal which allows conformal Killing vectors to be introduced in geometries withoutsymmetries. Discussion
The techniques described in this review provide a unified and largely non-perturbative formalism with which to understand how objects move in classicalfield theories. Although these techniques have thus far been applied only to ahandful of specific theories — Newtonian gravity, Klein-Gordon theory, electro-magnetism, and general relativity — they are easily generalized.One of the central concepts employed here is what we have called the “gen-eralized momentum.” This is used as a convenient observable with which todescribe an object’s motion in the large, and represents a body’s momentumnot as a tensor either in the interior of the spacetime or at infinity, but insteadas a linear map over a more abstract vector space. This automatically takesinto account the nonlocality inherent in the momentum concept and also makesexplicit how particular components of the momentum can be “conjugate to,”e.g., symmetry-generating vector fields.Another important property of the generalized momentum is that it unifiesa body’s linear and angular momenta into a single object. Given a generalizedmomentum, linear and angular components can easily be extracted. This pro-cess depends, however, on extra information, namely a choice of “observer” inthe sense of a preferred origin. This origin is arbitrary. It affects the angularmomentum even in elementary discussions of Newtonian mechanics, but moregenerally influences an object’s linear momentum as well. This has an inter-esting physical consequence:
Mathisson-Papapetrou terms arise in the evolutionequations governing a body’s linear and angular momenta due to the motionof the origin used to extract these components of the generalized momentum .Mathisson-Papapetrou effects are therefore kinematic in nature, arising fromthe changing “character” of each generalized (or genuine) Killing field at differ-ent points.Once a generalized momentum has been defined as a particular linear mapon a particular vector space, stress-energy conservation may be used to deriveits rate of change. The resulting generalized force is another linear map on thesame vector space. Letting ξ a be a particular element of that space, generalizedforces typically have the form F = (cid:90) B s ρ L ξ φ d S, (232)where φ represents some (not necessarily scalar) long-range field and ρ its source. B s is an appropriate 3-volume and d S an associated volume element. In-tegrals like these can be difficult to evaluate directly, so it is important toseek approximations in practical problems. The simplest such approximationsinvolve some combination of test body and smallness conditions which guar-antee that L ξ φ “varies slowly” throughout B s . A multipole expansion canthen be performed to express F in terms of φ and its derivatives as well asthe multipole moments of ρ computed on a (largely arbitrary) worldline { z s } : F = q ( z s , s ) L ξ φ ( z s ) + q a ( z s , s ) L ξ ∇ a φ ( z s ) + . . . .62t is far more difficult to obtain useful multipole expansions when an object’sself-field can no longer be ignored. The potentially-complicated nature of ρ isthen inherited by φ via the field equation, and there is typically no sense in which L ξ φ can be approximated using Taylor-like expansions inside B s . Coping withthis is perhaps the main theoretical problem associated with self-interaction inthe classical theory of motion.It is dealt with here by considering methods which alter the integrand in(232) without affecting the integral as a whole (or which alter the integral onlyvia terms which can be interpreted as renormalizations). Particularly useful forthis purpose are nonlocal deformations φ → ˆ φ generated by appropriate classesof propagators. Although the cases discussed here have used 2-point Greenfunctions associated with the physical field equations, other types of propagatorscan be more useful in other contexts. Regardless, the large variety of possibledeformations may be tailored to optimally simplify whichever problem is athand.In particular, it is often possible to find a deformation φ → ˆ φ such that L ξ ˆ φ varies slowly even when L ξ φ does not. Appropriately-modified multipoleexpansions can then be applied much more generally than might have beenexpected. This leads to the main physical principle which dictates motion ineach of the theories we have considered: Laws governing the motion of self-interacting masses are structurally identical to laws governing the motion oftest bodies . The fields appearing in these laws are nontrivial, however. Objectsgenerally act as though they were accelerated not by the physical field (i.e., φ ),but instead by the “effective field” ˆ φ .The use of effective fields to understand problems of motion is not new perse. Standard formulations of Newtonian celestial mechanics heavily rely, for ex-ample, on external gravitational potentials which are distinct from the physicalpotentials. What has been been stressed here is that generalizing the externalfield concept (where the “external” label is replaced by the more appropriate“effective”) is similarly essential for a simple understanding of motion even inhighly non-Newtonian regimes. All classical results on the self-force can easilybe recovered, for example, once the appropriately-formulated laws of motionhave been derived. Even point particle limits of these laws are well-definedprecisely as stated; they require no independent postulates or regularizations.The standard deformation φ → ˆ φ of the physical Newtonian gravitationalpotential into its effective counterpart leaves forces and torques completely un-affected: The Newtonian self-force and self-torque both vanish. Other theoriesare not so simple. Writing generalized forces in terms of effective fields gener-ally requires the introduction of compensating counterterms. It is only whenthese counterterms have a particularly simple form that the associated effectivefield is likely to be useful. Indeed, we have considered systems where theseterms act only to make a body’s momenta or other multipole moments appearto be shifted from those moments which might have been deduced using knowl-edge of a body’s internal structure. The details of this structure are rarelyknown in practice, in which case it is natural to “remove” residual forces and63orques by appropriately redefining an object’s momenta or other multipole mo-ments. These are renormalizations. They affect generic extended objects, andare always finite in this context. Considerable effort has been devoted here toidentifying renormalizations and interpreting them physically.The resulting techniques have shown that a large variety of renormalizationsare possible even in simple theories. The effective 4-momentum of an electriccharge may differ from its bare momentum not only in length (i.e., mass), butalso in direction. Spins and center of mass positions can be renormalized as well.Adding the additional complication of a curved spacetime, even the quadrupoleand higher multipole moments associated with a body’s stress-energy tensormay be dynamically shifted via the forces exerted by its self-field.Two general mechanisms have been shown responsible for these effects. Bothof these are associated with generalizations of — or failures to generalize —Newton’s third law. One mechanism relates from a direct violation of thislaw, while the other arises from an inability to fully take advantage of “action-reaction cancellations.” The second of these is simpler and affects a body’s linearand angular momenta. It is associated with self-fields which are nonlocal in time,in which case forces are sourced in four dimensions but act on matter only inthree-dimensional slices. If the propagators associated with these statementssatisfy certain minimal constraints, the inability to construct action-reactionpairs in this context conspires to dynamically shift an object’s momenta. Sucheffects are essentially universal in relativistic theories, but can also be relevantfor some non-relativistic systems.The second renormalization mechanism discussed here stems from more di-rect violations of Newton’s third law. Mathematically, it is related to the behav-ior of the relevant propagators under Lie dragging. If, say, a self-field is definedin terms of a propagator G , and L ξ G depends only on L ξ φ for some field φ ,the multipole moments coupling to φ are renormalized by the self-force. In thecases considered here, φ was the metric and the relevant moments were thoseassociated with a body’s stress-energy tensor. The same mechanism appliedto, e.g., a nonlinear scalar theory would instead renormalize a body’s chargemoments.Despite the generality of these results, much remains to be learned. Besidesthe various technical details which remain open — some of which have beenmentioned in the text — it would also be interesting to understand how thetechniques developed here can be applied in new ways. It may be possible, forexample, to adapt these techniques to systems where long-range fields couple toan object’s surface instead of its volume. Such problems arise when consideringthe motion of solid objects through fluids, among other cases. More generally, itmight be possible to investigate problems which are not related to motion at all.Quantities similar to (232) occur in many fields of physics and mathematics, asdo various types of regularizations and renormalizations. It appears likely thatthe methods developed here can be applied to better understand at least someof these systems. 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