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 selfforce that prevents the particle from moving on a geodesic of the background spacetime. The selfforce contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the selfforce 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 smooth field that is fully responsible for the selfforce. Because this field satisfies a homogeneous wave equation, it can be thought of as a free (radiative) field that interacts with the particle; it is this interaction that gives rise to the selfforce.
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 (Section 2). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle’s word line (Section 3). It continues with a thorough discussion of Green’s functions in curved spacetime (Section 4). The review concludes with a detailed derivation of each of the three equations of motion (Section 5).
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, and Poincaré, until Dirac [25] produced a proper relativistic derivation of the equations of motion in 1938. (The field’s early history is well related in [52].) In 1960 DeWitt and Brehme [24] generalized Dirac’s result to curved spacetimes, and their calculation was corrected by Hobbs [29] several years later. In 1997 the motion of a point mass in a curved background spacetime was investigated by Mino, Sasaki, and Tanaka [39], 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 [49] using an axiomatic approach. The case of a point scalar charge was finally considered by Quinn in 2000 [48], 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 nearzone field which acts directly on the particle and produces a selfforce. In curved spacetime the selfforce contains a radiationreaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angularmomentum transport. The selfforce 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 I derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely selfcontained, 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 (Section 2), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line (Section 3). One must next formulate a theory of Green’s functions in curved spacetimes (Section 4), and finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how they should act on the particle (Section 5). The review is very long, but the payoff, I hope, will be commensurate.
In this introductory section I 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.
My second key observation is that while the potential of Equation (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 Equation (1), whose source term is infinite on the world line. So while the wavezone 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’.
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 ≥ υ.
The physically relevant solution to Equation (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 smooth radiative part that produces the entire selfforce.
To decompose the retarded Green’s function into singular and radiative parts is not a straightforward task in curved spacetime. The flatspacetime definition for the singular Green’s function, Equation (9), cannot be adopted without modification: While the combination halfretarded plus halfadvanced Green’s functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Equation (13), this candidate for the singular Green’s function would produce a selfforce with an unacceptable dependence on the particle’s future history. For suppose that we made this choice. Then the radiative Green’s function would be given by the combination halfretarded minus halfadvanced Green’s functions, just as in flat spacetime. The resulting radiative potential would satisfy the homogeneous wave equation, and it would be smooth 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 radiative 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 radiative potential is generated by the complete history of the particle. A selfforce constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and radiative Green’s functions.
1.5 World line and retarded coordinates
To flesh out the ideas contained in the preceding Section 1.4 I 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 Section 1.6 I will display explicit expressions for the retarded, singular, and radiative 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 \({{\dot a}^\mu} \equiv D{a^\mu}/d\tau\).
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 twopoint function σ(x, z), known as Synge’s world function [55], 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 to x).
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 futuredirected null geodesic (this geodesic is directed from x′ to x); I 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′, α′, …; 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 \( {\sigma ^{{\alpha {\prime}}}}(x,{x{\prime}})\) 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 radiative 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 F_{a0} 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 (33) is the result that was first derived by DeWitt and Brehme [24] and later corrected by Hobbs [29]. (The original equation did not include the Riccitensor 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 Equation (33) reduces to Dirac’s result of Equation (5). In curved spacetime the selfforce 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 selfforce 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 Equation (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 Riccitensor term and the tail integral disappear, and Equation (40) takes the form of Equation (5) with q^{2}/(3m) replacing the factor of 2e^{2}/(3m). In this simple case Equation (41) reduces to dm/dτ = 0 and the mass is in fact a constant. This property remains true in a conformallyflat spacetime when the wave equation is conformally invariant (ξ = 1/6): In this case the Green’s function possesses only a lightcone part, and the righthand side of Equation (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 black hole, 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_{αβ}(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 Equation (48) were first derived by Mino, Sasaki, and Tanaka [39], and then reproduced with a different analysis by Quinn and Wald [49]. They are now known as the MiSaTaQuWa equations of motion. Detweiler and Whiting [23] have contributed the compelling interpretation that the motion is actually geodesic in a spacetime with metric \({{\rm{g}}_{\alpha \beta}} = {g_{\alpha \beta}} + {h_{\alpha \beta}}\). This metric satisfies the Einstein field equations in vacuum and is perfectly smooth on the world line. This spacetime can thus be viewed as the background spacetime perturbed by a free gravitational wave produced by the particle at an earlier stage of its history.
While Equation (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? Fortunately, Mino, Sasaki, and Tanaka [39] gave two different derivations of their result, and the second derivation was concerned not with the motion of a point mass, but with the motion of a small nonrotating black hole. In this alternative derivation of the MiSaTaQuWa equations, 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 Equation (48). This alternative derivation is entirely free of conceptual and technical pitfalls, and we conclude 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).
1.10 Evaluation of the selfforce
To concretely evaluate the selfforce, whether it be for a scalar charge, an electric charge, or a point mass, is a difficult undertaking. The difficulty resides mostly with the computation of the retarded Green’s function for the spacetime under consideration. Because Green’s functions are known for a very limited number of spacetimes, the selfforce has so far been evaluated in a rather limited number of situations.
The intriguing phenomenon of mass loss by a scalar charge was studied by Burko, Harte, and Poisson [15] in the simple context of a particle at rest in an expanding universe. For the special cases of a de Sitter cosmology, or a spatiallyflat matterdominated universe, the retarded Green’s function could be computed, and the action of the scalar field on the particle determined, without approximations. In de Sitter spacetime the particle is found to radiate all of its rest mass into monopole scalar waves. In the matterdominated cosmology this happens only if the charge of the particle is sufficiently large; for smaller charges the particle first loses a fraction of its mass, but then regains it eventually.
In recent years a large effort has been devoted to the elaboration of a practical method to compute the (scalar, electromagnetic, and gravitational) selfforce in the Schwarzschild spacetime. This work originated with Barack and Ori [7] and was pursued by Barack [2, 3] until it was put in its definitive form by Barack, Mino, Nakano, Ori, and Sasaki [6, 9, 11, 38]. The idea is to take advantage of the spherical symmetry of the Schwarzschild solution by decomposing the retarded Green’s function G_{+}(x, x′) into sphericalharmonic modes which can be computed individually. (To be concrete I refer here to the scalar case, but the method works just as well for the electromagnetic and gravitational cases.) From the modedecomposition of the Green’s function one obtains a modedecomposition of the field gradient ∇_{α}Φ, and from this subtracts a modedecomposition of the singular field ∇_{α}Φ_{S}, for which a local expression is known. This results in the radiative field ∇_{α}Φ_{R} decomposed into modes, and since this field is well behaved on the world line, it can be directly evaluated at the position of the particle by summing over all modes. (This sum converges because the radiative field is smooth; the mode sums for the retarded or singular fields, on the other hand, do not converge.) An extension of this method to the Kerr spacetime has recently been presented [44, 34, 10], and Mino [37] has devised a surprisingly simple prescription to calculate the timeaveraged evolution of a generic orbit around a Kerr black hole.
The modesum method was applied to a number of different situations. Burko computed the selfforce acting on an electric charge in circular motion in flat spacetime [12], as well as on a scalar and electric charge kept stationary in a Schwarzschild spacetime [14], in a spacetime that contains a spherical matter shell (Burko, Liu, and Soren [17]), and in a Kerr spacetime (Burko and Liu [16]). Burko also computed the scalar selfforce acting on a particle in circular motion around a Schwarzschild black hole [13], a calculation that was recently revisited by Detweiler, Messaritaki, and Whiting [21]. Barack and Burko considered the case of a particle falling radially into a Schwarzschild black hole, and evaluated the scalar selfforce acting on such a particle [4]; Lousto [33] and Barack and Lousto [5], on the other hand, calculated the gravitational selfforce.
1.11 Organization of this review
The main body of the review begins in Section 2 with a description of the general theory of bitensors, the name designating tensorial functions of two points in spacetime. I introduce Synge’s world function σ(x, x′) and its derivatives in Section 2.1, the parallel propagator \({g^\alpha}_{{\alpha {\prime}}}(x,{x{\prime}})\) in Section 2.3, and the van Vleck determinant Δ(x, x′) in Section 2.5. An important portion of the theory (covered in Sections 2.2 and 2.4) is concerned with the expansion of bitensors when x is very close to x′; expansions such as those displayed in Equations (23) and (24) are based on these techniques. The presentation in Section 2 borrows heavily from Synge’s book [55] and the article by DeWitt and Brehme [24]. These two sources use different conventions for the Riemann tensor, and I have adopted Synge’s conventions (which agree with those of Misner, Thorne, and Wheeler [40]). The reader is therefore warned that formulae derived in Section 2 may look superficially different from what can be found in DeWitt and Brehme.
In Section 3 I introduce a number of coordinate systems that play an important role in later parts of the review. As a warmup exercise I first construct (in Section 3.1) Riemann normal coordinates in a neighbourhood of a reference point x′. I then move on (in Section 3.2) to Fermi normal coordinates [36], 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 3.3. The relationship between Fermi and retarded coordinates is worked out in Section 3.4, which also locates the advanced point z(υ) associated with a field point x. The presentation in Section 3 borrows heavily from Synge’s book [55]. In fact, I am much indebted to Synge for initiating the construction of retarded coordinates in a neighbourhood of a world line. I have implemented his program quite differently (Synge was interested in a large neighbourhood of the world line in a weakly curved spacetime, while I am interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally his.
In Section 4 I review the theory of Green’s functions for (scalar, vectorial, and tensorial) wave equations in curved spacetime. I begin in Section 4.1 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 allimportant Hadamard decomposition [28] of the Green’s function into “lightcone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in Section 4.2 along with its restrictions on the past and future null cones of a reference point x′. The retarded, advanced, singular, and radiative Green’s functions for the scalar wave equation are introduced in Section 4.3. In Sections 4.4 and 4.5 I cover the vectorial and tensorial wave equations, respectively. The presentation in Section 4 is based partly on the paper by DeWitt and Brehme [24], but it is inspired mostly by Friedlander’s book [27]. The reader should be warned that in one important aspect, my notation differs from the notation of DeWitt and Brehme: While they denote the tail part of the Green’s function by −v(x, x′), I have taken the liberty of eliminating the silly minus sign and I call it instead +V(x, x′). The reader should also note that all my Green’s functions are normalized in the same way, with a factor of −4π multiplying a fourdimensional Dirac functional of the righthand side of the wave equation. (The gravitational Green’s function is sometimes normalized with a −16π on the righthand side.)
In Section 5 I compute the retarded, singular, and radiative fields associated with a point scalar charge (Section 5.1), a point electric charge (Section 5.2), and a point mass (Section 5.3). I provide two different derivations for each of the equations of motion. The first type of derivation was outlined previously: I follow Detweiler and Whiting [23] and postulate that only the radiative field exerts a force on the particle. In the second type of derivation I take guidance from Quinn and Wald [49] 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 restframe 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 Section 5.4 I present an alternative derivation of the MiSaTaQuWa equations of motion based on the method of matched asymptotic expansions [35, 31, 58, 19, 1, 20]; the derivation applies to a small nonrotating black hole instead of a point mass. The ideas behind this derivation were contained in the original paper by Mino, Sasaki, and Tanaka [39], but the implementation given here, which involves the retarded coordinates of Section 3.3 and displays explicitly the transformation between external and internal coordinates, is original work.
Concluding remarks are presented in Section 5.5. Throughout this review I use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [40].
2 General Theory of Bitensors
2.1 Synge’s world function
2.1.1 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 ½ε(λ_{1} − λ_{0})^{2}. If the geodesic is timelike, then λ can be set equal to the proper time τ, which implies that ε = −1 and σ = −½(Δτ)^{2}. If the geodesic is spacelike, then λ can be set equal to the proper distance s, which implies that ε = 1 and σ = ½(Δs)^{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 σ = ½η_{αβ}(x − x′)^{α}(x − x′)^{β} in Lorentzian coordinates.
2.1.2 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 \({\sigma _{\alpha \prime}} = \partial \sigma /\partial {x^{\alpha \prime}}\) 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 rank2 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 \({\sigma _{\alpha {\beta {\prime}}}} \equiv {\partial _{{\beta {\prime}}}}{\sigma _\alpha} = {\partial ^2}\sigma/\partial {x^{{\beta {\prime}}}}\partial {x^\alpha}\) 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 \({\sigma _{{\alpha {\prime}}\beta}} \equiv {\partial _\beta}{\sigma _{{\alpha {\prime}}}} = {\partial ^2}\sigma/\partial {x^\beta}\partial {x^{{\alpha {\prime}}}}\) 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 \({\sigma _{{\alpha {\prime}}{\beta {\prime}}}} \equiv {\nabla _{{\beta {\prime}}}}{\sigma _{{\alpha {\prime}}}}\) be the covariant derivative of σ_{α′} with respect to x′; this is a symmetric rank2 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 rank3 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 Equation (54), 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′ε} = σ_{α′β′δ′γε} = σ_{γεα′β′ε′}.
2.1.3 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 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 σ_{α′} = −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^{α′β′} σ_{α′β′}.
2.1.4 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 Equation (57), this function is a solution to the nonlinear differential equation ½g^{αβ}σ_{α}σ_{β} = σ. 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 ½g^{αβ}σ_{α}σ_{β} = σ, are clearly compatible with our preceding explicit construction of the world function.
2.2 Coincidence limits
2.2.1 Computation of coincidence limits
2.2.2 Derivation of Synge’s rule
2.3 Parallel propagator
2.3.1 Tetrad on β
2.3.2 Definition and properties of the parallel propagator
2.3.3 Coincidence limits
2.4 Expansion of bitensors near coincidence
2.4.1 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 flatspacetime quantity (x − x′)^{α}. For concreteness we shall consider the case of rank2 bitensor, and for the moment we will assume that the bitensor’s indices all refer to the base point x′.
2.4.2 Special cases
2.4.3 Expansion of tensors
2.5 Van Vleck determinant
2.5.1 Definition and properties
2.5.2 Derivations
3 Coordinate Systems
3.1 Riemann normal coordinates
3.1.1 Definition and coordinate transformation
3.1.2 Metric near x′
It is obvious from Equation (110) that g_{ab}(x′) = η_{ab} and \({\Gamma ^{\rm{a}}}_{{\rm{bc}}}({x{\prime}}) = 0\), where \({\Gamma ^{\rm{a}}}_{{\rm{bc}}} =  {1 \over 3}({R^{\rm{a}}}_{{\rm{bcd}}} + {R^{\rm{a}}}_{{\rm{cbd}}}){{\hat x}^{\rm{d}}} + {\mathcal O}({x^2})\) is the connection compatible with the metric g_{ab}. The Riemann normal coordinates therefore provide a constructive proof of the local flatness theorem.
3.2 Fermi normal coordinates
3.2.1 FermiWalker 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 α^{μ} = Du^{μ}/dτ be its acceleration vector.
The operation of FermiWalker (FW) transport satisfies two important properties. The first is that u^{μ} is automatically FW transported along γ; this follows at once from Equation (112) and the fact that u^{μ} is orthogonal to α^{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 Equation (112).
3.2.2 Tetrad and dual tetrad on γ
3.2.3 Fermi normal coordinates
3.2.4 Coordinate displacements near γ
3.2.5 Metric near γ
Notice that on γ, the metric of Equations (125, 126, 127) reduces to g_{tt} = −1 and g_{ab} = δ_{ab}. On the other hand, the nonvanishing Christoffel symbols (on γ) are \({\Gamma ^a}_{ta} = {\Gamma ^a}_{tt} = {a_a}\); these are zero (and the FNC enforce local flatness on the entire curve) when γ is a geodesic.
3.2.6 ThorneHartle coordinates
3.3 Retarded coordinates
3.3.1 Geometrical elements
The Fermi normal coordinates of Section 3.2 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 futuredirected 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 propertime parameter at this point.
3.3.2 Definition of the retarded coordinates
3.3.3 The scalar field r(x) and the vector field k^{α}(x)
3.3.4 Frame components of tensor fields on the world line
In Section 3.2 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.
3.3.5 Coordinate displacements near γ
3.3.6 Metric near γ
3.3.7 Transformation to angular coordinates
3.3.8 Specialization to a^{μ} = 0 = R_{μν}
In this section 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, and for simplicity we also set ω_{ab} to zero.
3.4 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 3.2), or it can be assigned a set of retarded coordinates (see Section 3.3). These coordinate systems can obviously be related to one another, and our first task in this section (which will occupy us in Sections 3.4.1, 3.4.2, and 3.4.3) 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 futuredirected 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, rΩ^{a}) be the retarded coordinates of x, with u denoting the value of γ’s propertime parameter at x′, r = σ_{α′}u^{α′} representing the affineparameter 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 \({\mathcal S}{\omega ^a} =  e_{\bar \alpha}^a{\sigma ^{\bar \alpha}}\) and σ(x, x′) = 0. Finally, we denote by \((\bar e_0^\alpha, \bar e_a^\alpha)\) the tetrad at x that is obtained by parallel transport of \(({u^{\bar \alpha}},e_a^{\bar \alpha})\) on the null geodesic.

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 Equations (218, 219, 220) below;

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 Equations (221, 222, 223);

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 Equations (224) and (225); and

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 Fermi normal coordinates by Equations (226) and (227).
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 3.4.4. Throughout this section we shall set ω_{ab} = 0, where ω_{ab} is the rotation tensor defined by Equation (138) — the tetrad vectors \(e_a^\mu\) will be assumed to be FermiWalker transported on γ.
3.4.1 From retarded to Fermi coordinates
Quantities at \(\bar x \equiv x(t)\) 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 2.5.)
3.4.2 From Fermi to retarded coordinates
The techniques developed in the preceding Section 3.4.2 can easily be adapted to the task of relating the retarded coordinates of x to its Fermi normal coordinates. Here we use \(\bar x \equiv z(t)\) as the reference point and express all quantities at x′ ≡ z(u) as Taylor expansions about τ = t.
3.4.3 Transformation of the tetrads at x
3.4.4 Advanced point
4 Green’s Functions
4.1 Scalar Green’s functions in flat spacetime
4.1.1 Green’s equation for a massive scalar field
4.1.2 Integration over the source
4.1.3 Singular part of g(σ)
We have seen that Equation (239) 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 Equation (236) must therefore be such that Equation (239) is satisfied. It follows immediately that g(σ) must be a singular function, because for a smooth function the integral of Equation (239) would be of order ϵ, and the lefthand side of Equation (239) 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 δfunctions and derivatives.
4.1.4 Smooth part of g(σ)
To summarize, the retarded and advanced solutions to Equation (235) are given by Equation (236) with g(σ) given by Equation (242) and V(σ) given by Equation (245).
4.1.5 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 Section, 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 δ_{±}(σ + ϵ) in some sensitive computations, and then take the limit ϵ → 0^{+}. Notice that the equation σ + ϵ = 0 describes a twobranch 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, \(\theta _ + {\prime}( \sigma  \epsilon) = {\theta _ +}(x,\Sigma){\theta {\prime}}( \sigma  \epsilon) =  {\theta _ +}(x,\Sigma)\delta (\sigma + \epsilon) =  {\delta _ +}(\sigma + \epsilon)\). This manipulation shows that after the shift from σ to σ + ϵ, the distributions of Equations (246) and (247) can be straightforwardly differentiated with respect to σ.
4.1.6 Alternative computation of the Green’s functions
4.2 Distributions in curved spacetime
The distributions introduced in Section 4.1.5 can also be defined in a fourdimensional spacetime with metric g_{αβ}. Here we produce the relevant generalizations of the results derived in that section.
4.2.1 Invariant Dirac distribution
4.2.2 Lightcone distributions
For the same reasons as those mentioned in Section 4.1.5, it is sometimes convenient to shift the argument of the step and δfunctions from σ to σ + ϵ, where ϵ is a small positive quantity. With this shift, the lightcone distributions can be straightforwardly differentiated with respect to σ. For example, \({\delta _ \pm}(\sigma + \epsilon) =  \theta _ \pm {\prime}( \sigma  \epsilon)\), with a prime indicating differentiation with respect to σ.
4.3 Scalar Green’s functions in curved spacetime
4.3.1 Green’s equation for a massless scalar field in curved spacetime
We let G_{+}(x, x′) be the retarded solution to Equation (268), and G_{−}(x, x′) be 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.
4.3.2 Hadamard construction of the Green’s functions
To summarize: We have shown that with U(x, x′) given by Equation (273) and V(x, x′) determined uniquely by the wave equation of Equation (279) and the characteristic data constructed with Equations (274) and (278), the retarded and advanced Green’s functions of Equation (269) do indeed satisfy Equation (268). It should be emphasized that the construction provided in this section is restricted to \({\mathcal N}({x{\prime}})\), the normal convex neighbourhood of the reference point x′.
4.3.3 Reciprocity
4.3.4 Kirchhoff representation
4.3.5 Singular and radiative Green’s functions
In Section 5 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 outgoingwave boundary conditions at infinity — the advanced solution would come instead with incomingwave 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_{S}(x, x′) ≡ ½ G_{+}(x, x′) + G_{−}(x, x′) and a radiative Green’s function \({G_{\rm{R}}}(x,{x{\prime}}) \equiv {1 \over 2}[{G_ +}(x,{x{\prime}})  {G_ }(x,{x{\prime}})]\). 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 radiative Green’s function, on the other hand, takes its name from the fact that it satisfies the homogeneous wave equation, without the Dirac functional on the righthand side; it produces a field that is smooth on the world line of the moving scalar charge.
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 standing waves at infinity. Removing G_{S}(x, x′) from the retarded Green’s function will therefore 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 radiative Green’s function, which produces a field that is smooth 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 outgoingwave boundary conditions: The particle will lose energy to the radiation.
In this section we attempt a decomposition of the curvedspacetime retarded Green’s function into singular and radiative Green’s functions. The flatspacetime 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 radiative field (which acts on the particle and determines its motion) if the naive definition G_{R}(x, x′) ≡ ½G_{+}(x, x′) − G_{−}(x, x′) were to be adopted.
 Sc.S1: G_{S}(x, x′) satisfies the inhomogeneous scalar wave equation,$$(\square  \xi R){G_{\rm{S}}}(x,x{\prime}) =  4\pi {\delta _4}(x,x{\prime});$$(284)
 Sc.S2: G_{S}(x, x′) is symmetric in its arguments,$${G_{\rm{S}}}(x{\prime},x) = {G_{\rm{S}}}(x,x{\prime});$$(285)
 Sc.S3: G_{S}(x, x′) vanishes if x is in the chronological past or future of x′,$${G_{\rm{S}}}(x,x{\prime}) = 0\quad {\rm{when}}\,x \in {I^ \pm}(x{\prime}).$$(286)
 Sc.R1: G_{R}(x, x′) satisfies the homogeneous wave equation,$$(\square  \xi R){G_{\rm{R}}}(x,x{\prime}) = 0;$$(288)
 Sc.R2: G_{R}(x, x′) agrees with the retarded Green’s function if x is in the chronological future of$${G_{\rm{R}}}(x,x{\prime}) = {G_ +}(x,x{\prime})\quad {\rm{when}}\,x \in {I^ +}(x{\prime});$$(289)
 Sc.R3: G_{R}(x, x′) vanishes if x is in the chronological past of x′,$${G_{\rm{R}}}(x,x{\prime}) = 0\quad {\rm{when}}\,x \in {I^ }(x{\prime}).$$(290)
 Sc.H1: H(x, x′) satisfies the homogeneous wave equation,$$(\square  \xi R)H(x,x{\prime}) = 0;$$(291)
 Sc.H2: H(x, x′) is symmetric in its arguments,$$H(x{\prime},x) = H(x,x{\prime});$$(292)
 Sc.H3: H(x, x′) agrees with the retarded Green’s function if x is in the chronological future of$$H(x,x{\prime}) = {G_ +}(x,x{\prime})\quad {\rm{when}}\,x \in {I^ +}(x{\prime});$$(293)
 Sc.H4: H(x, x′) agrees with the advanced Green’s function if x is in the chronological past of$$H(x,x{\prime}) = {G_ }(x,x{\prime})\quad {\rm{when}}\,x \in {I^ }(x{\prime}).$$(294)
The question is now: Does such a function H(x, x′) exist? I 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 4.3.6 we will see that there exist particular spacetimes for which H(x, x′) can be defined globally.
To satisfy all of Properties Sc.H4, Sc.H2, Sc.H3, and Sc.H4 might seem a tall order, but it should be possible. We first note that Property Sc.H4 is not independent from the rest: It follows from Property Sc.H2, Property Sc.H3, and the reciprocity relation (280) 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 Property Sc.H2, and by Property Sc.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 Property Sc.H4. Alternatively, and this shall be our point of view in the next paragraph, we can think of Property Sc.H3 as following from Properties Sc.H2 and Sc.H4.
4.3.6 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 Equations (309) and (310).
4.4 Electromagnetic Green’s functions
4.4.1 Equations of electromagnetism
We will assume that the retarded Green’s function \(G_{+ \beta}^\alpha (x,{x{\prime}})\), 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.
4.4.2 Hadamard construction of the Green’s functions
 the equationsand$$[{U^\alpha}_{\beta {\prime}}] = [{g^\alpha}_{\beta {\prime}}] = {\delta ^{\alpha {\prime}}}_{\beta {\prime}}$$(318)that determine \({U^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\);$$2{U^\alpha}_{\beta {\prime};\gamma}{\sigma ^\gamma} + ({\sigma ^\gamma}_\gamma  4){U^\alpha}_{\beta {\prime}} = 0$$(319)
 the equationthat determines \({{\check V}^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\), the restriction of \(V_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) on the light cone σ(x, x′) = 0; and$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over V} ^\alpha}_{\beta {\prime};\gamma}{\sigma ^\gamma} + {1 \over 2}({\sigma ^\gamma}_\gamma  2){\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over V} ^\alpha}_{\beta {\prime}} = {1 \over 2}{\left. {\left({\square{U^\alpha}_{\beta {\prime}}  {R^\alpha}_\beta {U^\beta}_{\beta {\prime}}} \right)} \right\vert _{\sigma = 0}}$$(320)
 the wave equationthat determines \(V_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) inside the light cone.$$\square{V^\alpha}_{\beta {\prime}}  {R^\alpha}_\beta {V^\beta}_{\beta {\prime}} = 0$$(321)
To summarize, the retarded and advanced electromagnetic Green’s functions are given by Equation (317) with \({U^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\) given by Equation (322) and \(V_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) determined by Equation (321) and the characteristic data constructed with Equations (320) and (327). It should be emphasized that the construction provided in this section is restricted to \({\mathcal N}({x{\prime}})\), the normal convex neighbourhood of the reference point x′.
4.4.3 Reciprocity and Kirchhoff representation
4.4.4 Singular and radiative Green’s functions
We shall now construct singular and radiative Green’s functions for the electromagnetic field. The treatment here parallels closely what was presented in Section 4.3.5, and the reader is referred to that section for a more complete discussion.
 Em.H1: \({H^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\) satisfies the homogeneous wave equation,$$\square{H^\alpha}_{\beta {\prime}}(x,x{\prime})  {R^\alpha}_\beta (x){H^\beta}_{\beta {\prime}}(x,x{\prime}) = 0;$$(331)
 Em.H2: \({H^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\) is symmetric in its indices and arguments,$${H_{\beta {\prime}\alpha}}(x{\prime},x) = {H_{\alpha \beta {\prime}}}(x,x{\prime});$$(332)
 Em.H3: \({H^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\) agrees with the retarded Green’s function if x is in the chronological future of x′,$${H^\alpha}_{\beta {\prime}}(x,x{\prime}) = G_{+ \beta {\prime}}^{\,\,\alpha}(x,x{\prime})\quad \quad {\rm{when}}\,x \in {I^ +}(x{\prime});$$(333)
 Em.H4: \({H^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\) agrees with the advanced Green’s function if x is in the chronological past of x′,$${H^\alpha}_{\beta {\prime}}(x,x{\prime}) = G_{ \beta {\prime}}^{\,\,\alpha}(x,x{\prime})\quad \quad {\rm{when}}\,x \in {I^ }(x{\prime}).$$(334)
 Em.S1: \({G_{\rm{S}}}_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) satisfies the inhomogeneous wave equation,$$\square G_{{\rm{S}}\beta {\rm{{\prime}}}}^{\,\,\alpha}(x,x{\prime})  {R^\alpha}_\beta (x)G_{{\rm{S}}\beta {\rm{{\prime}}}}^{\,\,\beta}(x,x{\prime}) =  4\pi {g^\alpha}_{\beta {\prime}}(x,x{\prime}){\delta _4}(x,x{\prime});$$(336)
 Em.S2: \({G_{\rm{S}}}_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) is symmetric in its indices and arguments,$$G_{\beta {\prime}\alpha}^{\rm{S}}(x,x{\prime}) = G_{\alpha \beta {\prime}}^{\rm{S}}(x,x{\prime});$$(337)
 Em.S3: \({G_{\rm{S}}}_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) vanishes if x is in the chronological past or future of x′,$$G_{{\rm{S}}\beta {\prime}}^{\,\,\alpha}(x,x{\prime}) = 0\quad \quad {\rm{when}}\,x \in {I^ \pm}(x{\prime}).$$(338)
 Em.R1: \({G_{\rm{R}}}_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) satisfies the homogeneous wave equation,$$\square G_{{\rm{R}}\beta {\prime}}^{\,\,\alpha}(x,x{\prime})  {R^\alpha}_\beta (x)G_{{\rm{R}}\beta {\prime}}^{\,\,\beta}(x,x{\prime}) = 0;$$(340)
 Em.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′$$G_{{\rm{R}}\beta {\prime}}^{\,\,\alpha}(x,x{\prime}) = G_{{\rm{+}}\beta {\prime}}^{\,\,\alpha}(x,x{\prime})\quad \quad {\rm{when}}\,x \in {I^ +}(x{\prime});$$(341)
 Em.R3: \({G_{\rm{R}}}_{{\beta {\prime}}}^\alpha (x,{x{\prime}})\) vanishes if x is in the chronological past of x′,$$G_{{\rm{R}}\beta {\prime}}^{\,\,\alpha}(x,x{\prime}) = 0\quad \quad {\rm{when}}\,x \in {I^ }(x{\prime}).$$(342)
4.5 Gravitational Green’s functions
4.5.1 Equations of linearized gravity
We will assume that the retarded Green’s function \(G{{_ + ^{\alpha \beta }}_{\gamma '\delta '}}(x,x')\), which is nonzero if x is in the causal future of x′, and the advanced Green’s function \(G{{_ + ^{\alpha \beta }}_{\gamma '\delta '}}(x,x')\), 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.
4.5.2 Hadamard construction of the Green’s functions
 the equationsand$$\left[ {{U^{\alpha \beta}}_{\gamma {\prime}\delta {\prime}}} \right] = \left[ {g_{\gamma {\prime}}^{(\alpha}g_{\delta {\prime}}^{\beta)}} \right] = {\delta ^{(\alpha {\prime}}}_{\gamma {\prime}}{\delta ^{\beta {\prime})}}_{\delta {\prime}}$$(353)that determine \({U^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}})\);$$2{U^{\alpha \beta}}_{\gamma {\prime}\delta {\prime};\gamma}{\sigma ^\gamma} + ({\sigma ^\gamma}_\gamma  4){U^{\alpha \beta}}_{\gamma {\prime}\delta {\prime}} = 0$$(354)
 the equationthat determines \({\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{V}} _{\;\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\), the restriction of \({V^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}})\) on the light cone σ(x, x′) = 0; and$$\check{V}_{\;\;\gamma \prime \delta \prime ;\gamma}^{\alpha \beta}{\sigma ^\gamma} + {1 \over 2}(\sigma _{\;\;\gamma}^\gamma  2)\check{V}_{\;\;\;\;\gamma \prime \delta \prime}^{\alpha \beta} = {\left. {{1 \over 2}\left({\square U_{\;\;\gamma \prime \delta \prime}^{\alpha \beta} + 2R_{\gamma \;\;\delta}^{\alpha \;\;\beta}U_{\;\;\gamma \prime \delta \prime}^{\gamma \delta}} \right)} \right\vert_{\sigma = 0}}$$(355)
 the wave equationthat determines \({V^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}})\) inside the light cone.$$\square {V^{\alpha \beta}}_{\gamma {\prime}\delta {\prime}} + 2{R_\gamma}{^\alpha _\delta}^\beta {V^{\gamma \delta}}_{\gamma {\prime}\delta {\prime}} = 0$$(356)
To summarize, the retarded and advanced gravitational Green’s functions are given by Equation (352) with \({U^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}})\) given by Equation (357) and \({V^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}})\) determined by Equation (356), and the characteristic data constructed with Equations (355) and (362). It should be emphasized that the construction provided in this section is restricted to \({\mathcal N}({x{\prime}})\), the normal convex neighbourhood of the reference point x′.
4.5.3 Reciprocity and Kirchhoff representation
4.5.4 Singular and radiative Green’s functions
We shall now construct singular and radiative Green’s functions for the linearized gravitational field. The treatment here parallels closely what was presented in Sections 4.3.5 and 4.4.4.
 Gr.H1: \({H^{\alpha \beta}}_{{\gamma {\prime}}{\delta {\prime}}}(x,{x{\prime}})\) satisfies the homogeneous wave equation,$$\square {H^{\alpha \beta}}_{\gamma {\prime}\delta {\prime}}(x,x{\prime}) + 2{R_\gamma}{^\alpha _\delta}^\beta (x){H^{\gamma \delta}}_{\gamma {\prime}\delta {\prime}}(x,x{\prime}) = 0;$$(366)
 Gr.H2: \({H^{\alpha \beta}}_{{\gamma {\prime}}{\delta {\prime}}}(x,{x{\prime}})\) is symmetric in its indices and arguments,$${H_{\gamma {\prime}\delta {\prime}\alpha \beta}}(x,x{\prime}) = {H_{\alpha \beta \gamma {\prime}\delta {\prime}}}(x,x{\prime});$$(367)
 Gr.H3: \({H^{\alpha \beta}}_{{\gamma {\prime}}{\delta {\prime}}}(x,{x{\prime}})\) agrees with the retarded Green’s function if x is in the chronological future of x′,$${H^{\alpha \beta}}_{\gamma {\prime}\delta {\prime}}(x,x{\prime}) = {G^{\alpha \beta}}_{+ \gamma {\prime}\delta {\prime}}(x,x{\prime})\quad \quad {\rm{when}}\;x \in {I^ +}(x{\prime});$$(368)
 Gr.H4: \({H^{\alpha \beta}}_{{\gamma {\prime}}{\delta {\prime}}}(x,{x{\prime}})\) agrees with the advanced Green’s function if x is in the chronological past of x′,$${H^{\alpha \beta}}_{\gamma {\prime}\delta {\prime}}(x,x{\prime}) = {G^{\alpha \beta}}_{ \gamma {\prime}\delta {\prime}}(x,x{\prime})\quad \quad {\rm{when}}\;x \in {I^ }(x{\prime}).$$(369)
 Gr.S1: \({G_S}_{\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\) satisfies the inhomogeneous wave equation,$$\square {G_{\rm{S}}}{^{\alpha \beta}_{\gamma {\prime}\delta {\prime}}}(x,x{\prime}) + 2{R_\gamma}{^\alpha _\delta}^\beta (x){G_{\rm{S}}}{^{\gamma \delta}_{\gamma {\prime}\delta {\prime}}}(x,x{\prime}) =  4\pi {g^{(\alpha}}_{\gamma {\prime}}(x,x{\prime})g_{\delta {\prime}}^{\beta)}(x,x{\prime}){\delta _4}(x,x{\prime});$$(371)
 Gr.S2: \({G_S}_{\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\) is symmetric in its indices and arguments,$$G_{\gamma {\prime}\delta {\prime}\alpha \beta}^{\rm{S}}(x,x{\prime}) = G_{\alpha \beta \gamma {\prime}\delta {\prime}}^{\rm{S}}(x,x{\prime});$$(372)
 Gr.S3: \({G_S}_{\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\) vanishes if x is in the chronological past or future of x′,$$G_{{\rm{S}}\;\;\gamma {\prime}\delta {\prime}\alpha \beta}^{\alpha \beta}(x,x{\prime}) = 0\quad \quad {\rm{when}}\;x \in {I^ \pm}(x{\prime}).$$(373)
 Gr.R1: \({G_R}_{\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\) satisfies the homogeneous wave equation,$$\square {G_{\rm{R}}}{^{\alpha \beta}_{\gamma {\prime}\delta {\prime}}}(x,x{\prime}) + 2{R_\gamma}{^\alpha _\delta}^\beta (x){G_{\rm{R}}}{^{\gamma \delta}_{\gamma {\prime}\delta {\prime}}}(x,x{\prime}) = 0;$$(375)
 Gr.R2: \({G_R}_{\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\) agrees with the retarded Green’s function if x is in the chronological future of x′,$${G_{\rm{R}}}{^{\alpha \beta}_{\gamma {\prime}\delta {\prime}}}(x,x{\prime}) = {G_{\rm{+}}}{^{\alpha \beta}_{\gamma {\prime}\delta {\prime}}}(x,x{\prime})\quad \quad {\rm{when}}\;x \in {I^ +}(x{\prime});$$(376)
 Gr.R3: \({G_R}_{\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\) vanishes if x is in the chronological past of x′,$${G_{\rm{R}}}{^{\alpha \beta}_{\gamma {\prime}\delta {\prime}}}(x,x{\prime}) = 0\quad \quad {\rm{when}}\;x \in {I^ }(x{\prime}).$$(377)
5 Motion of Point Particles
5.1 Motion of a scalar charge
5.1.1 Dynamics of a point scalar charge
5.1.2 Retarded potential near the world line
5.1.3 Field of a scalar charge in retarded coordinates
5.1.4 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 3.2. To effect this translation we make \(\bar x \equiv z(t)\) the new reference point on the world line. We resume here the notation of Section 3.4 and assign indices \(\bar \alpha, \, \bar \beta\), … to tensors at \({\bar x}\). The Fermi normal coordinates are denoted (t, s, ω^{a}), and we let \((\bar e_0^\alpha, \bar e_a^\alpha)\) 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 \({\bar x}\).
5.1.5 Singular and radiative fields
We recall first that a relation between retarded and advanced times was worked out in Equation (229), that an expression for the advanced distance was displayed in Equation (230), and that Equations (231) and (232) give expansions for ∂_{α}v and ∂_{α}r_{adv}, respectively.
5.1.6 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 5.1.3 and 5.1.4, and in Section 5.1.5 it was shown to originate from the singular field \(\Phi _\alpha ^{\rm{S}}(x)\); the radiative field \(\Phi _\alpha ^{\rm{R}}(x) = {\Phi _\alpha}(x)  \Phi _\alpha ^{\rm{S}}(x)\) was then shown to be smooth on the world line.
The equations of motion displayed in Equations (427) and (428) are thirdorder 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 [25, 30, 47]. 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 [47, 26] that within the context of this approximation, it is consistent to replace, on the righthand 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 second order.
5.2 Motion of an electric charge
5.2.1 Dynamics of a point electric charge
5.2.2 Retarded potential near the world line
5.2.3 Electromagnetic field in retarded coordinates
5.2.4 Electromagnetic field in Fermi normal coordinates
We now wish to express the electromagnetic field in the Fermi normal coordinates of Section 3.2; as before those will be denoted (t, s, ω). The translation will be carried out as in Section 5.1.4, 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 \(\bar x \equiv z(t)\).
5.2.5 Singular and radiative fields
5.2.6 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 5.2.3 and 5.2.4, and in Section 5.2.5 it was shown to originate from the singular field \(\gamma _{\alpha \beta; \gamma}^{\rm{S}}\); the radiative field \(\gamma _{\alpha \beta; \gamma}^{\rm{R}}\) was then shown to be smooth on the world line.
5.3 Motion of a point mass
5.3.1 Dynamics of a point mass
In this section we consider the motion of a point particle of mass m subjected to its own gravitational 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; it will obey a linear wave equation in the background spacetime. This linearization of the field equations will allow us to fit the problem of determining the motion of a point mass within the framework developed in Sections 5.1 and 5.2, and we shall obtain the equations of motion by following the same general line of reasoning. We shall find that γ is not a geodesic of the background spacetime because h_{αβ} acts on the particle and induces an acceleration of order m; the motion is geodesic in the testmass limit only.
Our discussion in this first section is largely formal: As in Sections 5.1.1 and 5.2.1 we insert the point particle in the background spacetime and ignore the fact that the field it produces is singular on the world line. To make sense of the formal equations of motion will be our goal in the following Sections 5.3.2, 5.3.3, 5.3.4, 5.3.5, 5.3.6, and 5.3.7. The problem of determining the motion of a small mass in a background spacetime will be reconsidered in Section 5.4 from a different and more satisfying premise: There the small body will be modeled as a black hole instead of as a point particle, and the singular behaviour of the perturbation will automatically be eliminated.
Let a point particle of mass m move on a world line γ in a curved spacetime with metric g_{αβ} This is the total metric of the perturbed spacetime, and it depends on m as well as all other relevant parameters. At a later stage of the discussion the total metric will be broken down into a “background” part g_{αβ} that is independent of m, and a “perturbation” part h_{αβ} that is proportional to 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. In this and the following sections we will use sansserif symbols to denote tensors that refer to the perturbed spacetime; tensors in the background spacetime will be denoted, as usual, by italic symbols.
It should be clear that Equation (494) is valid only in a formal sense, because the potentials obtained from Equations (493) diverge on the world line. The nonlinearity of the Einstein field equations makes this problem even worse here than for the scalar and electromagnetic cases, because the singular behaviour of the perturbation might render meaningless a formal expansion of g_{αβ} in powers of m. Ignoring this issue for the time being (we shall return to it in Section 5.4), we will proceed as in Sections 5.1 and 5.2 and attempt, with a careful analysis of the field’s singularity structure, to make sense of these equations.
5.3.2 Retarded potentials near the world line
In the following Sections 5.3.3, 5.3.4, 5.3.5, 5.3.6, and 5.3.7, 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.
5.3.3 Gravitational field in retarded coordinates
5.3.4 Gravitational field in Fermi normal coordinates
The translation of the results contained in Equations (505, 506, 507, 508, 509, 510) into the Fermi normal coordinates of Section 3.2 proceeds as in Sections 5.1.4 and 5.2.4, but is simplified by the fact that here the world line can be taken to be a geodesic. We may thus set \({a_a} = {{\dot a}_0} = {{\dot a}_a} = 0\) in Equations (224) and (225) that relate the tetrad \(\bar e_0^\alpha, \bar e_a^\alpha)\) to \((e_0^\alpha, e_a^\alpha)\), as well as in Equations (221, 222, 223) 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 \(\bar x \equiv z(t)\) on the world line that is linked to x by a spacelike geodesic that intersects γ orthogonally.
5.3.5 Singular and radiative fields
5.3.6 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 5.3.3 and 5.3.4, and in Section 5.3.5 it was shown to originate from the singular field \(\gamma _{\alpha \beta; \gamma}^{\rm{S}}\); the radiative field \(\gamma _{\alpha \beta; \gamma}^{\rm{R}}\) was then shown to be smooth on the world line.
Equation (550) was first derived by Yasushi Mino, Misao Sasaki, and Takahiro Tanaka in 1997 [39]. (An incomplete treatment had been given previously by MoretteDeWitt and Ging [42].) An alternative derivation was then produced, also in 1997, by Theodore C. Quinn and Robert M. Wald [49]. These equations are now known as the MiSaTaQuWa equations of motion. It should be noted that Equation (550) 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 radiative metric perturbation obtained by tracereversal of the potentials \(\gamma _{\alpha \beta}^{\rm{R}} \equiv {\gamma _{\alpha \beta}}  \gamma _{\alpha \beta}^{\rm{S}}\); this perturbed metric is smooth on the world line, and it is a solution to the vacuum field equations. This elegant interpretation of the MiSaTaQuWa equations was proposed in 2002 by Steven Detweiler and Bernard F. Whiting [23]. Quinn and Wald [50] have shown that under some conditions, the total work done by the gravitational selfforce is equal to the energy radiated (in gravitational waves) by the particle.
5.3.7 Gauge dependence of the equations of motion
The equations of motion derived in the preceding Section 5.3.6 refer to a specific choice of gauge for the metric perturbation h_{αβ} produced by a point particle of mass m. We indeed recall that back at Equation (492) we imposed the Lorenz gauge condition \({\gamma ^{\alpha \beta}}_{;\beta} = 0\) on the gravitational potentials \({\gamma _{\alpha \beta}} \equiv {h_{\alpha \beta}}  {1 \over 2}({g^{\gamma \delta}}{h_{\gamma \delta}}){g_{\alpha \beta}}\). By virtue of this condition we found that the potentials satisfy the wave equation of Equation (493) in a background spacetime with metric g_{αβ}. The hyperbolic nature of this equation allowed us to identify the retarded solution as the physically relevant solution, and the equations of motion were obtained by removing the singular part of the retarded field. It seems clear that the Lorenz condition is a most appropriate choice of gauge.
Once the equations of motion have been formulated, however, the freedom of performing a gauge transformation (either away from the Lorenz gauge, or within the class of Lorenz gauges) should be explored. A gauge transformation will affect the form of the equations of motion: These must depend on the choice of coordinates, and there is no reason to expect Equation (550) to be invariant under a gauge transformation. Our purpose in this section is to work out how the equations of motion change under such a transformation. This issue was first examined by Barack and Ori [8].
5.4 Motion of a small black hole
5.4.1 Matched asymptotic expansions
The derivation of the MiSaTaQuWa equations of motion presented in Section 5.3 was framed within the paradigm introduced in Sections 5.1 and 5.2 to describe the motion of a point scalar charge, and a point electric charge, respectively. While this paradigm is well suited to fields that satisfy linear wave equations, it is not the best conceptual starting point in the nonlinear context of general relativity. The linearization of the Einstein field equations with respect to the small parameter m did allow us to use the same mathematical techniques as in Sections 5.1 and 5.2, but the validity of the perturbative method must be critically examined when the gravitational potentials are allowed to be singular. So while Equation (550) does indeed give the correct equations of motion when m is small, its previous derivation leaves much to be desired. In this section I provide another derivation that is entirely free of conceptual and technical pitfalls. Here the point mass will be replaced by a nonrotating black hole, and the perturbation’s singular behaviour on the world line will be replaced by a wellbehaved metric at the event horizon. We will use the powerful technique of matched asymptotic expansions [35, 31, 58, 19, 1, 20].
The problem presents itself with a clean separation of length scales, and the method relies entirely on this. On the one hand we have the length scale associated with the small black hole, which is set by its mass m. On the other hand we have the length scale associated with the background spacetime in which the black hole moves, which is set by the radius of curvature \({\mathcal R}\); formally this is defined so that a typical component of the background spacetime’s Riemann tensor is equal to \(1/{{\mathcal R}^2}\) up to a numerical factor of order unity. We demand that \(m/{\mathcal R} \ll 1\). As before we assume that the background spacetime contains no matter, so that its metric is a solution to the Einstein field equations in vacuum.
For example, suppose that our small black hole of mass m is on an orbit of radius b around another black hole of mass M. Then \({\mathcal R} \sim b\sqrt {b/M} > b\) and we take m to be much smaller than the orbital separation. Notice that the time scale over which the background geometry changes is of the order of the orbital period \(b\sqrt {b/M} \sim{\mathcal R}\), so that this does not constitute a separate scale. Similarly, the inhomogeneity scale — the length scale over which the Riemann tensor of the background spacetime changes — is of order \(b\sim {\mathcal R}\sqrt {M/b} < {\mathcal R}\) and also does not constitute an independent scale. (In this discussion we have considered b/M to be of order unity, so as to represent a strongfield, fastmotion situation.)
The metric g(external zone) returned by the procedure described in the preceding paragraph is a functional of a world line γ that represents the motion of the small black hole in the background spacetime. Our goal is to obtain a description of this world line, in the form of equations of motion to be satisfied by the black hole; these equations will be formulated in the background spacetime. It is important to understand that fundamentally, γ exists only as an externalzone construct: It is only in the external zone that the black hole can be thought of as moving on a world line; in the internal zone the black hole is revealed as an extended object and the notion of a world line describing its motion is no longer meaningful.
Equations (555) and (556) give two different expressions for the metric of the same spacetime; the first is valid in the internal zone \(r < {r_i} \ll {\mathcal R}\), while the second is valid in the external zone r > r_{e} ≫ m. The fact that \({\mathcal R} \gg m\) allows us to define a buffer zone in which r is restricted to the interval r_{e} < r < r_{i}. In the buffer zone r is simultaneously much larger than m and much smaller than \({\mathcal R}\) — a typical value might be \(\sqrt {m{\mathcal R}}\) — and Equations (555, 556) are simultaneously valid. Since the two metrics are the same up to a diffeomorphism, these expressions must agree. And since g(external zone) is a functional of a world line γ while g(internal zone) contains no such information, matching the metrics necessarily determines the motion of the small black hole in the background spacetime. What we have here is a beautiful implementation of the general observation that the motion of selfgravitating bodies is determined by the Einstein field equations.
Matching the metrics of Equations (555) and (556) in the buffer zone can be carried out in practice only after performing a transformation from the external coordinates used to express g(external zone) to the internal coordinates employed for g(internal zone). The details of this coordinate transformation will be described in Section 5.4.4, and the end result of matching — the MiSaTaQuWa equations of motion — will be revealed in Section 5.4.5.
5.4.2 Metric in the internal zone
To flesh out the ideas contained in the previous Section 5.4.1 we first calculate the internalzone metric and replace Equation (555) by a more concrete expression. We recall that the internal zone is defined by \(r < {r_i} \ll {\mathcal R}\) where r is a suitable measure of distance from the black hole.
Why is the assumption of no acceleration justified? As I shall explain in the next paragraph (and you might also refer back to the discussion of Section 5.3.7), the reason is simply that it reflects a choice of coordinate system: Setting the acceleration to zero amounts to adopting a specific — and convenient — gauge condition. This gauge differs from the Lorenz gauge adopted in Section 5.3, and it will be our choice in this section only; in the following Section 5.4.3 we will return to the Lorenz gauge, and the acceleration will be seen to return to its standard MiSaTaQuWa expression.
Inspection of Equations (560) and (561) reveals that the angular dependence of the metric perturbation is generated entirely by scalar, vectorial, and tensorial spherical harmonics of degree ℓ = 2. In particular, H_{2} contains no ℓ = 0 and ℓ =1 modes, and this statement reflects a choice of gauge condition. Zerilli has shown [63] that a perturbation of the Schwarzschild spacetime with ℓ = 0 corresponds to a shift in the mass parameter. As Thorne and Hartle have shown [58], a black hole interacting with its environment will undergo a change of mass, but this effect is of order \({m^3}/{{\mathcal R}^2}\) and thus beyond the level of accuracy of our calculations. There is therefore no need to include \(\ell = 0\) terms in H_{2}. Similarly, it was shown by Zerilli that oddparity perturbations of degree ℓ =1 correspond to a shift in the black hole’s angularmomentum parameters. As Thorne and Hartle have shown, a change of angular momentum is quadratic in the hole’s angular momentum, and we can ignore this effect when dealing with a nonrotating black hole. There is therefore no need to include oddparity, ℓ =1 terms in H_{2}. Finally, Zerilli has shown that in a vacuum spacetime, evenparity perturbations of degree ℓ = 1 correspond to a change of coordinate system — these modes are pure gauge. Since we have the freedom to adopt any gauge condition, we can exclude evenparity, ℓ = 1 terms from the perturbed metric. This leads us to Equations (562, 563, 564, 565), which contain only ℓ = 2 perturbation modes; the evenparity modes are contained in those terms that involve \({{\mathcal E}_{ab}}\), while the oddparity modes are associated with \({{\mathcal B}_{ab}}\). The perturbed metric contains also higher multipoles, but those come at a higher order in \(1/{\mathcal R}\); for example, the terms of order \(1/{{\mathcal R}^3}\) include ℓ = 3 modes. We conclude that Equations (562, 563, 564, 565) is a sufficiently general ansatz for the perturbed metric in the internal zone.
5.4.3 Metric in the external zone
We now place ourselves in the buffer zone (where \(m \ll r \ll {\mathcal R}\) and where the matching will take place) and work toward expressing g(external zone) as an expansion in powers of \(r/{\mathcal R}\). For this purpose we will adopt the retarded coordinates (u, rΩ^{a}) of Section 3.3 and rely on the machinery developed there.
5.4.4 Transformation from external to internal coordinates
Comparison of Equations (568, 569, 570) and Equations (596, 597, 598) reveals that the internalzone and externalzone metrics do no match in the buffer zone. But as the metrics are expressed in two different coordinate systems, this mismatch is hardly surprising. A meaningful comparison of the two metrics must therefore come after a transformation from the external coordinates (u, rΩ^{a}) to the internal coordinates \(\left({\bar u,\,\bar r{{\bar \Omega}^a}} \right)\). Our task in this section is to construct this coordinate transformation. We shall proceed in three stages. The first stage of the transformation, (u, rΩ^{a}) → (u′, r′Ω′^{a}) will be seen to remove unwanted terms of order m/r in g_{αβ}. The second stage, (u′, r′Ω′a) → (u″,r″Ω″^{a}), will remove all terms of order \(m/{\mathcal R}\) in g_{αβ}. Finally, the third stage \(\left({u{\prime\prime},\,r{\prime\prime}{{\Omega {\prime\prime}}^a}} \right) \rightarrow \left({\bar u,\,\bar r{{\bar \Omega}^a}} \right)\) will produce the desired internal coordinates.
5.4.5 Motion of the black hole in the background spacetime
5.5 Concluding remarks
I have presented a number of derivations of the equations that determine the motion of a point scalar charge q, a point electric charge e, and a point mass m in a specified background spacetime. In this concluding section I summarize these derivations, and identify their strengths and weaknesses. I also describe the challenges that lie ahead in the concrete evaluation of the selfforces, most especially in the gravitational case.
5.5.1 Conservation of energymomentum
For each of the three cases (scalar, electromagnetic, and gravitational) I have presented two different derivations of the equations of motion. The first derivation is based on a spatial averaging of the retarded field, and the second is based on a decomposition of the retarded field into singular and radiative fields. In the gravitational case, a third derivation, based on matched asymptotic expansions, was also presented. These derivations will be reviewed below, but I want first to explain why I have omitted to present a fourth derivation, based on energymomentum conservation, in spite of the fact that historically, it is one of the most important.
Conservation of energymomentum was used by Dirac [25] to derive the equations of motion of a point electric charge in flat spacetime, and the same method was adopted by DeWitt and Brehme [24] in their generalization of Dirac’s work to curved spacetimes. This method was also one of the starting points of Mino, Sasaki, and Tanaka [39] in their calculation of the gravitational selfforce. I have not discussed this method for two reasons. First, it is technically more difficult to implement than the methods presented in this review (considerably longer computations are involved). Second, it is difficult to endow this method with an adequate level of rigour, to the point that it is perhaps less convincing than the methods presented in this review. While the level of rigour achieved in flat spacetime is now quite satisfactory [56], I do not believe the same can be said of the generalization to curved spacetimes. (But it should be possible to improve on this matter.)
There is no obstacle in evaluating the wall integral, for which T^{αβ} reduces to the field’s stressenergy tensor; for a wall of radius r the integral scales as 1/r^{2}. The integrations over the caps, however, are problematic: While the particle’s contribution to the stressenergy tensor is integrable, the integration over the field’s contribution goes as \(\int\nolimits_0^r {{{\left({r{\prime}} \right)}^{ 2}}dr{\prime}}\) and diverges. To properly regularize this integral requires great care, and the removal of all singular terms can be achieved by mass renormalization [24]. This issue arises also in flat spacetime [25], and while it is plausible that the rigourous distributional methods presented in [56] could be generalized to curved spacetimes, this remains to be done. More troublesome, however, is the interior integral, which does not appear in flat spacetime. Because \({g^\mu}_{\alpha; \beta}\) scales as r, this integral goes as \(\int\nolimits_0^r {{{\left({r{\prime}} \right)}^{ 1}}dr{\prime}}\) and it also diverges, albeit less strongly than the caps integration. While simply discarding this integral produces the correct equations of motion, it would be desirable to go through a careful regularization of the interior integration, and provide a convincing reason to discard it altogether. To the best of my knowledge, this has not been done.
5.5.2 Averaging method
The averaging method is sound, but it is not immune to criticism. A first source of criticism concerns the specifics of the averaging procedure, in particular, the choice of a spherical surface over any other conceivable shape. Another source is a slight inconsistency of the method that gives rise to the famous “4/3 problem” [52]: The mass shift δm is related to the shell’s electrostatic energy \(E = {e^2}/\left({2s} \right)\) by \(\delta m = {4 \over 3}E\) instead of the expected δm = E. This problem is likely due [45] to the fact that the field that is averaged over the surface of the shell is sourced by a point particle and not by the shell itself. It is plausible that a more careful treatment of the nearsource field will eliminate both sources of criticism: We can expect that the field produced by an extended spherical object will give rise to a mass shift that equals the object’s electrostatic energy, and the object’s spherical shape would then fully justify a spherical averaging. (Considering other shapes might also be possible, but one would prefer to keep the object’s structure simple and avoid introducing additional multipole moments.) Further work is required to clean up these details.
The averaging method is at the core of the approach followed by Quinn and Wald [49], who also average the retarded field over a spherical surface surrounding the particle. Their approach, however, also incorporates a “comparison axiom” that allows them to avoid renormalizing the mass.
5.5.3 DetweilerWhiting axiom
This axiom, which is motivated by the symmetric nature of the singular field, and also its causal structure, gives rise to the equations of motion \(m{a_\mu} = eF_{\mu \nu}^{\rm{R}}{u^\nu}\), in agreement with the averaging method (but with an implicit, instead of explicit, mass shift). In this picture, the particle simply interacts with a free radiative field (whose origin can be traced to the particle’s past), and the procedure of mass renormalization is sidestepped. In the scalar and electromagnetic cases, the picture of a particle interacting with a radiative field removes any tension between the nongeodesic motion of the charge and the principle of equivalence. In the gravitational case the DetweilerWhiting axiom produces the statement that the point mass m moves on a geodesic in a spacetime whose metric \({g_{\alpha \beta}} + h_{\alpha \beta}^{\rm{R}}\) is nonsingular and a solution to the vacuum field equations. This is a conceptually powerful, and elegant, formulation of the MiSaTaQuWa equations of motion.the singular field exerts no force on the particle (it merely contributes to the particle’s inertia); the entire selfforce arises from the action of the radiative field.
5.5.4 Matched asymptotic expansions
It is well known that in general relativity the motion of gravitating bodies is determined, along with the spacetime metric, by the Einstein field equations; the equations of motion are not separately imposed. This observation provides a means of deriving the MiSaTaQuWa equations without having to rely on the fiction of a point mass. In the method of matched asymptotic expansions, the small body is taken to be a nonrotating black hole, and its metric perturbed by the tidal gravitational field of the external universe is matched to the metric of the external universe perturbed by the black hole. The equations of motion are then recovered by demanding that the metric be a valid solution to the vacuum field equations. This method, which was the second starting point of Mino, Sasaki, and Tanaka [39], gives what is by far the most compelling derivation of the MiSaTaQuWa equations. Indeed, the method is entirely free of conceptual and technical pitfalls — there are no singularities (except deep inside the black hole) and only retarded fields are employed.
The introduction of a point mass in a nonlinear theory of gravitation would appear at first sight to be severely misguided. The lesson learned here is that one can in fact get away with it. The derivation of the MiSaTaQuWa equations of motion based on the method of matched asymptotic expansions does indeed show that results obtained on the basis of a pointparticle description can be reliable, in spite of all their questionable aspects. This is a remarkable observation, and one that carries a lot of convenience: It is much easier to implement the pointmass description than to perform the matching of two metrics in two coordinate systems.
5.5.5 Evaluation of the gravitational selfforce
The concrete evaluation of the scalar, electromagnetic, and gravitational selfforces is made challenging by the need to first obtain the relevant retarded Green’s function. Successes achieved in the past were reviewed in Section 1.10, and here I want to describe the challenges that lie ahead. I will focus on the specific task of computing the gravitational selfforce acting on a point mass that moves in a background Kerr spacetime. This case is especially important because the motion of a small compact object around a massive (galactic) black hole is a promising source of lowfrequency gravitational waves for the Laser Interferometer Space Antenna (LISA) [32]; to calculate these waves requires an accurate description of the motion, beyond the testmass approximation which ignores the object’s radiation reaction.

solving the Teukolsky equation for one of the NewmanPenrose quantities ψ_{0} and ψ_{4} (which are complex components of the Weyl tensor) produced by the point particle;

obtaining from ψ_{0} or ψ_{4} a related (Hertz) potential Ψ by integrating an ordinary differential equation;

applying to Ψ a number of differential operators to obtain the metric perturbation in a radiation gauge that differs from the Lorenz gauge; and

performing a gauge transformation from the radiation gauge to the Lorenz gauge.
The next sequence of steps is concerned with the regularization of each a^{μ}[h_{l}] by removing the contribution from \(h_{\alpha \beta}^{\rm{S}}\) [6, 7, 9, 11, 38, 21]. The singular field can be constructed locally in a neighbourhood of the particle, and then decomposed into modes of multipole order l. This gives rise to modes \({a^\mu}\left[ {h_l^{\rm{S}}} \right]\) for the singular part of the selfacceleration; these are also finite and discontinuous, and their sum over l also diverges. But the true modes \({a^\mu}\left[ {h_l^{\rm{R}}} \right] = {a^\mu}\left[ {{h_l}} \right]  {a^\mu}\left[ {h_l^{\rm{S}}} \right]\) of the selfacceleration are continuous at the radial position of the particle, and their sum does converge to the particle’s acceleration. (It might be noted that obtaining a mode decomposition of the singular field involves providing an extension of \(h_{\alpha \beta}^{\rm{S}}\) on a sphere of constant radial coordinate, and then integrating over the angular coordinates. The arbitrariness of the extension introduces ambiguities in each \({a^\mu}\left[ {h_l^{\rm{S}}} \right]\), but the ambiguity disappears after summing over l.)
The selfacceleration is thus obtained by first computing a^{μ}[h_{l}] from the metric perturbation derived from ψ_{0} or ψ_{4}, then computing the counterterms \({a^\mu}\left[ {h_l^{\rm{S}}} \right]\) by modedecomposing the singular field, and finally summing over all \({a^\mu}\left[ {h_l^{\rm{R}}} \right] = {a^\mu}\left[ {{h_l}} \right]  {a^\mu}\left[ {h_l^{\rm{S}}} \right]\). This procedure is lengthy and involved, and thus far it has not been brought to completion, except for the special case of a particle falling radially toward a nonrotating black hole [5]. In this regard it should be noted that the replacement of the central Kerr black hole by a Schwarzschild black hole simplifies the task considerably. In particular, because there exists a practical and welldeveloped formalism to describe the metric perturbations of a Schwarzschild spacetime [51, 59, 63], there is no necessity to rely on the Teukolsky formalism and the complicated reconstruction of the metric variables.
The procedure described above is lengthy and involved, but it is also incomplete. The reason is that the metric perturbations \(h_{\alpha \beta}^l\) that can be recovered from ψ_{0} or ψ_{4} do not by themselves sum up to the complete gravitational perturbation produced by the moving particle. Missing are the perturbations derived from the other NewmanPenrose quantities: ψ_{1}, ψ_{2} and ψ_{3} While ψ_{1} and ψ_{3} can always be set to zero by an appropriate choice of null tetrad, ψ_{2} contains such important physical information as the shifts in mass and angularmomentum parameters produced by the particle [60]. Because the mode decompositions of ψ_{0} and ψ_{4} start at l = 2, we might colloquially say that what is missing from the above procedure are the “l = 0 and l = 1” modes of the metric perturbations. It is not currently known how the procedure can be completed so as to incorporate all modes of the metric perturbations. Specializing to a Schwarzschild spacetime eliminates this difficulty, and in this context the low multipole modes have been studied for the special case of circular orbits [43, 22].
In view of these many difficulties (and I choose to stay silent on others, for example, the issue of relating metric perturbations in different gauges when the gauge transformation is singular on the world line), it is perhaps not too surprising that such a small number of concrete calculations have been presented to date. But progress in dealing with these difficulties has been steady, and the situation should change dramatically in the next few years.
5.5.6 Beyond the selfforce
The successful computation of the gravitational selfforce is not the end of the road. After the difficulties reviewed in the preceding Section 5.5.5 have all been removed and the motion of the small body is finally calculated to order m, it will still be necessary to obtain gaugeinvariant information associated with the body’s corrected motion. Because the MiSaTaQuWa equations of motion are not by themselves gaugeinvariant, this step will necessitate going beyond the selfforce.
To see how this might be done, imagine that the small body is a pulsar, and that it emits light pulses at regular propertime intervals. The motion of the pulsar around the central black hole modulates the pulse frequencies as measured at infinity, and information about the body’s corrected motion is encoded in the timesofarrival of the pulses. Because these can be measured directly by a distant observer, they clearly constitute gaugeinvariant information. But the timesofarrival are determined not only by the pulsar’s motion, but also by the propagation of radiation in the perturbed spacetime. This example shows that to obtain gaugeinvariant information, one must properly combine the MiSaTaQuWa equations of motion with the metric perturbations.
In the context of the Laser Interferometer Space Antenna, the relevant observable is the instrument’s response to a gravitational wave, which is determined by gaugeinvariant waveforms, h_{+} and h_{×} To calculate these is the ultimate goal of this research programme, and the challenges that lie ahead go well beyond what I have described thus far. To obtain the waveforms it will be necessary to solve the Einstein field equations to second order in perturbation theory.
To understand this, consider first the formulation of the firstorder problem. Schematically, one introduces a perturbation h that satisfies a wave equation □h = T[z] in the background spacetime, where T[z] is the stressenergy tensor of the moving body, which is a functional of the world line z(τ). In firstorder perturbation theory, the stressenergy tensor must be conserved in the background spacetime, and z(τ) must describe a geodesic. It follows that in firstorder perturbation theory, the waveforms constructed from the perturbation h contain no information about the body’s corrected motion.
The firstorder perturbation, however, can be used to correct the motion, which is now described by the world line z(τ) + δz(τ). In a naive implementation of the selfforce, one would now resolve the wave equation with a corrected stressenergy tensor, □h = T[z + δz], and the new waveforms constructed from h would then incorporate information about the corrected motion. This implementation is naive because this information would not be gaugeinvariant. In fact, to be consistent one would have to include all secondorder terms in the wave equation, not just the ones that come from the corrected motion. Schematically, the new wave equation would have the form of □h = (1 + h)T[z + δz] + (∇h)^{2}, and this is much more difficult to solve than the naive problem (if only because the source term is now much more singular than the distributional singularity contained in the stressenergy tensor). But provided one can find a way to make this secondorder problem well posed, and provided one can solve it (or at least the relevant part of it), the waveforms constructed from the secondorder perturbation h will be gauge invariant. In this way, information about the body’s corrected motion will have properly been incorporated into the gravitational waveforms.
The story is far from being over.
Footnotes
 1.
Dirac’s original expression actually involved the rate of change of the acceleration vector on the righthand side. The resulting equation gives rise to the wellknown problem of runaway solutions. To avoid such unphysical behaviour I have submitted Dirac’s equation to a reductionoforder procedure whereby da^{ν}/dτ is replaced with \({m^{ 1}}df_{{\rm{ext}}}^\nu/d\tau\). This procedure is explained and justified, for example, in [47, 26].
 2.
His analysis was restricted to a minimallycoupled scalar field, so that ξ = 0 in his expressions. The extension to an arbitrary coupling constant was carried out by myself for this review.
 3.
Note that I use sansserif 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.
Notes
Acknowledgments
My understanding of the work presented in this review was shaped by a series of annual meetings named after the movie director Frank Capra. The first of these meetings took place in 1998 and was held at Capra’s ranch in Southern California; the ranch now belongs to Caltech, Capra’s alma mater. Subsequent meetings were held in Dublin, Pasadena, Potsdam, State College PA, and Kyoto. At these meetings and elsewhere I have enjoyed many instructive conversations with Warren Anderson, Patrick Brady, Claude Barrabès, Leor Barack, Lior Burko, Manuella Campanelli, Steve Detweiler, Eanna Flanagan, Scott Hughes, Werner Israel, Carlos Lousto, Yasushi Mino, Hiroyuki Nakano, Amos Ori, Misao Sasaki, Takahiro Tanaka, Bill Unruh, Bob Wald, Alan Wiseman, and Bernard Whiting. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
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