The Motion of Point Particles in Curved Spacetime
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Abstract
This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle.
The field’s action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field’s singular part and show that it exerts no force on the particle — its only effect is to contribute to the particle’s inertia. What remains after subtraction is a regular field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free field that interacts with the particle; it is this interaction that gives rise to the self-force.
The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors (Part I). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle’s word line (Part II). It continues with a thorough discussion of Green’s functions in curved spacetime (Part III). The review presents a detailed derivation of each of the three equations of motion (Part IV). Because the notion of a point mass is problematic in general relativity, the review concludes (Part V) with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.
1 Introduction and summary
1.1 Invitation
The motion of a point electric charge in flat spacetime was the subject of active investigation since the early work of Lorentz, Abrahams, Poincaré, and Dirac [56], until Gralla, Harte, and Wald produced a definitive derivation of the equations motion [82] with all the rigour that one should demand, without recourse to postulates and renormalization procedures. (The field’s early history is well related in Ref. [154].) In 1960 DeWitt and Brehme [54] generalized Dirac’s result to curved spacetimes, and their calculation was corrected by Hobbs [95] several years later. In 1997 the motion of a point mass in a curved background spacetime was investigated by Mino, Sasaki, and Tanaka [130], who derived an expression for the particle’s acceleration (which is not zero unless the particle is a test mass); the same equations of motion were later obtained by Quinn and Wald [150] using an axiomatic approach. The case of a point scalar charge was finally considered by Quinn in 2000 [149], and this led to the realization that the mass of a scalar particle is not necessarily a constant of the motion.
This article reviews the achievements described in the preceding paragraph; it is concerned with the motion of a point scalar charge q, a point electric charge e, and a point mass m in a specified background spacetime with metric g_{ αβ }. These particles carry with them fields that behave as outgoing radiation in the wave zone. The radiation removes energy and angular momentum from the particle, which then undergoes a radiation reaction — its world line cannot be simply a geodesic of the background spacetime. The particle’s motion is affected by the near-zone field which acts directly on the particle and produces a self-force. In curved spacetime the self-force contains a radiation-reaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angular-momentum transport. The self-force is proportional to q^{2} in the case of a scalar charge, proportional to e^{2} in the case of an electric charge, and proportional to m^{2} in the case of a point mass.
In this review we derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely self-contained, and all relevant materials are developed ab initio. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. One must first assimilate the basic theory of bitensors (Part I), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line (Part II). One must next formulate a theory of Green’s functions in curved spacetimes (Part III), and finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how they should act on the particle (Part IV). A dedicated reader, correctly skeptical that sense can be made of a point mass in general relativity, will also want to work through the last portion of the review (Part V), which provides a derivation of the equations of motion for a small, but physically extended, body; this reader will be reassured to find that the extended body follows the same motion as the point mass. The review is very long, but the satisfaction derived, we hope, will be commensurate.
In this introductory section we set the stage and present an impressionistic survey of what the review contains. This should help the reader get oriented and acquainted with some of the ideas and some of the notation. Enjoy!
1.2 Radiation reaction in flat spacetime
An immediate difficulty presents itself: the vector potential, and also the electromagnetic field tensor, diverge on the particle’s world line, because the field of a point charge is necessarily infinite at the charge’s position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.
Our second key observation is that while the potential of Eq. (1.2) does not exert a force on the charged particle, it is just as singular as the retarded potential in the vicinity of the world line. This follows from the fact that \(A_{{\rm{ret}}}^\alpha,\; A_{{\rm{adv}}}^\alpha\), and \(A_{\rm{S}}^\alpha\) all satisfy Eq. (1.1), whose source term is infinite on the world line. So while the wave-zone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line — all three electromagnetic fields are dominated by the particle’s Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript ‘S’: it stands for ‘singular’ as well as ‘symmetric’.
To establish that the singular field exerts no force on the particle requires a careful analysis that is presented in the bulk of the paper. What really happens is that, because the particle is a monopole source for the electromagnetic field, the singular field is locally isotropic around the particle; it therefore exerts no force, but contributes to the particle’s inertia and renormalizes its mass. In fact, one could do without a decomposition of the field into singular and regular solutions, and instead construct the force by using the retarded field and averaging it over a small sphere around the particle, as was done by Quinn and Wald [150]. In the body of this review we will use both methods and emphasize the equivalence of the results. We will, however, give some emphasis to the decomposition because it provides a compelling physical interpretation of the self-force as an interaction with a free electromagnetic field.
1.3 Green’s functions in flat spacetime
1.4 Green’s functions in curved spacetime
Similar statements can be made about the advanced Green’s function and the advanced solution to the wave equation. While in flat spacetime the advanced Green’s function has support only on the past light cone of x′, in curved spacetime its support extends inside the light cone, and \(G_{- \beta {\prime}}^{\;\alpha}(x,x{\prime})\) is nonzero when x ∈ I −(x′), which denotes the chronological past of x′. This implies that the advanced potential at x is generated by the point charge during its entire future history following the advanced time v associated with x: the potential depends on the particle’s state of motion for all times τ ≥ v.
The physically relevant solution to Eq. (1.13) is obviously the retarded potential \(A_{{\rm{ret}}}^{\alpha}(x)\), and as in flat spacetime, this diverges on the world line. The cause of this singular behaviour is still the pointlike nature of the source, and the presence of spacetime curvature does not change the fact that the potential diverges at the position of the particle. Once more this behaviour makes it difficult to figure out how the retarded field is supposed to act on the particle and determine its motion. As in flat spacetime we shall attempt to decompose the retarded solution into a singular part that exerts no force, and a regular part that produces the entire self-force.
To decompose the retarded Green’s function into singular and regular parts is not a straightforward task in curved spacetime. The flat-spacetime definition for the singular Green’s function, Eq. (1.9), cannot be adopted without modification: While the combination half-retarded plus half-advanced Green’s functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Eq. (1.13), this candidate for the singular Green’s function would produce a self-force with an unacceptable dependence on the particle’s future history. For suppose that we made this choice. Then the regular two-point function would be given by the combination half-retarded minus half-advanced Green’s functions, just as in flat spacetime. The resulting potential would satisfy the homogeneous wave equation, and it would be regular on the world line, but it would also depend on the particle’s entire history, both past (through the retarded Green’s function) and future (through the advanced Green’s function). More precisely stated, we would find that the regular potential at x depends on the particle’s state of motion at all times τ outside the interval u < τ < v; in the limit where x approaches the world line, this interval shrinks to nothing, and we would find that the regular potential is generated by the complete history of the particle. A self-force constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and regular Green’s functions.
1.5 World line and retarded coordinates
To flesh out the ideas contained in the preceding subsection we add yet another layer of mathematical formalism and construct a convenient coordinate system to chart a neighbourhood of the particle’s world line. In the next subsection we will display explicit expressions for the retarded, singular, and regular fields of a point electric charge.
Let γ be the world line of a point particle in a curved spacetime. It is described by parametric relations z^{ μ } (τ) in which τ is proper time. Its tangent vector is u^{ μ } = dz^{ μ }/dτ and its acceleration is a^{ μ } = Du^{ μ }/dτ; we shall also encounter ȧ^{ μ }: = Da^{ μ }/dτ.
Consider a point x in a neighbourhood of the world line γ. We assume that x is sufficiently close to the world line that a unique geodesic links x to any neighbouring point z on γ. The two-point function σ (x,z), known as Synge’s world function [169], is numerically equal to half the squared geodesic distance between z and x; it is positive if x and z are spacelike related, negative if they are timelike related, and σ (x, z) is zero if x and z are linked by a null geodesic. We denote its gradient ∂σ/∂z^{ μ } by σ_{ μ } (x,z), and −σ^{ μ } gives a meaningful notion of a separation vector (pointing from z tox).
To construct a coordinate system in this neighbourhood we locate the unique point x′:= z (u) on γ which is linked to x by a future-directed null geodesic (this geodesic is directed from x′ to x); we shall refer to x′ as the retarded point associated with x, and u will be called the retarded time. To tensors at x′ we assign indices α′, β′, …; this will distinguish them from tensors at a generic point z (τ) on the world line, to which we have assigned indices μ, ν, …. We have σ (x, x′) = 0 and −σ^{α′} (x, x′) is a null vector that can be interpreted as the separation between x′ and x.
To tensors at x we assign indices α, β, …. These tensors will be decomposed in a tetrad \((e_0^{\alpha},e_a^{\alpha})\) that is constructed as follows: Given x we locate its associated retarded point x′ on the world line, as well as the null geodesic that links these two points; we then take the tetrad \(({u^{\alpha{\prime}}},e_a^{\alpha{\prime}})\) at x′ and parallel transport it to x along the null geodesic to obtain \((e_0^{\alpha},e_a^{\alpha})\).
1.6 Retarded, singular, and regular electromagnetic fields of a point electric charge
The expansion of F_{ αβ } (x) near the world line does indeed reveal many singular terms. We first recognize terms that diverge when r → 0; for example the Coulomb field Fa0 diverges as r−^{2} when we approach the world line. But there are also terms that, though they stay bounded in the limit, possess a directional ambiguity at r = 0; for example F_{ ab } contains a term proportional to R_{a0bc}Ω^{ c } whose limit depends on the direction of approach.
1.7 Motion of an electric charge in curved spacetime
Equation (1.33) is the result that was first derived by DeWitt and Brehme [54] and later corrected by Hobbs [95]. (The original version of the equation did not include the Ricci-tensor term.) In flat spacetime the Ricci tensor is zero, the tail integral disappears (because the Green’s function vanishes everywhere within the domain of integration), and Eq. (1.33) reduces to Dirac’s result of Eq. (1.5). In curved spacetime the self-force does not vanish even when the electric charge is moving freely, in the absence of an external force: it is then given by the tail integral, which represents radiation emitted earlier and coming back to the particle after interacting with the spacetime curvature. This delayed action implies that in general, the self-force is nonlocal in time: it depends not only on the current state of motion of the particle, but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that the physical process that leads to Eq. (1.33) is simply an interaction between the charge and a free electromagnetic field \(F_{\alpha \beta}^{\rm{R}}\); it is this field that carries the information about the charge’s past.
1.8 Motion of a scalar charge in curved spacetime
In flat spacetime the Ricci-tensor term and the tail integral disappear and Eq. (1.40) takes the form of Eq. (1.5) with q^{2}/(3m) replacing the factor of 2e^{2}/(3m). In this simple case Eq. (1.41) reduces to dm/dτ = 0 and the mass is in fact a constant. This property remains true in a conformally flat spacetime when the wave equation is conformally invariant (ξ = 1/6): in this case the Green’s function possesses only a light-cone part and the right-hand side of Eq. (1.41) vanishes. In generic situations the mass of a point scalar charge will vary with proper time.
1.9 Motion of a point mass, or a small body, in a background spacetime
The case of a point mass moving in a specified background spacetime presents itself with a serious conceptual challenge, as the fundamental equations of the theory are nonlinear and the very notion of a “point mass” is somewhat misguided. Nevertheless, to the extent that the perturbation h_{ aβ } (x) created by the point mass can be considered to be “small”, the problem can be formulated in close analogy with what was presented before.
The equations of motion of Eq. (1.48) were first derived by Mino, Sasaki, and Tanaka [130], and then reproduced with a different analysis by Quinn and Wald [150]. They are now known as the MiSaTaQuWa equations of motion. As noted by these authors, the MiSaTaQuWa equation has the appearance of the geodesic equation in a metric \({g_{\alpha \beta}} + h_{\alpha \beta}^{{\rm{tail}}}\). Detweiler and Whiting [53] have contributed the more compelling interpretation that the motion is actually geodesic in a spacetime with metric \({g_{\alpha \beta}} + h_{\alpha \beta}^{\rm{R}}\). The distinction is important: Unlike the first version of the metric, the Detweiler-Whiting metric is regular on the world line and satisfies the Einstein field equations in vacuum; and because it is a solution to the field equations, it can be viewed as a physical metric — specifically, the metric of the background spacetime perturbed by a free field produced by the particle at an earlier stage of its history.
While Eq. (1.48) does indeed give the correct equations of motion for a small mass m moving in a background spacetime with metric g_{ αβ }, the derivation outlined here leaves much to be desired — to what extent should we trust an analysis based on the existence of a point mass? As a partial answer to this question, Mino, Sasaki, and Tanaka [130] produced an alternative derivation of their result, which involved a small nonrotating black hole instead of a point mass. In this alternative derivation, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: it must move according to Eq. (1.48). This alternative derivation (which was given a different implementation in Ref. [142]) is entirely free of singularities (except deep within the black hole), and it suggests that that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body’s internal structure can be ignored). This derivation, however, was limited to the case of a non-rotating black hole, and it relied on a number of unjustified and sometimes unstated assumptions [83, 144, 145]. The conclusion was made firm by the more rigorous analysis of Gralla and Wald [83] (as extended by Pound [144]), who showed that the MiSaTaQuWa equations apply to any sufficiently compact body of arbitrary internal structure.
The gravitational self-force possesses a physical significance that is not shared by its scalar and electromagnetic analogues, because the motion of a small body in the strong gravitational field of a much larger body is a problem of direct relevance to gravitational-wave astronomy. Indeed, extreme-mass-ratio inspirals, involving solar-mass compact objects moving around massive black holes of the sort found in galactic cores, have been identified as promising sources of low-frequency gravitational waves for space-based interferometric detectors such as the proposed Laser Interferometer Space Antenna (LISA [115]). These systems involve highly eccentric, nonequatorial, and relativistic orbits around rapidly rotating black holes, and the waves produced by such orbital motions are rich in information concerning the strongest gravitational fields in the Universe. This information will be extractable from the LISA data stream, but the extraction depends on sophisticated data-analysis strategies that require a detailed and accurate modeling of the source. This modeling involves formulating the equations of motion for the small body in the field of the rotating black hole, as well as a consistent incorporation of the motion into a wave-generation formalism. In short, the extraction of this wealth of information relies on a successful evaluation of the gravitational self-force.
The finite-mass corrections to the orbital motion are important. For concreteness, let us assume that the orbiting body is a black hole of mass m = 10 M_{⊙} and that the central black hole has a mass M = 10^{6} M_{⊙}. Let us also assume that the small black hole is in the deep field of the large hole, near the innermost stable circular orbit, so that its orbital period P is of the order of minutes. The gravitational waves produced by the orbital motion have frequencies f of the order of the mHz, which is well within LISA’s frequency band. The radiative losses drive the orbital motion toward a final plunge into the large black hole; this occurs over a radiation-reaction timescale (M/m)P of the order of a year, during which the system will go through a number of wave cycles of the order of M/m = 10^{5}. The role of the gravitational self-force is precisely to describe this orbital evolution toward the final plunge. While at any given time the self-force provides fractional corrections of order m/M = 10^{−5} to the motion of the small black hole, these build up over a number of orbital cycles of order M/m = 10^{5} to produce a large cumulative effect. As will be discussed in some detail in Section 2.6, the gravitational self-force is important, because it drives large secular changes in the orbital motion of an extreme-mass-ratio binary.
1.10 Case study: static electric charge in Schwarzschild spacetime
We should remark that the identification of φ^{ S } and φ^{ R } with the Detweiler-Whiting singular and regular fields, respectively, is a matter of conjecture. Although φ^{ S } and φ^{ R } satisfy the essential properties of the Detweiler-Whiting decomposition — being, respectively, a regular homogenous solution and a singular solution sourced by the particle — one should accept the possibility that they may not be the actual Detweiler-Whiting fields. It is a topic for future research to investigate the precise relation between the Copson field and the Detweiler-Whiting singular field.
It is instructive to compare the electromagnetic self-force produced by the presence of a grounded conductor to the self-force produced by the presence of a black hole. In the case of a conductor, the total induced charge on the conducting surface is e′ = −eR/r_{0}, and it is this charge that is responsible for the attractive self-force; the induced charge is supplied by the electrodes that keep the conductor grounded. In the case of a black hole, there is no external apparatus that can supply such a charge, and the total induced charge on the horizon necessarily vanishes. The origin of the self-force is therefore very different in this case. As we have seen, the self-force is produced by a fictitious charge eM/r_{0} situated at the centre of black hole; and because this charge is positive, the self-force is repulsive.
1.11 Organization of this review
After a detailed review of the literature in Section 2, the main body of the review begins in Part I (Sections 3 to 7) with a description of the general theory of bitensors, the name designating tensorial functions of two points in spacetime. We introduce Synge’s world function σ (x, x′) and its derivatives in Section 3, the parallel propagator \(g_{\;\alpha {\prime}}^\alpha (x,x{\prime})\) in Section 5, and the van Vleck determinant Δ(x, x′) in Section 7. An important portion of the theory (covered in Sections 4 and 6) is concerned with the expansion of bitensors when x is very close to x′; expansions such as those displayed in Eqs. (1.23) and (1.24) are based on these techniques. The presentation in Part I borrows heavily from Synge’s book [169] and the article by DeWitt and Brehme [54]. These two sources use different conventions for the Riemann tensor, and we have adopted Synge’s conventions (which agree with those of Misner, Thorne, and Wheeler [131]). The reader is therefore warned that formulae derived in Part I may look superficially different from those found in De Witt and Brehme.
In Part II (Sections 8 to 11) we introduce a number of coordinate systems that play an important role in later parts of the review. As a warmup exercise we first construct (in Section 8) Riemann normal coordinates in a neighbourhood of a reference point x′. We then move on (in Section 9) to Fermi normal coordinates [122], which are defined in a neighbourhood of a world line γ. The retarded coordinates, which are also based at a world line and which were briefly introduced in Section 1.5, are covered systematically in Section 10. The relationship between Fermi and retarded coordinates is worked out in Section 11, which also locates the advanced point z (v) associated with a field point x. The presentation in Part II borrows heavily from Synge’s book [169]. In fact, we are much indebted to Synge for initiating the construction of retarded coordinates in a neighbourhood of a world line. We have implemented his program quite differently (Synge was interested in a large neighbourhood of the world line in a weakly curved spacetime, while we are interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally his.
In Part III (Sections 12 to 16) we review the theory of Green’s functions for (scalar, vectorial, and tensorial) wave equations in curved spacetime. We begin in Section 12 with a pedagogical introduction to the retarded and advanced Green’s functions for a massive scalar field in flat spacetime; in this simple context the all-important Hadamard decomposition [88] of the Green’s function into “light-cone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in Section 13 along with its restrictions on the past and future null cones of a reference point x′. The retarded, advanced, singular, and regular Green’s functions for the scalar wave equation are introduced in Section 14. In Sections 15 and 16 we cover the vectorial and tensorial wave equations, respectively. The presentation in Part III is based partly on the paper by DeWitt and Brehme [54], but it is inspired mostly by Friedlander’s book [71]. The reader should be warned that in one important aspect, our notation differs from the notation of DeWitt and Brehme: While they denote the tail part of the Green’s function by −v (x, x′), we have taken the liberty of eliminating the silly minus sign and call it instead +V (x, x′). The reader should also note that all our Green’s functions are normalized in the same way, with a factor of −4π multiplying a four-dimensional Dirac functional of the right-hand side of the wave equation. (The gravitational Green’s function is sometimes normalized with a −16π on the right-hand side.)
In Part IV (Sections 17 to 19) we compute the retarded, singular, and regular fields associated with a point scalar charge (Section 17), a point electric charge (Section 18), and a point mass (Section 19). We provide two different derivations for each of the equations of motion. The first type of derivation was outlined previously: We follow Detweiler and Whiting [53] and postulate that only the regular field exerts a force on the particle. In the second type of derivation we take guidance from Quinn and Wald [150] and postulate that the net force exerted on a point particle is given by an average of the retarded field over a surface of constant proper distance orthogonal to the world line — this rest-frame average is easily carried out in Fermi normal coordinates. The averaged field is still infinite on the world line, but the divergence points in the direction of the acceleration vector and it can thus be removed by mass renormalization. Such calculations show that while the singular field does not affect the motion of the particle, it nonetheless contributes to its inertia.
In Part V (Sections 20 to 23), we show that at linear order in the body’s mass m, an extended body behaves just as a point mass, and except for the effects of the body’s spin, the world line representing its mean motion is governed by the MiSaTaQuWa equation. At this order, therefore, the picture of a point particle interacting with its own field, and the results obtained from this picture, is justified. Our derivation utilizes the method of matched asymptotic expansions, with an inner expansion accurate near the body and an outer expansion accurate everywhere else. The equation of motion of the body’s world line, suitably defined, is calculated by solving the Einstein equation in a buffer region around the body, where both expansions are accurate.
Concluding remarks are presented in Section 24, and technical developments that are required in Part V are relegated to Appendices. Throughout this review we use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [131].
2 Computing the self-force: a 2010 literature survey
Much progress has been achieved in the development of practical methods for computing the self-force. We briefly summarize these efforts in this section, with the goal of introducing the main ideas and some key issues. A more detailed coverage of the various implementations can be found in Barack’s excellent review [9]. The 2005 collection of reviews published in Classical and Quantum Gravity [118] is also recommended for an introduction to the various aspects of self-force theory and numerics. Among our favourites in this collection are the reviews by Detweiler [49] and Whiting [183].
An important point to bear in mind is that all the methods covered here mainly compute the self-force on a particle moving on a fixed world line of the background spacetime. A few numerical codes based on the radiative approximation have allowed orbits to evolve according to energy and angular-momentum balance. As will be emphasized below, however, these calculations miss out on important conservative effects that are only accounted for by the full self-force. Work is currently underway to develop methods to let the self-force alter the motion of the particle in a self-consistent manner.
2.1 Early work: DeWitt and DeWitt; Smith and Will
2.2 Mode-sum method
Self-force calculations involving a sum over modes were pioneered by Barack and Ori [16, 7], and the method was further developed by Barack, Ori, Mino, Nakano, and Sasaki [15, 8, 18, 20, 19, 127]; a somewhat related approach was also considered by Lousto [117]. It has now emerged as the method of choice for self-force calculations in spacetimes such as Schwarzschild and Kerr. Our understanding of the method was greatly improved by the Detweiler-Whiting decomposition [53] of the retarded field into singular and regular pieces, as outlined in Sections 1.4 and 1.8, and subsequent work by Detweiler, Whiting, and their collaborators [51].
2.2.1 Detweiler-Whiting decomposition; mode decomposition; regularization parameters
From the point of view of Eq. (2.5), the task of computing the self-force appears conceptually straightforward: Either (i) compute the retarded and singular potentials, subtract them, and take a gradient of the difference; or (ii) compute the gradients of the retarded and singular potentials, and then subtract the gradients. Indeed, this is the basic idea for most methods of self-force computations. However, the apparent simplicity of this sequence of steps is complicated by the following facts: (i) except for a very limited number of cases, the retarded potential of a point particle cannot be computed analytically and must therefore be obtained by numerical means; and (ii) both the retarded and singular potential diverge at the particle’s position. Thus, any sort of subtraction will generally have to be performed numerically, and for this to be possible, one requires representations of the retarded and singular potentials (and/or their gradients) in terms of finite quantities.
Fortunately, there is a piece of each l-mode that does not contribute to the self-force, and that can be subtracted out; this piece is the corresponding l-mode of the singular field ∇_{ α } Φ_{S}. Because the retarded and singular fields share the same singularity structure near the particle’s world line (as described in Section 1.6), the subtraction produces a mode decomposition of the regular field ∇_{α}Φ_{R}. And because this field is regular at the particle’s position, the sum over all modes q (∇_{ a } Φ_{R},)_{ l } is guaranteed to converge to the correct value for the self-force. The key to the mode-sum method, therefore, is the ability to express the singular field as a mode decomposition.
2.2.2 Mode sum
2.2.3 Case study: static electric charge in extreme Reissner-Nordström spacetime
The practical use of the mode-sum method can be illustrated with the help of a specific example that can be worked out fully and exactly. We consider, as in Section 1.10, an electric charge e held in place at position r = r_{0} in the spacetime of an extreme Reissner-Nordström black hole of mass M and charge Q = M. The reason for selecting this spacetime resides in the resulting simplicity of the spherical-harmonic modes for the electromagnetic field.
2.2.4 Computations in Schwarzschild spacetime
The mode-sum method was successfully implemented in Schwarzschild spacetime to compute the scalar and electromagnetic self-forces on a static particle [31, 36]. It was used to calculate the scalar self-force on a particle moving on a radial trajectory [10], circular orbit [30, 51, 87, 37], and a generic bound orbit [84]. It was also developed to compute the electromagnetic self-force on a particle moving on a generic bound orbit [85], as well as the gravitational self-force on a point mass moving on circular [21, 1] and eccentric orbits [22]. The mode-sum method was also used to compute unambiguous physical effects associated with the gravitational self-force [50, 157, 11], and these results were involved in detailed comparisons with post-Newtonian theory [50, 29, 28, 44, 11]. These achievements will be described in more detail in Section 2.6.
An issue that arises in computations of the electromagnetic and gravitational self-forces is the choice of gauge. While the self-force formalism is solidly grounded in the Lorenz gauge (which allows the formulation of a wave equation for the potentials, the decomposition of the retarded field into singular and regular pieces, and the computation of regularization parameters), it is often convenient to carry out the numerical computations in other gauges, such as the popular Regge-Wheeler gauge and the Chrzanowski radiation gauge described below. Compatibility of calculations carried out in different gauges has been debated in the literature. It is clear that the singular field is gauge invariant when the transformation between the Lorenz gauge and the adopted gauge is smooth on the particle’s world line; in such cases the regularization parameters also are gauge invariant [17], the transformation affects the regular field only, and the self-force changes according to Eq. (1.49). The transformations between the Lorenz gauge and the Regge-Wheeler and radiation gauges are not regular on the world line, however, and in such cases the regularization of the retarded field must be handled with extreme care.
2.2.5 Computations in Kerr spacetime; metric reconstruction
The reliance of the mode-sum method on a spherical-harmonic decomposition makes it generally impractical to apply to self-force computations in Kerr spacetime. Wave equations in this spacetime are better analyzed in terms of a spheroidal-harmonic decomposition, which simultaneously requires a Fourier decomposition of the field’s time dependence. (The eigenvalue equation for the angular functions depends on the mode’s frequency.) For a static particle, however, the situation simplifies, and Burko and Liu [35] were able to apply the method to calculate the self-force on a static scalar charge in Kerr spacetime. More recently, Warburton and Barack [181] carried out a mode-sum calculations of the scalar self-force on a particle moving on equatorial orbits of a Kerr black hole. They first solve for the spheroidal multipoles of the retarded potential, and then re-express them in terms of spherical-harmonic multipoles. Fortunately, they find that a spheroidal multipole is well represented by summing over a limited number of spherical multipoles. The Warburton-Barack work represents the first successful computations of the self-force in Kerr spacetime, and it reveals the interesting effect of the black hole’s spin on the behaviour of the self-force.
The analysis of the scalar wave equation in terms of spheroidal functions and a Fourier decomposition permits a complete separation of the variables. For decoupling and separation to occur in the case of a gravitational perturbation, it is necessary to formulate the perturbation equations in terms of Newman-Penrose (NP) quantities [172], and to work with the Teukolsky equation that governs their behaviour. Several computer codes are now available that are capable of integrating the Teukolsky equation when the source is a point mass moving on an arbitrary geodesic of the Kerr spacetime. (A survey of these codes is given below.) Once a solution to the Teukolsky equation is at hand, however, there still remains the additional task of recovering the metric perturbation from this solution, a problem referred to as metric reconstruction.
Reconstruction of the metric perturbation from solutions to the Teukolsky equation was tackled in the past in the pioneering efforts of Chrzanowski [41], Cohen and Kegeles [42, 105], Stewart [166], and Wald [179]. These works have established a procedure, typically attributed to Chrzanowski, that returns the metric perturbation in a so-called radiation gauge. An important limitation of this method, however, is that it applies only to vacuum solutions to the Teukolsky equation. This makes the standard Chrzanowski procedure inapplicable in the self-force context, because a point particle must necessarily act as a source of the perturbation. Some methods were devised to extend the Chrzanowski procedure to accommodate point sources in specific circumstances [121, 134], but these were not developed sufficiently to permit the computation of a self-force. See Ref. [184] for a review of metric reconstruction from the perspective of self-force calculations.
A remarkable breakthrough in the application of metric-reconstruction methods in self-force calculations was achieved by Keidl, Wiseman, and Friedman [107, 106, 108], who were able to compute a self-force starting from a Teukolsky equation sourced by a point particle. They did it first for the case of an electric charge and a point mass held at a fixed position in a Schwarzschild space-time [107], and then for the case of a point mass moving on a circular orbit around a Schwarzschild black hole [108]. The key conceptual advance is the realization that, according to the Detweiler-Whiting perspective, the self-force is produced by a regularized field that satisfies vacuum field equations in a neighbourhood of the particle. The regular field can therefore be submitted to the Chrzanowski procedure and reconstructed from a source-free solution to the Teukolsky equation.
More concretely, suppose that we have access to the Weyl scalar ψ_{0} produced by a point mass moving on a geodesic of a Kerr spacetime. To compute the self-force from this, one first calculates the singular Weyl scalar \(\psi _0^{\rm{S}}\) from the Detweiler-Whiting singular field \(h_{\alpha \beta}^{\rm{S}}\), and subtracts it from ψ_{0}. The result is a regularized Weyl scalar \(\psi _0^{\rm{R}}\), which is a solution to the homogeneous Teukolsky equation. This sets the stage for the metric-reconstruction procedure, which returns (a piece of) the regular field \(h_{\alpha \beta}^{\rm{R}}\) in the radiation gauge selected by Chrzanowski. The computation must be completed by adding the pieces of the metric perturbation that are not contained in ψ_{0}; these are referred to either as the nonradiative degrees of freedom (since ψ_{0} is purely radiative), or as the l = 0 and l = 1 field multipoles (because the sum over multipoles that make up ψ_{0} begins at l = 2). A method to complete the Chrzanowski reconstruction of \(h_{\alpha \beta}^{\rm{R}}\) was devised by Keidl et al. [107, 108], and the end result leads directly to the gravitational self-force. The relevance of the l = 0 and l = 1 modes to the gravitational self-force was emphasized by Detweiler and Poisson [52].
2.2.6 Time-domain versus frequency-domain methods
When calculating the spherical-harmonic components Φ^{ lm } (t, r) of the retarded potential Φ — refer back to Eq. (2.6) — one can choose to work either directly in the time domain, or perform a Fourier decomposition of the time dependence and work instead in the frequency domain. While the time-domain method requires the integration of a partial differential equation in t and r, the frequency-domain method gives rise to set of ordinary differential equations in r, one for each frequency ω. For particles moving on circular or slightly eccentric orbits in Schwarzschild spacetime, the frequency spectrum is limited to a small number of discrete frequencies, and a frequency-domain method is easy to implement and yields highly accurate results. As the orbital eccentricity increases, however, the frequency spectrum broadens, and the computational burden of summing over all frequency components becomes more significant. Frequency-domain methods are less efficient for large eccentricities, the case of most relevance for extreme-mass-ratio inspirals, and it becomes advantageous to replace them with time-domain methods. (See Ref. [25] for a quantitative study of this claim.) This observation has motivated the development of accurate evolution codes for wave equations in 1+1 dimensions.
Such codes must be able to accommodate point-particle sources, and various strategies have been pursued to represent a Dirac distribution on a numerical grid, including the use of very narrow Gaussian pulses [116, 110, 34] and of “finite impulse representations” [168]. These methods do a good job with waveform and radiative flux calculations far away from the particle, but are of very limited accuracy when computing the potential in a neighborhood of the particle. A numerical method designed to provide an exact representation of a Dirac distribution in a time-domain computation was devised by Lousto and Price [120] (see also Ref. [123]). It was implemented by Haas [84, 85] for the specific purpose of evaluating Φ^{ lm } (t, r) at the position of particle and computing the self-force. Similar codes were developed by other workers for scalar [176] and gravitational [21, 22] self-force calculations.
Most extant time-domain codes are based on finite-difference techniques, but codes based on pseudo-spectral methods have also been developed [67, 68, 37, 38]. Spectral codes are a powerful alternative to finite-difference codes, especially when dealing with smooth functions, because they produce much faster convergence. The fact that self-force calculations deal with point sources and field modes that are not differentiable might suggest that spectral convergence should not be expected in this case. This objection can be countered, however, by placing the particle at the boundary between two spectral domains. Functions are then smooth in each domain, and discontinuities are handled by formulating appropriate boundary conditions; spectral convergence is thereby achieved.
2.3 Effective-source method
The mode-sum methods reviewed in the preceding subsection have been developed and applied extensively, but they do not exhaust the range of approaches that may be exploited to compute a self-force. Another set of methods, devised by Barack and his collaborators [12, 13, 60] as well as Vega and his collaborators [176, 177, 175], begin by recognizing that an approximation to the exact singular potential can be used to regularize the delta-function source term of the original field equation. We shall explain this idea in the simple context of a scalar potential Φ.
2.4 Quasilocal approach with “matched expansions”
As was seen in Eqs. (1.33), (1.40), and (1.47), the self-force can be expressed as an integral over the past world line of the particle, the integrand involving the Green’s function for the appropriate wave equation. Attempts have been made to compute the Green’s function directly [132, 141, 33, 86], and to evaluate the world-line integral. The quasilocal approach, first introduced by Anderson and his collaborators [4, 3, 6, 5], is based on the expectation that the world-line integral might be dominated by the particle’s recent past, so that the Green’s function can be represented by its Hadamard expansion, which is restricted to the normal convex neighbourhood of the particle’s current position. To help with this enterprise, Ottewill and his collaborators [136, 182, 137, 39] have pushed the Hadamard expansion to a very high order of accuracy, building on earlier work by Décanini and Folacci [48].
The weak-field calculations performed by DeWitt and DeWitt [132] and Pfenning and Poisson [141] suggest that the world-line integral is not, in fact, dominated by the recent past. Instead, most of the self-force is produced by signals that leave the particle at some time in the past, scatter off the central mass, and reconnect with the particle at the current time; such signals mark the boundary of the normal convex neighbourhood. The quasilocal evaluation of the world-line integral must therefore be supplemented with contributions from the distant past, and this requires a representation of the Green’s function that is not limited to the normal convex neighbourhood. In some spacetimes it is possible to express the Green’s function as an expansion in quasi-normal modes, as was demonstrated by Casals and his collaborators for a static scalar charge in the Nariai spacetime [40]. Their study provided significant insights into the geometrical structure of Green’s functions in curved spacetime, and increased our understanding of the non-local character of the self-force.
2.5 Adiabatic approximations
The accurate computation of long-term waveforms from extreme-mass-ratio inspirals (EMRIs) involves a lengthy sequence of calculations that include the calculation of the self-force. One can already imagine the difficulty of numerically integrating the coupled linearized Einstein equation for durations much longer than has ever been attempted by existing numerical codes. While doing so, the code would also have to evaluate the self-force reasonably often (if not at each time step) in order to remain close to the true dynamics of the point mass. Moreover, gravitational-wave data analysis via matched filtering require full evolutions of the sort just described for a large sample of systems parameters. All these considerations underlie the desire for simplified approximations to fully self-consistent self-force EMRI models.
The adiabatic approximation refers to one such class of potentially useful approximations. The basic assumption is that the secular effects of the self-force occur on a timescale that is much longer than the orbital period. In an extreme-mass-ratio binary, this assumption is valid during the early stage of inspiral; it breaks down in the final moments, when the orbit transitions to a quasi-radial infall called the plunge. From the adiabaticity assumption, numerous approximations have been formulated: For example, (i) since the particle’s orbit deviates only slowly from geodesic motion, the self-force can be calculated from a field sourced by a geodesic; (ii) since the radiation-reaction timescale t_{ rr }, over which a significant shrinking of the orbit occurs due to the self-force, is much longer than the orbital period, periodic effects of the self-force can be neglected; and (iii) conservative effects of the self-force can be neglected (the radiative approximation).
A seminal example of an adiabatic approximation is the Peters-Mathews formalism [140, 139], which determines the long-term evolution of a binary orbit by equating the time-averaged rate of change of the orbital energy E and angular momentum L to, respectively, the flux of gravitational-wave energy and angular momentum at infinity. This formalism was used to successfully predict the decreasing orbital period of the Hulse-Taylor pulsar, before more sophisticated methods, based on post-Newtonian equations of motion expanded to 2.5pn order, were incorporated in times-of-arrival formulae.
In the hope of achieving similar success in the context of the self-force, considerable work has been done to formulate a similar approximation for the case of an extreme-mass-ratio inspiral [124, 125, 126, 98, 61, 62, 159, 158, 78, 128, 94]. Bound geodesics in Kerr spacetime are specified by three constants of motion — the energy E, angular momentum L, and Carter constant C. If one could easily calculate the rates of change of these quantities, using a method analogous to the Peters-Mathews formalism, then one could determine an approximation to the long-term orbital evolution of the small body in an EMRI, avoiding the lengthy process of regularization involved in the direct integration of the self-forced equation of motion. In the early 1980s, Gal’tsov [77] showed that the average rates of change of E and L, as calculated from balance equations that assume geodesic source motion, agree with the averaged rates of change induced by a self-force constructed from a radiative Green’s function defined as \({G_{{\rm{rad}}}} := {{1 \over 2}}({G_ -} - {G_ +})\). As discussed in Section 1.4, this is equal to the regular two-point function Gr in flat spacetime, but G_{rad} ≠ G_{R} in curved spacetime; because of its time-asymmetry, it is purely dissipative. Mino [124], building on the work of Gal’tsov, was able to show that the true self-force and the dissipative force constructed from G_{rad} cause the same averaged rates of change of all three constants of motion, lending credence to the radiative approximation. Since then, the radiative Green’s function was used to derive explicit expressions for the rates of change of E, L, and C in terms of the particle’s orbit and wave amplitudes at infinity [159, 158, 78], and radiative approximations based on such expressions have been concretely implemented by Drasco, Hughes and their collaborators [99, 61, 62].
The relevance of the conservative part of the self-force — the part left out when using G_{rad} — was analyzed in numerous recent publications [32, 148, 146, 147, 94, 97]. As was shown by Pound et al. [148, 146, 147], neglect of the conservative effects of the self-force generically leads to long-term errors in the phase of an orbit and the gravitational wave it produces. These phasing errors are due to orbital precession and a direct shift in orbital frequency. This shift can be understood by considering a conservative force acting on a circular orbit: the force is radial, it alters the centripetal acceleration, and the frequency associated with a given orbital radius is affected. Despite these errors, a radiative approximation may still suffice for gravitational-wave detection [94]; for circular orbits, which have minimal conservative effects, radiative approximations may suffice even for parameter-estimation [97]. However, at this point in time, these analyses remain inconclusive because they all rely on extrapolations from post-Newtonian results for the conservative part of the self-force. For a more comprehensive discussion of these issues, the reader is referred to Ref. [94, 143].
Hinderer and Flanagan performed the most comprehensive study of these issues [69], utilizing a two-timescale expansion [109, 145] of the field equations and self-forced equations of motion in an EMRI. In this method, all dynamical variables are written in terms of two time coordinates: a fast time t and a slow time \(\tilde t := (m/M)t\). In the case of an EMRI, the dynamical variables are the metric and the phase-space variables of the world line. The fast-time dependence captures evolution on the orbital timescale ∼ M, while the slow-time dependence captures evolution on the radiation-reaction timescale ∼ M^{2}/m. One obtains a sequence of fast-time and slow-time equations by expanding the full equations in the limit of small m while treating the two time coordinates as independent. Solving the leading-order fast-time equation, in which \(\tilde {t}\) is held fixed, yields a metric perturbation sourced by a geodesic, as one would expect from the linearized field equations for a point particle. The leading-order effects of the self-force are incorporated by solving the slow-time equation and letting \(\tilde {t}\) vary. (Solving the next-higher-order slow-time equation determines similar effects, but also the backreaction that causes the parameters of the large black hole to change slowly.)
2.6 Physical consequences of the self-force
To be of relevance to gravitational-wave astronomy, the paramount goal of the self-force community remains the computation of waveforms that properly encode the long-term dynamical evolution of an extreme-mass-ratio binary. This requires a fully consistent orbital evolution fed to a wave-generation formalism, and to this day the completion of this program remains as a future challenge. In the meantime, a somewhat less ambitious, though no less compelling, undertaking is that of probing the physical consequences of the self-force on the motion of point particles.
2.6.1 Scalar charge in cosmological spacetimes
The intriguing phenomenon of a scalar charge changing its rest mass because of an interaction with its self-field was studied by Burko, Harte, and Poisson [33] and Haas and Poisson [86] in the simple context of a particle at rest in an expanding universe. The scalar Green’s function could be computed explicitly for a wide class of cosmological spacetimes, and the action of the field on the particle determined without approximations. It is found that for certain cosmological models, the mass decreases and then increases back to its original value. For other models, the mass is restored only to a fraction of its original value. For de Sitter spacetime, the particle radiates all of its rest mass into monopole scalar waves.
2.6.2 Physical consequences of the gravitational self-force
The earliest calculation of a gravitational self-force was performed by Barack and Lousto for the case of a point mass plunging radially into a Schwarzschild black hole [14]. The calculation, however, depended on a specific choice of gauge and did not identify unambiguous physical consequences of the self-force. To obtain such consequences, it is necessary to combine the self-force (computed in whatever gauge) with the metric perturbation (computed in the same gauge) in the calculation of a well-defined observable that could in principle be measured. For example, the conservative pieces of the self-force and metric perturbation can be combined to calculate the shifts in orbital frequencies that originate from the gravitational effects of the small body; an application of such a calculation would be to determine the shift (as measured by frequency) in the innermost stable circular orbit of an extreme-mass-ratio binary, or the shift in the rate of periastron advance for eccentric orbits. Such calculations, however, must exclude all dissipative aspects of the self-force, because these introduce an inherent ambiguity in the determination of orbital frequencies.
A calculation of this kind was recently achieved by Barack and Sago [22, 23], who computed the shift in the innermost stable circular orbit of a Schwarzschild black hole caused by the conservative piece of the gravitational self-force. The shift in orbital radius is gauge dependent (and was reported in the Lorenz gauge by Barack and Sago), but the shift in orbital frequency is measurable and therefore gauge invariant. Their main result — a genuine milestone in self-force computations — is that the fractional shift in frequency is equal to 0.4870m/M; the frequency is driven upward by the gravitational self-force. Barack and Sago compare this shift to the ambiguity created by the dissipative piece of the self-force, which was previously investigated by Ori and Thorne [135] and Sundararajan [167]; they find that the conservative shift is very small compared with the dissipative ambiguity. In a follow-up analysis, Barack, Damour, and Sago [11] computed the conservative shift in the rate of periastron advance of slightly eccentric orbits in Schwarzschild spacetime.
Conservative shifts in the innermost stable circular orbit of a Schwarzschild black hole were first obtained in the context of the scalar self-force by Diaz-Rivera et al. [55]; in this case they obtain a fractional shift of 0.0291657q^{2}/(mM), and here also the frequency is driven upward.
2.6.3 Detweiler’s redshift factor
In another effort to extract physical consequences from the gravitational self-force on a particle in circular motion in Schwarzschild spacetime, Detweiler discovered [50] that u^{ t }, the time component of the velocity vector in Schwarzschild coordinates, is invariant with respect to a class of gauge transformations that preserve the helical symmetry of the perturbed spacetime. Detweiler further showed that 1/ut is an observable: it is the redshift that a photon suffers when it propagates from the orbiting body to an observer situated at a large distance on the orbital axis. This gaugeinvariant quantity can be calculated together with the orbital frequency Ω, which is a second gaugeinvariant quantity that can be constructed for circular orbits in Schwarzschild spacetime. Both u^{ t } and Ω acquire corrections of fractional order m/M from the self-force and the metric perturbation. While the functions u^{ t } (r) and Ω(r) are still gauge dependent, because of the dependence on the radial coordinate r, elimination of r from these relations permits the construction of u^{ t } (Ω), which is gauge invariant. A plot of u^{ t } as a function of Ω therefore contains physically unambiguous information about the gravitational self-force.
The computation of the gauge-invariant relation ut(Ω) opened the door to a detailed comparison between the predictions of the self-force formalism to those of post-Newtonian theory. This was first pursued by Detweiler [50], who compared u^{ t } (Ω) as determined accurately through second post-Newtonian order, to self-force results obtained numerically; he reported full consistency at the expected level of accuracy. This comparison was pushed to the third post-Newtonian order [29, 28, 44, 11]. Agreement is remarkable, and it conveys a rather deep point about the methods of calculation. The computation of ut(Ω), in the context of both the self-force and post-Newtonian theory, requires regularization of the metric perturbation created by the point mass. In the self-force calculation, removal of the singular field is achieved with the Detweiler-Whiting prescription, while in post-Newtonian theory it is performed with a very different prescription based on dimensional regularization. Each prescription could have returned a different regularized field, and therefore a different expression for ut(Ω). This, remarkably, does not happen; the singular fields are “physically the same” in the self-force and post-Newtonian calculations.
A generalization of Detweiler’s redshift invariant to eccentric orbits was recently proposed and computed by Barack and Sago [24], who report consistency with corresponding post-Newtonian results in the weak-field regime. They also computed the influence of the conservative gravitational self-force on the periastron advance of slightly eccentric orbits, and compared their results with full numerical relativity simulations for modest mass-ratio binaries. Thus, in spite of the unavailability of self-consistent waveforms, it is becoming clear that self-force calculations are already proving to be of value: they inform post-Newtonian calculations and serve as benchmarks for numerical relativity.
3 Part I: General Theory of Bitensors
4 Synge’s world function
4.1 Definition
By virtue of the geodesic equation, the quantity ε: = g_{ μν }t^{ μ }t^{ ν } is constant on the geodesic. The world function is therefore numerically equal to \({1 \over 2}\varepsilon {({\lambda _1} - {\lambda _0})^2}\). If the geodesic is timelike, then λ can be set equal to the proper time τ, which implies that ε = −1 and \(\sigma = - {1 \over 2}{(\Delta \tau)^2}\). If the geodesic is spacelike, then λ can be set equal to the proper distance s, which implies that ε = 1 and \(\sigma = {1 \over 2}{(\Delta s)^2}\). If the geodesic is null, then σ = 0. Quite generally, therefore, the world function is half the squared geodesic distance between the points x′ and x.
In flat spacetime, the geodesic linking x to x^{′} is a straight line, and \(\sigma = {1 \over 2}{\eta _{\alpha \beta}}{(x - x{\prime})^\alpha}{(x - x{\prime})^\beta}\) in Lorentzian coordinates.
4.2 Differentiation of the world function
The world function σ (x, x′) can be differentiated with respect to either argument. We let σ_{ α } = ∂σ/∂x^{ α } be its partial derivative with respect to x, and σ_{α′} = ∂σ/∂x^{α′} its partial derivative with respect to x′. It is clear that σ_{ α } behaves as a dual vector with respect to tensorial operations carried out at x, but as a scalar with respect to operations carried out x′. Similarly, σ_{α′} is a scalar at x but a dual vector at x′.
We let σ_{ αβ }:= ∇_{ β }σ_{ α } be the covariant derivative of σ_{ α } with respect to x; this is a rank-2 tensor at x and a scalar at x′. Because σ is a scalar at x, we have that this tensor is symmetric: σ_{ βα } = σ_{ αβ }. Similarly, we let σ_{αβ′} := ∂_{β′}σ_{ α } = ∂^{ 2 }σ/∂x^{β′} ∂x^{ α } be the partial derivative of σ_{ α } with respect to x′; this is a dual vector both at x and x′. We can also define σ_{α′β} := ∂_{ β }σ_{ α } _{′} = ∂^{2}σ/∂x^{ β }∂x^{α′} to be the partial derivative of σ_{α′} with respect to x. Because partial derivatives commute, these bitensors are equal: σ_{β′α} = σ_{ αβ } _{′}. Finally, we let σ_{ σα′β } := ∇_{β′}σ_{α′} be the covariant derivative of σ_{α′} with respect to x′; this is a symmetric rank-2 tensor at x′ and a scalar at x.
The notation is easily extended to any number of derivatives. For example, we let σ_{σαβγ′}:= ∇_{δ′}∇_{γ}∇_{ β } ∇_{ α }σ, which is a rank-3 tensor at x and a dual vector at x′. This bitensor is symmetric in the pair of indices α and β, but not in the pairs α and γ, nor β and γ. Because ∇_{δ′} is here an ordinary partial derivative with respect to x′, the bitensor is symmetric in any pair of indices involving δ′. The ordering of the primed index relative to the unprimed indices is therefore irrelevant: the same bitensor can be written as σ_{δ′αβγ} or σ_{αδ′βγ} or σ_{αβδ′γ}, making sure that the ordering of the unprimed indices is not altered.
The message of Eq. (3.2), when applied to derivatives of the world function, is that while the ordering of the primed and unprimed indices relative to themselves is important, their ordering with respect to each other is arbitrary. For example, σ_{α′β′γδ′δ} = σ_{α′β′δ′γδ} = σ_{γϵα′β′δ′}.
4.3 Evaluation of first derivatives
We can compute σ_{ α } by examining how σ varies when the field point x moves. We let the new field point be x + δx, and δσ:= σ (x + δx, x′) − σ (x, x′) is the corresponding variation of the world function. We let β + δβ be the unique geodesic segment that links x + δx to x′; it is described by relations z^{ μ } (λ) + δz^{ μ } (λ), in which the affine parameter is scaled in such a way that it runs from λ_{0} to λ_{1} also on the new geodesic. We note that δz (λ_{0}) = δx′ = 0 and δz (λ_{1}) = δx.
We note that in flat spacetime, σ_{ α } = η_{ αβ } (x − x^{′})^{ β } and σα′ = −η_{ αβ } (x − x′)^{ β } in Lorentzian coordinates. From this it follows that σ_{ αβ } = σ_{α′β′} = −σ_{αβ′} = −σ_{ α′β } = η_{ αβ }, and finally, g^{ αβ }σ_{ αβ } = 4 = g^{a′β′}σ_{a′β′}.
4.4 Congruence of geodesics emanating from x′
If the base point x′ is kept fixed, σ can be considered to be an ordinary scalar function of x. According to Eq. (3.5), this function is a solution to the nonlinear differential equation \({1 \over 2}{g^{\alpha \beta}}{\sigma _\alpha}{\sigma _\beta} = \sigma\). Suppose that we are presented with such a scalar field. What can we say about it?
These considerations, which all follow from a postulated relation \({1 \over 2}{g^{\alpha \beta}}{\sigma _\alpha}{\sigma _\beta} = \sigma\), are clearly compatible with our preceding explicit construction of the world function.
5 Coincidence limits
5.1 Computation of coincidence limits
5.2 Derivation of Synge’s rule
6 Parallel propagator
6.1 Tetrad on β
(You will have noticed that we use sans-serif symbols for the frame indices. This is to distinguish them from another set of frame indices that will appear below. The frame indices introduced here run from 0 to 3; those to be introduced later will run from 1 to 3.)
6.2 Definition and properties of the parallel propagator
6.3 Coincidence limits
7 Expansion of bitensors near coincidence
7.1 General method
We would like to express a bitensor Ω_{α′β′} (x, x′) near coincidence as an expansion in powers of −σ^{α′} (x, x′), the closest analogue in curved spacetime to the flat-spacetime quantity (x − x′)^{ α }. For concreteness we shall consider the case of rank-2 bitensor, and for the moment we will assume that the tensorial indices all refer to the base point x′.
7.2 Special cases
7.3 Expansion of tensors
8 van Vleck determinant
8.1 Definition and properties
8.2 Derivations
9 Part II: Coordinate Systems
10 Riemann normal coordinates
10.1 Definition and coordinate transformation
10.2 Metric near x′
It is obvious from Eq. (8.8) that \({g_{{\rm{ab}}}}(x\prime) = {\eta _{{\rm{ab}}}}\) and \(\Gamma _{\;{\rm{bc}}}^{\rm{a}}(x {\prime}) = 0\), where \(\Gamma _{\;{\rm{bc}}}^{\rm{a}} = - {1 \over 3}(R_{\;{\rm{bcd}}}^{\rm{a}} + R_{\;{\rm{cbd}}}^{\rm{a}}){\hat x^{\rm{d}}} + O({x^2})\) is the connection compatible with the metric g_{ab}. The Riemann normal coordinates therefore provide a constructive proof of the local flatness theorem.
11 Fermi normal coordinates
11.1 Fermi-Walker transport
Let γ be a timelike curve described by parametric relations z^{ μ } (τ) in which τ is proper time. Let u^{ μ } = dz^{ μ }/dτ be the curve’s normalized tangent vector, and let a^{ μ } = Du^{ μ }/dτ be its acceleration vector.
The operation of Fermi-Walker (FW) transport satisfies two important properties. The first is that u^{ μ } is automatically FW transported along γ; this follows at once from Eq. (9.1) and the fact that u^{ μ } is orthogonal to a^{ μ }. The second is that if the vectors v^{ μ } and w^{ μ } are both FW transported along γ, then their inner product v_{ μ }w^{ μ } is constant on γ: D (v_{ μ }w^{ μ })/dτ = 0; this also follows immediately from Eq. (9.1).
11.2 Tetrad and dual tetrad on γ
11.3 Fermi normal coordinates
11.4 Coordinate displacements near γ
11.5 Metric near γ
Notice that on γ, the metric of Eqs. (9.14) — (9.16) reduces to g_{ tt } = −1 and g_{ ab } = δ_{ ab }. On the other hand, the nonvanishing Christoffel symbols (on γ) are \(\Gamma _{\;ta}^t = \Gamma _{\;tt}^a = {a_a}\); these are zero (and the FNC enforce local flatness on the entire curve) when γ is a geodesic.
11.6 Thorne—Hartle—Zhang coordinates
12 Retarded coordinates
12.1 Geometrical elements
The Fermi normal coordinates of Section 9 were constructed on the basis of a spacelike geodesic connecting a field point x to the world line. The retarded coordinates are based instead on a null geodesic going from the world line to the field point. We thus let x be within the normal convex neighbourhood of γ, β be the unique future-directed null geodesic that goes from the world line to x, and x′ := z (u) be the point at which β intersects the world line, with u denoting the value of the proper-time parameter at this point.
12.2 Definition of the retarded coordinates
12.3 The scalar field r (x) and the vector field k^{ α }(x)
12.4 Frame components of tensor fields on the world line
In Section 9 we saw that the frame components of a given tensor were also the components of this tensor (evaluated on the world line) in the Fermi normal coordinates. We should not expect this property to be true also in the case of the retarded coordinates: the frame components of a tensor are not to be identified with the components of this tensor in the retarded coordinates. The reason is that the retarded coordinates are in fact singular on the world line. As we shall see, they give rise to a metric that possesses a directional ambiguity at r = 0. (This can easily be seen in Minkowski spacetime by performing the coordinate transformation \(u = t - \sqrt {{x^2} + {y^2} + {z^2}}\).) Components of tensors are therefore not defined on the world line, although they are perfectly well defined for r ≠ 0. Frame components, on the other hand, are well defined both off and on the world line, and working with them will eliminate any difficulty associated with the singular nature of the retarded coordinates.
12.5 Coordinate displacements near γ
12.6 Metric near γ
12.7 Transformation to angular coordinates
12.8 Specialization to a^{ μ } = 0 = R_{ μν }
In this subsection we specialize our previous results to a situation where γ is a geodesic on which the Ricci tensor vanishes. We therefore set a^{ μ } = 0 = R_{ μν } everywhere on γ.
13 Transformation between Fermi and retarded coordinates; advanced point
A point x in the normal convex neighbourhood of a world line γ can be assigned a set of Fermi normal coordinates (as in Section 9), or it can be assigned a set of retarded coordinates (Section 10). These coordinate systems can obviously be related to one another, and our first task in this section (which will occupy us in Sections 11.1–11.3) will be to derive the transformation rules. We begin by refining our notation so as to eliminate any danger of ambiguity.
The retarded coordinates of x refer to a point x′ := z (u) on γ that is linked to x by a future-directed null geodesic; see Figure 8. We refer to this point as x’s retarded point, and to tensors at x′ we assign indices α′, β′, etc. We let (u, r Ω^{ a }) be the retarded coordinates of x, with u denoting the value of γ’s proper-time parameter at x″ representing the affine-parameter distance from x′ to x along the null geodesic, and Ω^{ a } denoting a unit vector (δ_{ ab } Ω^{ a } Ω^{ b } = 1) that determines the direction of the geodesic. The retarded coordinates are defined by \(r{\Omega ^a} = - e_{\alpha \prime}^a{\sigma ^{\alpha \prime}}\) and σ (x, x′) = 0. Finally, we denote by \((e_0^\alpha ,\; e_a^\alpha)\) the tetrad at x that is obtained by parallel transport of \(({u^{\alpha \prime}},\; e_a^{\alpha \prime})\) on the null geodesic.
The reader who does not wish to follow the details of this discussion can be informed that: (i) our results concerning the transformation from the retarded coordinates (u, r, Ω^{ a }) to the Fermi normal coordinates (t, s, ω^{ a }) are contained in Eqs. (11.1) — (11.3) below; (ii) our results concerning the transformation from the Fermi normal coordinates (t, s, ω^{ a }) to the retarded coordinates (u, r, Ω^{ a }) are contained in Eqs. (11.4) — (11.6); (iii) the decomposition of each member of \((\bar e_0^\alpha ,\bar e_a^\alpha)\) in the tetrad \((e_0^\alpha ,\; e_a^\alpha)\) is given in retarded coordinates by Eqs. (11.7) and (11.8); and (iv) the decomposition of each member of \((e_0^\alpha ,\; e_a^\alpha)\) in the tetrad \((\bar e_0^\alpha ,\bar e_a^\alpha)\) is given in Fermi normal coordinates by Eqs. (11.9) and (11.10).
Our final task will be to define, along with the retarded and simultaneous points, an advanced point x″ on the world line γ; see Figure 8. This is taken on in Section 11.4.
13.1 From retarded to Fermi coordinates
Quantities at \(\bar{x} := z(t)\) can be related to quantities at x′ := z (u) by Taylor expansion along the world line γ. To implement this strategy we must first find an expression for Δ := t − u. (Although we use the same notation, this should not be confused with the van Vleck determinant introduced in Section 7.)
13.2 From Fermi to retarded coordinates
The techniques developed in the preceding subsection can easily be adapted to the task of relating the retarded coordinates of x to its Fermi normal coordinates. Here we use \(\bar{x} := z(t)\) as the reference point and express all quantities at x^{′} := z (u) as Taylor expansions about τ = t.
13.3 Transformation of the tetrads at x
13.4 Advanced point
14 Part III: Green’s Functions
15 Scalar Green’s functions in flat spacetime
15.1 Green’s equation for a massive scalar field
15.2 Integration over the source
15.3 Singular part of g (σ)
We have seen that Eq. (12.7) properly encodes the influence of the singular source term on both the retarded and advanced Green’s function. The function g (σ) that enters into the expressions of Eq. (12.4) must therefore be such that Eq. (12.7) is satisfied. It follows immediately that g (σ) must be a singular function, because for a smooth function the integral of Eq. (12.7) would be of order ϵ and the left-hand side of Eq. (12.7) could never be made equal to −1. The singularity, however, must be integrable, and this leads us to assume that g′(σ) must be made out of Dirac δ-functions and derivatives.
15.4 Smooth part of g (σ)
To summarize, the retarded and advanced solutions to Eq. (12.3) are given by Eq. (12.4) with g (σ) given by Eq. (12.10) and V (σ) given by Eq. (12.13).
15.5 Advanced distributional methods
The techniques developed previously to find Green’s functions for the scalar wave equation are limited to flat spacetime, and they would not be very useful for curved spacetimes. To pursue this generalization we must introduce more powerful distributional methods. We do so in this subsection, and in the next we shall use them to recover our previous results.
The distributions θ_{±}(−σ) and δ_{±}(σ) are not defined at x = x′ and they cannot be differentiated there. This pathology can be avoided if we shift σ by a small positive quantity ϵ. We can therefore use the distributions θ_{±}(−σ − ϵ) and θ_{±}(σ + ϵ) in some sensitive computations, and then take the limit ϵ → 0^{+}. Notice that the equation σ + ϵ = 0 describes a two-branch hyperboloid that is located just within the light cone of the reference point x′. The hyperboloid does not include x′, and θ_{+} (x, Σ) is one everywhere on its future branch, while θ −(x, Σ) is one everywhere on its past branch. These factors, therefore, become invisible to differential operators. For example, θ′+ (−σ − ϵ) = θ_{+} (x, Σ)θ′(−σ − ϵ) = −θ_{+}(x, Σ)δ (σ + ϵ) = − δ_{+} (σ + ϵ). This manipulation shows that after the shift from σ to σ + ϵ, the distributions of Eqs. (12.14) and (12.15) can be straightforwardly differentiated with respect to σ.
15.6 Alternative computation of the Green’s functions
16 Distributions in curved spacetime
The distributions introduced in Section 12.5 can also be defined in a four-dimensional spacetime with metric g_{ αβ }. Here we produce the relevant generalizations of the results derived in that section.
16.1 Invariant Dirac distribution
16.2 Light-cone distributions
For the same reasons as those mentioned in Section 12.5, it is sometimes convenient to shift the argument of the step and δ-functions from σ to σ + ϵ, where ϵ is a small positive quantity. With this shift, the light-cone distributions can be straightforwardly differentiated with respect to σ. For example, \(\delta_{\pm}(\sigma + \epsilon) = -\theta\prime_{\pm}(-\sigma-\epsilon)\), with a prime indicating differentiation with respect to σ.
17 Scalar Green’s functions in curved spacetime
17.1 Green’s equation for a massless scalar field in curved spacetime
We let G +(x,x′) be the retarded solution to Eq. (14.3), and G_{ − } (x,x′) is the advanced solution; when viewed as functions of x, G_{+}(x,x′) is nonzero in the causal future of x′, while G_{−}(x,x′) is nonzero in its causal past. We assume that the retarded and advanced Green’s functions exist as distributions and can be defined globally in the entire spacetime.
17.2 Hadamard construction of the Green’s functions
To summarize: We have shown that with U (x, x′) given by Eq. (14.8) and V (x,x′) determined uniquely by the wave equation of Eq. (14.14) and the characteristic data constructed with Eqs. (14.9) and (14.13), the retarded and advanced Green’s functions of Eq. (14.4) do indeed satisfy Eq. (14.3). It should be emphasized that the construction provided in this subsection is restricted to \({\mathcal N}(x \prime)\), the normal convex neighbourhood of the reference point x′.
17.3 Reciprocity
17.4 Kirchhoff representation
17.5 Singular and regular Green’s functions
In Part IV of this review we will compute the retarded field of a moving scalar charge, and we will analyze its singularity structure near the world line; this will be part of our effort to understand the effect of the field on the particle’s motion. The retarded solution to the scalar wave equation is the physically relevant solution because it properly incorporates outgoing-wave boundary conditions at infinity — the advanced solution would come instead with incoming-wave boundary conditions. The retarded field is singular on the world line because a point particle produces a Coulomb field that diverges at the particle’s position. In view of this singular behaviour, it is a subtle matter to describe the field’s action on the particle, and to formulate meaningful equations of motion.
When facing this problem in flat spacetime (recall the discussion of Section 1.3) it is convenient to decompose the retarded Green’s function G_{+} (x, x′) into a singular Green’s function \({G_{\rm{S}}}(x,x\prime) := {1 \over 2}[{G_ +}(x,x\prime) + {G_ -}(x,x\prime)]\) and a regular two-point function \({G_{\rm{R}}}(x,x\prime) := {1 \over 2}[{G_ +}(x,x\prime) - {G_ -}(x,x\prime)]\). The singular Green’s function takes its name from the fact that it produces a field with the same singularity structure as the retarded solution: the diverging field near the particle is insensitive to the boundary conditions imposed at infinity. We note also that G_{S}(x,x′) satisfies the same wave equation as the retarded Green’s function (with a Dirac functional as a source), and that by virtue of the reciprocity relations, it is symmetric in its arguments. The regular two-point function, on the other hand, takes its name from the fact that it satisfies the homogeneous wave equation, without the Dirac functional on the right-hand side; it produces a field that is regular on the world line of the moving scalar charge. (We reserve the term “Green’s function” to a two-point function that satisfies the wave equation with a Dirac distribution on the right-hand side; when the source term is absent, the object is called a “two-point function”.)
Because the singular Green’s function is symmetric in its argument, it does not distinguish between past and future, and it produces a field that contains equal amounts of outgoing and incoming radiation — the singular solution describes a standing wave at infinity. Removing G_{S}(x, x′) from the retarded Green’s function will have the effect of removing the singular behaviour of the field without affecting the motion of the particle. The motion is not affected because it is intimately tied to the boundary conditions: If the waves are outgoing, the particle loses energy to the radiation and its motion is affected; if the waves are incoming, the particle gains energy from the radiation and its motion is affected differently. With equal amounts of outgoing and incoming radiation, the particle neither loses nor gains energy and its interaction with the scalar field cannot affect its motion. Thus, subtracting G_{S}(x,x′) from the retarded Green’s function eliminates the singular part of the field without affecting the motion of the scalar charge. The subtraction leaves behind the regular two-point function, which produces a field that is regular on the world line; it is this field that will govern the motion of the particle. The action of this field is well defined, and it properly encodes the outgoing-wave boundary conditions: the particle will lose energy to the radiation.
In this subsection we attempt a decomposition of the curved-spacetime retarded Green’s function into singular and regular pieces. The flat-spacetime relations will have to be amended, however, because of the fact that in a curved spacetime, the advanced Green’s function is generally nonzero when x′ is in the chronological future of x. This implies that the value of the advanced field at x depends on events x′ that will unfold in the future; this dependence would be inherited by the regular field (which acts on the particle and determines its motion) if the naive definition \({G_{\rm{R}}}(x,x\prime) := {1 \over 2}[{G_ +}(x,x\prime) - {G_ -}(x,x\prime)]\) were to be adopted.
- S1: G_{S}(x,x′) satisfies the inhomogeneous scalar wave equation,$$(\square \, - \xi R){G_{\rm{S}}}(x,x\prime) = - 4\pi {\delta _4}(x,x\prime);$$(14.19)
- S2: G_{S}(x,x′) is symmetric in its arguments,$${G_{\rm{S}}}(x\prime ,x) = {G_{\rm{S}}}(x,x\prime);$$(14.20)
- S3: G_{S}(x,x′) vanishes if x is in the chronological past or future of x′,$${G_{\rm{S}}}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ \pm}(x\prime).$$(14.21)
- R1: G_{R}(x,x′) satisfies the homogeneous wave equation,$$(\square\, - \xi R){G_{\rm{R}}}(x,x\prime) = 0;$$(14.23)
- R2: G_{R}(x,x′) agrees with the retarded Green’s function if x is in the chronological future of$${G_{\rm{R}}}(x,x\prime) = {G_ +}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(14.24)
- R3: G_{R}(x,x′) vanishes if x is in the chronological past of x′,$${G_{\rm{R}}}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ -}(x\prime).$$(14.25)
- H1: H (x,x′) satisfies the homogeneous wave equation,$$(\square\, - \xi R)H(x,x\prime) = 0;$$(14.26)
- H2: H (x,x′) is symmetric in its arguments,$$H(x\prime ,x) = H(x,x\prime);$$(14.27)
- H3: H (x,x′) agrees with the retarded Green’s function if x is in the chronological future of$$H(x,x\prime) = {G_ +}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(14.28)
- H4: H (x,x′) agrees with the advanced Green’s function if x is in the chronological past of$$H(x,x\prime) = {G_ -}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ -}(x\prime).$$(14.29)
The question is now: does such a function H (x, x′) exist? We will present a plausibility argument for an affirmative answer. Later in this section we will see that H (x,x′) is guaranteed to exist in the local convex neighbourhood of x′, where it is equal to V (x,x′). And in Section 14.6 we will see that there exist particular spacetimes for which H (x, x′) can be defined globally.
To satisfy all of H1–H4 might seem a tall order, but it should be possible. We first note that property H4 is not independent from the rest: it follows from H2, H3, and the reciprocity relation (14.15) satisfied by the retarded and advanced Green’s functions. Let x ∈ I^{−}(x′), so that x′ ∈ I^{+} (x). Then H (x,x′) = H (x′,x) by H2, and by H3 this is equal to G_{+} (x′,x). But by the reciprocity relation this is also equal to G_{−}(x,x′), and we have obtained H4. Alternatively, and this shall be our point of view in the next paragraph, we can think of H3 as following from H2 and H4.
17.6 Example: Cosmological Green’s functions
As a final observation we note that for this cosmological spacetime, the normal convex neighbourhood of any point x consists of the whole spacetime manifold (which excludes the cosmological singularity at a = 0). The Hadamard construction of the Green’s functions is therefore valid globally, a fact that is immediately revealed by Eqs. (14.44) and (14.45).
18 Electromagnetic Green’s functions
18.1 Equations of electromagnetism
We will assume that the retarded Green’s function \(G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\), which is nonzero if x is in the causal future of x′, and the advanced Green’s function \(G_{- \beta \prime}^{\;\alpha}(x,x\prime)\), which is nonzero if x is in the causal past of x′, exist as distributions and can be defined globally in the entire spacetime.
18.2 Hadamard construction of the Green’s functions
To summarize, the retarded and advanced electromagnetic Green’s functions are given by Eq. (15.4) with \(U_{\;\beta \prime}^\alpha (x,x\prime)\) given by Eq. (15.9) and \(V_{\;\beta \prime}^\alpha (x,x\prime)\) determined by Eq. (15.8) and the characteristic data constructed with Eqs. (15.7) and (15.14). It should be emphasized that the construction provided in this subsection is restricted to \({\mathcal N}(x\prime)\), the normal convex neighbourhood of the reference point x′.
18.3 Reciprocity and Kirchhoff representation
18.4 Relation with scalar Green’s functions
18.5 Singular and regular Green’s functions
We shall now construct singular and regular Green’s functions for the electromagnetic field. The treatment here parallels closely what was presented in Section 14.5, and the reader is referred to that section for a more complete discussion.
- H1: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) satisfies the homogeneous wave equation,$$\square H_{\;\,\beta \prime}^\alpha (x,x\prime) - R_{\;\,\beta}^\alpha (x)H_{\;\,\beta \prime}^\beta (x,x\prime) = 0;$$(15.20)
- H2: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) is symmetric in its indices and arguments,$${H_{\beta \prime \alpha}}(x\prime ,x) = {H_{\alpha \beta \prime}}(x,x\prime);$$(15.21)
- H3: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,$$H_{\;\,\beta \prime}^\alpha (x,x\prime) = G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(15.22)
- H4: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) agrees with the advanced Green’s function if x is in the chronological past of x′,$$H_{\;\,\beta \prime}^\alpha (x,x\prime) = G_{- \beta \prime}^{\;\alpha}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ -}(x\prime).$$(15.23)
- S1: \(G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime)\) satisfies the inhomogeneous wave equation,$$\square G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime) - R_{\;\,\beta}^\alpha (x)G_{{\rm{S}}\,\beta \prime}^{\;\beta}(x,x\prime) = - 4\pi g_{\;\,\beta \prime}^\alpha (x,x\prime){\delta _4}(x,x\prime);$$(15.25)
- S2: \(G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime)\) is symmetric in its indices and arguments,$$G_{\beta \prime \alpha}^{\rm{S}}(x\prime ,x) = G_{\alpha \beta \prime}^{\rm{S}}(x,x\prime);$$(15.26)
- S3: \(G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime)\) vanishes if x is in the chronological past or future of x′,$$G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ \pm}(x\prime).$$(15.27)
- R1: \(G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime)\) satisfies the homogeneous wave equation,$$\square G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime) - R_{\;\,\beta}^\alpha (x)G_{{\rm{R}}\,\beta \prime}^{\;\,\beta}(x,x\prime) = 0;$$(15.29)
- R2: \(G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,$$G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime) = G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(15.30)
- R3: \(G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime)\) vanishes if x is in the chronological past of x′,$$G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ -}(x\prime).$$(15.31)
19 Gravitational Green’s functions
19.1 Equations of linearized gravity
We will assume that the retarded Green’s function \(G_{+ \;\,\gamma \prime\delta \prime}^{\;\alpha \beta}(x,x\prime)\), which is nonzero if x is in the causal future of x′, and the advanced Green’s function \(G_{- \;\,\gamma \prime\delta \prime}^{\;\alpha \beta}(x,x\prime)\), which is nonzero if x is in the causal past of x′, exist as distributions and can be defined globally in the entire background spacetime.
19.2 Hadamard construction of the Green’s functions
To summarize, the retarded and advanced gravitational Green’s functions are given by Eq. (16.7) with \(U_{\;\;\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) given by Eq. (16.12) and \(V_{\;\;\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) determined by Eq. (16.11) and the characteristic data constructed with Eqs. (16.10) and (16.17). It should be emphasized that the construction provided in this subsection is restricted to \({\mathcal N}(x\prime)\), the normal convex neighbourhood of the reference point x′.
19.3 Reciprocity and Kirchhoff representation
19.4 Relation with electromagnetic and scalar Green’s functions
The identity of Eq. (16.22) follows simply from the fact that \({g^{\gamma \prime\delta \prime}}G_{\,\,\,\,\gamma \prime\delta \prime}^{\alpha \beta}\) and g^{ αβ }G satisfy the same tensorial wave equation in a Ricci-flat spacetime.
19.5 Singular and regular Green’s functions
We shall now construct singular and regular Green’s functions for the linearized gravitational field. The treatment here parallels closely what was presented in Sections 14.5 and 15.5.
- H1: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) satisfies the homogeneous wave equation,$$\square H_{\;\;\,\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime) + 2R_{\gamma \;\,\delta}^{\;\alpha \;\,\beta}(x)H_{\;\;\gamma \prime \delta \prime}^{\gamma \delta}(x,x\prime) = 0;$$(16.24)
- H2: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) is symmetric in its indices and arguments,$${H_{\gamma \prime \delta \prime \alpha \beta}}(x\prime ,x) = {H_{\alpha \beta \gamma \prime \delta \prime}}(x,x\prime);$$(16.25)
- H3: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,$$H_{\;\;\,\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime) = G_{+ \;\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(16.26)
- H4: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) agrees with the advanced Green’s function if x is in the chronological past of x′,$$H_{\;\;\,\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime) = G_{- \;\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ -}(x\prime)$$(16.27)
- S1: \(G_{{\rm{S}}\,\,\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\) satisfies the inhomogeneous wave equation,$$\square G_{{\rm{S}}\;\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime) + 2R_{\gamma \;\,\delta}^{\;\alpha \;\beta}(x)G_{{\rm{S}}\;\,\gamma \prime \delta \prime}^{\;\gamma \delta}(x,x\prime) = - 4\pi g_{\;\gamma \prime}^{(\alpha}(x,x\prime)g_{\;\delta \prime}^{\beta)}(x,x\prime){\delta _4}(x,x\prime);$$(16.29)
- S2: \(G_{{\rm{S}}\,\,\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\) is symmetric in its indices and arguments,$$G_{\gamma \prime \delta \prime \alpha \beta}^{\rm{S}}(x\prime ,x) = G_{\alpha \beta \gamma \prime \delta \prime}^{\rm{S}}(x,x\prime);$$(16.30)
- S3: \(G_{{\rm{S}}\,\,\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\) vanishes if x is in the chronological past or future of x′,$$G_{{\rm{S}}\;\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ \pm}(x\prime)$$(16.31)
- R1: \(G_{{\rm{R}}\;\,\gamma \prime\delta \prime}^{\;\,\alpha \beta}(x,x\prime)\) satisfies the homogeneous wave equation,$$\square G_{{\rm{R}}\;\,\,\gamma \prime \delta \prime}^{\;\,\alpha \beta}(x,x\prime) + 2R_{\gamma \;\,\delta}^{\;\alpha \;\,\beta}(x)G_{{\rm{R}}\;\,\gamma \prime \delta \prime}^{\;\,\gamma \delta}(x,x\prime) = 0;$$(16.33)
- R2: \(G_{{\rm{R}}\;\,\gamma \prime\delta \prime}^{\;\,\alpha \beta}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,$$G_{{\rm{R}}\;\,\gamma \prime \delta \prime}^{\;\,\alpha \beta}(x,x\prime) = G_{+ \;\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(16.34)
- R3: \(G_{{\rm{R}}\;\,\gamma \prime\delta \prime}^{\;\,\alpha \beta}(x,x\prime)\) vanishes if x is in the chronological past of x′,$$G_{{\rm{R}}\;\,\,\gamma \prime \delta \prime}^{\;\,\alpha \beta}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ -}(x\prime)$$(16.35)
20 Part IV: Motion of Point Particles
21 Motion of a scalar charge
21.1 Dynamics of a point scalar charge
21.2 Retarded potential near the world line
21.3 Field of a scalar charge in retarded coordinates
21.4 Field of a scalar charge in Fermi normal coordinates
The gradient of the scalar potential can also be expressed in the Fermi normal coordinates of Section 9. To effect this translation we make x:= z(t) the new reference point on the world line. We resume here the notation of Section 11 and assign indices \(\bar \alpha ,\,\,\bar \beta ,\,\, \ldots\) to tensors at \(\bar{x}\). The Fermi normal coordinates are denoted (t, s, ω^{ a }), and we let \((\bar{e}^\alpha_0, \bar{e}^\alpha_a)\) be the tetrad at x that is obtained by parallel transport of \(({u^{\bar \alpha}},e_a^{\bar \alpha})\) on the spacelike geodesic that links x to \(\bar{x}\).
21.5 Singular and regular fields
We recall first that a relation between retarded and advanced times was worked out in Eq. (11.12), that an expression for the advanced distance was displayed in Eq. (11.13), and that Eqs. (11.14) and (11.15) give expansions for ∂_{ α }v and ∂_{ α }r_{adv}, respectively.
The tensors appearing in Eq. (17.42) all refer to the retarded point x′ := z(u), which now stands for an arbitrary point on the world line γ.
21.6 Equations of motion
The retarded field Φ_{ α }(x) of a point scalar charge is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and affect its motion. The field’s singularity structure was analyzed in Sections 17.3 and 17.4, and in Section 17.5 it was shown to originate from the singular field \(\Phi _\alpha ^{\rm{S}}(x)\); the regular field \(\Phi _\alpha ^{\rm{R}}(x) = {\Phi _\alpha}(x) - \Phi _\alpha ^{\rm{S}}(x)\) was then shown to be regular on the world line.
We must confess that the derivation of the equations of motion outlined above returns the wrong expression for the self-energy of a spherical shell of scalar charge. We obtained δm = q^{2}/(3s_{0}), while the correct expression is δm = q^{2}/(2s_{0}); we are wrong by a factor of 2/3. We believe that this discrepancy originates in a previously stated assumption, that the field on the shell (as produced by the shell itself) is equal to the field of a point particle evaluated at s = s_{0}. We believe that this assumption is in fact wrong, and that a calculation of the field actually produced by a spherical shell would return the correct expression for δm. We also believe, however, that except for the diverging terms that determine δm, the difference between the shell’s field and the particle’s field should vanish in the limit s_{0} → 0. Our conclusion is therefore that while our expression for δm is admittedly incorrect, the statement of the equations of motion is reliable.
The equations of motion displayed in Eqs. (17.47) and (17.48) are third-order differential equations for the functions z^{ μ }(τ). It is well known that such a system of equations admits many unphysical solutions, such as runaway situations in which the particle’s acceleration increases exponentially with τ, even in the absence of any external force [56, 101]. And indeed, our equations of motion do not yet incorporate an external force which presumably is mostly responsible for the particle’s acceleration. Both defects can be cured in one stroke. We shall take the point of view, the only admissible one in a classical treatment, that a point particle is merely an idealization for an extended object whose internal structure — the details of its charge distribution — can be considered to be irrelevant. This view automatically implies that our equations are meant to provide only an approximate description of the object’s motion. It can then be shown [112, 70] that within the context of this approximation, it is consistent to replace, on the right-hand side of the equations of motion, any occurrence of the acceleration vector by \(f_{{\rm{ext}}}^\mu/m\), where \(f_{{\rm{ext}}}^\mu\) is the external force acting on the particle. Because \(f_{{\rm{ext}}}^\mu\) is a prescribed quantity, differentiation of the external force does not produce higher derivatives of the functions z^{ μ }(τ), and the equations of motion are properly of the second order.
22 Motion of an electric charge
22.1 Dynamics of a point electric charge
22.2 Retarded potential near the world line
22.3 Electromagnetic field in retarded coordinates
22.4 Electromagnetic field in Fermi normal coordinates
We now wish to express the electromagnetic field in the Fermi normal coordinates of Section 9; as before those will be denoted (t, s, ω^{ a }). The translation will be carried out as in Section 17.4, and we will decompose the field in the tetrad \((\bar e_0^\alpha ,\bar e_a^\alpha)\) that is obtained by parallel transport of \(({u^{\bar \alpha}},e_a^{\bar \alpha})\) on the spacelike geodesic that links x to the simultaneous point x:= z(t).
22.5 Singular and regular fields
The tensors appearing in Eq. (18.41) all refer to the retarded point x′ := z(u), which now stands for an arbitrary point on the world line γ.
22.6 Equations of motion
The retarded field F_{ αβ } of a point electric charge is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and exert a force. The field’s singularity structure was analyzed in Sections 18.3 and 18.4, and in Section 18.5 it was shown to originate from the singular field \(F_{\alpha \beta}^{\rm{S}}\); the regular field \(F_{\alpha \beta}^{\rm{R}} = {F_{\alpha \beta}} - F_{\alpha \beta}^{\rm{S}}\) was then shown to be regular on the world line.
We must confess, as we did in the case of the scalar self-force, that the derivation of the equations of motion outlined above returns the wrong expression for the self-energy of a spherical shell of electric charge. We obtained δm = 2e^{2}/(3s_{0}), while the correct expression is δm = e^{2}/(2s_{0}); we are wrong by a factor of 4/3. As before we believe that this discrepancy originates in a previously stated assumption, that the field on the shell (as produced by the shell itself) is equal to the field of a point particle evaluated at s = s_{0}. We believe that this assumption is in fact wrong, and that a calculation of the field actually produced by a spherical shell would return the correct expression for δm. We also believe, however, that except for the diverging terms that determine δm, the difference between the shell’s field and the particle’s field should vanish in the limit s_{0} → 0. Our conclusion is therefore that while our expression for δm is admittedly incorrect, the statement of the equations of motion is reliable.
23 Motion of a point mass
23.1 Dynamics of a point mass
23.1.1 Introduction
In this section we consider the motion of a point particle of mass m subjected to its own gravitational field in addition to an external field. The particle moves on a world line γ in a curved spacetime whose background metric g_{ αβ } is assumed to be a vacuum solution to the Einstein field equations. We shall suppose that m is small, so that the perturbation h_{ αβ } created by the particle can also be considered to be small. In the final analysis we shall find that h_{ αβ } obeys a linear wave equation in the background spacetime, and this linearization of the field equations will allow us to fit the problem of determining the motion of a point mass within the general framework developed in Sections 17 and 18. We shall find that γ is not a geodesic of the background spacetime because h_{ αβ } acts on the particle and produces an acceleration proportional to m; the motion is geodesic in the test-mass limit only.
While we can make the problem fit within the general framework, it is important to understand that the problem of motion in gravitation is conceptually very different from the versions encountered previously in the case of a scalar or electromagnetic field. In these cases, the field equations satisfied by the scalar potential Φ or the vector potential A^{ α } are fundamentally linear; in general relativity the field equations satisfied by h_{ αβ } are fundamentally nonlinear, and this makes a major impact on the formulation of the problem. (In all cases the coupled problem of determining the field and the motion of the particle is nonlinear.) Another difference resides with the fact that in the previous cases, the field equations and the law of motion could be formulated independently of each other (because the action functional could be varied independently with respect to the field and the world line); in general relativity the law of motion follows from energy-momentum conservation, which is itself a consequence of the field equations.
The dynamics of a point mass in general relativity must therefore be formulated with care. We shall describe a formal approach to this problem, based on the fiction that the spacetime of a point particle can be constructed exactly in general relativity. (This is indeed a fiction, because it is known [80] that the metric of a point particle, as described by a Dirac distribution on a world line, is much too singular to be defined as a distribution in spacetime. The construction, however, makes distributional sense at the level of the linearized theory.) The outcome of this approach will be an approximate formulation of the equations of motion that relies on a linearization of the field equations, and which turns out to be closely analogous to the scalar and electromagnetic cases encountered previously. We shall put the motion of a small mass on a much sounder foundation in Part V, where we take m to be a (small) extended body instead of a point particle.
23.1.2 Exact formulation
Let a point particle of mass m move on a world line γ in a curved spacetime with metric g_{ αβ }. This is the exact metric of the perturbed spacetime, and it depends on m as well as all other relevant parameters. At a later stage of the discussion g_{ αβ } will be expressed as sum of a “background” part g_{ αβ } that is independent of m, and a “perturbation” part h_{ αβ } that contains the dependence on m. The world line is described by relations z^{ μ }(λ) in which λ is an arbitrary parameter — this will later be identified with proper time τ in the background spacetime. We use sans-serif symbols to denote tensors that refer to the perturbed spacetime; tensors in the background spacetime will be denoted, as usual, by italic symbols.
23.1.3 Decomposition into background and perturbation
23.1.4 Field equations and conservation statement
23.1.5 Integration of the field equations
The split of the Einstein field equations into a wave equation and a gauge condition directly tied to the conservation of the effective energy-momentum tensor is a most powerful tool, because it allows us to disentangle the problems of obtaining h_{ αβ } and determining the motion of the particle. This comes about because the wave equation can be solved first, independently of the gauge condition, for a particle moving on an arbitrary world line γ; the world line is determined next, by imposing the Lorenz gauge condition on the solution to the wave equation. More precisely stated, the source term \(T^{\alpha\beta}_{\rm{eff}}\) for the wave equation can be evaluated for any world line γ, without demanding that the effective energy-momentum tensor be conserved, and without demanding that γ be a geodesic of the perturbed spacetime. Solving the wave equation then returns h_{ αβ }[γ] as a functional of the arbitrary world line, and the metric is not yet fully specified. Because imposing the Lorenz gauge condition is equivalent to imposing conservation of the effective energy-momentum tensor, inserting h_{ αβ }[γ] within Eq. (19.15) finally determines γ, and forces it to be a geodesic of the perturbed spacetime. At this stage the full set of Einstein field equations is accounted for, and the metric is fully specified as a tensor field in spacetime. The split of the field equations into a wave equation and a gauge condition is key to the formulation of the gravitational self-force; in this specific context the Lorenz gauge is conferred a preferred status among all choices of gauge.
An important question to be addressed is how the wave equation is to be integrated. A method of principle, based on the assumed smallness of m and h_{ αβ }, is suggested by post-Minkowskian theory [180, 26]. One proceeds by iterations. In the first iterative stage, one fixes γ and substitutes \(h_0^{\alpha \beta} = 0\) within \(T_{{\rm{eff}}}^{\alpha \beta}\); evaluation of the integral in Eq. (19.17) returns the first-order approximation \(h_1^{\alpha \beta}[\gamma ] = O(m)\) for the perturbation. In the second stage \(h_1^{\alpha \beta}\) is inserted within \(T_{{\rm{eff}}}^{\alpha \beta}\) and Eq. (19.17) returns the second-order approximation \(h_2^{\alpha \beta}[\gamma ] = O(m,{m^2})\) for the perturbation. Assuming that this procedure can be repeated at will and produces an adequate asymptotic series for the exact perturbation, the iterations are stopped when the n^{th}-order approximation \(h_n^{\alpha \beta}[\gamma ] = O(m,{m^2}, \cdots ,{m^n})\) is deemed to be sufficiently accurate. The world line is then determined, to order m^{ n }, by subjecting the approximated field to the Lorenz gauge condition. It is to be noted that the procedure necessarily produces an approximation of the field, and an approximation of the motion, because the number of iterations is necessarily finite. This is the only source of approximation in our formulation of the dynamics of a point mass.
23.1.6 Equations of motion
23.1.7 Implementation to first order in m
While our formulation of the dynamics of a point mass is in principle exact, any practical implementation will rely on an approximation method. As we saw previously, the most immediate source of approximation concerns the number of iterations involved in the integration of the wave equation. Here we perform a single iteration and obtain the perturbation h_{ αβ } and the equations of motion to first order in the mass m.
It should be clear that Eq. (19.25) is valid only in a formal sense, because the potentials obtained from Eqs. (19.23) diverge on the world line. To make sense of these equations we will proceed as in Sections 17 and 18 with a careful analysis of the field’s singularity structure; regularization will produce a well-defined version of Eq. (19.25). Our formulation of the dynamics of a point mass makes it clear that a proper implementation requires that the wave equation of Eq. (19.22) and the equations of motion of Eq. (19.25) be integrated simultaneously, in a self-consistent manner.
23.1.8 Failure of a strictly linearized formulation
In the preceding discussion we started off with an exact formulation of the problem of motion for a small mass m in a background spacetime with metric g_{ αβ }, but eventually boiled it down to an implementation accurate to first order in m. Would it not be simpler and more expedient to formulate the problem directly to first order? The answer is a resounding no: By doing so we would be driven toward a grave inconsistency; the nonlinear formulation is absolutely necessary if one wishes to contemplate a self-consistent integration of Eqs. (19.22) and (19.25).
A strictly linearized formulation of the problem would be based on the field equations δG^{ αβ } = 8πT^{ αβ }, where T^{ αβ } is the energy-momentum tensor of Eq. (19.21). The Bianchi-like identities δG^{ αβ }_{;β} = 0 dictate that T^{ αβ } must be conserved in the background spacetime, and a calculation identical to the one leading to Eq. (19.5) would reveal that the particle’s motion must be geodesic in the background spacetime. In the strictly linearized formulation, therefore, the gravitational potentials of Eq. (19.23) must be sourced by a particle moving on a geodesic, and there is no opportunity for these potentials to exert a self-force. To get the self-force, one must provide a formulation that extends beyond linear order. To be sure, one could persist in adopting the linearized formulation and “save the phenomenon” by relaxing the conservation equation. In practice this could be done by adopting the solutions of Eq. (19.23), demanding that the motion be geodesic in the perturbed spacetime, and relaxing the linearized gauge condition to γ^{ αβ }_{;β} = O(m^{2}). While this prescription would produce the correct answer, it is largely ad hoc and does not come with a clear justification. Our exact formulation provides much more control, at least in a formal sense. We shall do even better in Part V.
An alternative formulation of the problem provided by Gralla and Wald [83] avoids the inconsistency by refraining from performing a self-consistent integration of Eqs. (19.22) and (19.25). Instead of an expansion of the acceleration in powers of m, their approach is based on an expansion of the world line itself, and it returns the equations of motion for a deviation vector which describes the offset of the true world line relative to a reference geodesic. While this approach is mathematically sound, it eventually breaks down as the deviation vector becomes large, and it does not provide a justification of the self-consistent treatment of the equations.
23.1.9 Vacuum background spacetime
23.2 Retarded potentials near the world line
In the following subsections we shall refer to γ_{ αβ }(x) as the gravitational potentials at x produced by a particle of mass m moving on the world line γ, and to γ_{αβ;γ}(x) as the gravitational field at x. To compute this is our next task.
23.3 Gravitational field in retarded coordinates
23.4 Gravitational field in Fermi normal coordinates
The translation of the results contained in Eqs. (19.39)–(19.44) into the Fermi normal coordinates of Section 9 proceeds as in Sections 17.4 and 18.4, but is simplified by setting a_{ a } = ȧ_{0} = ȧ_{ a } = 0 in Eqs. (11.7), (11.8), (11.4), (11.5), and (11.6) that relate the Fermi normal coordinates (t, s, ω^{ a }) to the retarded coordinates. We recall that the Fermi normal coordinates refer to a point x:= z(t) on the world line that is linked to x by a spacelike geodesic that intersects γ orthogonally.
23.5 Singular and regular fields
23.6 Equations of motion
The retarded gravitational field γ_{αβ;γ} of a point particle is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and influence its motion. The field’s singularity structure was analyzed in Sections 19.3 and 19.4, and in Section 19.5 it was shown to originate from the singular field \(\gamma _{\alpha \beta ;\gamma}^{\rm{S}}\); the regular field \(\gamma _{\alpha \beta ;\gamma}^{\rm{R}}\) was then shown to be regular on the world line.
Eq. (19.84) was first derived by Yasushi Mino, Misao Sasaki, and Takahiro Tanaka in 1997 [130]. (An incomplete treatment had been given previously by Morette-DeWitt and Ging [133].) An alternative derivation was then produced, also in 1997, by Theodore C. Quinn and Robert M. Wald [150]. These equations are now known as the MiSaTaQuWa equations of motion, and other derivations [83, 144], based on an extended-body approach, will be reviewed below in Part V. It should be noted that Eq. (19.84) is formally equivalent to the statement that the point particle moves on a geodesic in a spacetime with metric \({g_{\alpha \beta}} + h_{\alpha \beta}^{\rm{R}}\), where \(h_{\alpha \beta}^{\rm{R}}\) is the regular metric perturbation obtained by trace-reversal of the potentials \(\gamma _{\alpha \beta}^{\rm{R}}: = {\gamma _{\alpha \beta}} - \gamma _{\alpha \beta}^{\rm{S}}\); this perturbed metric is regular on the world line, and it is a solution to the vacuum field equations. This elegant interpretation of the MiSaTaQuWa equations was proposed in 2003 by Steven Detweiler and Bernard F. Whiting [53]. Quinn and Wald [151] have shown that under some conditions, the total work done by the gravitational self-force is equal to the energy radiated (in gravitational waves) by the particle.
24 Part V: Motion of a Small Body
25 Point-particle limits and matched asymptotic expansions
The expansion presented in the previous section is based on an exact point-particle source. But in the full, nonlinear theory, no distributional solution would exist for such a source [80]. Although the expansion nevertheless yields a well-behaved linear approximation, it is ill-behaved beyond that order, since the second- and higher-order Einstein tensors will contain products of delta functions, which have no meaning as distributions. It may be possible to overcome this limitation using more advanced methods such as Colombeau algebras [164], which allow for the multiplication of distributions, but little work has been done to that end. Instead, the common approach, and the one we shall pursue here, has been to abandon the fiction of a point particle in favor of considering an asymptotically small body. As we shall see, we can readily generalize the self-consistent expansion scheme to this case. Furthermore, we shall find that the results of the previous section are justified by this approach: at linear order, the metric perturbation due to an asymptotically small body is precisely that of a point particle moving on a world line with an acceleration given by the MiSaTaQuWa equation (plus higher-order corrections).
In order for the body to be considered “small”, its mass and size must be much smaller than all external lengthscales. We denote these external scales collectively as ℛ, which we may define to be the radius of curvature of the spacetime (were the small body removed from it) in the region in which we seek an approximation. Given this definition, a typical component of the spacetime’s Riemann tensor is equal to 1/ℛ^{2} up to a numerical factor of order unity. Now, we consider a family of metrics g_{ αβ }(ε) containing a body whose mass scales as ε in the limit ε → 0; that is, ε ∼ m/ℛ. If each member of the family is to contain a body of the same type, then the size of the body must also approach zero with ε. The precise scaling of size with ε is determined by the type of body, but it is not generally relevant. What is relevant is the “gravitational size” — the length scale determining the metric outside the body — and this size always scales linearly with the mass. If the body is compact, as is a neutron star or a black hole, then its gravitational size is also its actual linear size. In what remains, we assume that all lengths have been scaled by ℛ, such that we can write, for example, m ≪ 1. Our goal is to determine the metric perturbation and the equation of motion produced by the body in this limit.
Point-particle limits such as this have been used to derive equations of motion many times in the past, including in derivations of geodesic motion at leading order [100, 79, 64] and in constructing post-Newtonian limits [74]. Perhaps the most obvious means of approaching the problem is to first work nonperturbatively, with a body of arbitrary size, and then take the limit. Using this approach (though with some restrictions on the body’s size and compactness) and generalized definitions of momenta, Harte has calculated the self-force in the case of scalar [89] and electromagnetic [90] charge distributions in fixed backgrounds, following the earlier work of Dixon [57, 58, 59]. However, while this approach is conceptually compelling, at this stage it applies only to material bodies, not black holes, and has not yet been presented as part of a systematic expansion of the Einstein equation. Here, we focus instead on a more general method.
Alternatively, one could take the opposite approach, essentially taking the limit first and then trying to recover the higher-order effects, by treating the body as an effective point particle at leading order, with finite size effects introduced as higher-order effective fields, as done by Galley and Hu [76, 75]. However, while this approach is computationally efficient, allowing one to perform high-order calculations with (relative) ease, it requires methods such as dimensional regularization and mass renormalization in order to arrive at meaningful results. Because of these undesirable requirements, we will not consider it here.
In the approach we review, we make use of the method of matched asymptotic expansions [46, 47, 102, 103, 174, 130, 129, 2, 142, 49, 74, 170, 83, 82, 144, 145]. Broadly speaking, this method consists of constructing two different asymptotic expansions, each valid in a specific region, and combining them to form a global expansion. In the present context, the method begins with two types of point-particle limits: an outer limit, in which ε → 0 at fixed coordinate values (we will slightly modify this in a moment); and an inner limit, in which ε → 0 at fixed values of \(\tilde R := R/\varepsilon\), where R is a measure of radial distance from the body. In the outer limit, the body shrinks toward zero size as all other distances remain roughly constant; in the inner limit, the small body keeps a constant size while all other distances blow up toward infinity. Thus, the inner limit serves to “zoom in” on a small region around the body. The outer limit can be expected to be valid in regions where R ∼ 1, while the inner limit can be expected to be valid in regions where \(\tilde{R} \sim 1\) (or R ∼ ε), though both of these regions can be extended into larger domains.
More precisely, consider an exact solution g_{ αβ } on a manifold \({{\mathcal M}_\varepsilon}\) with two coordinate systems: a local coordinate system X^{ α } = (T, R, Θ^{ A }) that is centered (in some approximate sense) on the small body, and a global coordinate system x^{ α }. For example, in an extreme-mass-ratio inspiral, the local coordinates might be the Schwarzschild-type coordinates of the small body, and the global coordinates might be the Boyer-Lindquist coordinates of the supermassive Kerr black hole. In the outer limit, we expand g_{ αβ } for small ε while holding x^{ α } fixed. The leading-order solution in this case is the background metric g_{ αβ } on a manifold \({{\mathcal M}_E}\); this is the external spacetime, which contains no small body. It might, for example, be the spacetime of the supermassive black hole. In the inner limit, we expand g_{ αβ } for small ε while holding \((T,\tilde {R},{\Theta ^{A}})\) fixed. The leading-order solution in this case is the metric \(g_{\alpha \beta}^{{\rm{body}}}\) on a manifold \({{\mathcal M}_I}\); this is the spacetime of the small body if it were isolated (though it may include slow evolution due to its interaction with the external spacetime — this will be discussed below). Note that \({{\mathcal M}_E}\) and \({{\mathcal M}_I}\) generically differ: in an extreme-mass-ratio inspiral, for example, if the small body is a black hole, then \({{\mathcal M}_I}\) will contain a spacelike singularity in the black hole’s interior, while \({{\mathcal M}_E}\) will be smooth at the “position” where the small black hole would be. What we are interested in is that “position” — the world line in the smooth external spacetime \({{\mathcal M}_E}\) that represents the motion of the small body. Note that this world line generically appears only in the external spacetime, rather than as a curve in the exact spacetime \({{\rm{g}}_{\alpha \beta}},{{\mathcal M}_\varepsilon})\); in fact, if the small body is a black hole, then obviously no such curve exists.
Determining this world line presents a fundamental problem. In the outer limit, the body vanishes at ε = 0, leaving only a remnant, ε-independent curve in \({{\mathcal M}_E}\). (Outside any small body, the metric will contain terms such as m/R, such that in the limit m → 0, the limit exists everywhere except at R = 0, which leaves a removable discontinuity in the external spacetime; the removal of this discontinuity defines the remnant world line of the small body.) But the true motion of the body will generically be ε-dependent. If we begin with the remnant world line and correct it with the effects of the self-force, for example, then the corrections must be small: they are small deviation vectors defined on the remnant world line. Put another way, if we expand g_{ αβ } in powers of ε, then all functions in it must similarly be expanded, including any representation of the motion, and in particular, any representative world line. We would then have a representation of the form \({z^\alpha}(t,\varepsilon) = z_{(0)}^\alpha (t) + \varepsilon z_{(1)}^\alpha (t) + \ldots\) where \(z_{(1)}^\alpha (t)\) is a vector defined on the remnant curve described by \(z_{(0)}^\alpha (t)\). The remnant curve would be a geodesic, and the small corrections would incorporate the self-force and finite-size effects [83] (see also [102]). However, because the body will generically drift away from any such geodesic, the small corrections will generically grow large with time, leading to the failure of the regular expansion. So we will modify this approach by performing a self-consistent expansion in the outer limit, following the same scheme as presented in the point-particle case. Refs. [144, 145, 143] contain far more detailed discussions of these points.
In this calculation, the structure of the body is left unspecified. Our only condition is that part of the buffer region must lie outside the body, because we wish to solve the Einstein field equations in vacuum. This requires the body to be sufficiently compact. For example, our calculation would fail for a diffuse body such as our Sun; likewise, it would fail if a body became tidally disrupted. Although we will detail only the case of an uncharged body, the same techniques would apply to charged bodies; Gralla et al. [82] have recently performed a similar calculation for the electromagnetic self-force on an asymptotically small body in a flat background spacetime. Using very different methods, Futamase et al. [73] have calculated equations of motion for an asymptotically small charged black hole.
The structure of our discussion is as follows: In Section 21, we present the self-consistent expansion of the Einstein equation. Next, in Section 22, we solve the equations in the buffer region up to second order in the outer expansion. Last, in Section 23, we discuss the global solution in the outer expansion and show that it is that of a point particle at first order. Over the course of this calculation, we will take the opportunity to incorporate several details that we could have accounted for in the point-particle case but opted to neglect for simplicity: an explicit expansion of the acceleration vector that makes the self-consistent expansion properly systematic, and a finite time domain that accounts for the fact that large errors eventually accumulate if the approximation is truncated at any finite order. For more formal discussions of matched asymptotic expansions in general relativity, see Refs. [104, 145]; the latter reference, in particular, discusses the method as it pertains to the motion of small bodies. For background on the use of matched asymptotic expansions in applied mathematics, see Refs. [63, 96, 109, 111, 178]; the text by Eckhaus [63] provides the most rigorous treatment.
26 Self-consistent expansion
26.1 Introduction
We wish to represent the motion of the body through the external background spacetime \(({g_{\alpha \beta}},{{\mathcal M}_\varepsilon})\), rather than through the exact spacetime \(({{\rm{g}}_{\alpha \beta}},{{\mathcal M}_\varepsilon})\). In order to achieve this, we begin by surrounding the body with a (hollow, three-dimensional) world tube Γ embedded in the buffer region. We define the tube to be a surface of constant radius \(s = {\mathcal R}(\varepsilon)\) in Fermi normal coordinates centered on a world line \(\gamma \subset {{\mathcal M}_E}\), though the exact definition of the tube is immaterial. Since there exists a diffeomorphism between \({{\mathcal M}_E}\) and \({{\mathcal M}_I}\) in the buffer region, this defines a tube \({\Gamma _I} \subset {{\mathcal M}_I}\). Now, the problem is the following: what equation of motion must γ satisfy in order for Γ_{ I } to be “centered” about the body?
How shall we determine if the body lies at the centre of the tube’s interior? Since the tube is close to the small body (relative to all external length scales), the metric on the tube is primarily determined by the small body’s structure. Recall that the buffer region corresponds to an asymptotically large spatial distance in the inner expansion. Hence, on the tube, we can construct a multipole expansion of the body’s field, with the form ∑R^{−n} (or ∑s^{−n} — we will assume s ∼ R in the buffer region). Although alternative definitions could be used, we define the tube to be centered about the body if the mass dipole moment vanishes in this expansion. Note that this is the typical approach in general relativity: Whereas in Newtonian mechanics one directly finds the equation of motion for the centre of mass of a body, in general relativity one typically seeks a world line about which the mass dipole of the body vanishes (or an equation of motion for the mass dipole relative to a given nearby world line) [66, 152, 83, 144]. This definition of the world line is sufficiently general to apply to a black hole. If the body is material, one could instead imagine a centre-of-mass world line that lies in the interior of the body in the exact spacetime. This world line would then be the basis of our self-consistent expansion. We use our more general definition to cover both cases. See Ref. [173] and references therein for discussion of multipole expansions in general relativity, see Refs. [173, 174] for discussions of mass-centered coordinates in the buffer region, and see, e.g., Refs. [160, 65] for alternative definitions of centre of mass in general relativity.
In the remainder of this section, we present a sequence of perturbation equations that arise in this expansion scheme, along with a complementary sequence for the inner expansion.
26.2 Field equations in outer expansion
Historically, in derivations of the self-force, solutions to the perturbative field equations were taken to be global in time, with tail integrals extended to negative infinity, as we wrote them in the preceding sections. But as was first noted in Ref. [144], because the self-force drives long-term, cumulative changes, any approximation truncated at a given order will be accurate to that order only for a finite time; and this necessites working in a finite region such as Ω. This is also true in the case of point charges and masses. For simplicity, we neglected this detail in the preceding sections, but for completeness, we account for it here.
26.2.1 Field equations
One should note several important properties of these integral representations: First, x must lie in the interior of Ω; an alternative expression must be derived if x lies on the boundary [153]. Second, the integral over the boundary is, in each case, a homogeneous solution to the wave equation, while the integral over the volume is an inhomogeneous solution. Third, if the field at the boundary satisfies the Lorenz gauge condition, then by virtue of the wave equation, it satisfies the gauge condition everywhere; hence, imposing the gauge condition to some order in the buffer region ensures that it is imposed to the same order everywhere.
While the integral representation is satisfied by any solution to the associated wave equation, it does not provide a solution. That is, one cannot prescribe arbitrary boundary values on Γ and then arrive at a solution. The reason is that the tube is a timelike boundary, which means that field data on it can propagate forward in time and interfere with the data at a later time. However, by applying the wave operator E_{ αβ } onto Eq. (21.8), we see that the integral representation of \(h_{\alpha \beta}^{(n)}\) is guaranteed to satisfy the wave equation at each point x ∈ Ω. In other words, the problem arises not in satisfying the wave equation in a pointwise sense, but in simultaneously satisfying the boundary conditions. But since the tube is chosen to lie in the buffer region, these boundary conditions can be supplied by the buffer-region expansion. And as we will discuss in Section 23, because of the asymptotic smallness of the tube, the pieces of the buffer-region expansion diverging as s^{−n} are sufficient boundary data to fully determine the global solution.
26.2.2 Gauge transformations and the Lorenz condition
The outer expansion is defined not only by holding x^{ α } fixed, but also by demanding that the mass dipole of the body vanishes when calculated in coordinates centered on γ. If we perform a gauge transformation generated by a vector ξ^{(1)α}(x; γ), then the mass dipole will no longer vanish in those coordinates. Hence, a new world line γ′ must be constructed, such that the mass dipole vanishes when calculated in coordinates centered on that new world line. In other words, in the outer expansion we have the usual gauge freedom of regular perturbation theory, so long as the world line is appropriately transformed as well: (h_{ αβ }, γ) → (h′_{ αβ }, γ′). The transformation law for the world line was first derived by Barack and Ori [17]; it was displayed in Eq. (1.49), and it will be worked out again in Section 22.6.
26.3 Field equations in inner expansion
For the inner expansion, we assume the existence of some local polar coordinates X^{ α } = (T, R, Θ^{ A }), such that the metric can be expanded for ε → 0 while holding fixed \(\tilde R: = R/\varepsilon ,{\Theta ^A}\), and T; to relate the inner and outer expansions, we assume R ∼ s, but otherwise leave the inner expansion completely general.
Since we are interested in the inner expansion only insofar as it informs the outer expansion, we shall not seek to explicitly solve the perturbative Einstein equation in the inner expansion. See Ref. [144] for the forms of the equations and an example of an explicit solution in the case of a perturbed black hole.
27 General expansion in the buffer region
We now seek the general solution to the equations of the outer expansion in the buffer region. To perform the expansion, we adopt Fermi coordinates centered about γ and expand for small s. In solving the first-order equations, we will determine a^{(0)μ}; in solving the second-order equations, we will determine a^{(1)μ}, including the self-force on the body. Although we perform this calculation in the Lorenz gauge, the choice of gauge is not essential for our purposes here — the essential aspect is our assumed expansion of the acceleration of the world line γ.
27.1 Metric expansions
One can make use of this fact by first determining the inner and outer expansions as fully as possible, then fixing any unknown functions in them by matching them term by term in the buffer region; this was the route taken in, e.g., Refs. [130, 142, 49, 170]. However, such an approach is complicated by the subtleties of matching in a diffeomorphism-invariant theory, where the inner and outer expansions are generically in different coordinate systems. See Ref. [145] for an analysis of the limitations of this approach as it has typically been implemented. Alternatively, one can take the opposite approach, working in the buffer region first, constraining the forms of the two expansions by making use of their matching, then using the buffer-region information to construct a global solution; this was the route taken in, e.g., Refs. [102, 83, 144]. In general, some mixture of these two approaches can be taken. Our calculation follows Ref. [144]. The only information we take from the inner expansion is its general form, which is characterized by the multipole moments of the body. From this information, we determine the external expansion, and thence the equation of motion of the world line.
Over the course of our calculation, we will find that the external metric perturbation in the buffer region is expressed as the sum of two solutions: one that formally diverges at s = 0 and is entirely determined from a combination of (i) the multipole moments of the internal background metric \(g_{\alpha \beta}^{{\rm{body}}}\), (ii) the Riemann tensor of the external background g_{ αβ }, and (iii) the acceleration of the world line γ; and a second solution that is formally regular at s = 0 and depends on the past history of the body and the initial conditions of the field. At leading order, these two solutions are identified as the Detweiler-Whiting singular and regular fields \(h_{\alpha \beta}^{\rm{S}}\) and \(h_{\alpha \beta}^{\rm{R}}\), respectively, and the self-force is determined entirely by \(h_{\alpha \beta}^{\rm{R}}\). Along with the self-force, the acceleration of the world line includes the Papapetrou spin force [138]. This calculation leaves us with the self-force in terms of the the metric perturbation in the neighbourhood of the body. In Section 23, we use the local information from the buffer region to construct a global solution for the metric perturbation, completing the solution of the problem.
27.2 The form of the expansion
Before proceeding, we define some notation. We use the multi-index notation \({\omega ^L}: = {\omega ^{{i_1}}} \cdots {\omega ^{{i_\ell}}} := {\omega ^{{i_1} \cdots {i_\ell}}}\). Angular brackets denote the STF combination of the enclosed indices, and a tensor bearing a hat is an STF tensor. To accommodate this, we now write the Fermi spatial coordinates as x^{ a }, instead of \({\hat x^a}\) as they were written in previous sections. Finally, we define the one-forms t_{ α } := ∂_{ α }t and \(x_\alpha ^a := {\partial _\alpha}{x^a}\).
One should note that the coordinate transformation x^{ α }(t, x^{ a }) between Fermi coordinates and the global coordinates is ε-dependent, since Fermi coordinates are tethered to an ε-dependent world line. If one were using a regular expansion, then this coordinate transformation would devolve into a background coordinate transformation to a Fermi coordinate system centered on a geodesic world line, combined with a gauge transformation to account for the ε-dependence. But in the self-consistent expansion, the transformation is purely a background transformation, because the ε-dependence in it is reducible to that of the fixed world line.
Because the dependence on ε in the coordinate transformation cannot be reduced to a gauge transformation, in Fermi coordinates the components g_{ αβ } of the background metric become ε-dependent. This dependence takes the explicit form of factors of the acceleration a^{ μ }(t, ε) and its derivatives, for which we have assumed the expansion a^{ i }(t, ε) = a^{(0)i}(t) + a^{(1)i}(t; γ) + O(ε^{2}). There is also an implicit dependence on ε in that the proper time t on the world line depends on ε if written as a function of the global coordinates; but this dependence can be ignored so long as we work consistently with Fermi coordinates.
In what follows, the reader may safely assume that all calculations are lengthy unless noted otherwise.
27.3 First-order solution in the buffer region
In principle, solving the first-order Einstein equation in the buffer region is straightforward. One need simply substitute the expansion of \(h_{\alpha \beta}^{(1)}\), given in Eq. (22.11), into the linearized wave equation (22.16) and the gauge condition (22.18). Equating powers of s in the resulting expansions then yields a sequence of equations that can be solved for successively higher-order terms in \(h_{\alpha \beta}^{(1)}\). Solving these equations consists primarily of expressing each quantity in its irreducible STF form, using the decompositions (B.3) and (B.7); since the terms in this STF decomposition are linearly independent, we can solve each equation term by term. This calculation is aided by the fact that \({\nabla _\alpha} = x_\alpha ^a{\partial _a} + O({s^0})\), so that, for example, the wave operator E_{ αβ } consists of a flat-space Laplacian ∂^{ a }∂_{ a } plus corrections of order 1/s. Appendix B also lists many useful identities, particularly \({\partial ^a}s = {\omega ^a} := {x^a}/s,\,\,{\omega ^a}{\partial _a}{\hat \omega ^L} = 0\), and the fact that \({\hat \omega ^L}\) is an eigenvector of the flat-space Laplacian: \({s^2}{\partial ^a}{\partial _a}{\hat \omega ^L} = - \ell (\ell + 1){\hat \omega ^L}\).
27.3.1 Summary of results
27.3.2 Order (1,-1)
27.3.3 Order (1,0)
27.3.4 Order (1,1)
27.3.5 First-order solution
To summarize the results of this section, we have \(h_{\alpha \beta}^{(1)} = {s^{- 1}}{\rm{}}h_{\alpha \beta}^{(1, - 1)} + h_{\alpha \beta}^{(1,0)} + sh_{\alpha \beta}^{(1,1)} + O({s^2})\), where \(h_{\alpha \beta}^{(1, - 1)}\) is given in Eq. (22.27), \(h_{\alpha \beta}^{(1,0)}\) is given in Eq. (22.29), and \(h_{\alpha \beta}^{(1,1)}\) is given in Eq. (22.38). In addition, we have determined that the ADM mass of the internal background spacetime is time-independent, and that the acceleration of the body’s world line vanishes at leading order.
27.4 Second-order solution in the buffer region
Though the calculations are much lengthier, solving the second-order Einstein equation in the buffer region is essentially no different from solving the first. We seek to solve the approximate wave equation (22.17), along with the gauge condition (22.19), for the second-order perturbation \(h_{B\alpha \beta}^{(2)} := h_{\alpha \beta}^{(2)}{\vert_{a = {a_0}}}\); doing so will also, more importantly, determine the acceleration \(a_{(1)}^\mu\). In this calculation, the acceleration is set to \({a^i} = a_{(0)}^i = 0\) everywhere except in the left-hand side of the gauge condition, \(L_\mu ^{(1)}[{h^{(1)}}]\), which is linear in \(a_{(1)}^\mu\).