# The Motion of Point Particles in Curved Spacetime

## Abstract

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle.

The field’s action on the particle is difficult to calculate because of its singular nature: The field diverges at the position of the particle. But it is possible to isolate the field’s singular part and show that it exerts no force on the particle — its only effect is to contribute to the particle’s inertia. What remains after subtraction is a smooth field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free (radiative) field that interacts with the particle; it is this interaction that gives rise to the self-force.

The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors (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 near-zone field which acts directly on the particle and produces a *self-force*. In curved spacetime the self-force contains a radiation-reaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angular-momentum transport. The self-force is proportional to *q*^{2} in the case of a scalar charge, proportional to *e*^{2} in the case of an electric charge, and proportional to *m*^{2} in the case of a point mass.

In this review I derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely self-contained, and all relevant materials are developed *ab initio*. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. One must first assimilate the basic theory of bitensors (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

*e*moving in flat spacetime [52, 30, 56, 47]. The charge produces an electromagnetic vector potential

*A*

^{α}that satisfies the wave equation

_{α}

*A*

^{α}= 0. (On page 294 in [30] Jackson explains why the term “Lorenz gauge” is preferable to “Lorentz gauge”.) The vector

*j*

^{α}is the charge’s current density, which is formally written in terms of a four-dimensional Dirac functional supported on the charge’s world line: The density is zero everywhere, except at the particle’s position where it is infinite. For concreteness we will imagine that the particle moves around a centre (perhaps another charge, which is taken to be fixed) and that it emits outgoing radiation. We expect that the charge will undergo a radiation reaction and that it will spiral down toward the centre. This effect must be accounted for by the equations of motion, and these must therefore include the action of the charge’s own field, which is the only available agent that could be responsible for the radiation reaction. We seek to determine this self-force acting on the particle.

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.

*the situation considered here, in which the radiation is propagating outward and the charge is spiraling inward, breaks the time-reversal invariance of Maxwell’s theory*. A specific time direction was adopted when, among all possible solutions to the wave equation, we chose \(A_{{\rm{ret}}}^\alpha\), the

*retarded solution*, as the physically-relevant solution. Choosing instead the

*advanced solution*\(A_{{\rm{adv}}}^\alpha\) would produce a time-reversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition

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 wave-zone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line — all three electromagnetic fields are dominated by the particle’s Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript ‘S’: It stands for ‘singular’ as well as ‘symmetric’.

*this new potential must be entirely responsible for the radiation reaction*. We therefore introduce the new potential

*u*

^{μ}=

*dz*

^{μ}/

*dτ*is the charge’s four-velocity (

*z*

^{μ}(

*τ*) gives the description of the world line and

*τ*is proper time),

*a*

^{μ}=

*du*

^{μ}/

*dτ*its acceleration, m its (renormalized) mass, and \(f_{{\rm{ext}}}^\mu\) an external force also acting on the particle. Calculation of the radiative field yields the more concrete expression

*e*

^{2}, it is orthogonal to the four-velocity, and it depends on the rate of change of the external force. This is the result that was first derived by Dirac [25]

^{1}.

### 1.3 Green’s functions in flat spacetime

*x*= (

*t*,

*x*) is an arbitrary field point,

*x*′ = (

*t*′,

*x*′) is a source point, and

*dV*′ ≡

*d*

^{4}

*x*′; tensors at

*x*are identified with unprimed indices, while primed indices refer to tensors at

*x*′. Similarly, the advanced solution can be expressed as

*x*lies outside of the future light cone of

*x*′, and \(G_{+ {\beta {\prime}}}^\alpha (x,{x{\prime}})\) is infinite at these points. On the other hand, the advanced Green’s function is zero whenever

*x*lies outside of the past light cone of

*x*′, and \(G_{- {\beta {\prime}}}^\alpha (x,{x{\prime}})\) is infinite at these points. The retarded and advanced Green’s functions satisfy the reciprocity relation

*x*and

*x*′ are interchanged.

*A*

^{α}= 0 and is well behaved on the world line.

*x*is generated by a single event in spacetime: the intersection of the world line and the past light cone of

*x*’ (see Figure 1). I shall call this the

*retarded point*associated with

*x*and denote it

*z*(

*u*);

*u*is the

*retarded time*, the value of the proper-time parameter at the retarded point. Similarly we find that the advanced potential of Equation (7) is generated by the intersection of the world line and the future light cone of the field point

*x*. I shall call this the

*advanced point*associated with

*x*and denote it

*z*(

*u*);

*v*is the

*advanced time*, the value of the proper-time parameter at the advanced point.

### 1.4 Green’s functions in curved spacetime

*g*

_{αβ}the wave equation for the vector potential becomes

*g*

^{αβ}∇

_{α}∇

_{β}is the covariant wave operator and

*R*

_{αβ}is the spacetime’s Ricci tensor; the Lorenz gauge conditions becomes ∇

_{α}

*A*

^{α}= 0, and ∇

_{α}denotes covariant differentiation. Retarded and advanced Green’s functions can be defined for this equation, and solutions to Equation (13) take the same form as in Equations (6) and (7), except that

*dV*′ now stands for \(\sqrt {- g({x{\prime}})} {d^4}{x{\prime}}\).

*x*′, in curved spacetime its support extends

*inside*the light cone as well; \(G_{+ {\beta {\prime}}}^\alpha (x,{x{\prime}})\) is therefore nonzero when

*x*∊

*I*

^{+}(

*x*′), which denotes the chronological future of

*x*′. This property reflects the fact that in curved spacetime, electromagnetic waves propagate not just at the speed of light, but at

*all speeds smaller than or equal to the speed of light*; the delay is caused by an interaction between the radiation and the spacetime curvature. A direct implication of this property is that the retarded potential at

*x*is now generated by the point charge during its entire history prior to the retarded time

*u*associated with

*x*: The potential depends on the particle’s state of motion for all times

*τ*≤

*u*(see Figure 2).

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 self-force.

To decompose the retarded Green’s function into singular and radiative parts is not a straightforward task in curved spacetime. The flat-spacetime definition for the singular Green’s function, Equation (9), cannot be adopted without modification: While the combination half-retarded plus half-advanced Green’s functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Equation (13), this candidate for the singular Green’s function would produce a self-force with an unacceptable dependence on the particle’s future history. For suppose that we made this choice. Then the radiative Green’s function would be given by the combination half-retarded minus half-advanced 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 self-force constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and radiative Green’s functions.

*x*is in the chronological past of

*x*′: \({x{\prime}}:{H^\alpha}_{{\beta {\prime}}}(x,{x{\prime}}) = G_{- {\beta {\prime}}}^\alpha (x,{x{\prime}})\) when

*x*∊

*I*

^{−}(

*x*′). This ensures that \(G_{{\rm{S}}{\beta {\prime}}}^\alpha (x,{x{\prime}})\) vanishes when

*x*is in the chronological past of

*x*′. In fact, reciprocity implies that \({H^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\) will also agree with the

*retarded*Green’s function when

*x*is in the chronological future of

*x*′, and it follows that the symmetric Green’s function vanishes also when

*x*is in the chronological future of

*x*′.

*τ*restricted to the interval

*u*≤

*τ*≤

*v*(see Figure 3). Because this potential satisfies Equation (13), it is just as singular as the retarded potential in the vicinity of the world line. And because the singular Green’s function is symmetric in its arguments, the singular potential can be shown to exert no force on the charged particle. (This requires a lengthy analysis that will be presented in the bulk of the paper.)

*τ*prior to the advanced time

*v*:

*τ*≤

*v*. Because this potential satisfies the homogeneous wave equation, it is well behaved on the world line and its action on the point charge is well defined. And because the singular potential \(A_{\rm{S}}^\alpha (x)\) can be shown to exert no force on the particle, we conclude that \(A_{\rm{R}}^\alpha (x)\) alone is responsible for the self-force.

*u*

^{μ}=

*dz*

^{μ}/

*dτ*is again the charge’s four-velocity, but

*a*

^{μ}=

*Du*

^{μ}/

*dτ*is now its covariant acceleration.

### 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 \({{\dot a}^\mu} \equiv D{a^\mu}/d\tau\).

*γ*we erect an orthonormal basis that consists of the four-velocity

*u*

^{μ}and three spatial vectors \(e_a^\mu\) labelled by a frame index

*a*= (1, 2, 3). These vectors satisfy the relations

*g*

_{μν}

*u*

^{μ}

*u*

^{ν}= −1, \({g_{\mu \nu}}{u^\mu}e_a^\nu = 0\), and \({g_{\mu \nu}}e_a^\mu e_b^\nu = {\delta _{ab}}\). We take the spatial vectors to be Fermi-Walker transported on the world line: \(De_a^\mu/d\tau = {a_a}{u^\mu}\), where

*δ*

_{ab}= diag(1, 1, 1) and its inverse

*δ*

^{ab}= diag(1, 1, 1) to lower and raise frame indices, respectively.

Consider a point *x* in a neighbourhood of the world line *γ*. We assume that *x* is sufficiently close to the world line that a unique geodesic links *x* to any neighbouring point *z* on *γ*. The two-point function *σ*(*x*, *z*), known as *Synge’s world function* [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 future-directed 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*.

*retarded coordinates*of the point

*x*are (

*u*, \({{\hat x}^a}\)), where \({{\hat x}^a} = - e_{{\alpha {\prime}}}^a{\sigma ^{{\alpha {\prime}}}}\) are the frame components of the separation vector. They come with a straightforward interpretation (see Figure 4). The invariant quantity

*x*to

*x*′; it can be loosely interpreted as the time delay between

*x*and

*x*′ as measured by an observer moving with the particle. This therefore gives a meaningful notion of distance between

*x*and the retarded point, and I shall call

*r*the

*retarded distance*between

*x*and the world line. The unit vector

*x*to

*x*′. Because Ω

^{α}is a different constant on each null geodesic that emanates from

*x*′, keeping

*u*fixed and varying Ω

^{α}produces a congruence of null geodesics that generate the future light cone of the point

*x*′ (the congruence is hypersurface orthogonal). Each light cone can thus be labelled by its retarded time

*u*, each generator on a given light cone can be labelled by its direction vector Ω

^{α}, and each point on a given generator can be labelled by its retarded distance

*r*. We therefore have a good coordinate system in a neighbourhood of

*γ*.

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

*F*

_{αβ}, which we decompose in the tetrad \((e_0^\alpha, e_a^\alpha)\) introduced at the end of Section 1.5. We then express the frame components of the field tensor in retarded coordinates, in the form of an expansion in powers of

*r*. This gives

*x*′ =

*z*(

*u*) associated with

*x*; for example, \({a_a} \equiv {a_a}(u) \equiv {a_{{\alpha {\prime}}}}e_a^\alpha\). The tail part of the electromagnetic field tensor is written as an integral over the portion of the world line that corresponds to the interval −∞ <

*τ*≤

*u*

^{−}≡

*u*− 0

^{+}; this represents the past history of the particle. The integral is cut short at

*u*

^{−}to avoid the singular behaviour of the retarded Green’s function when

*z*(

*τ*) coincides with

*x*′; the portion of the Green’s function involved in the tail integral is smooth, and the singularity at coincidence is completely accounted for by the other terms in Equations (23) and (24).

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.

*x*′ the radiative field becomes

### 1.7 Motion of an electric charge in curved spacetime

*z*(

*τ*), the current position of the particle on the world line. The primed indices in the tail integral refer to a point

*z*(

*τ*′) which represents a prior position; the integration is cut short at

*τ*′ =

*τ*

^{−}≡

*τ*− 0

^{+}to avoid the singular behaviour of the retarded Green’s function at coincidence. To get Equation (33) I have reduced the order of the differential equation by replacing

*ȧ*

^{ν}with \({m^{- 1}}\dot f_{{\rm{ext}}}^\nu\) on the right-hand side; this procedure was explained at the end of Section 1.2.

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 Ricci-tensor term.) In flat spacetime the Ricci tensor is zero, the tail integral disappears (because the Green’s function vanishes everywhere within the domain of integration), and Equation (33) reduces to Dirac’s result of Equation (5). In curved spacetime the self-force does not vanish even when the electric charge is moving freely, in the absence of an external force: It is then given by the tail integral, which represents radiation emitted earlier and coming back to the particle after interacting with the spacetime curvature. This delayed action implies that, in general, the self-force is nonlocal in time: It depends not only on the current state of motion of the particle, but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that the physical process that leads to 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

*q*produces a scalar field Φ(

*x*), which satisfies a wave equation

*R*is the spacetime’s Ricci scalar, and

*ξ*is an arbitrary coupling constant; the scalar charge density

*μ*(

*x*) is given by a four-dimensional Dirac functional supported on the particle’s world line

*γ*. The retarded solution to the wave equation is

*G*

_{+}(

*x*,

*z*) is the retarded Green’s function associated with Equation (34). The field exerts a force on the particle, whose equations of motion are

*m*is the particle’s mass; this equation is very similar to the Lorentz-force law. But the dynamics of a scalar charge comes with a twist: If Equations (34) and (36) are to follow from a variational principle,

*the particle’s mass should not be expected to be a constant of the motion*. It is found instead to satisfy the differential equation

*m*will vary with proper time. This phenomenon is linked to the fact that a scalar field has zero spin: The particle can radiate monopole waves and the radiated energy can come at the expense of the rest mass.

_{S}(

*x*) must be removed before Equations (36) and (37) can be evaluated. This procedure produces the radiative field Φ

_{R}(

*x*), and it is this field (which satisfies the homogeneous wave equation) that gives rise to a self-force. The gradient of the radiative field takes the form of

^{2}.

In flat spacetime the Ricci-tensor term and the tail integral disappear, and Equation (40) takes the form of Equation (5) with *q*^{2}/(3*m*) replacing the factor of 2*e*^{2}/(3*m*). 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 conformally-flat spacetime when the wave equation is conformally invariant (*ξ* = 1/6): In this case the Green’s function possesses only a light-cone part, and the right-hand side of 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.

*g*

_{αβ}of the background spacetime to be a solution of the Einstein field equations in vacuum. (We impose this condition globally.) We describe the gravitational perturbation produced by a point particle of mass

*m*in terms of trace-reversed potentials

*γ*

_{αβ}defined by

*h*

_{αβ}is the difference between g

_{αβ}, the actual metric of the perturbed spacetime, and

*g*

_{αβ}. The potentials satisfy the wave equation

*g*

^{αβ}∇

_{α}∇

_{β}is the wave operator for the background spacetime, and

*T*

^{αβ}is the stress-energy tensor of the point mass; this is given by a Dirac distribution supported on the particle’s world line

*γ*. The retarded solution is

*h*

_{αβ}(

*x*) can be recovered by inverting Equation (42).

_{αβ}=

*g*

_{αβ}+

*h*

_{αβ}. After a mapping to the background spacetime, the equations of motion take the form of

*m*; in the test-mass limit the world line of the particle is a geodesic of the background spacetime.

*z*(

*τ*) of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions

*z*(

*τ*′), at which tensors with primed indices are evaluated. As before the integral is cut short at

*τ*′ =

*τ*

^{−}≡

*τ*− 0

^{+}to avoid the singular behaviour of the retarded Green’s function at coincidence.

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).

*the MiSaTaQuWa equations are not gauge invariant*, and they depend upon the Lorenz gauge condition \({\gamma ^{\alpha \beta}}_{;\beta} = 0\). Barack and Ori [8] have shown that under a coordinate transformation of the form

*x*

^{α}→

*x*

^{α}+

*ξ*

^{α}, where

*x*

^{α}are the coordinates of the background spacetime and

*ξ*

^{α}is a smooth vector field of order

*m*, the particle’s acceleration changes according to

*a*

^{μ}→

*a*

^{μ}+

*a*[

*ξ*]

^{μ}, where

*μ*

^{ν}in the direction of the world line. This implies that the particle’s acceleration can be altered at will by a gauge transformation;

*ξ*

^{α}could even be chosen so as to produce

*a*

^{μ}= 0, making the motion geodesic after all. This observation provides a dramatic illustration of the following point:

*The MiSaTaQuWa equations of motion are not gauge invariant and they cannot by themselves produce a meaningful answer to a well-posed physical question; to obtain such answers it shall always be necessary to combine the equations of motion with the metric perturbation h*

_{αβ}

*so as to form gauge-invariant quantities that will correspond to direct observables*. This point is very important and cannot be over-emphasized.

### 1.10 Evaluation of the self-force

To concretely evaluate the self-force, 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 self-force has so far been evaluated in a rather limited number of situations.

*M*is the total mass contained in the spacetime,

*r*= ∣

*∣ is the distance from the centre of mass, \(\hat r = x/r\), and*

**x***= −*

**g****∇**Φ is the Newtonian gravitational field. (In these expressions the bold-faced symbols represent vectors in three-dimensional flat space.) The first term on the right-hand side of Equation (50) is a conservative correction to the Newtonian force

*m*

*. The second term is the standard radiation-reaction force; although it comes from the tail integral, this is the same result that would be obtained in flat spacetime if an external force*

**g***m*

*were acting on the particle. This agreement is necessary, but remarkable!*

**g***ξ*is the coupling of the scalar field to the spacetime curvature; the conservative term disappears when the field is minimally coupled. Pfenning and Poisson also computed the gravitational self-force acting on a massive particle moving in a weakly curved spacetime. The expression they obtained is in complete agreement (within its domain of validity) with the standard post-Newtonian equations of motion.

*m*is the mass of the charge,

*M*is the mass of the black hole, and

*r*is the charge’s radial coordinate (the expression is valid in Schwarzschild coordinates). The first term on the right-hand side of Equation (52) is the force required to keep a neutral test particle stationary in a Schwarzschild spacetime; the second term is the negative of the electromagnetic self-force, and its expression agrees with the weak-field result of Equation (50). Wiseman [62] performed a similar calculation for a scalar charge. He found that in this case the self-force vanishes. This result is not incompatible with Equation (51), even for nonminimal coupling, because the computation of the weak-field self-force requires the presence of matter, while Wiseman’s scalar charge lives in a purely vacuum spacetime.

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 spatially-flat matter-dominated 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 matter-dominated 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) self-force 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 spherical-harmonic 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 mode-decomposition of the Green’s function one obtains a mode-decomposition of the field gradient ∇_{α}Φ, and from this subtracts a mode-decomposition 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 time-averaged evolution of a generic orbit around a Kerr black hole.

The mode-sum method was applied to a number of different situations. Burko computed the self-force 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 self-force 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 self-force acting on such a particle [4]; Lousto [33] and Barack and Lousto [5], on the other hand, calculated the gravitational self-force.

### 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 all-important Hadamard decomposition [28] of the Green’s function into “light-cone” 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 −4*π* multiplying a four-dimensional Dirac functional of the right-hand side of the wave equation. (The gravitational Green’s function is sometimes normalized with a −16*π* on the right-hand 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 rest-frame average is easily carried out in Fermi normal coordinates. The averaged field is still infinite on the world line, but the divergence points in the direction of the acceleration vector and it can thus be removed by mass renormalization. Such calculations show that while the singular field does not affect the motion of the particle, it nonetheless contributes to its inertia. In 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

*bitensors*, tensorial functions of two points in spacetime. The first is

*x*′, to which we refer as the “base point”, and to which we assign indices

*α*′,

*β*′, etc. The second is

*x*, to which we refer as the “field point”, and to which we assign indices

*α*,

*β*, etc. We assume that

*x*belongs to \({\mathcal N}({x{\prime}})\), the

*normal convex neighbourhood*of

*x*′; this is the set of points that are linked to

*x*′ by a

*unique*geodesic. The geodesic

*β*that links

*x*to

*x*′ is described by relations

*z*

^{μ}(λ) in which λ is an affine parameter that ranges from λ

_{0}to λ

_{1}; we have

*z*(λ

_{0}) ≡

*x*′ and

*z*(λ

_{1}) ≡

*x*. To an arbitrary point

*z*on the geodesic we assign indices

*μ*,

*ν*, etc. The vector

*t*

^{μ}=

*dz*

^{μ}/

*d*λ is tangent to the geodesic, and it obeys the geodesic equation

*Dt*

^{μ}/

*dλ*= 0. The situation is illustrated in Figure 5.

*x*′ and the field point

*x*. It is defined by

*β*that links

*x*to

*x*′. You may notice that

*σ*is invariant under a constant rescaling of the affine parameter, \(\lambda \rightarrow \bar \lambda = a\lambda + b\), where

*a*and

*b*are constants.

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 ½*ε*(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(Δ*τ*)^{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(Δ*s*)^{2}. If the geodesic is null, then *σ* = 0. Quite generally, therefore, the world function is half the squared geodesic distance between the points *x*′ and *x*.

In flat spacetime, the geodesic linking *x* to *x*′ is a straight line, and σ = ½*η*_{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 rank-2 tensor at *x* and a scalar at *x*′. Because *σ* is a scalar at *x*, we have that this tensor is symmetric: *σ*_{βα} = *σ*_{αβ}. Similarly, we let \({\sigma _{\alpha {\beta {\prime}}}} \equiv {\partial _{{\beta {\prime}}}}{\sigma _\alpha} = {\partial ^2}\sigma/\partial {x^{{\beta {\prime}}}}\partial {x^\alpha}\) 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 rank-2 tensor at *x*′ and a scalar at *x*.

The notation is easily extended to any number of derivatives. For example, we let *σ*_{σαβγ′} ≡ ∇_{δ′} ∇_{γ}∇_{β}∇_{α}*σ*, which is a rank-3 tensor at *x* and a dual vector at *x*′. This bitensor is symmetric in the pair of indices *α* and *β*, but not in the pairs *α* and *γ*, nor *β* and *γ*. Because ∇_{δ′} is here an ordinary partial derivative with respect to *x*′, the bitensor is symmetric in any pair of indices involving *δ*′. The ordering of the primed index relative to the unprimed indices is therefore irrelevant: The same bitensor can be written as *σ*_{δ′αβγ} or *σ*_{αδ′βγ} or *σ*_{αβδ′γ}, making sure that the ordering of the unprimed indices is not altered.

_{…}(

*x*,

*x*′) satisfy the property

_{…;αβ′}with respect to the pair

*α*and

*β*′. This is most easily done by adopting Fermi normal coordinates (see Section 3.2) adapted to the geodesic

*β*, and setting the connection to zero both at

*x*and

*x*′. In these coordinates, the bitensor Ω

_{…;α}is the partial derivative of Ω… with respect to

*x*

^{α}, and Ω

_{…;αβ′}is obtained by taking an additional partial derivative with respect to

*x*

^{β′}. These two operations commute, and Ω

_{…;β′α}= Ω

_{…;αβ′}follows as a bitensorial identity. Equation (54) then follows by further differentiation with respect to either

*x*or

*x*′.

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}) = δ*x*.

_{1}− λ

_{0}, an overdot indicates differentiation with respect to λ, and the metric and its derivatives are evaluated on

*β*. Integrating the first term by parts gives

*z*

^{μ}(λ) satisfies the geodesic equation. The boundary term at λ

_{0}is zero because the variation

*δz*

^{μ}vanishes there. We are left with

*δσ*= Δ

*λg*

_{αβ}

*t*

^{α}

*δx*

^{β}, or

*x*. Apart from a factor Δλ, we see that

*σ*

^{α}(

*x*,

*x*′) is equal to the geodesic’s tangent vector at

*x*. If in Equation (55) we replace

*x*by a generic point

*z*(λ) on

*β*, and if we correspondingly replace λ

_{1}by λ, we obtain

*σ*

^{μ}(

*z*,

*x*′) = (λ − λ

_{0})

*t*

^{μ}; we therefore see that

*σ*

^{μ}(

*z, x*

^{′}) is a rescaled tangent vector on the geodesic.

*σ*varies under a change of base point

*x*′. Here the variation of the geodesic is such that

*δz*(λ

_{0}) =

*δx*′ and

*δz*(λ

_{1}) =

*δx*= 0, and we obtain \(\delta \sigma = - \Delta \lambda {g_{{\alpha {\prime}}{\beta {\prime}}}}{t^{{\alpha {\prime}}}}\delta {x^{{\beta {\prime}}}}\). This shows that

*x*′. Apart from a factor Δλ, we see that

*σ*

^{α′}(

*x*,

*x*′) is minus the geodesic’s tangent vector at

*x*′.

*σ*

_{α}. According to Equation (55) we have

*g*

_{αβ}

*σ*

^{α}

*σ*

^{β}= (Δλ)

^{2}

*g*

_{αβ}

*t*

^{α}

*t*

^{β}= (Δλ)

^{2}

*ε*. According to Equation (53), this is equal to 2

*σ*. We have obtained

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 ½*g*^{αβ}*σ*_{α}*σ*_{β} = *σ*. Suppose that we are presented with such a scalar field. What can we say about it?

*σ*

^{α}≡

*σ*

^{;α}satisfies

*σ*< 0, they are spacelike where

*σ*> 0, and they are null where

*σ*= 0. Here, for concreteness, we shall consider only the timelike subset of the congruence.

*u*

^{α}

_{;β}

*u*

^{β}= 0. The parameter λ is then proper time

*τ*. If λ⇒t denotes the original parameterization of the geodesics, we have that

*d*λ⇒t/

*dτ*= ∣2

*σ*∣

^{−1/2}, and we see that the original parameterization is singular at

*σ*= 0.

*θ*⇒t = (

*δV*)

^{−1}(

*d/d*λ⇒t)(

*δV*) is the expansion in the original parameterization (

*δV*is the congruence’s cross-sectional volume). While

*θ*⇒t is well behaved in the limit

*σ*→ 0 (we shall see below that

*θ*⇒t → 3), we have that

*θ*→ ∞. This means that the point

*x*′ at which

*σ*= 0 is a caustic of the congruence: All geodesics emanate from this point.

These considerations, which all follow from a postulated relation ½*g*^{αβ}*σ*_{α}*σ*_{β} = *σ*, are clearly compatible with our preceding explicit construction of the world function.

### 2.2 Coincidence limits

*σ*… as

*x*approaches

*x*′. We introduce the notation

_{…}(

*x*,

*x*′) as

*x*approaches

*x*′; this is called the

*coincidence limit*of the bitensor. We assume that the coincidence limit is a unique tensorial function of the base point

*x*′, independent of the direction in which the limit is taken. In other words, if the limit is computed by letting λ → λ

_{0}after evaluating Ω… (

*z*,

*x*′) as a function of λ on a specified geodesic

*β*, it is assumed that the answer does not depend on the choice of geodesic.

#### 2.2.1 Computation of coincidence limits

*σ*

_{γ}=

*g*

^{αβ}

*σ*

_{α}

*σ*

_{βγ}=

*σ*

^{β}

*σ*

_{βγ}, or (

*g*

_{βγ}−

*σ*

_{βγ})

*t*

^{β}= 0 after using Equation (55). From the assumption stated in the preceding paragraph,

*σ*

_{βγ}becomes independent of

*t*

^{β}in the limit

*x*→

*x*′, and we arrive at [

*σ*

_{αβ}] =

*g*

_{α′β′}. By very similar calculations we obtain all other coincidence limits for the second derivatives of the world function. The results are

*θ*⇒t] = 3, where

*θ*⇒t was defined in Equation (61).

*σ*

_{αβ′}] = [

*σ*

_{α}]

_{;β′}− [

*σ*

_{αβ}], and since the coincidence limit of

*σ*

_{α}is zero, this gives us [

*σ*

_{αβ′}] = − [

*σ*

_{αβ}] = −

*g*

_{α′β′}, as was stated in Equation (63). Similarly, [

*σ*

_{α′β′}] = [

*σ*

_{α′}]

_{;β′}− [

*σ*

_{α′β}] = −[

*σ*

_{βα′}] =

*g*

_{α′β′}. The results of Equation (63) can thus all be generated from the known result for [

*σ*

_{αβ}].

*σ*

_{γαβ}] + [

*σ*

_{βαγ}] = 0 if we recognize that the operations of raising or lowering indices and taking the limit

*x*→

*x*′ commute. Noting the symmetries of

*σ*

_{αβ}, this gives us [

*σ*

_{αγβ}] + [

*σ*

_{αβγ}] = 0, or \(2[{\sigma _{\alpha \beta \gamma}}] - [{R^\delta}_{\alpha \beta \gamma}{\sigma _\delta}] = 0,\,{\rm{or}}\,{\rm{2[}}{\sigma _{\alpha \beta \gamma}}] = {R^{{\delta {\prime}}}}_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}}[{\sigma _{{\delta {\prime}}}}]\). Since the last factor is zero, we arrive at

*σ*

_{αβγ}] = 0 by employing Synge’s rule.

*σ*

_{σαβγ}] + [

*σ*

_{σαδβ}] + [

*σ*

_{αγβδ}] = 0. To simplify the third term we differentiate Ricci’s identity \({\sigma _{\alpha \gamma \beta}} = {\sigma _{\alpha \beta \gamma}} - {R^\epsilon}_{\alpha \beta \gamma}{\sigma _\epsilon}\) with respect to

*x*

^{δ}and then take the coincidence limit. This gives us \([{\sigma _{\alpha \gamma \beta \delta}}] = [{\sigma _{\alpha \beta \gamma \delta}}] + {R_{{\alpha {\prime}}{\delta {\prime}}{\beta {\prime}}{\gamma {\prime}}}}\). The same manipulations on the second term give \([{\sigma _{\alpha \delta \beta \gamma}}] = [{\sigma _{\alpha \beta \delta \gamma}}] + {R_{{\alpha {\prime}}{\gamma {\prime}}{\beta {\prime}}{\delta {\prime}}}}\). Using the identity \({\sigma _{\alpha \beta \delta \gamma}} = {\sigma _{\alpha \beta \gamma \delta}} - {R^\epsilon}_{\alpha \gamma \delta}{\sigma _{\epsilon \beta}} - {R^\epsilon}_{\beta \gamma \delta}{\sigma _{\alpha \epsilon}}\) and the symmetries of the Riemann tensor, it is then easy to show that [

*σ*

_{σαβδ}] = [

*σ*

_{σαβγ}]. Gathering the results, we obtain \(3[{\sigma _{\alpha \beta \gamma \delta}}] + {R_{{\alpha {\prime}}{\gamma {\prime}}{\beta {\prime}}{\delta {\prime}}}} + {R_{{\alpha {\prime}}{\delta {\prime}}{\beta {\prime}}{\gamma {\prime}}}} = 0\), and Synge’s rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are

#### 2.2.2 Derivation of Synge’s rule

*any*bitensor Ω

_{AB′}(

*x*,

*x*′) in which

*A*=

*α*…

*β*is a multi-index that represents any number of unprimed indices, and

*B*′ =

*γ*′…

*δ*′ a multi-index that represents any number of primed indices. (It does not matter whether the primed and unprimed indices are segregated or mixed.) On the geodesic

*β*that links

*x*to

*x*′ we introduce an ordinary tensor

*P*

^{M}(

*z*) where

*M*is a multi-index that contains the same number of indices as

*A*. This tensor is arbitrary, but we assume that it is parallel transported on

*β*; this means that it satisfies

*P*

^{A}

_{;α}

*t*

^{α}= 0 at

*x*. Similarly, we introduce an ordinary tensor

*Q*

^{N}(

*z*) in which

*N*contains the same number of indices as

*B*′. This tensor is arbitrary, but we assume that it is parallel transported on

*β*; at

*x*′ it satisfies \({Q^{{B{\prime}}}}_{;{\alpha {\prime}}}{t^{{\alpha {\prime}}}} = 0\). With Ω,

*P*, and

*Q*we form a biscalar

*H*(

*x, x*′) defined by

*x*to

*x*′, we can consider

*H*to be a function of λ

_{0}and λ

_{1}. If λ

_{1}is not much larger than λ

_{0}(so that

*x*is not far from

*x*′), we can express

*H*(λ

_{1}, λ

_{0}) as

*H*(λ

_{1}, λ

_{1}) −

*H*(λ

_{0}, λ

_{0})]/(λ

_{1}− λ

_{0}) when λ

_{1}→ λ

_{0}. The partial derivative of

*H*with respect to λ

_{0}is equal to \({\Omega _{A{B{\prime}};{\alpha {\prime}}}}{t^{{\alpha {\prime}}}}{P^A}{Q^{{B{\prime}}}}\), and in the limit this becomes \([{\Omega _{A{B{\prime}};{\alpha {\prime}}}}]{t^{{\alpha {\prime}}}}{P^{{A{\prime}}}}{Q^{{B{\prime}}}}\). Similarly, the partial derivative of

*H*with respect to λ

_{1}is Ω

_{AB′;α}

*t*

^{α}

*P*

^{A}

*Q*

^{B′}, and in the limit λ

_{1}λ

_{0}this becomes [Ω

_{AB′;α}]

*t*

^{α′}

*P*

^{A′}

*Q*

^{B′}. Finally,

*H*(λ

_{0}, λ

_{0}) = [Ω

_{AB′}]

*P*

^{A′}

*Q*

^{B′}, and its derivative with respect to λ

_{0}is [Ω

_{AB′}]

_{;α′}

*t*

^{α′}

*P*

^{A′}

*Q*

^{B′}. Gathering the results we find that

*P*

^{M}and

*Q*

^{N}, and the direction of the selected geodesic

*β*, are all arbitrary. Equation (67) reduces to Equation (64) when

*σ*… is substituted in place of Ω

_{AB′}.

### 2.3 Parallel propagator

#### 2.3.1 Tetrad on *β*

*β*that links

*x*to

*x*′ we introduce an orthonormal basis \(e_{\rm{a}}^\mu (z)\) that is parallel transported on the geodesic. The frame indices

^{3}a, b, …, run from 0 to 3, and the frame vectors satisfy

*η*

_{ab}= diag(−1, 1, 1, 1) is the Minkowski metric (which we shall use to raise and lower frame indices). We have the completeness relations

*β*. In terms of the dual tetrad the completeness relations take the form

*β*. In particular, with an appropriate change of notation they hold at

*x′*and

*x*; for example, \({g_{\alpha \beta}} = {\eta _{{\rm{ab}}}}e_\alpha ^{\rm{a}}e_\beta ^{\rm{b}}\) is the metric at

*x*.

#### 2.3.2 Definition and properties of the parallel propagator

*A*

^{μ}(

*z*) on

*β*can be decomposed in the basis \(e_{\rm{a}}^\mu : {A^\mu} = {A^{\rm{a}}}e_{\rm{a}}^\mu\), and the vector’s frame components are given by \({A^{\rm{a}}} = {A^\mu}e_\mu ^{\rm{a}}\). If

*A*

^{μ}is parallel transported on the geodesic, then the coefficients

*A*

^{a}are constants. The vector at

*x*can then be expressed as \({A^\alpha} = ({A^{{\alpha {\prime}}}}e_{{\alpha {\prime}}}^{\rm{a}})e_{\rm{a}}^\alpha\), or

*parallel propagator*: It takes a vector at

*x*′ and parallel-transports it to

*x*along the unique geodesic that links these points.

*x*and parallel-transports it back to

*x*′. Clearly,

*β*. Then the frame components \({p_{\rm{a}}} = {p_\mu}e_{\rm{a}}^\mu\) are constants, and the dual vector at

*x*can be expressed as \({p_\alpha} = ({p_{{\alpha {\prime}}}}e_{\rm{a}}^{{\alpha {\prime}}})e_{\rm{a}}^\alpha\), or

*x*′ and parallel-transports it to

*x*. As another example, it is easy to show that a tensor \({A^{\alpha \beta}}\) at

*x*obtained by parallel transport from

*x*′ must be given by

*β*, and a special case of Equation (77) is therefore \({g_{\alpha \beta}} = {g^{{\alpha {\prime}}}}_\alpha {g^{{\beta {\prime}}}}_\beta {g_{{\alpha {\prime}}{\beta {\prime}}}}\).

*β*, they satisfy \(e_{{\rm{a;}}\beta}^\alpha {\sigma ^\beta} = 0\) at

*x*and \(e_{{\rm{a;}}{\beta {\prime}}}^{{\alpha {\prime}}}{\sigma ^{{\beta {\prime}}}} = 0\) at

*x*′. This immediately implies that the parallel propagators must satisfy

*t*

^{μ}=

*dz*

^{μ}/

*dλ*is tangent to the geodesic connecting

*x*to

*x*′, then \({t^\alpha} = {g^\alpha}_{{\alpha {\prime}}}{t^{{\alpha {\prime}}}}\). Using Equations (55) and (56), this observation gives us the relations

#### 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 flat-spacetime quantity (*x* − *x*′)^{α}. For concreteness we shall consider the case of rank-2 bitensor, and for the moment we will assume that the bitensor’s indices all refer to the base point *x*′.

*A*

_{α′β′},

*A*

_{α′β′γ′}, and

*A*

_{α′β′γ′δ′}are all ordinary tensors at

*x*′; this last tensor is symmetric in the pair of indices

*γ*′ and

*δ*′, and

*ϵ*measures the size of a typical component of

*σ*

^{α′}.

_{α′β′}(

*x*,

*x*′) to second order in

*ϵ*.

_{α′β}, with one index referring to

*x*′ and the other to

*x*. The previous procedure can be applied directly if we introduce an auxiliary bitensor \({{\tilde \Omega}_{{\alpha {\prime}}{\beta {\prime}}}} \equiv {g^\beta}_{{\beta {\prime}}}{\Omega _{{\alpha {\prime}}\beta}}\) whose indices all refer to the point

*x*′. Then \({{\tilde \Omega}_{{\alpha {\prime}}{\beta {\prime}}}}\) can be expanded as in Equation (83), and the original bitensor is reconstructed as \({\Omega _{{\alpha {\prime}}\beta}} \equiv {g^{{\beta {\prime}}}}_\beta {{\tilde \Omega}_{{\alpha {\prime}}{\beta {\prime}}}}\), or

_{α′β}by substituting the relation \({{\tilde \Omega}_{{\alpha {\prime}}{\beta {\prime}}}} = {g^\beta}_{{\beta {\prime}}}{\Omega _{{\alpha {\prime}}\beta}}\) and its derivatives. After using the results of Equation (80, 81, 82) we find

*B*

_{α′β′γ′δ′}.

_{αβ}, whose indices all refer to

*x*. Much as we did before, we introduce an auxiliary bitensor \({{\tilde \Omega}_{{\alpha {\prime}}{\beta {\prime}}}} \equiv {g^\beta}_{{\beta {\prime}}}{\Omega _{{\alpha {\prime}}\beta}}\) whose indices all refer to

*x*′, we expand \({{\tilde \Omega}_{{\alpha {\prime}}{\beta {\prime}}}}\) as in Equation (83), and we then reconstruct the original bitensor. This gives us

*C*

_{α′β′γ′δ′}.

#### 2.4.2 Special cases

*σ*

_{α′β′},

*σ*

_{α′β}, and

*σ*

_{αβ}. In the first instance we have

*A*

_{α′β′}=

*g*

_{α′β′},

*A*

_{α′β′γ′}= 0, and \({A_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}{\delta {\prime}}}} = - {1 \over 3}({R_{{\alpha {\prime}}{\gamma {\prime}}{\beta {\prime}}{\delta {\prime}}}} + {R_{{\alpha {\prime}}{\delta {\prime}}{\beta {\prime}}{\gamma {\prime}}}})\). In the second instance we have \({B_{{\alpha {\prime}}{\beta {\prime}}}} = - {g_{{\alpha {\prime}}{\beta {\prime}}}},{B_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}}} = 0\), and \({B_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}{\delta {\prime}}}} = - {1 \over 3}({R_{{\beta {\prime}}{\alpha {\prime}}{\gamma {\prime}}{\delta {\prime}}}} + {R_{{\beta {\prime}}{\gamma {\prime}}{\alpha {\prime}}{\delta {\prime}}}}) - {1 \over 2}{R_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}{\delta {\prime}}}} = {1 \over 3}{R_{{\alpha {\prime}}{\delta {\prime}}{\beta {\prime}}{\gamma {\prime}}}} - {1 \over 6}{R_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}{\delta {\prime}}}}\). In the third instance we have \({C_{{\alpha {\prime}}{\beta {\prime}}}} = {g_{{\alpha {\prime}}{\beta {\prime}}}},{C_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}}} = 0\), and \({C_{{\alpha {\prime}}{\beta {\prime}}{\gamma {\prime}}{\delta {\prime}}}} = - {1 \over 3}({R_{{\alpha {\prime}}{\gamma {\prime}}{\beta {\prime}}{\delta {\prime}}}} + {R_{{\alpha {\prime}}{\delta {\prime}}{\beta {\prime}}{\gamma {\prime}}}})\). This gives us the expansions

*x*

^{′}. Equation (91) reveals that timelike geodesics are focused if the Ricci tensor is nonzero and the strong energy condition holds: When

*R*

_{α′β′}

*σ*

^{α′}

*σ*

^{β′}> 0 we see that

*θ*⇒t is smaller than 3, the value it would take in flat spacetime.

#### 2.4.3 Expansion of tensors

*A*

_{αβ}at a point

*x*in terms of its values (and that of its covariant derivatives) at a neighbouring point

*x*

^{′}. The tensor can be written as an expansion in powers of −

*σ*

^{α′}(

*x*,

*x*

^{′}), and in this case we have

*β*that links

*x*to

*x*′, then Equation (93) reduces to Equation (77). The extension of this formula to tensors of other ranks is obvious.

*A*

_{μν}(

*z*), the restriction of the tensor field on

*β*, in terms of its tetrad components \({A_{{\rm{ab}}}}(\lambda) = {A_{\mu \nu}}e_{\rm{a}}^\mu e_{\rm{b}}^\nu\). Recall from Section 2.3.1 that \(e_{\rm{a}}^\mu\) is an orthonormal basis that is parallel transported on

*β*; recall also that the affine parameter

*λ*ranges from

*λ*

_{0}(its value at

*x*′) to

*λ*

_{1}(its value at

*x*). We have \({A_{{\alpha {\prime}}{\beta {\prime}}}}({x{\prime}}) = {A_{{\rm{ab}}}}({\lambda _0})e_{{\alpha {\prime}}}^{\rm{a}}e_{{\beta {\prime}}}^{\rm{b}}{A_{\alpha \beta}}(x) = {A_{{\rm{ab}}}}({\lambda _1})e_\alpha ^{\rm{a}}e_\beta ^{\rm{b}}\), and

*A*

_{ab}(

*λ*

_{1}) can be expressed in terms of quantities at

*λ*=

*λ*

_{0}by straightforward Taylor expansion. Since, for example,

### 2.5 Van Vleck determinant

#### 2.5.1 Definition and properties

*x*,

*x*′) is defined by

*g*is the metric determinant at

*x*and

*g*′ the metric determinant at

*x*′.

*, \(\det (1 + a) = 1 + {\rm{tr(}}a{\rm{) + {\mathcal O}(}}{a^2})\).*

**a***x*′. If

*θ*⇒t < 3, then the congruence expands less rapidly than it would in flat spacetime, and Δ

*increases*along the geodesics. If, on the other hand,

*θ*⇒t > 3, then the congruence expands more rapidly than it would in flat spacetime, and Δ

*decreases*along the geodesics. Thus, Δ > 1 indicates that the geodesics are undergoing focusing, while Δ < 1 indicates that the geodesics are undergoing defocusing. The connection between the van Vleck determinant and the strong energy condition is well illustrated by Equation (97): The sign of Δ − 1 near

*x*′ is determined by the sign of

*R*

_{α′β′}

*σ*

^{α′}

*σ*

^{β′}.

#### 2.5.2 Derivations

**g**^{−1}=

**EηE**^{T}, where

*denotes the 4 × 4 matrix whose entries correspond to \(e_{\rm{a}}^\alpha\). (In this translation we put tensor and frame indices on equal footing.) With*

**E***e*denoting the determinant of this matrix, we have 1/

*g*= −

*e*

^{2}, or \(e = 1/\sqrt {- g}\). Similarly, we rewrite the completeness relations at

*x*′, \({g^{{\alpha {\prime}}{\beta {\prime}}}} = {\eta ^{{\rm{ab}}}}e_{\rm{a}}^{{\alpha {\prime}}}e_{\rm{b}}^{{\beta {\prime}}}\), in the matrix form

*′*

**g**^{−1}=

*′*

**E***′*

**ηE**^{T}, where

*′ is the matrix corresponding to \(e_{\rm{a}}^{{\alpha {\prime}}}\). With*

**E***e*′ denoting its determinant, we have 1/

*g*′ = −

*e*′

^{2}, or \({e{\prime}} = 1/\sqrt {- {g{\prime}}}\). Now, the parallel propagator is defined by \({g^\alpha}_{{\alpha {\prime}}} = {\eta ^{{\rm{ab}}}}{g_{{a{\prime}}{\beta {\prime}}}}e_{\rm{a}}^\alpha e_{\rm{b}}^{{\beta {\prime}}}\), and the matrix form of this equation is

*=*

**ĝ***′*

**EηE**^{T}

*′*

**g**^{T}. The determinant of the parallel propagator is therefore \(\hat g = - e{e{\prime}}{g{\prime}} = \sqrt {- {g{\prime}}}/\sqrt {- g}\). So we have

**ĝ**^{−1}

**g**^{−1}

*, where*

**σ***is the matrix corresponding to*

**σ***σ*

_{αβ′}.

*σ*= ½

*σ*

^{γ}

*σ*

_{γ}twice and obtain \({\sigma _{\alpha {\beta {\prime}}}} = {\sigma ^\gamma}_{^\alpha}{\sigma _{\gamma {\beta {\prime}}}} + {\sigma ^\gamma}{\sigma _{\gamma \alpha {\beta {\prime}}}}\). If we replace the last factor by

*σ*

_{αβ′γ}and multiply both sides by −

*g*

^{α′α}we find

*δ*ln det

*= tr(*

**M**

**M**^{−1}

*δ*

*), which relates the variation of a determinant to the variation of the matrix elements. It implies, in particular, that \({({\Delta ^{- 1}})^{{\beta {\prime}}}}_{{\alpha {\prime}}}{\Delta ^{{\alpha {\prime}}}}_{{\beta {\prime}};\gamma} = {(\ln \Delta)_{,\gamma}}\), and we finally obtain*

**M**## 3 Coordinate Systems

### 3.1 Riemann normal coordinates

#### 3.1.1 Definition and coordinate transformation

*x*′ and a tetrad \(e_{\rm{a}}^{{\alpha {\prime}}}({x{\prime}})\), we assign to a neighbouring point

*x*the four coordinates

*x*′. The new coordinates \({{\hat x}^{\rm{a}}}\) are called

*Riemann normal coordinates*(RNC), and they are such that \({\eta _{{\rm{ab}}}}{{\hat x}^{\rm{a}}}{{\hat x}^{\rm{b}}} = {\eta _{{\rm{ab}}}}e_{{\alpha {\prime}}}^{\rm{a}}e_{{\beta {\prime}}}^{\rm{b}}{\sigma ^{{\alpha {\prime}}}}{\sigma ^{{\beta {\prime}}}} = {g_{{\alpha {\prime}}{\beta {\prime}}}}{\sigma ^{{\alpha {\prime}}}}{\sigma ^{{\beta {\prime}}}}\), or

*x*and the base point

*x*′. It is obvious that x′ is at the origin of the RNC, where \({{\hat x}^{\rm{a}}} = 0\).

*x*to

*x*+

*δx*, the new coordinates change to \({{\hat x}^{\rm{a}}} + \delta {{\hat x}^{\rm{a}}} = - e_{{\alpha {\prime}}}^{\rm{a}}{\sigma ^{{\alpha {\prime}}}}(x + \delta x,{x{\prime}}) = {{\hat x}^{\rm{a}}} - e_{{\alpha {\prime}}}^{\rm{a}}{\sigma ^{{\alpha {\prime}}}}_\beta \delta {x^\beta}\), so that

*g*is the determinant of the metric in the original coordinates, and Δ(

*x*,

*x*′) is the van Vleck determinant of Equation (95). This result follows simply by writing the coordinate transformation in the form \(\partial {{\hat x}^{\rm{a}}}/\partial {x^\beta} = - {\eta ^{{\rm{ab}}}}e_{\rm{b}}^{{\alpha {\prime}}}{\sigma _{{\alpha {\prime}}\beta}}\) and computing the product of the determinants. It allows us to deduce that in the RNC, the determinant of the metric is given by

*x*′ are straight lines in the RNC. The proper volume of a small comoving region is then equal to \(dV = {\Delta ^{- 1}}{d^4}\hat x\), and this is smaller than the flat-spacetime value of \({d^4}\hat x\) if Δ > 1, that is, if the geodesics are focused by the spacetime curvature.

#### 3.1.2 Metric near *x*′

*x*′. We recall the expansion of Equation (89),

*x*obtained by parallel transport of \(e_{{\beta {\prime}}}^{\rm{b}}({x{\prime}})\). After some algebra we obtain

*x*in the RNC.

*R*

_{acbd}appearing in Equation (110) are the frame components of the Riemann tensor evaluated at the base point

*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 Fermi-Walker transport

Let *γ* be a timelike curve described by parametric relations *z*^{μ}(*τ*) in which *τ* is proper time. Let *u*^{μ} = *dz*^{μ}/*dτ* be the curve’s normalized tangent vector, and let *α*^{μ} = *Du*^{μ}/*dτ* be its acceleration vector.

*v*

^{μ}is said to be

*Fermi-Walker transported*on

*γ*if it is a solution to the differential equation

*a*

^{μ}= 0 and

*γ*is a geodesic.

The operation of Fermi-Walker (FW) transport satisfies two important properties. The first is that *u*^{μ} is automatically FW transported along *γ*; this follows at once from 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*

*γ*. At this point we erect an orthonormal tetrad \(({u^{\bar \mu}},e_a^{\bar \mu})\) where, contrary to former usage, the frame index

*a*runs from 1 to 3. We then propagate each frame vector on

*γ*by FW transport; this guarantees that the tetrad remains orthonormal everywhere on

*γ*. At a generic point

*z*(

*τ*) we have

*γ*we define a dual tetrad \((e_\mu ^0,e_\mu ^a)\) by the relations

*γ*. The tetrad and its dual give rise to the completeness relations

#### 3.2.3 Fermi normal coordinates

*x*in the normal convex neighbourhood of

*γ*, we locate the unique

*spacelike geodesic β*that passes through

*x*and intersects

*γ orthogonally*. We denote the intersection point by \(\bar x \equiv z(t)\), with

*t*denoting the value of the proper-time parameter at this point. To tensors at \({\bar x}\) we assign indices \(\bar \alpha, \bar \beta\) and so on. The FNC of

*x*are defined by

*β*at \({\bar x}\), be orthogonal to \({u^{\bar \alpha}}\), the vector tangent to

*γ*. From the definition of the FNC and the completeness relations of Equation (115) it follows that

*s*is the spatial distance between \({\bar x}\) and

*x*along the geodesic

*β*. This statement gives an immediate meaning to \({{\hat x}^a}\), the spatial Fermi normal coordinates; and the time coordinate \({{\hat x}^0}\) is simply proper time at the intersection point \({\bar x}\). The situation is illustrated in Figure 6.

*x*+

*δx*. This typically induces a change in the spacelike geodesic

*β*, which moves to

*β*+

*δβ*, and a corresponding change in the intersection point \({\bar x}\), which moves to \(x\prime\prime \equiv \bar x + \delta \bar x\), with \(\delta {x^{\bar \alpha}} = {u^{\bar \alpha}}\delta t\). The FNC of the new point are then \({{\hat x}^0}(x + \delta x) = t + \delta t\) and \({{\hat x}^a}(x + \delta x) = - e_{^{{\alpha ^{{\prime\prime}}}}}^a({x^{{\prime\prime}}}){\sigma ^{{\alpha ^{{\prime\prime}}}}}(x + \delta x,{x^{{\prime\prime}}})\), with

*x*″ determined by

*σ*

_{α″}(

*x*+

*δx*,

*x*″)

*u*

^{α″}(

*x*″) = 0. Expanding these relations to first order in the displacements, and simplifying using Equations (113), yields

*μ*is determined by \({\mu ^{- 1}} = - ({\sigma _{\bar \alpha \bar \beta}}{u^{\bar \alpha}}{u^{\bar \beta}} + {\sigma _{\bar \alpha}}{a^{\bar \alpha}})\).

#### 3.2.4 Coordinate displacements near *γ*

*s*, the spatial distance from \({\bar x}\) to

*x*. For this we use the expansions of Equations (88) and (89), in which we substitute \({\sigma ^{\bar \alpha}} = - e_a^{\bar \alpha}{{\hat x}^a}\), and \({g^{\bar \alpha}}_\alpha = {u^{\bar \alpha}}\bar e_\alpha ^0 + e_a^{\bar \alpha}\bar e_\alpha ^a\) where \((\bar e_\alpha ^0,\bar e_\alpha ^a)\) is a dual tetrad at

*x*obtained by parallel transport of \((- {u_{\bar \alpha}},e_{\bar \alpha}^a)\) on the spacelike geodesic

*β*. After some algebra we obtain

*γ*. This last result is easily inverted to give

*γ*. (Note that frame indices are raised with

*δ*

^{ab}.)

*x*coincides with the dual tetrad \((- {u_{\bar \alpha}},e_{\bar \alpha}^a)\) at \({\bar x}\). It follows that on

*γ*, the transformation matrix between the original coordinates

*x*

^{α}and the FNC \((t,{{\hat x}^a})\) is formed by the Fermi-Walker transported tetrad:

*a*

_{a}(

*t*) are also the

*components*of the acceleration vector in the FNC; orthogonality between \({u^{\bar \alpha}}\) and \({a^{\bar \alpha}}\) means that

*a*

_{0}= 0. Similarly,

*R*

_{0c0d}(

*t*),

*R*

_{0cbd}(

*t*), and

*R*

_{acbd}(

*t*) are the

*components*of the Riemann tensor (evaluated on

*γ*) in the Fermi normal coordinates.

#### 3.2.5 Metric near *γ*

*x*. They can also be used to compute the metric at

*x*, after invoking the completeness relations \({g_{\alpha \beta}} = - \bar e_\alpha ^0\bar e_\beta ^0 + {\delta _{ab}}\bar e_\alpha ^a\bar e_\beta ^b\). This gives

*γ*in the Fermi normal coordinates. Recall that

*a*

_{a}(

*t*) are the components of the acceleration vector of

*γ*— the timelike curve described by \({{\hat x}^a} = 0\) — while

*R*

_{0c0d}(

*t*),

*R*

_{0cbd}(t), and

*R*

_{acbd}(

*t*) are the components of the Riemann tensor evaluated on

*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 Thorne-Hartle coordinates

*Thorne-Hartle coordinates*\((t,{{\hat y}^a})\) brings the metric to the form

_{tt}and

*g*

_{ta}unchanged, but that it diagonalizes

*g*

_{ab}. This metric was first displayed in [58] and the coordinate transformation was later produced by Zhang [64].

*R*

_{acbd}of the Riemann tensor are completely determined by \({{\mathcal E}_{ab}} \equiv {R_{0a0b}}\), which in this special case is a symmetric-tracefree tensor. To prove this we invoke the completeness relations of Equation (115) and take frame components of Equation (128). This produces the three independent equations

*δ*

^{ab}

*δ*

^{cd}

*R*

_{acbd}= 0, and this implies that

*R*

_{acbd}has five independent components. Since this is also the number of independent components of \({{\mathcal E}_{ab}}\), we see that the first equation can be inverted −

*R*

_{acbd}can be expressed in terms of \({{\mathcal E}_{ab}}\). A complete listing of the relevant relations is \({R_{1212}} = {{\mathcal E}_{11}} + {{\mathcal E}_{22}} = - {{\mathcal E}_{33}},{R_{1213}} = {{\mathcal E}_{23}},{R_{1223}} = - {{\mathcal E}_{13}},{R_{1313}} = {{\mathcal E}_{11}} + {{\mathcal E}_{33}} = - {{\mathcal E}_{22}},\,{R_{1323}} = {{\mathcal E}_{12}}\), and \({R_{2323}} = {{\mathcal E}_{22}} + {{\mathcal E}_{33}} = - {{\mathcal E}_{11}}\). These are summarized by

*δ*

^{cd}

*R*

_{0cad}= 0, together with the usual symmetries of the Riemann tensor, imply that

*R*

_{0cad}too possesses five independent components. These may thus be related to another symmetric-tracefree tensor \({{\mathcal B}_{ab}}\). We take the independent components to be \({R_{0112}} \equiv - {{\mathcal B}_{13}},\,{R_{0113}} \equiv {{\mathcal B}_{12}},\,{R_{0123}} \equiv - {{\mathcal B}_{11}},\,{R_{0212}} \equiv - {{\mathcal B}_{23}}\), and \({R_{0213}} \equiv {{\mathcal B}_{22}}\), and it is easy to see that all other components can be expressed in terms of these. For example, \({R_{0223}} = - {R_{0113}} = - {{\mathcal B}_{12}},\,{R_{0312}} = - {R_{0123}} + {R_{0213}} = {{\mathcal B}_{11}} + {{\mathcal B}_{22}} = - {{\mathcal B}_{33}},\,{R_{0313}} = - {R_{0212}} = {{\mathcal B}_{23}}\), and \({R_{0323}} = {R_{0112}} = - {{\mathcal B}_{13}}\). These relations are summarized by

*ε*

_{abc}is the three-dimensional permutation symbol. The inverse relation is \({{\mathcal B}^a}_b = {1 \over 2}{{\mathcal E}^{acd}}{R_{0bcd}}\).

*ŷ*

^{a}. This is given by

*g*

_{tt}nor

*g*

_{ta}at orders s and

*s*

^{2}. The remaining components of the metric, however, transform according to

*g*

_{ab}(THC) =

*g*

_{ab}(FNC) −

*ξ*

_{a;b}−

*ξ*

_{b;a}, where

### 3.3 Retarded coordinates

#### 3.3.1 Geometrical elements

*γ*described by relations

*z*

^{μ}(

*τ*), its normalized tangent vector

*u*

^{μ}=

*dz*

^{μ}/

*dτ*, and its acceleration vector

*a*

^{μ}=

*Du*

^{μ}/

*dτ*. We also have an orthonormal triad \(e_a^\mu\) that is transported on the world line according to

*ω*

_{ab}(

*τ)*= −

*ω*

_{ba}(

*τ*) is a prescribed rotation tensor. Here the triad is

*not*Fermi-Walker transported: For added generality we allow the spatial vectors to rotate as they are transported on the world line. While

*ω*

_{ab}will be set to zero in most sections of this paper, the freedom to perform such a rotation can be useful and will be exploited in Section 5.4. It is easy to check that Equation (138) is compatible with the requirement that the tetrad \(({u^\mu},e_a^\mu)\) be orthonormal everywhere on

*γ*. Finally, we have a dual tetrad \((e_\mu ^0,e_\mu ^a)\), with \(e_\mu ^0 = {\delta ^{ab}}{g_{\mu \nu}}e_b^\nu\) and \(e_\mu ^a = {\delta ^{ab}}{g_{\mu \nu}}e_b^\nu\). The tetrad and its dual give rise to the completeness relations

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 future-directed null geodesic that goes from the world line to *x*, and *x*′ = *z*(*u*) be the point at which *β* intersects the world line, with *u* denoting the value of the proper-time parameter at this point.

*x*′ we obtain another tetrad \((e_0^\alpha, e_a^\alpha)\) at

*x*by parallel transport on

*β*. By raising the frame index and lowering the vectorial index we also obtain a dual tetrad at \(x:e_\alpha ^0 = - {g_{\alpha \beta}}e_0^\beta\) and \(e_\alpha ^a = {\delta ^{ab}}{g_{\alpha \beta}}e_b^\beta\). The metric at

*x*can be then be expressed as

*x*′ to

*x*is given by

#### 3.3.2 Definition of the retarded coordinates

*x*′ and

*x*are linked by a null geodesic. From the fact that

*σ*

^{α′}is a null vector we obtain

*r*is a positive quantity by virtue of the fact that

*β*is a future-directed null geodesic — this makes

*σ*

^{α′}past-directed. In flat spacetime,

*σ*

^{α′}= − (

*x*−

*x*′)

^{α}, and in a Lorentz frame that is momentarily comoving with the world line,

*r*=

*t*−

*t*

^{′}> 0; with the speed of light set equal to unity,

*r*is also the spatial distance between

*x*′ and

*x*as measured in this frame. In curved spacetime, the quantity

*r*=

*u*

_{α′}

*σ*

^{α′}can still be called the

*retarded distance*between the point

*x*and the world line. Another consequence of Equation (142) is that

*δ*

_{ab}Ω

^{a}Ω

^{b}= 1.

*x*, the retarded coordinates change according to

*k*

_{α}=

*σ*

_{α}/

*r*is a future-directed null vector at

*x*that is tangent to the geodesic

*β*. To obtain these results we must keep in mind that a displacement of

*x*typically induces a simultaneous displacement of

*x*′ because the new points

*x*+

*δx*and

*x*′ +

*δx*′ must also be linked by a null geodesic. We therefore have 0 =

*σ*(

*x*+

*δx*,

*x*′ +

*δx*′) =

*σ*

_{α}

*δx*

^{α}+

*σ*

_{α′}

*δx*

^{α′}, and the first relation of Equation (145) follows from the fact that a displacement along the world line is described by

*δx*

^{α′}=

*u*

^{α′}

*δu*.

#### 3.3.3 The scalar field *r*(*x*) and the vector field *k*^{α}(*x*)

*x*′ linked to

*x*by the relation

*σ*(

*x*,

*x*′) = 0, then the quantity

*γ*. We can compute the gradient of

*r*by finding how

*r*changes under a displacement of

*x*(which again induces a displacement of

*x*′). The result is

*x*′. It is easy to check that this vector satisfies the identities

*x*is therefore obtained by parallel transport of \({u^{{\alpha {\prime}}}} + {\Omega ^a}e_a^{{\alpha {\prime}}}\) on

*β*. From this and Equation (141) we get the alternative expression

*k*

^{α}is a future-directed null vector field (recall that \({\Omega ^a} = {{\hat x}^a}/r\) is a unit vector).

*k*

_{α}can be computed by finding how the vector changes under a displacement of

*x*. (It is in fact easier to first calculate how

*rk*

_{α}changes, and then substitute our previous expression for

*∂*

_{β}

*r*.) The result is

*k*

^{α}satisfies the geodesic equation in affine-parameter form, \({k^\alpha}_{;\beta}{k^\beta} = 0\), and Equation (150) informs us that the affine parameter is in fact

*r*. A displacement along a member of the congruence is therefore given by

*dx*

^{α}=

*k*

^{α}

*dr*. Specializing to retarded coordinates, and using Equations (145) and (149), we find that this statement becomes

*du*= 0 and \(d{{\hat x}^a} = ({{\hat x}^a}/r)dr\), which integrate to

*u*= const and \({{\hat x}^a} = r{\Omega ^a}\), respectively, with Ω

^{a}still denoting a constant unit vector. We have found that the congruence of null geodesics emanating from

*x*′ is described by

*θ*

^{A}(

*A*= 1, 2) serve to parameterize the unit vector

*Ω*

^{α}, which is independent of

*r*.

*x*′. This result confirms that the congruence is singular at

*r*= 0, because

*θ*diverges as 2/

*r*in this limit; the caustic coincides with the point

*x*′.

*k*

^{α}is hypersurface orthogonal. This, together with the property that

*k*

^{α}satisfies the geodesic equation in affine-parameter form, implies that there exists a scalar field

*u*(

*x*) such that

*x*′. We conclude that the geodesics to which

*k*

^{α}is tangent are the generators of the null cone

*u*= const As Equation (154) indicates, a specific generator is selected by choosing a direction Ω

^{α}(which can be parameterized by two angles

*θ*

^{A}), and

*r*is an affine parameter on each generator. The geometrical meaning of the retarded coordinates is now completely clear; it is illustrated in Figure 7.

#### 3.3.4 Frame components of tensor fields on the world line

*x*in the retarded coordinates will be expressed in terms of frame components of vectors and tensors evaluated on the world line

*γ*. For example, if

*a*

^{α′}is the acceleration vector at

*x*′, then as we have seen,

*u*.

*γ*. From these we form the useful combinations

*θ*

^{A}only — they are independent of

*u*and

*r*.

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*

*x*are given by Equation (145). In these we substitute the standard expansions for

*σ*

_{α′β′}and

*σ*

_{α′β}, as given by Equations (88) and (89), as well as Equations (144) and (151). After a straightforward (but fairly lengthy) calculation, we obtain the following expressions for the coordinate displacements:

*du*is exact, but that \(d{{\hat x}^a}\) is expressed as an expansion in powers of

*r*.

#### 3.3.6 Metric near *γ*

*x*. The metric is then computed by using the completeness relations of Equation (140). We find

*r*= 0 still depends on the vector \({\Omega ^a} = {{\hat x}^a}/r\) that specifies the direction to the point

*x*. The retarded coordinates are therefore singular on the world line, and tensor components cannot be defined on

*γ*.

*S*

_{ab}=

*S*

_{a}=

*S*= 0 in Equations (171, 172, 173) we obtain the metric of flat spacetime in the retarded coordinates. This we express as

*g*

^{uu}and

*g*

^{ua}are exact, and they follow from the general relations

*g*

^{αβ}(

*d*

_{α}

*u*)(

*d*

_{β}

*u*) = 0 and

*g*

^{αβ}(

*∂*

_{α}

*u*)(∂

_{β}

*r*) = −1 that are derived from Equations (150) and (157).

*θ*is the expansion of the generators of the null cones

*u*= const.

#### 3.3.7 Transformation to angular coordinates

*δ*

_{ab}Ω

^{a}Ω

^{b}= 1, it can be parameterized by two angles

*θ*

^{A}. A canonical choice for the parameterization is Ω

^{a}= (sin

*θ*cos

*ϕ*, sin

*θ*sin

*ϕ*, cos

*θ*). It is then convenient to perform a coordinate transformation from \({{\hat x}^a}\) to (

*r*,

*θ*

^{A}), using the relations \({{\hat x}^a} = r{\Omega ^a}({\theta ^A})\). (Recall from Section 3.3.3 that the angles

*θ*

^{A}are constant on the generators of the null cones

*u*= const., and that

*r*is an affine parameter on these generators. The relations \({{\hat x}^a} = r{\Omega ^a}\) therefore describe the behaviour of the generators.) The differential form of the coordinate transformation is

_{ab}= diag(1, sin

^{2}

*θ*). We use the inverse of Ω

_{ab}, denoted Ω

^{AB}, to raise upper-case latin indices. We then define the new object

*α*and

*b*, orthogonal to Ω

_{a}and Ω

^{b}, and have the same trace.

*r*,

*θ*

^{A}) is accomplished by

*r*,

*θ*

^{A}) to \((r,{\theta ^A})\) is accomplished by

*g*

_{ru}= −1,

*g*

_{rr}= 0, and

*g*

_{rA}= 0 are exact, and they follow from the fact that in the retarded coordinates,

*k*

_{α}

*dx*

^{α}= −

*du*and

*k*

^{α}

*∂*

_{α}=

*∂*

_{r}.

*g*

^{uu}= 0,

*g*

^{ur}= −1, and

*g*

^{uA}= 0 are exact, and they follow from the same reasoning as before.

_{ab}; in the canonical parameterization, \(\sqrt \Omega = \sin \theta\).

#### 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.

*γ*, all frame components of the Riemann tensor can be expressed in terms of the symmetric-tracefree tensors \({{\mathcal E}_{ab}}(u)\) and \({{\mathcal B}_{ab}}(u)\). The relations are \({R_{a0b0}} = {{\mathcal E}_{ab}},\,{R_{a0bc}} = {\epsilon _{bcd}}{{\mathcal B}^d}_a\), and \({R_{acbd}} = {\delta _{ab}}{{\mathcal E}_{cd}} + {\delta _{cd}}{{\mathcal E}_{ab}} - {\delta _{ad}}{{\mathcal E}_{bc}} - {\delta _{bc}}{{\mathcal E}_{ad}}\). These can be substituted into Equations (160, 161, 162) to give

*u*is contained in \({{\mathcal E}_{ab}}\) and \({{\mathcal B}_{ab}}\), while the angular dependence is encoded in the unit vector Ω

^{a}.

*a*

_{a}and

*ω*

_{ab}are both set equal to zero — we obtain the compact expressions

### 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.

*x*refer to a point \(\bar x \equiv z(t)\) on

*γ*that is related to

*x*by a spacelike geodesic that intersects

*γ*orthogonally (see Figure 8). We refer to this point as

*x*’s

*simultaneous point*, and to tensors at \({\bar x}\) we assign indices \({\bar \alpha}\), \({\bar \beta}\) etc. We let (

*t*,

*sω*

^{a}) be the Fermi normal coordinates of

*x*, with

*t*denoting the value of

*γ*’s proper-time parameter at \(\bar x,s = \sqrt {2\sigma (x,\bar x)}\) representing the proper distance from \({\bar x}\) to

*x*along the spacelike geodesic, and

*ω*

^{a}denoting a unit vector (

*δ*

_{ab}

*ω*

^{a}

*ω*

^{b}= 1) that determines the direction of the geodesic. The Fermi normal coordinates are defined by \({\mathcal S}{\omega ^a} = - e_{\bar \alpha}^a{\sigma ^{\bar \alpha}}\) and \({\sigma _{\bar \alpha}}{u^{\bar \alpha}} = 0\). Finally, we denote by \((\bar e_0^\alpha, \bar e_a^\alpha)\), the tetrad at

*x*that is obtained by parallel transport of \(({u^{\bar \alpha}},\bar e_a^{\bar \alpha})\) on the spacelike geodesic.

The retarded coordinates of *x* refer to a point *x*′ ≡ *z*(*u*) on *γ* that is linked to *x* by a future-directed null geodesic (see Figure 8). We refer to this point as *x*’s *retarded point*, and to tensors at *x*′ we assign indices *α*′, *β*′, etc. We let (*u*, *rΩ*^{a}) be the retarded coordinates of *x*, with *u* denoting the value of *γ*’s proper-time parameter at *x*′, *r* = *σ*_{α′}*u*^{α′} representing the affine-parameter distance from *x*′ to *x* along the null geodesic, and *Ω*^{a} denoting a unit vector (*δ*_{ab}Ω^{a}Ω^{b} = 1) that determines the direction of the geodesic. The retarded coordinates are defined by \({\mathcal S}{\omega ^a} = - e_{\bar \alpha}^a{\sigma ^{\bar \alpha}}\) 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 Fermi-Walker 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.)

*p*(

*τ*) of the proper-time parameter

*τ*defined by

*x*is kept fixed and in which

*z*(

*τ*) is an arbitrary point on the world line. We have that

*p*(

*u*) =

*r*and

*p*(

*t*) = 0, and Δ can ultimately be obtained by expressing

*p*(

*t*) as

*p*(

*u*+ Δ) and expanding in powers of Δ. Formally,

*τ*. We have

*σ*

_{α′β′}(as well as additional expansions for the higher derivatives of the world function, obtained by further differentiation of this result) and the relation \({\sigma ^{{\alpha {\prime}}}} = - r({u^{{\alpha {\prime}}}} + {\Omega ^a}e_a^{{\alpha {\prime}}})\) first derived in Equation (144). With a degree of accuracy sufficient for our purposes we obtain

*S*=

*R*

_{a0b0}Ω

^{a}Ω

^{b}was first introduced in Equation (162), and where \({{\dot a}_0} \equiv {{\dot a}_{{\alpha {\prime}}}}{u^{{\alpha {\prime}}}},\,\,{{\dot a}_a} \equiv {{\dot a}_{{\alpha {\prime}}}}e_a^{{\alpha {\prime}}}\) are the frame components of the covariant derivative of the acceleration vector. To arrive at these results we made use of the identity

*a*

_{α′}

*a*

^{α′}+

*a*

_{α′}

*u*

^{α′}= 0 that follows from the fact that

*a*

^{μ}is orthogonal to

*u*

^{μ}. Notice that there is no distinction between the two possible interpretations

*ȧ*

_{a}≡

*da*

_{a}/

*dτ*and \({{\dot a}_a} \equiv {{\dot a}_\mu}e_a^\mu\) for the quantity

*ȧ*

_{a}(

*τ*); their equality follows at once from the substitution of \(De_a^\mu/d\tau = {a_a}{u^\mu}\) (which states that the basis vectors are Fermi-Walker transported on the world line) into the identity \(d{a_a}/d\tau = D({a_\nu}e_a^\nu)/d\tau\).

*t*−

*u*expressed as an expansion in powers of

*r*. The final result is

*z*(

*u*).

*x*is fixed and

*z*(

*τ*) is an arbitrary point on

*γ*. We have that the retarded coordinates are given by

*r*Ω

^{a}=

*p*

^{a}(

*u*), while the Fermi coordinates are given instead by

*sω*

^{a}=

*p*

^{a}(

*t*) =

*p*

^{a}(

*u*+ Δ). This last expression can be expanded in powers of Δ, producing

*S*

_{a}=

*R*

_{a0b0}

*Ω*

^{b}−

*R*

_{ab0c}Ω

^{b}Ω

^{c}, as it was first defined in Equation (161).

*t*−

*u*. From squaring Equation (219) and using the identity

*δ*

_{ab}

*ω*

^{a}

*ω*

^{b}= 1 we can also deduce

*x*and

*z*(

*t*).

#### 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*.

*τ*on

*γ*. We have that

*σ*(

*t*) = ½

*s*

^{2}and

*σ*(

*u*) = 0, and Δ ≡

*t*−

*u*is now obtained by expressing

*σ*(

*u*) as

*σ*(

*t*− Δ) and expanding in powers of Δ. Using the fact that \(\dot \sigma (\tau) = p(\tau)\), we have

*p*(

*τ*) evaluated at

*τ*=

*t*can be constructed from results derived previously in Section 3.4.1: it suffices to replace all primed indices by barred indices and then substitute the relation \({\sigma ^{\bar \alpha}} = - {\mathcal S}{\omega ^a}e_a^{\bar \alpha}\) that follows immediately from Equation (116). This gives

*p*(

*t*) = 0. Solving for Δ as an expansion in powers of

*s*returns

*z*(

*t*).

*r*=

*p*(

*u*) can be obtained by expanding

*p*(

*t*− Δ) in powers of Δ. We have

*x*and

*z*(

*u*).

*r*Ω

^{a}=

*p*

^{a}(

*u*) can be related to the Fermi coordinates by expanding

*p*

^{a}(

*t*− Δ) in powers of Δ, so that

*δ*

_{ab}Ω

^{a}Ω

^{b}= 1 returns the same result as Equation (222).

#### 3.4.3 Transformation of the tetrads at *x*

*x*. The first set is 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 \(\bar x \equiv z(t)\). The second set is 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 null geodesic that links

*x*to the retarded point

*x*′ ≡

*z*(

*u*). Since each tetrad forms a complete set of basis vectors, each member of \((\bar e_0^\alpha, \bar e_a^\alpha)\) can be decomposed in the tetrad \((e_0^\alpha, e_a^\alpha)\), and correspondingly, each member of \((e_0^\alpha, e_a^\alpha)\) can be decomposed in the tetrad \((\bar e_0^\alpha, \bar e_a^\alpha)\). These decompositions are worked out in this Section. For this purpose we shall consider the functions

*x*is a fixed point in a neighbourhood of

*γ*,

*z*(

*τ*) is an arbitrary point on the world line, and \({g^\alpha}_\mu (x,z)\) is the parallel propagator on the unique geodesic that links

*x*to

*z*. We have \(\bar e_0^\alpha = {p^\alpha}(t),\bar e_a^\alpha = p_a^\alpha (t),\,e_0^\alpha = {p^\alpha}(u)\), and \(e_a^\alpha = p_a^\alpha (u)\).

*z*(

*u*). This decomposition will be expressed in the retarded coordinates as an expansion in powers of

*r*. As in Section 3.2.1 we express quantities at

*z*(

*t*) in terms of quantities at

*z*(

*u*) by expanding in powers of Δ ≡

*t*−

*u*. We have

*z*(

*t*). This decomposition will be expressed in the Fermi normal coordinates as an expansion in powers of

*s*. Here, as in Section 3.2.2, we shall express quantities at

*z*(

*u*) in terms of quantities at

*z*(

*t*). We begin with

*p*

^{α}(

*τ*) at

*τ*=

*t*. To accomplish this we rely on our previous results (replacing primed indices with barred indices), on the expansions of Equation (92), and on the decomposition of \({g^\alpha}_{\bar \alpha}(x,\bar x)\) in the tetrads at

*x*and \({\bar x}\). This gives

*t*−

*u*. Our final result is

#### 3.4.4 Advanced point

*advanced point*associated with the field point

*x*. The advanced point will be denoted

*x*″ ≡

*z*(

*v*), with

*v*denoting the value of the proper-time parameter at

*x*″; to tensors at this point we assign indices

*α*″,

*β*″, etc. The advanced point is linked to

*x*by a

*past-directed null geodesic*(refer back to Figure 8), and it can be located by solving

*σ*(

*x*,

*x*″) = 0 together with the requirement that

*σ*

^{α″}(

*x*,

*x*″) be a future-directed null vector. The affine-parameter distance between

*x*and

*x*″ along the null geodesic is given by

*advanced distance*between

*x*and the world line. Notice that

*r*

_{adv}is a positive quantity.

*v*in terms of the retarded coordinates of

*x*. For this purpose we define Δ′ ≡

*v*−

*u*and re-introduce the function

*σ*(

*τ*) =

*σ*(

*x*,

*z*(

*τ*)) first considered in Section 3.4.2. We have that

*σ*(

*v*) =

*σ*(

*u*) = 0, and Δ

^{′}can ultimately be obtained by expressing

*σ*(

*v*) as

*σ*(

*u*+ Δ′) and expanding in powers of Δ′. Recalling that \(\dot \sigma (\tau) = p(\tau)\), we have

*p*(

*τ*) that were first obtained in Section 3.4.1, we write this as

*r*, we obtain

*z*(

*u*).

*r*

_{adv}. For this purpose we observe that

*r*

_{adv}= −

*p*(

*v*) = −

*p*(

*u*+ Δ′), which we can expand in powers of Δ′ ≡

*v*−

*u*. This gives

*v*is given by

*r*was truncated to a sufficient number of terms. Similarly, Equations (167, 168, 230) imply that the gradient of the advanced distance is given by

*S*

_{a}and

*S*were first introduced in Equations (161) and (162), respectively. We emphasize that in Equations (231) and (232), all frame components are evaluated at the retarded point

*z*(

*u*).

## 4 Green’s Functions

### 4.1 Scalar Green’s functions in flat spacetime

#### 4.1.1 Green’s equation for a massive scalar field

*x*) in flat spacetime. This field satisfies the wave equation

*η*

^{αβ}

*∂*

_{α}

*∂*

_{β}is the wave operator,

*μ*(

*x*) a prescribed source, and where the mass parameter

*k*has a dimension of inverse length. We seek a Green’s function

*G*(

*x*,

*x*′) such that a solution to Equation (233) can be expressed as

*δ*

_{4}(

*x*−

*x*′) =

*δ*(

*t*−

*t*′)

*δ*(

*x*−

*x*′)

*δ*(

*y*−

*y*′)

*δ*(

*z*−

*z*′) is a four-dimensional Dirac distribution in flat spacetime. Two types of Green’s functions will be of particular interest: the retarded Green’s function, a solution to Equation (235) with the property that it vanishes if

*x*is in the past of

*x*′, and the advanced Green’s function, which vanishes when

*x*is in the future of

*x*′.

*σ*= ½

*η*

_{αβ}(

*x*−

*x*′)

^{α}(

*x*−

*x*′)

^{β}is Synge’s world function in flat spacetime, and where

*g*(

*σ*) is a function to be determined. For the remainder of this section we set

*x*′ = 0 without loss of generality.

#### 4.1.2 Integration over the source

*V*that contains

*x*′ ≡ 0 This volume is bounded by a closed hypersurface

*∂V*. After using Gauss’ theorem on the first term of Equation (235), we obtain \(\oint\nolimits_{\partial V} {{G^{;\alpha}}d{\Sigma _\alpha} - {k^2}\int\nolimits_V G} \;dV = - 4\pi\), where

*d*Σ

*α*is a surface element on

*∂V*. Assuming that the integral of

*G*over

*V*goes to zero in the limit

*V*→ 0, we have

*V*must contain the point

*x*′.

*w*,

*χ*,

*θ*,

*φ*) defined by

*∂V*be a surface of constant

*w*. The metric of flat spacetime is given by

*d*Ω

^{2}=

*dθ*

^{2}+

*sin*

^{2}

*θdϕ*

^{2}Notice that

*w*is a timelike coordinate when cos2

*χ*> 0, and that

*χ*is then a spacelike coordinate; the roles are reversed when cos 2

*χ*< 0. Straightforward computations reveal that in these coordinates

*σ*= − ½

*w*

^{2}cos 2

*χ*, \(\sqrt {- g} = {w^3}{\sin ^2}\chi \sin \theta\),

*g*

^{ww}= −cos2

*χ*,

*g*

^{wχ}=

*w*

^{−1}sin

^{2}

*χ*,

*g*

^{χχ}=

*w*

^{−2}cos

^{2}

*χ*, and the only nonvanishing component of the surface element is

*d*Σ

_{w}=

*w*

^{3}sin

^{2}

*χ dχ d*Ω, where

*d*Ω = sin

*θ dθ dϕ*.To calculate the gradient of the Green’s function we express it as

*G*=

*θ*(±

*t*)

*g*(

*σ*) =

*θ*(

*±*

*w*cos

*χ*)

*g*(−½

*w*

^{2}cos2

*χ*), with the upper (lower) sign belonging to the retarded (advanced) Green’s function. Calculation gives \({G^{;\alpha}}d{\Sigma _\alpha} = \theta (\pm \cos \chi){w^4}{\sin ^2}\chi {g{\prime}}(\sigma)\,d\chi \,d\Omega\), with a prime indicating differentiation with respect to

*σ*it should be noted that derivatives of the step function do not appear in this expression.

*G*

^{;α}

*d*Σ

_{α}with respect to

*d*Ω is immediate, and we find that Equation (237) reduces to

*χ*<

*π*/2 in which

*σ*increases from −½

*w*

^{2}to ½

*w*

^{2}. Changing the variable of integration from

*χ*to

*σ*transforms Equation (238) into

*ϵ*≡ ½

*w*

^{2}. For the advanced Green’s function, the domain of integration is

*π*/2 <

*χ*<

*χ*, in which

*σ*decreases from ½

*w*

^{2}to −½

*w*

^{2}. Changing the variable of integration from

*χ*to

*σ*also produces Equation (239).

#### 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 left-hand 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 *δ*-functions and derivatives.

*V*(

*σ*) is a smooth function, and

*A*,

*B*,

*C*, … are constants. The first term represents a function supported within the past and future light cones of

*x*′ ≡ 0; we exclude a term proportional to

*θ*(

*σ*) for reasons of causality. The other terms are supported on the past and future light cones. It is sufficient to take the coefficients in front of the

*δ*-functions to be constants. To see this we invoke the distributional identities

*σ*

^{2}

*δ*

^{′}(

*σ*) =

*σ*

^{3}

*δ*″(

*σ*) = … = 0. A term like

*f*(

*σ*)

*δ*(

*σ*) is then distribution-ally equal to

*f*(0)

*δ*′(

*σ*), while a term like

*f*(

*σ*)

*δ*′(

*σ*) is distributionally equal to

*f*(0)

*δ*′(

*σ*) −

*f*′(0)

*δ*(

*σ*), and a term like

*f*(

*σ*)

*δ*″(

*σ*) is distributionally equal to

*f*(0)

*δ*″(

*σ)*− 2

*f*′(0)

*δ*′(

*σ*) + 2

*f*″(0)

*δ*(

*σ*); here

*f*(

*σ*) is an arbitrary test function. Summing over such terms, we recover an expression of the form of Equation (241), and there is no need to make

*A*,

*B*,

*C*, … functions of

*σ*.

*ξ*≡

*σ/ϵ*The limit

*ϵ*→ 0 exists if and only if

*B*=

*C*= … = 0 In the limit we must then have

*Aẇ*(0) = 1, which implies

*A*= 1. We conclude that

*g*(

*σ*) must have the form of

*V*(

*σ*) being a smooth function that cannot be determined from Equation (239) alone.

#### 4.1.4 Smooth part of *g*(*σ*)

*V*(

*σ*) we must go back to the differential equation of Equation (235). Because the singular structure of the Green’s function is now under control, we can safely set

*x*≠

*x*′≡ 0 in the forthcoming operations. This means that the equation to solve is in fact (□ −

*k*

^{2})

*g*(

*σ*) = 0, the homogeneous version of Equation (235). We have ∇

_{αg}=

*g*′

*σ*

_{α}, ∇

_{α}∇

_{βg}=

*g*″

*σ*

_{α}

*σ*

_{β}+

*g*′

*σ*

_{αβ}, and □

*g*= 2

*σg*″ + 4

*g*′, so that Green’s equation reduces to the ordinary differential equation

*V*(

*σ*) uniquely, even in the absence of a second boundary condition at

*σ*= 0, because the differential equation is singular at

*σ*= 0 and

*V*is known to be smooth.

*V*=

*F*(

*z*)/

*z*, with \(z \equiv k\sqrt {- 2\sigma}\). This gives rise to Bessel’s equation for the new function

*F*:

*z*= 0 is

*F*= a

*J*

_{1}(

*z*), where a is a constant to be determined. We have that

*J*

_{1}(

*z*) ∼ ½

*z*for small values of

*z*, and it follows that

*V*∼

*a*/2. From Equation (244) we see that

*a*= −

*k*

^{2}. So we have found that the only acceptable solution to Equation (244) is

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.

*θ*

_{+}(

*x*, Σ) be a generalized step function, defined to be one if

*x*is in the future of the spacelike hypersurface Σ, and defined to be zero otherwise. Similarly, define

*θ*

_{−}(

*x*, Σ) ≡ 1 −

*θ*

_{+}(

*x*, Σ) to be one if

*x*is in the past of the spacelike hypersurface Σ, and zero otherwise. Then define the light-cone step functions

*θ*

_{+}(−

*σ*) is one if

*x*is an element of

*I*

^{+}(

*x*′), the chronological future of

*x*′, and zero otherwise, and

*θ*

_{−}(−

*σ*) is one if

*x*is an element of

*I*

^{−}(

*x*′), the chronological past of

*x*′, and zero otherwise; the choice of hypersurface is immaterial so long as Σ is spacelike and contains the reference point

*x*′. Notice that

*θ*

_{+}(−

*σ*) +

*θ*

_{−}(−

*θ*) =

*θ*(−

*σ*) Define also the light-cone Dirac functionals

*δ*

_{+}(

*σ*), when viewed as a function of

*x*, is supported on the future light cone of

*x*′, while

*δ*

_{−}(

*σ*) is supported on its past light cone. Notice that

*δ*

_{+}(

*σ*) +

*δ*

_{−}(

*σ*) =

*δ*(

*σ*). In Equations (246) and (247),

*σ*is the world function for flat spacetime; it is negative if

*x*and

*x*′ are timelike related, and positive if they are spacelike related.

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 two-branch hyperboloid that is located just *within* the light cone of the reference point *x*′. The hyperboloid does not include *x*′, and *θ*_{+}(*x*, Σ) is one everywhere on its future branch, while *θ*_{−}(*x*, Σ) is one everywhere on its past branch. These factors, therefore, become invisible to differential operators. For example, \(\theta _ + {\prime}(- \sigma - \epsilon) = {\theta _ +}(x,\Sigma){\theta {\prime}}(- \sigma - \epsilon) = - {\theta _ +}(x,\Sigma)\delta (\sigma + \epsilon) = - {\delta _ +}(\sigma + \epsilon)\). 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*.

^{2}

*r*

^{−1}= −4

*πδ*

_{3}(

*x*), in which we write the left-hand side as lim

_{ϵ=0+}∇

^{2}

*R*

^{−1}; since

*R*

^{−1}is nonsingular at

*= 0, it can be straightforwardly differentiated, and the result is ∇*

**x**^{2}

*R*

^{−1}= −6

*ϵ*/

*R*

^{5}from which Equation (251) follows.

*ϵδ*

_{±}(

*σ*+

*ϵ*) vanishes as a distribution in the limit

*ϵ*→ 0

^{+}. For this we must prove that a functional of the form

*f*(

*x*) =

*f*(

*t*,

*) is a smooth test function, vanishes for all such functions*

**x***f*. Our first task will be to find a more convenient expression for

*δ*

_{±}(

*σ*+

*ϵ*). Once more we set

*x*′ = 0 (without loss of generality) and we note that 2(

*σ*+

*ϵ*) = −

*t*

^{2}+

*r*

^{2}+ 2

*ϵ*= −(

*t*−

*R*)(

*t*+

*R*) where we have used Equation (251). It follows that

*f*. We have

*t*, and we have used

*∂R/∂ϵ*= 1/

*R*.

*πf*(0, 0). We have

#### 4.1.6 Alternative computation of the Green’s functions

*θ*

_{±}and

*δ*

_{±}were defined in the preceding Section 4.1.5, and we assume that

*V*(

*σ*) is a smooth function of

*σ*(

*x*,

*x*′) = ½

*η*

_{αβ}(

*x*−

*x*′)

^{α}(

*x*−

*x*′)

^{β}; because this function is smooth, it is not necessary to evaluate

*V*at

*σ*+

*ϵ*in Equation (253). We recall also that

*θ*

_{+}and

*δ*

_{+}are nonzero when

*x*is in the future of

*x*′, while

*θ*

_{−}and

*δ*

_{−}are nonzero when

*x*is in the past of

*x*′. We will therefore prove that the retarded and advanced Green’s functions are of the form

*x*′. We will also determine the form of the function

*V*(

*σ*).

*k*

^{2})

*G*= 2

*σG*″ + 4

*G*′ −

*k*

^{2}

*G*, with a prime indicating differentiation with respect to

*σ*. From Equation (253) we obtain

*G*′ =

*δ*′ −

*Vδ*+

*V*′

*θ*and

*G*″ =

*δ*″ −

*Vδ*′ − 2

*V*′

*δ*+

*V*″

*θ*. The identities of Equation (241) can be expressed as (

*σ*+

*ϵ*)

*δ*′(

*σ*+

*ϵ*) = −

*δ*(

*σ*+

*ϵ*) and (

*σ*+

*ϵ*)

*δ*″(

*σ*+

*ϵ*) = −2

*δ*′(

*σ*+

*ϵ*), and combining this with our previous results gives

*ϵ*→ 0

^{+}, and the third term becomes −4

*πδ*

_{4}(

*x*−

*x*′). Provided that the first two terms vanish also, we recover (□ −

*k*

^{2})

*G*(

*x*−

*x*′) = −4

*πδ*

_{4}(

*x*−

*x*′). in the limit, as required. Thus, the limit of \(G_ \pm ^\epsilon (x,{x{\prime}})\) as

*ϵ*→ 0

^{+}will indeed satisfy Green’s equation provided that

*V*(

*σ*) is a solution to

### 4.2 Distributions in curved spacetime

The distributions introduced in Section 4.1.5 can also be defined in a four-dimensional spacetime with metric *g*_{αβ}. Here we produce the relevant generalizations of the results derived in that section.

#### 4.2.1 Invariant Dirac distribution

*δ*

_{4}(

*x*,

*x*′), an

*invariant*Dirac functional in a four-dimensional curved spacetime. This is defined by the relations

*f*(

*x*) is a smooth test function,

*V*any four-dimensional region that contains

*x*′, and

*V*′ any four-dimensional region that contains

*x*. These relations imply that

*δ*

_{4}(

*x*,

*x*′) is symmetric in its arguments, and it is easy to see that

*δ*

_{4}(

*x*−

*x′*) =

*δ*(

*x*

^{0}−

*x*′

^{0})

*δ*(

*x*

^{1}−

*x*′

^{1})

*δ*(

*x*

^{2}−

*x′*

^{2})

*δ*(

*x*

^{3}−

*x′*

^{3}) is the ordinary (coordinate) four-dimensional Dirac functional. The relations of Equation (259) are all equivalent because

*f*(

*x*)

*δ*

_{4}(

*x*,

*x′*) =

*f*(

*x′*)

*δ*

_{4}(

*x*,

*x′*) is a distributional identity; the last form is manifestly symmetric in

*x*and

*x′*.

*x*,

*x′*) is any bitensor and \({g^\alpha}_{{\alpha {\prime}}}(x,{x{\prime}}),{g^{{\alpha {\prime}}}}_\alpha (x,{x{\prime}})\) are parallel propagators. The first identity follows immediately from the definition of the

*δ*-function. The second and third identities are established by showing that integration against a test function

*f*(

*x*) gives the same result from both sides. For example, the first of the Equations (258) implies

*scalar*functions of the coordinates

*x*; the fact that these functions are also vectors with respect to

*x*′ does not invalidate the procedure. The third identity of Equation (260) is proved in a similar way.

#### 4.2.2 Light-cone distributions

*β*links these two points. We then let

*σ*(

*x*,

*x*′) be the curved spacetime world function, and we define light-cone step functions by

*θ*

_{+}(

*x*, Σ) is one if

*x*is in the future of the spacelike hypersurface Σ and zero otherwise, and

*θ*

_{−}(

*x*, Σ) = 1 −

*θ*

_{+}(

*x*, Σ). These are immediate generalizations to curved spacetime of the objects defined in flat spacetime by Equation (246). We have that

*θ*

_{+}(−

*σ*) is one if

*x*is an element of

*I*

^{+}(

*x′*), the chronological future of

*x′*, and zero otherwise, and

*θ*

_{−}(−

*σ*) is one if

*x*is an element of

*I*

^{−}(

*x′*), the chronological past of

*x′*, and zero otherwise. We also have

*θ*

_{+}(−

*σ*) +

*θ*

_{−}(−

*σ*) =

*θ*(−

*σ*).

*δ*

_{+}(

*σ*), when viewed as a function of

*x*, is supported on the future light cone of

*x*′, while

*δ*

_{−}(

*σ*) is supported on its past light cone. We also have

*δ*

_{+}(

*σ*) +

*δ*

_{−}(

*σ*) = δ(

*σ*), and we recall that a is negative if

*x*and

*x*′ are timelike related, and positive if they are spacelike related.

For the same reasons as those mentioned in Section 4.1.5, it is sometimes convenient to shift the argument of the step and *δ*-functions from *σ* to *σ* + *ϵ*, where *ϵ* is a small positive quantity. With this shift, the light-cone distributions can be straightforwardly differentiated with respect to *σ*. For example, \({\delta _ \pm}(\sigma + \epsilon) = - \theta _ \pm {\prime}(- \sigma - \epsilon)\), with a prime indicating differentiation with respect to *σ*.

*x*′. We have that \(\sigma (x,{x{\prime}}) = {1 \over 2}{\eta _{\alpha \beta}}{\hat x^\alpha}{\hat x^\beta}\) and \({\delta _4}(x,{x{\prime}}) = \Delta (x,{x{\prime}}){\delta _4}(x - {x{\prime}}) = {\delta _4}(x - {x{\prime}})\) where Δ(

*x*,

*x*′) is the van Vleck determinant, whose coincidence limit is unity. In Riemann normal coordinates, therefore, Equations (263, 264, 265) take exactly the same form as Equations (248, 264, 250). Because the identities are true in flat spacetime, they must be true also in curved spacetime (in Riemann normal coordinates based at

*x*′); and because these are scalar relations, they must be valid in any coordinate system.

### 4.3 Scalar Green’s functions in curved spacetime

#### 4.3.1 Green’s equation for a massless scalar field in curved spacetime

*x*) in a curved spacetime with metric

*g*

_{αβ}. The field satisfies the wave equation

*g*

^{αβ}∇

_{α}∇

_{β}is the wave operator,

*R*the Ricci scalar,

*ξ*an arbitrary coupling constant, and

*μ*(

*x*) is a prescribed source. We seek a Green’s function

*G*(

*x*,

*x′*) such that a solution to Equation (266) can be expressed as

*δ*

_{4}(

*x*,

*x′*) is the invariant Dirac functional introduced in Section 4.2.1. It is easy to verify that the field defined by Equation (267) is truly a solution to Equation (266).

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

*x*is restricted to the normal convex neighbourhood of

*x*′, we make the ansatz

*U*(

*x*,

*x*′) and

*V*(

*x*,

*x*′) are smooth biscalars; the fact that the spacetime is no longer homogeneous means that these functions cannot depend on

*σ*alone.

*ϵ*. We shall therefore consider the distributions

*ϵ*→ 0

^{+}. Differentiation of these objects is straightforward, and in the following manipulations we will repeatedly use the relation

*σ*

^{α}

*σ*

_{α}= 2

*σ*satisfied by the world function. We will also use the distributional identities \(\sigma {\delta _ \pm}(\sigma + \epsilon) = - \epsilon {\delta _ \pm}(\sigma + \epsilon),\,\sigma \delta _ \pm {\prime}(\sigma + \epsilon) = - {\delta _ \pm}(\sigma + \epsilon) - \epsilon \delta _ \pm {\prime}(\sigma + \epsilon)\) and \(\sigma \delta _ \pm ^{{\prime\prime}}(\sigma + \epsilon) = - 2{\delta {\prime}}(\sigma + \epsilon) - \epsilon {\delta ^{{\prime\prime}}}(\sigma + \epsilon)\). After a routine calculation we obtain

*ϵ*→ 0

^{+}, after using the identities of Equations (263, 264, 265).

*πδ*

_{4}(

*x, x*′). This immediately gives us the coincidence condition

*U*(

*x, x*′). To eliminate the \(\delta _ \pm {\prime}\) term we make its coefficient vanish:

*U*(

*x, x*′) uniquely.

*σ*

^{α}is a vector at

*x*that is tangent to the unique geodesic

*β*that connects

*x*to

*x*′. This geodesic is affinely parameterized by λ and a displacement along

*β*is described by

*dx*

^{α}= (

*σ*

^{α}/

*λ*)

*d*λ. The first term of Equation (272) therefore represents the rate of change of

*U*(

*x, x*′) along

*β*, and this can be expressed as 2λ

*dU/d*λ. For the second term we recall from Section 2.5.1 the differential equation Δ

^{−1}(Δ

*σ*

^{α})

_{;α}= 4 satisfied by Δ(

*x, x*′), the van Vleck determinant. This gives us \({\sigma ^\alpha}_\alpha - 4 = {\Delta ^{- 1}}{\Delta _{,\alpha}}{\sigma ^\alpha} = {\Delta ^{- 1}}\lambda d\Delta/d\lambda\) and Equation (272) becomes

*U*

^{2}/Δ is constant on

*β*, and it must therefore be equal to its value at the starting point

*x*′:

*U*

^{2}/Δ = [

*U*

^{2}/Δ] = 1, by virtue of Equation (271) and the property [Δ] = 1 of the van Vleck determinant. Since this statement must be true for all geodesics

*β*that emanate from

*x*′, we have found that the unique solution to Equations (271) and (272) is

*δ*

_{±}term can be eliminated by demanding that its coefficient vanish when σ = 0. This, however, does not constrain its value away from the light cone, and we thus obtain information about

*V*∣

_{σ=0}only. Denoting this by \({\check V}(x,{x{\prime}})\) the restriction of

*V*(

*x*,

*x*′) on the light cone

*σ*(

*x*,

*x*′) = 0 — we have

*σ*(

*x*,

*x*′) = 0. For these integrations to be well posed, however, we must provide initial values at

*x*=

*x*′. As we shall now see, these can be inferred from Equation (274) and the fact that

*V*(

*x*,

*x*′) must be smooth at coincidence.

*U*(

*x*,

*x*′) admits the expansion

*R*

_{α′β′}is the Ricci tensor at

*x*′ and λ is the affine-parameter distance to

*x*(which can be either on or off the light cone). Differentiation of this relation gives

*V*(

*x, x*′) on the null cone

*σ*(

*x, x*′) = 0. These values can then be used as characteristic data for the wave equation

*θ*

_{±}term in Equation (270). While this certainly does not constitute a practical method to compute the biscalar

*V*(

*x, x*′), these considerations show that

*V*(

*x, x*′) exists and is unique.

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

*V*(

*x*,

*x*′) must be symmetric in its arguments,

*I*

^{+}(

*x*′), then

*G*

_{+}(

*x*,

*x*′) =

*V*(

*x*,

*x*′) and

*G*

_{−}(

*x*′,

*x*) =

*V*(

*x*′,

*x*).

*V*that contains both

*x*′ and

*x*″, we obtain

*∂V*is the boundary of

*V*. Assuming that the Green’s functions fall off sufficiently rapidly at infinity (in the limit

*∂V*→ ∞; this statement imposes some restriction on the spacetime’s asymptotic structure), we have that the left-hand side of the equation evaluates to zero in the limit. This gives us the statement

*G*

_{+}(

*x*″,

*x*′) =

*G*

_{−}(

*x*′,

*x*″), which is just Equation (280) with

*x*″ replacing

*x*.

#### 4.3.4 Kirchhoff representation

*x*′) and its normal derivative

*n*

^{α′}∇

_{α′}Φ(

*x*′) are known on a spacelike hypersurface Σ. Suppose also that the scalar field satisfies the homogeneous wave equation

*x*in the future of Σ is given by Kirchhoff’s formula,

*d*Σ

_{α′}is the surface element on Σ. If

*n*

_{α′}is the future-directed unit normal, then

*d*Σ

_{α′}= −

*n*

_{α′}

*dV*, with

*dV*denoting the invariant volume element on Σ; notice that

*d*Σ

_{α′}is past directed.

*x*and

*x*′ refer to arbitrary points in spacetime. Taking their difference gives

*V*that is bounded in the past by the hyper-surface Σ. We suppose that

*V*contains

*x*and we obtain

*d*Σ

_{α′}is the outward-directed surface element on the boundary

*∂V*. Assuming that the Green’s function falls off sufficiently rapidly into the future, we have that the only contribution to the hypersurface integral is the one that comes from Σ. Since the surface element on Σ points in the direction opposite to the outward-directed surface element on

*∂V*, we must change the sign of the left-hand side to be consistent with the convention adopted previously. With this change we have

#### 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 outgoing-wave boundary conditions at infinity — the advanced solution would come instead with incoming-wave boundary conditions. The retarded field is singular on the world line because a point particle produces a Coulomb field that diverges at the particle’s position. In view of this singular behaviour, it is a subtle matter to describe the field’s action on the particle, and to formulate meaningful equations of motion.

When facing this problem in flat spacetime (recall the discussion of Section 1.3), it is convenient to decompose the retarded Green’s function *G*_{+}(*x*, *x*′) into a *singular* Green’s function *G*_{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 right-hand 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 outgoing-wave boundary conditions: The particle will lose energy to the radiation.

In this section we attempt a decomposition of the curved-spacetime retarded Green’s function into singular and radiative Green’s functions. The flat-spacetime relations will have to be amended, however, because of the fact that in a curved spacetime, the advanced Green’s function is generally nonzero when *x*′ is in the chronological future of *x*. This implies that the value of the advanced field at *x* depends on events *x*′ that will unfold *in the future*; this dependence would be inherited by the radiative field (which acts on the particle and determines its motion) if the naive definition *G*_{R}(*x*, *x*′) ≡ ½*G*_{+}(*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)

*x*.

*G*

_{+}(

*x*,

*x*′) is the retarded Green’s function. This comes with the properties

- 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)

*x*is in past of

*x*′. The properties of the radiative Green’s function ensure that the corresponding radiative field will be smooth at the world line, and will depend only on the past history of the scalar charge.

*H*(

*x*,

*x*′) that would possess the properties

- 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)

*H*(

*x*,

*x*′) satisfying these relations, a singular Green’s function defined by

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.

*H*(

*x*,

*x*′) satisfies the homogeneous wave equation (Property Sc.H1), it can be given the Kirkhoff representation of Equation (283): If Σ is a spacelike hypersurface in the past of both

*x*and

*x*′, then

*d*Σ

_{α″}is a surface element on Σ. The hypersurface can be partitioned into two segments, Σ

^{−}(

*x*′) and Σ − Σ

^{−}(

*x*′), with Σ

^{−}(

*x*′) denoting the intersection of Σ with

*I*

^{−}(

*x*′). To enforce Property Sc.H4 it suffices to choose for

*H*(

*x*,

*x*′) initial data on Σ

^{−}(

*x*′) that agree with the initial data for the advanced Green’s function; because both functions satisfy the homogeneous wave equation in

*I*

^{−}(

*x*′), the agreement will be preserved in all of the domain of dependence of Σ

^{−}(

*x*′). The data on Σ − Σ

^{−}(

*x*′) is still free, and it should be possible to choose it so as to make

*H*(

*x*,

*x*′) symmetric. Assuming that this can be done, we see that Property Sc.H2 is enforced and we conclude that the Properties Sc.H1, Sc.H2, Sc.H3, and Sc.H4 can all be satisfied.

*x*is restricted to the normal convex neighbourhood of

*x*′, Properties Sc.H1, Sc.H2, Sc.H3, and Sc.H4 imply that

*H*(

*x*,

*x*′) is assumed to be defined globally in the entire spacetime, the existence of

*V*(

*x*,

*x*′) is limited to \({\mathcal N}({x{\prime}})\). With Equations (269) and (295) we find that the singular Green’s function is given explicitly by

*I*

^{±}(

*x*′) in which

*θ*(

*σ*) = 0 (Property Sc.S3). From Equation (287) we get an analogous expression for the radiative Green’s function:

*G*

_{+}(

*x*,

*x*′) in

*I*

^{+}(

*x*′), in which

*θ*(

*σ*) = 0 and

*θ*

_{+}(−

*σ*) = 1 (Property Sc.R2), and that its support does not include

*I*

^{−}(

*x*′), in which

*θ*(

*σ*) =

*θ*

_{+}(−

*σ*) = 0 (Property Sc.R3).

#### 4.3.6 Example: Cosmological Green’s functions

*ξ*= 0) scalar field in a cosmological spacetime with metric

*a*(

*η*) is the scale factor expressed in terms of conformal time. For concreteness we take the universe to be matter dominated, so that

*a*(

*η*) =

*Cη*

^{2}, where

*C*is a constant. This spacetime is one of the very few for which Green’s functions can be explicitly constructed. The calculation presented here was first carried out by Burko, Harte, and Poisson [15]; it can be extended to other cosmologies.

*G*(

*x*,

*x*′) = −4

*πδ*

_{4}(

*x*,

*x*′) we first introduce a reduced Green’s function

*g*(

*x*,

*x*′) defined by

*x*= (

*x*,

*y*,

*z*) is a vector in three-dimensional flat space, and ∇

^{2}is the Laplacian operator in this space. We next expand

*g*(

*x*,

*x*′) in terms of plane-wave solutions to Laplace’s equation,

*δ*

_{3}(

*−*

**x***′), is an ordinary differential equation for \(\tilde g(\eta, {\eta {\prime}};k)\)*

**x***k*

^{2}=

*·*

**k***. To generate the retarded Green’s function we set*

**k***ĝ*depends only on the modulus of the vector

*. To generate the advanced Green’s function we would set instead \(\tilde g - (\eta, {\eta {\prime}};k) = \theta ({\eta {\prime}} - \eta)\hat g(\eta, {\eta {\prime}};k)\). The following manipulations will refer specifically to the retarded Green’s function; they are easily adapted to the case of the advanced Green’s function.*

**k***ĝ*must satisfy the homogeneous equation

*yields*

**k***η*≡

*η*−

*η*′ and

*R*≡ ∣

*x*−

*x*′ ∣.

*k*Δ

*η*) + (

*kη*)

^{−1}cos(

*k*Δ

*η*) as linearly independent solutions, and

*ĝ*(

*η, η′; k*) must be given by a linear superposition. The coefficients can be functions of

*η*′, and after imposing Equations (306) we find that the appropriate combination is

*θ*(Δ

*η*−

*R*).

*g*

_{±}(

*x, x*′) are solutions to the reduced Green’s equation of Equation (301). The actual Green’s functions are obtained by substituting Equations (309) and (310) into Equation (300). We note that the support of the retarded Green’s function is given by

*η*−

*η*′ ≥ ∣

*x*−

*x*′ ∣ while the support of the advanced Green’s function is given by

*η*−

*η*′ ≤ − ∣

*x*−

*x*′ ∣

*−*

**x***′ ∣ ≤*

**x***η*−

*η*′ ≤ ∣

*x*−

*x*′ ∣ According to Equation (287) the radiative Green’s function is given by

*η*−

*η*′ ≥ − ∣

*−*

**x***′ ∣ and for*

**x***η*−

*η*′ ≥− ∣

*−*

**x***′ ∣ the radiative Green’s function agrees with the retarded Green’s function.*

**x**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

*F*

_{αβ}= ∂

_{α}

*A*

_{β}− ∂

_{β}

*A*

_{α}is expressed in terms of a vector potential

*A*

_{α}. In the Lorenz gauge ∂

_{α}

*A*

^{α}= 0, the vector potential satisfies the wave equation

*g*

^{αβ}∂

_{α}∂

_{β}is the wave operator, \({R^\alpha}_\beta\) the Ricci tensor, and

*j*

^{α}a prescribed current density. The wave equation enforces the condition ∂

_{α}

*j*

^{α}= 0, which expresses charge conservation.

*δ*

_{4}(

*x*,

*x*′) an invariant Dirac distribution. The parallel propagator is inserted on the right-hand side of Equation (316) to keep the index structure of the equation consistent from side to side; because \({g^\alpha}_{{\beta {\prime}}}(x,{x{\prime}}){\delta _4}(x,{x{\prime}})\) is distributionally equal to \([{g^\alpha}_{{\beta {\prime}}}]{\delta _4}(x,{x{\prime}}) = {\delta ^{{\alpha {\prime}}}}_{^{{\beta {\prime}}}}{\delta _4}(x,{x{\prime}})\), it could have been replaced by either \({\delta ^{{\alpha {\prime}}}}_{{\beta {\prime}}}\) or \({\delta ^\alpha}_\beta\). It is easy to check that by virtue of Equation (316), the vector potential of Equation (315) satisfies the wave equation of Equation (314).

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

*x*is in the normal convex neighbourhood of

*x*′, we make the ansatz

*θ*

_{±}(−

*σ*),

*δ*

_{±}(

*σ*) are the light-cone distributions introduced in Section 4.2.2, and where \({U^\alpha}_{{\beta {\prime}}}(x,{x{\prime}}),\,{V^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})\) are smooth bitensors.

*σ*by a small positive quantity

*ϵ*. The Green’s functions are then recovered by the taking the limit of

*ϵ*→ 0

^{+}. When we substitute this into the left-hand side of Equation (316) and then take the limit, we obtain

- 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$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over V} ^\alpha}_{\beta {\prime};\gamma}{\sigma ^\gamma} + {1 \over 2}({\sigma ^\gamma}_\gamma - 2){\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over V} ^\alpha}_{\beta {\prime}} = {1 \over 2}{\left. {\left({\square{U^\alpha}_{\beta {\prime}} - {R^\alpha}_\beta {U^\beta}_{\beta {\prime}}} \right)} \right\vert _{\sigma = 0}}$$(320)
*σ*(*x*,*x*′) = 0; and - 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)

*β*that links

*x*′ to

*x*. The initial conditions are provided by Equation (318), and if we set \(U_{{\beta {\prime}}}^\alpha (x,{x{\prime}}) = {g^\alpha}_{{\beta {\prime}}}(x,{x{\prime}})U(x,{x{\prime}})\), we find that these equations reduce to Equations (272) and (271), respectively. According to Equation (273), then, we have

*λ*denoting the affine-parameter distance between

*x*′ and

*x*. Differentiation of this relation gives

*σ*(

*x*,

*x*′) = 0. The initial values are obtained by taking the coincidence limit of this equation, using Equations (318), (326), and the additional relation \([{\sigma ^\gamma}_\gamma ] = 4\) We arrive at

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

*A*

^{α}(

*x*) satisfies the

*homogeneous*version of Equation (314) and that initial values \({A^{{\alpha {\prime}}}}({x{\prime}}),{n^{{\beta {\prime}}}}{\nabla _{{\beta {\prime}}}}{A^{{\alpha {\prime}}}}({x{\prime}})\) are specified on a spacelike hypersurface Σ. Then the value of the potential at a point

*x*in the future of Σ is given by

*dV*is the invariant volume element on the hypersurface. The derivation of Equation (330) is virtually identical to what was presented in Section 4.3.4.

#### 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)

*x*is restricted to the normal convex neighbourhood of

*x*′, we have the explicit relations

*I*

^{+}(

*x*′) (Property Em.S3). We see also that the radiative Green’s function coincides with \(G_{+ {\beta {\prime}}}^\alpha (x,{x{\prime}})\) in

*I*

^{+}(

*x*′) (Property Em.R2), and that its support does not include

*I*

^{−}(

*x*′) (Property Em.R3).

### 4.5 Gravitational Green’s functions

#### 4.5.1 Equations of linearized gravity

*g*

_{αβ}satisfies the Einstein field equations

*in vacuum*. We then perturb the metric from

*g*

_{αβ}to

*h*

_{αβ}is assumed to be small, and when working out the Einstein field equations to be satisfied by the new metric g

_{αβ}, we work consistently to first order in

*h*

_{αβ}. To simplify the expressions we use the trace-reversed potentials

*γ*

_{αβ}defined by

*g*

_{αβ}.Similarly, the connection involved in Equation (348), and in all other equations below, is the one that is compatible with the background metric. If

*T*

^{αβ}is the perturbing stress-energy tensor, then by virtue of the linearized Einstein field equations the perturbation field obeys the wave equation

*g*

^{αβ}∂

_{α}∂

_{β}is the wave operator and

*R*

_{γαδβ}the Riemann tensor. In first-order perturbation theory, the stress-energy tensor must be conserved in the background spacetime: \({T^{\alpha \beta}}_{;\beta} = 0\).

*δ*

_{4}(

*x*,

*x*′) an invariant Dirac functional. The parallel propagators are inserted on the right-hand side of Equation (351) to keep the index structure of the equation consistent from side to side; in particular, both sides of the equation are symmetric in

*α*and

*β*, and in

*γ*′ and

*δ*′. It is easy to check that by virtue of Equation (351), the perturbation field of Equation (350) satisfies the wave equation of Equation (349). Once

*γ*

_{αβ}is known, the metric perturbation can be reconstructed from the relation \({h_{\alpha \beta}} = {\gamma _{\alpha \beta}} - {1 \over 2}({g^{\gamma \delta}}{\gamma _{\gamma \delta}}){g_{\alpha \beta}}\).

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

*x*is in the normal convex neighbourhood of

*x*′, we make the ansatz

*θ*

_{±}(−

*σ*),

*δ*

_{±}(

*σ*) are the light-cone distributions introduced in Section 4.2.2, and where \({U^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}}),\,{V^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}})\) are smooth bitensors.

*σ*by a small positive quantity

*ϵ*. The Green’s functions are then recovered by the taking the limit of

*ϵ*→ 0

^{+}. When we substitute this into the left-hand side of Equation (351) and then take the limit, we obtain

- 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$$\check{V}_{\;\;\gamma \prime \delta \prime ;\gamma}^{\alpha \beta}{\sigma ^\gamma} + {1 \over 2}(\sigma _{\;\;\gamma}^\gamma - 2)\check{V}_{\;\;\;\;\gamma \prime \delta \prime}^{\alpha \beta} = {\left. {{1 \over 2}\left({\square U_{\;\;\gamma \prime \delta \prime}^{\alpha \beta} + 2R_{\gamma \;\;\delta}^{\alpha \;\;\beta}U_{\;\;\gamma \prime \delta \prime}^{\gamma \delta}} \right)} \right\vert_{\sigma = 0}}$$(355)
*σ*(*x*,*x*′) = 0; and - 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)

*β*that links

*x*′ to

*x*. The initial conditions are provided by Equation (353), and if we set \({U^\alpha}_{{\gamma {\prime}}{\delta {\prime}}}^\beta (x,{x{\prime}}) = g_{{\gamma {\prime}}}^{(\alpha}g_{{\delta {\prime}}}^{\beta)}U(x,{x{\prime}})\), we find that these equations reduce to Equations (272) and (271), respectively. According to Equation (273), then, we have

*λ*denoting the affine-parameter distance between

*x*′ and

*x*; there is no term of order

*λ*

^{2}because, by assumption, the background Ricci tensor vanishes at

*x*′ (as it does in the entire spacetime). Differentiation of this relation gives

*σ*(

*x*,

*x*′) = 0. The initial values are obtained by taking the coincidence limit of this equation, using Equations (353), (361), and the additional relation \([{\sigma ^\gamma}_\gamma ] = 4\). We arrive at

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

*γ*

_{αβ}is formulated as follows. Suppose that

*γ*

^{αβ}(

*x*) satisfies the homogeneous version of Equation (349) and that initial values \({\gamma ^{{\alpha {\prime}}{\beta {\prime}}}}({x{\prime}}),\,{n^{{\gamma {\prime}}}}{\nabla _{{\gamma {\prime}}}}{\gamma ^{{\alpha {\prime}}{\beta {\prime}}}}({x{\prime}})\) are specified on a spacelike hypersurface Σ. Then the value of the perturbation field at a point

*x*in the future of Σ is given by

*dV*is the invariant volume element on the hypersurface. The derivation of Equation (365) is virtually identical to what was presented in Sections 4.3.4 and 4.4.3.

#### 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)

*x*is restricted to the normal convex neighbourhood of

*x*′, we have the explicit relations

*I*

^{+}(

*x*′) (Property Gr.S3). We see also that the radiative Green’s function coincides with \({G_+}_{\;\;\gamma '\delta '}^{\alpha \beta }(x,x')\) in

*I*

^{+}(

*x*′) (Property Gr.R2), and that its support does not include

*I*

^{−}(

*x*′) (Property Gr.R3).

## 5 Motion of Point Particles

### 5.1 Motion of a scalar charge

#### 5.1.1 Dynamics of a point scalar charge

*q*and moves on a world line

*γ*described by relations

*z*

^{μ}(

*λ*), in which

*λ*is an arbitrary parameter. The particle generates a scalar potential Φ(

*x*) and a field Φ

_{α}(

*x*) ≡ ∇

_{α}Φ(

*x*). The dynamics of the entire system is governed by the action

*S*

_{field}is an action functional for a free scalar field in a spacetime with metric

*gαβ*,

*S*

_{particle}is the action of a free particle moving on a world line

*γ*in this spacetime, and

*S*

_{interaction}is an interaction term that couples the field to the particle.

*R*by an arbitrary constant

*α*. The particle action is

*m*

_{0}is the bare mass of the particle and \(d\tau = \sqrt {- {g_{\mu \nu}}(z){{\dot z}^\mu}{{\dot z}^\nu}} d\lambda\) is the differential of proper time along the world line; we use an overdot on

*z*

^{μ}(

*λ*) to indicate differentiation with respect to the parameter

*λ*. Finally, the interaction term is given by

*S*

_{particle}and

*S*

_{interaction}are invariant under a reparameterization

*λ*→

*λ*′(

*λ*) of the world line.

*δ*Φ(

*x*) of the field configuration yields the wave equation

*x*) defined by

_{α}(

*x*) once the motion of the scalar charge is specified. On the other hand, demanding that the total action be stationary under a variation (

*δz*

^{μ}(

*λ*) of the world line yields the equations of motion for the scalar charge,

*τ*as the parameter on the world line, and introduced the four-velocity

*u*

^{μ}(

*τ*) ≡

*dz*

^{μ}/

*dτ*. The dynamical mass that appears in Equation (387) is defined by

*m*(

*τ*) =

*m*

_{0}−

*q*Φ(

*z*), which can also be written in differential form as

#### 5.1.2 Retarded potential near the world line

*G*

_{+}(

*x*,

*x*′) is the retarded Green’s function introduced in Section 4.3. After substitution of Equation (386) we obtain

*z*(

*τ*) gives the description of the world line

*γ*. Because the retarded Green’s function is defined globally in the entire spacetime, Equation (389) applies to any field point

*x*.

*x*near the world line (see Figure 9). We let \({\mathcal N}(x)\) be the normal convex neighbourhood of this point, and we assume that the world line traverses \({\mathcal N}(x)\). Let

*τ*

_{<}be the value of the proper-time parameter at which

*γ*enters \({\mathcal N}(x)\) from the past, and let

*τ*

_{>}be its value when the world line leaves \({\mathcal N}(x)\). Then Equation (389) can be broken down into the three integrals

*x*is then in the past of

*z*(

*τ*), and

*G*

_{+}(

*x*,

*z*) = 0. For the second integration,

*x*is the normal convex neighbourhood of

*z*(

*τ*), and the retarded Green’s function can be expressed in the Hadamard form produced in Section 4.3.2. This gives

*x*′ ≡

*z*(

*u*) be the retarded point associated with

*x*; these points are related by

*σ*(

*x*,

*x*′) = 0 and \(r \equiv {\sigma _{{\alpha {\prime}}}}{u^{{\alpha {\prime}}}}\) is the retarded distance between

*x*and the world line. We resume the index convention of Section 3.3: To tensors at

*x*we assign indices

*α*,

*β*, etc.; to tensors at

*x*′ we assign indices

*α*′,

*β*′, etc.; and to tensors at a generic point

*z*(

*τ*) on the world line we assign indices

*μ*,

*ν*, etc.

*τ*to

*σ*, noticing that a increases as

*z*(

*τ*) passes through

*x*′. The change of

*σ*on the world line is given by \(d\sigma \equiv \sigma (x,z + dz) - \sigma (x,z) = {\sigma _\mu}{u^\mu}d\tau\), and we find that the first integral evaluates to

*U*(

*x*,

*z*)/(

*σ*

_{μ}

*u*

^{σ}) with

*z*identified with

*x*′. The second integration is cut off at

*τ*=

*u*by the step function, and we obtain our final expression for the retarded potential of a point scalar charge:

*x*sufficiently close to the world line that there exists a nonempty intersection between \({\mathcal N}(x)\) and

*γ*.

#### 5.1.3 Field of a scalar charge in retarded coordinates

*x*induces a variation in

*x*′ because the new points

*x*+

*δ*′ and

*x*′ +

*δx*′ must also be linked by a null geodesic — you may refer back to Section 3.3.2 for a detailed discussion. This means, for example, that the total variation of

*U*(

*x*,

*x*′) is \(\delta U = U(x + \delta x,{x{\prime}} + \delta {x{\prime}}) - U(x,{x{\prime}}) = {U_{;\alpha}}\delta {x^\alpha} + {U_{;{\alpha {\prime}}}}{u^{{\alpha {\prime}}}}\delta u\). The gradient of the scalar potential is therefore given by

_{α}

*G*

_{+}(

*x*,

*z*) from

*τ*= −∞ to almost

*τ*=

*u*, but we cut the integration short at

*τ*=

*u*

^{−}=

*u*≡ 0

^{+}to avoid the singular behaviour of the retarded Green’s function at

*σ*= 0. This limiting procedure gives rise to the first form of the definition, with the advantage that the integral need not be broken down into contributions that refer to \({\mathcal N}(x)\) and its complement, respectively.

_{α}(

*x*) in powers of

*r*, and express the results in terms of the retarded coordinates (

*u*,

*r*, Ω

^{a}) introduced in Section 3.3. It will be convenient to decompose Φ

_{α}(

*x*) in the tetrad \((e_0^\alpha, e_a^\alpha)\) that is obtained by parallel transport of \(({u^{{\alpha {\prime}}}},e_a^{{\alpha {\prime}}})\) on the null geodesic that links

*x*to

*x*′ ≡

*z*(

*u*); this construction is detailed in Section 3.3. Note that throughout this section we set

*ω*

_{ab}= 0, where

*ω*

_{ab}is the rotation tensor defined by Equation (138): The tetrad vectors are taken to be Fermi-Walker transported on

*γ*. The expansion relies on Equation (166) for ∂

_{α}

*u*, Equation (168) for ∂

_{α}

*r*, and we shall need

*x*′. We shall also need the expansions

*R*≡

*R*(

*u*) is the Ricci scalar evaluated at

*x*′.

*x*′, and

*x*′. Equations (397) and (398) show clearly that Φ

_{α}(

*x*) is singular on the world line: The field diverges as

*r*

^{−2}when

*r*→ 0, and many of the terms that stay bounded in the limit depend on Ω

^{a}and therefore possess a directional ambiguity at

*r*= 0.

#### 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 \({\bar x}\).

_{α}(

*x*) in the tetrad \((\bar e_0^\alpha, \bar e_a^\alpha)\), thereby defining \({{\bar \Phi}_0} \equiv {\Phi _\alpha}\bar e_0^\alpha\) and \({{\bar \Phi}_a} \equiv {\Phi _\alpha}\bar e_a^\alpha\) For this purpose we use Equations (224, 225) and (397, 398) to obtain

*x*′, except for \(\bar \Phi _0^{{\rm{tail}}}\) and \(\bar \Phi _a^{{\rm{tail}}}\) which are evaluated at \({\bar x}\).

*t*,

*s*,

*ω*

^{a}). For this we involve Equations (221, 222, 223), from which we deduce, for example,

*a*

_{a}(

*u*) as \({a_a}(t) - {\mathcal S}{{\dot a}_a}(t) + {\mathcal O}({{\mathcal S}^2})\), since according to Equation (221), \(u = t - s + {\mathcal O}({{\mathcal S}^2})\).

*R*(

*t*) is the Ricci scalar evaluated at \({\bar x}\). Finally, we have that

*S*(

*t*,

*s*), a two-surface of constant

*t*and

*s*; these will represent the mean value of the field at a fixed proper distance away from the world line, as measured in a reference frame that is momentarily comoving with the particle. The two-surface is charted by angles

*θ*

^{A}(

*A*= 1, 2) and it is described, in the Fermi normal coordinates, by the parametric relations \({{\hat x}^a} = {\mathcal S}{\omega ^a}({\theta ^A})\); canonical choice of parameterization is

*ω*

^{a}= (sin

*θ*cos

*ϕ*sin

*θ*sin

*ϕ*, cos

*θ*). Introducing the transformation matrices \(\omega _A^a \equiv \partial {\omega ^a}/\partial {\theta ^A}\), we find from Equation (127) that the induced metric on

*S*(

*t*,

*s*) is given by

*t*and the angles

*θ*

^{A}. From this we infer that the element of surface area is given by

*dΩ*= sin

*θ dθdΦ*. Integration of Equation (404) produces the total surface area of

*S*(

*t*,

*s*), and \({\mathcal A} = 4\pi {{\mathcal S}^2}[1 - {1 \over {18}}{{\mathcal S}^2}{R^{ab}}_{ab} + {\mathcal O}({{\mathcal S}^3})]\).

*s*→ 0 of the expressions displayed in Equations (407) and (408). In the limit the tetrad \((\bar e_0^\alpha, \bar e_a^\alpha)\) reduces to \(({u^{\bar \alpha}},\bar e_a^{\bar \alpha})\), and we can reconstruct the field at \({\bar x}\) by invoking the completeness relations \({\delta ^{\bar \alpha}}_{\bar \beta} = - {u^{\bar \alpha}}{u_{\bar \beta}} + e_a^{\bar \alpha}e_\beta ^a\). We thus obtain

#### 5.1.5 Singular and radiative fields

*x*is sufficiently close to

*γ*that the world line traverses \({\mathcal N}(x)\) (refer back to Figure 9). As before we let

*τ*

_{<}and

*τ*

_{>}be the values of the proper-time parameter at which

*γ*enters and leaves \({\mathcal N}(x)\), respectively. Then Equation (411) can be broken down into the three integrals,

*x*is then in the chronological future of

*z*(

*τ*), and

*G*

_{S}(

*x*,

*z*) = 0 by Equation (286). Similarly, the third integration vanishes because

*x*is then in the chronological past of

*z*(

*τ*). For the second integration,

*x*is the normal convex neighbourhood of

*z*(

*τ*), the singular Green’s function can be expressed in the Hadamard form of Equation (297), and we have

*x*′ ≡

*z*(

*u*) and let

*x*″ ≡

*z*(

*v*) be the

*advanced point*associated with

*x*; we recall from Section 3.4.4 that these points are related by

*σ*(

*x*,

*x*″) = 0 and that \({r_{{\rm{adv}}}} \equiv - {\sigma _{{\alpha ^{{\prime\prime}}}}}{u^{{\alpha ^{{\prime\prime}}}}}\) is the advanced distance between

*x*and the world line.

*τ*to

*σ*, noticing that

*σ*increases as

*z*(

*τ*) passes through

*x*′; the integral evaluates to

*U*(

*x*,

*x*′)/

*r*. We do the same for the second integration, but we notice now that

*σ*decreases as

*z*(

*τ*) passes through

*x*″; the integral evaluates to \(U(x,{x^{{\prime\prime}}})/{r_{{\rm{adv}}}}\) The third integration is restricted to the interval

*u*≤

*τ*≤

*v*by the step function, and we obtain our final expression for the singular potential of a point scalar charge:

*u*and the advanced time

*v*; contrary to what was found in Section 5.1.2 for the retarded potential, there is no dependence on the particle’s remote past.

*r*. For this we shall rely on results already established in Section 5.1.3, as well as additional expansions that will involve the advanced point

*x*″. Those we develop now.

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.

*U*(

*x*,

*x*″) we follow the general method of Section 3.4.4 and define a function

*U*(

*τ*) ≡

*U*(

*x*,

*z*(

*τ*)) of the proper-time parameter on

*γ*. We have that

*τ*, and where Δ′ ≡

*v*−

*u*. The leading term

*U*(

*u*) ≡

*U*(

*x*,

*x*′) was worked out in Equation (393), and the derivatives of

*U*(

*τ*) are given by

*x*′ ≡

*z*(

*u*), and not at the advanced point. The preceding computation gives us also an expansion for \({U_{;{\alpha ^{{\prime\prime}}}}}{u^{{\alpha ^{{\prime\prime}}}}} \equiv \dot U(\upsilon) = \dot U(u) + \ddot U(u){\Delta {\prime}} + {\mathcal O}({\Delta ^{{\prime}2}})\). This becomes

*U*

_{;α}(

*x*,

*x*″). Here we introduce the functions

*U*

_{α}(

*τ*) ≡

*U*

_{;α}(

*x*,

*z*(

*τ*)) and express

*U*

_{;α}(

*x*,

*x*″) as \({U_\alpha}(\upsilon) = {U_\alpha}(u) + {{\dot U}_\alpha}(u){\Delta {\prime}} + {\mathcal O}({\Delta ^{{\prime}2}})\) The leading term

*U*

_{α}(

*u*) ≡

*U*

_{;α}(

*x*,

*x*″) was computed in Equation (394), and

*x*′.

*x*′ ≡

*z*(

*u*). Comparison of these expressions with Equations (397) and (398) reveals that the retarded and singular fields share the same singularity structure.

*x*=

*x*′, where the tetrad \((e_0^\alpha, e_a^\alpha)\) becomes \(({u^{{\alpha {\prime}}}},e_a^{{\alpha {\prime}}})\). Reconstructing the field at

*x*′ from its frame components, we obtain

*x*′ ≡

*z*(

*u*), which now stands for an arbitrary point on the world line

*γ*.

#### 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.

*s*

_{0}in Fermi normal coordinates, and it is independent of the angles contained in the unit vector

*ω*

^{a}. The

*net force*acting at proper time

*τ*on this hollow shell is the average of

_{q}Φ

_{α}(

*τ*,

*s*

_{0},

*ω*

^{a}) over the surface of the shell. This was worked out at the end of Section 5.1.4, and ignoring terms that disappear in the limit

*s*

_{0}→ 0, we obtain

*z*(

*τ*) on the world line.

*m*≡

*m*

_{0}−

*q*Φ(

*z*) denoting the (also formally divergent) dynamical mass of the particle. We see that m and

*δm*combine in Equation (427) to form the particle’s observed mass

*m*

_{obs}, which is taken to be finite and to give a true measure of the particle’s inertia. All diverging quantities have thus disappeared into the process of mass renormalization. Substituting Equations (424) and (426) into Equation (388), in which we replace

*m*by

*m*

_{obs}=

*m*+

*δm*, returns an expression for the rate of change of the observed mass,

*not*conserved is a remarkable property of the dynamics of a scalar charge in a curved spacetime. Physically, this corresponds to the fact that in a spacetime with a time-dependent metric, a scalar charge radiates monopole waves and the radiated energy comes at the expense of the particle’s inertial mass.

*δm*, the averaged field of Equation (424) has exactly the same form as the radiative field of Equation (422), which we re-express as

The equations of motion displayed in Equations (427) and (428) are third-order differential equations for the functions *z*^{μ}(*τ*). It is well known that such a system of equations admits many unphysical solutions, such as runaway situations in which the particle’s acceleration increases exponentially with *τ*, even in the absence of any external force [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 right-hand side of the equations of motion, any occurrence of the acceleration vector by \(f_{{\rm{ext}}}^\mu/m\), where \(f_{{\rm{ext}}}^\mu\) is the external force acting on the particle. Because \(f_{{\rm{ext}}}^\mu\) is a prescribed quantity, differentiation of the external force does not produce higher derivatives of the functions *z*^{μ}(*τ*), and the equations of motion are properly of second order.

*z*(

*τ*). We recall that the tail integration must be cut short at

*τ*′ =

*τ*

^{−}≡

*τ*− 0

^{+}to avoid the singular behaviour of the retarded Green’s function at coincidence; this procedure was justified at the beginning of Section 5.1.3. Equations (431) and (432) were first derived by Theodore C. Quinn in 2000 [48]. In his paper Quinn also establishes that the total work done by the scalar self-force matches the amount of energy radiated away by the particle.

### 5.2 Motion of an electric charge

#### 5.2.1 Dynamics of a point electric charge

*e*and moves on a world line

*γ*described by relations

*z*

^{μ}(

*λ*), in which

*λ*is an arbitrary parameter. The particle generates a vector potential

*A*

^{α}(

*x*) and an electromagnetic field

*F*

_{αβ}(

*x*) = ∂

_{α}

*A*

_{β}− ∂

_{β}

*A*

_{α}. The dynamics of the entire system is governed by the action

*S*

_{fie1d}is an action functional for a free electromagnetic field in a spacetime with metric

*g*

_{αβ},

*S*

_{partic1e}is the action of a free particle moving on a world line

*γ*in this spacetime, and

*S*

_{interaction}is an interaction term that couples the field to the particle.

*m*is the bare mass of the particle and \(d\tau = \sqrt {- {g_{\mu \nu}}(z){{\dot z}^\mu}{{\dot z}^\nu}} d\lambda\) is the differential of proper time along the world line; we use an overdot to indicate differentiation with respect to the parameter

*λ*. Finally, the interaction term is given by

*S*

_{partic1e}and

*S*

_{interaction}are invariant under a reparameterization

*λ*→

*λ*′(

*λ*) of the world line.

*δA*

^{α}(

*x*) of the vector potential yields Maxwell’s equations

*j*

^{α}(

*x*) defined by

*F*

_{αβ}, once the motion of the electric charge is specified. On the other hand, demanding that the total action be stationary under a variation

*δz*

^{μ}(λ) of the world line yields the equations of motion for the electric charge,

*τ*as the parameter on the world line, and introduced the four-velocity

*u*

^{μ}(

*τ*) ≡

*dz*

^{μ}/

*dτ*.

*F*

_{αβ}is invariant under a gauge transformation of the form

*A*

_{α}→

*A*

_{α}+ ∂

_{α}Λ, in which Λ(

*x*) is an arbitrary scalar function. This function can always be chosen so that the vector potential satisfies the Lorenz gauge condition,

*g*

^{αβ}∇

_{α}∇

_{β}is the wave operator and \({R^\alpha}_\beta\) is the Ricci tensor. Having adopted

*τ*as the parameter on the world line, we can re-express the current density of Equation (438) as

#### 5.2.2 Retarded potential near the world line

*z*

^{μ}(

*τ*) gives the description of the world line

*γ*and

*u*

^{μ}(

*τ*) =

*dz*

^{μ}/

*dτ*. Because the retarded Green’s function is defined globally in the entire spacetime, Equation (443) applies to any field point

*x*.

*x*close to the world line. We let \({\mathcal N}(x)\) be the normal convex neighbourhood of this point, and we assume that the world line traverses \({\mathcal N}(x)\) (refer back to Figure 9). As in Section 5.1.2 we let

*τ*

_{<}and

*τ*

_{>}be the values of the proper-time parameter at which

*γ*enters and leaves \({\mathcal N}(x)\), respectively. Then Equation (443) can be expressed as

*x*is then in the past of

*z*(

*τ*), and \(G_{+ \mu}^\alpha (x,z) = 0\). For the second integration,

*x*is the normal convex neighbourhood of

*z*(

*τ*), and the retarded Green’s function can be expressed in the Hadamard form produced in Section 4.4.2. This gives

*x*′ ≡

*z*(

*u*) be the retarded point associated with

*x*; these points are related by

*σ*(

*x*,

*x*′) = 0 and \(r \equiv {\sigma _{{\alpha {\prime}}}}{u^{{\alpha {\prime}}}}\) is the retarded distance between

*x*and the world line. To perform the first integration we change variables from

*τ*to

*σ*, noticing that

*σ*increases as

*z*(

*τ*) passes through

*x′*; the integral evaluates to \({U^\alpha}_{{\beta {\prime}}}{u^{{\beta {\prime}}}}/r\). The second integration is cut off at

*τ*=

*u*by the step function, and we obtain our final expression for the vector potential of a point electric charge:

*x*sufficiently close to the world line that there exists a nonempty intersection between \({\mathcal N}(x)\) and

*γ*.

#### 5.2.3 Electromagnetic field in retarded coordinates

*x*′, because the new points

*x*+

*δx*and

*x*′ +

*δx*′ must also be linked by a null geodesic. Taking this into account, we find that the gradient of the vector potential is given by

*τ*= −∞ to

*τ*=

*u*

^{−}≡

*u*≡ 0

^{+}to avoid the singular behaviour of the retarded Green’s function at

*σ*= 0, is equivalent to the first form.

*F*

_{αβ}= ∇

_{α}

*A*

_{β}− ∇

_{β}

*A*

_{α}in powers of

*r*, and express the result in terms of the retarded coordinates (

*u*,

*r*, Ω

^{a}) introduced in Section 3.3. It will be convenient to decompose the electromagnetic field in the tetrad \((e_0^\alpha, e_a^\alpha)\) that is obtained by parallel transport of \(({u^{{\alpha \prime}}},e_a^{{\alpha \prime}})\) on the null geodesic that links

*x*to

*x*′ ≡

*z*(

*u*); this construction is detailed in Section 3.3. Note that throughout this section we set

*ω*

_{ab}= 0, where

*ω*

_{ab}is the rotation tensor defined by Equation (138): The tetrad vectors \(e_a^{{\alpha \prime}}\) are taken to be Fermi-Walker transported on

*γ*. We recall from Equation (141) that the parallel propagator can be expressed as \({g^{{\alpha \prime}}}_\alpha = {u^{{\alpha \prime}}}e_\alpha ^0 + e_a^{{\alpha \prime}}e_\alpha ^a\). The expansion relies on Equation (166) for

*∂*

_{α}

*u*, Equation (168) for

*∂*

_{α}

*r*, and we shall need

*x*′:

*x*′ ≡

*z*(

*u*) associated with

*x*; for example, \({a_a} \equiv {a_a}(u) \equiv {a_{{\alpha \prime}}}e_a^{{\alpha \prime}}\). It is clear from these equations that the electromagnetic field

*F*

_{αβ}(

*x*) is singular on the world line.

#### 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 \(\bar x \equiv z(t)\).

*F*

_{αβ}(

*x*) in the tetrad \({F_{\alpha \beta}}(x)\), thereby defining \({{\bar F}_{a0}} = {F_{\alpha \beta}}\bar e_a^\alpha \bar e_0^\beta\) and \({{\bar F}_{ab}} = {F_{\alpha \beta}}\bar e_a^\alpha \bar e_b^\beta\). For this purpose we use Equations (224, 225) and (451, 452) to obtain

*x*′, except for

*t*,

*s*,

*ω*

^{a}). For this we involve Equations (221, 222, 223), and we recycle some computations that were first carried out in Section 5.1.4. After some algebra, we arrive at

*S*(

*t*,

*s*), a two-surface of constant

*t*and

*s*. These are defined by

*S*(

*t*,

*s*), and \({\mathcal A} = \oint {d{\mathcal A}}\). Using the methods developed in Section 5.1.4, we find

*s*→ 0 of the expressions displayed in Equations (458) and (459). In the limit the tetrad \((\bar e_0^\alpha, \bar e_a^\alpha)\) becomes \(({u^{\bar \alpha}},e_a^{\bar \alpha})\), and we can easily reconstruct the field at \({\bar x}\) from its frame components. We thus obtain

*γ*.

#### 5.2.5 Singular and radiative fields

*x*is sufficiently close to

*γ*that the world line traverses \({\mathcal N}(x)\) (refer back to Figure 9). As before we let

*τ*

_{<}and

*τ*

_{>}be the values of the proper-time parameter at which

*γ*enters and leaves \({\mathcal N}(x)\), respectively. Then Equation (462) becomes

*x*is then in the chronological future of

*z*(

*τ*), and \(G_{S\mu}^{\,\,\alpha}(x,z) = 0\) by Equation (338). Similarly, the third integration vanishes because

*x*is then in the chronological past of

*z*(

*τ*). For the second integration,

*x*is the normal convex neighbourhood of

*z*(

*τ*), the singular Green’s function can be expressed in the Hadamard form of Equation (344), and we have

*x*′ ≡

*z*(

*u*) and

*x*″ ≡

*z*(

*v*) be the retarded and advanced points associated with

*x*, respectively. To perform the first integration we change variables from

*τ*to a, noticing that

*σ*increases as

*z*(

*τ*) passes through

*x*′; the integral evaluates to \({U^\alpha}_{{\beta \prime}}{u^{{\beta \prime}}}/r\). We do the same for the second integration, but we notice now that

*σ*decreases as

*z*(

*τ*) passes through

*x*″; the integral evaluates to \({U^\alpha}_{{\beta ^{\prime \prime}}}{u^{{\beta ^{\prime \prime}}}}/{r_{{\rm{adv}}}}\) where \({r_{{\rm{adv}}}} \equiv - {\sigma _{{\alpha ^{\prime \prime}}}}{u^\alpha}^{\prime \prime}\) is the advanced distance between

*x*and the world line. The third integration is restricted to the interval

*u*≤

*τ*≤

*v*by the step function, and we obtain the expression

*r*. For this we will rely on results already established in Section 5.2.3, as well as additional expansions that will involve the advanced point

*x*″. We recall 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.

*U*

_{α}(

*τ*) ≡

*U*

_{αμ}(

*x*,

*z*)

*u*

^{μ}. We have that

*τ*, and Δ′ ≡

*v*−

*u*. The leading term \({U_\alpha}(u) \equiv {U_{\alpha {\beta \prime}}}{u^{{\beta \prime}}}\) was worked out in Equation (447), and the derivatives of

*U*

_{α}(

*τ*) are given by

*x*′, and not at the advanced point. The preceding computation gives us also an expansion for

*x*′. Comparison of these expressions with Equations (451) and (452) reveals that the retarded and singular fields share the same singularity structure.

*x*=

*x*′, where the tetrad \((e_0^\alpha, e_a^\alpha)\) becomes \(({u^{{\alpha \prime}}},e_a^{{\alpha \prime}})\). Reconstructing the field at

*x*′ from its frame components, we obtain

*x*′ ≡

*z*(

*u*), which now stands for an arbitrary point on the world line

*γ*.

#### 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.

*s*

_{0}in Fermi normal coordinates, and it is independent of the angles contained in the unit vector

*ω*

^{a}. The

*net force*acting at proper time

*τ*on this shell is proportional to the average of

*F*

_{αβ}(

*τ*,

*s*

_{0},

*ω*

^{a}) over the shell’s surface. This was worked out at the end of Section 5.2.4, and ignoring terms that disappear in the limit

*s*

_{0}→ 0, we obtain

*z*(

*τ*) on the world line.

*δm*combine in Equation (478) to form the particle’s observed mass

*m*

_{obs}, which is finite and gives a true measure of the particle’s inertia. All diverging quantities have thus disappeared into the procedure of mass renormalization.

*δm*, the averaged force of Equation (475) has exactly the same form as the force that arises from the radiative field of Equation (473), which we express as

*m*denotes the observed inertial mass of the electric charge and all tensors are evaluated at

*z*(

*τ*), the current position of the particle on the world line; the primed indices in the tail integral refer to the point

*z*(

*τ′*), which represents a prior position. We recall that the integration must be cut short at

*τ*′ =

*τ*

^{−}≡

*τ*− 0

^{+}to avoid the singular behaviour of the retarded Green’s function at coincidence; this procedure was justified at the beginning of Section 5.2.3. Equation (481) was first derived (without the Ricci-tensor term) by Bryce S. DeWitt and Robert W. Brehme in 1960 [24], and then corrected by J.M. Hobbs in 1968 [29]. An alternative derivation was produced by Theodore C. Quinn and Robert M. Wald in 1997 [49]. In a subsequent publication [50], Quinn and Wald proved that the total work done by the electromagnetic self-force matches the energy radiated away by the particle.

### 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 test-mass 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 **sans-serif** symbols to denote tensors that refer to the perturbed spacetime; tensors in the background spacetime will be denoted, as usual, by italic symbols.

*ż*

^{μ}=

*dz*

^{μ}/

*dλ*is tangent to the world line and the metric is evaluated at

*z*. We assume that the particle provides the

*only*source of matter in the spacetime — an explanation will be provided at the end of this section — so that the Einstein field equations take the form of

^{αβ}is the Einstein tensor constructed from g

_{αβ}and

*S*

_{partic1e}with respect to g

_{αβ}(

*x*); the parallel propagators appear naturally by expressing g

_{μν}as \({g^\alpha}_\mu {g^\beta}_\nu {g_{\alpha \beta}}\).

_{αβ}is obtained by solving the Einstein field equations, and the world line is determined by solving the equations of energy-momentum conservation, which follow from the field equations. From Equations (81, 260, 484) we obtain

*ż*

^{μ}/

*dλ*is the covariant acceleration and k is a scalar field on the world line. Energy-momentum conservation therefore produces the geodesic equation

*λ*to be an affine parameter on the geodesic

*γ*.

*m*as a formal expansion parameter, and we write

*g*

_{αβ}denoting the

*m*→ 0 limit of the total metric g

_{αβ}, and \({h_{\alpha \beta}} = {\mathcal O}(m)\) the first-order correction. We shall refer to

*g*

_{αβ}as the “metric of the background spacetime” and to

*h*

_{αβ}as the “perturbation” produced by the particle. We similarly write

*T*

^{αβ}(

*x*) describes the stress-energy tensor of a test particle of mass

*m*that moves on a world line

*γ*in a background spacetime with metric

*g*

_{αβ}. If we choose

*λ*to be proper time

*τ*as measured in this spacetime, then Equation (484) implies

*u*

^{μ}(

*τ*) =

*dz*

^{μ}/

*dτ*is the particle’s four-velocity.

*g*

_{αβ}must therefore be a solution to the vacuum field equations:

*G*

^{αβ}[

*g*] = 0. Equations (483, 488, (489) then imply

*H*

^{αβ}[

*g*;

*h*] = 8

*πT*

^{αβ}, in which both sides of the equation are of order

*m*. To simplify the expression of the first-order correction to the Einstein tensor we introduce the trace-reversed gravitational potentials

*g*

_{αβ}. We then have \({H^{\alpha \beta}} = - {1 \over 2}(\square{\gamma ^{\alpha \beta}} + 2R_{\gamma \;\delta}^{\alpha \;\beta}{\gamma ^{\gamma \delta}})\), and Equation (483) reduces to

*g*

^{αα}∇

_{α}∇

_{β}is the wave operator and

*T*

^{αβ}is defined by Equation (490). We have here a linear wave equation for the potentials

*γ*

_{αβ}, and this equation can be placed on an equal footing with Equation (385) for the potential Φ associated with a point scalar charge, and Equation (441) for the vector potential

*A*

^{α}associated with a point electric charge.

*τ*(as measured in the background spacetime) as the parameter on the world line. On the other hand, Equation (486) gives

*a*

^{λ}=

*Du*

^{λ}/

*dτ*is the particle’s acceleration vector. Since it is clear that the acceleration will be of order

*m*, the second term can be discarded and we obtain

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.

*somewhere*else.) Suppose also that the matter distribution is described by a collection of matter fields Ψ. Then the field equations satisfied by the matter have the schematic form

*E*[Ψ;

*g*] = 0, and the metric is determined by the Einstein field equations

*G*[

*g*] = 8

*πM*[Ψ;

*g*], in which

*M*[Ψ;

*g*] stands for the matter’s stress-energy tensor. We now insert the point particle in the spacetime, and recognize that this displaces the background solution (Ψ,

*g*) to a new solution (Ψ +

*δ*Ψ,

*g*+

*δg*). The perturbations are determined by the coupled set of equations

*E*[Ψ +

*δ*Ψ;

*g*+

*δg*] =0 and

*G*[

*g*+

*δg*] = 8

*πM*[Ψ +

*δ*Ψ;

*g*+

*δg*] + 8

*πT*[

*g*]. After linearization these take the form of

*E*

_{Ψ},

*E*

_{g},

*M*

_{Φ}, and

*M*

_{g}are suitable differential operators acting on the perturbations. This is a

*coupled set*of partial differential equations for the perturbations

*δ*Ψ and

*δg*. These equations are linear, but they are much more difficult to deal with than the single equation for

*δg*that was obtained in the vacuum case. And although it is still possible to solve the coupled set of equations via a Green’s function technique, the degree of difficulty is such that we will not attempt this here. We shall, therefore, continue to restrict our attention to the case of a point particle moving in a vacuum (globally Ricci-flat) background spacetime.

#### 5.3.2 Retarded potentials near the world line

*z*

^{μ}(

*τ*) gives the description of the world line

*γ*and

*u*

^{μ}=

*dz*

^{μ}/

*dτ*. Because the retarded Green’s function is defined globally in the entire background spacetime, Equation (495) describes the gravitational perturbation created by the particle at any point

*x*in that spacetime.

*x*to be in a neighbourhood of the world line. The following manipulations follow closely those performed in Section 5.1.2 for the case of a scalar charge, and in Section 5.2.2 for the case of an electric charge. Because these manipulations are by now familiar, it will be sufficient here to present only the main steps. There are two important simplifications that occur in the case of a massive particle. First, for the purposes of computing

*γ*

^{αβ}(

*x*) to first order in

*m*, it is sufficient to take the world line to be a

*geodesic*of the background spacetime: The deviations from geodesic motion that we are in the process of calculating are themselves of order m and would affect

*γ*

^{αβ}(

*x*) at order

*m*

^{2}only. We shall therefore be allowed to set

*g*

_{αβ}to be a solution to the vacuum field equations, we are also allowed to set

*x*is close to the world line (refer back to Figure 9), we substitute the Hadamard construction of Equation (352) into Equation (495) and integrate over the portion of

*γ*that is contained in \({\mathcal N}(x)\). The result is

*x*′ ≡

*z*(

*u*) associated with \(r \equiv {\sigma _{{\alpha \prime}}}{u^{{\alpha \prime}}}\) is the retarded distance from

*x*′ to

*x*, and

*τ*

_{<}is the proper time at which

*γ*enters \({\mathcal N}(x)\) from the past.

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

*x*′ and

*x*are related by

*σ*(

*x*,

*x*′) = 0, a straightforward computation reveals that the covariant derivatives of the gravitational potentials are given by

*τ*=

*u*

^{−}≡

*u*− 0

^{+}to avoid the singular behaviour of the retarded Green’s function at

*σ*= 0, is equivalent to the first form.

*α*

_{αβ;γ}(

*x*) in the retarded coordinates of Section 3.3, as an expansion in powers of

*r*. For this purpose we decompose the field in the tetrad \((e_0^\alpha, e_a^\alpha)\) that is obtained by parallel transport of \(({u^{{\alpha \prime}}},e_a^{{\alpha \prime}})\) on the null geodesic that links

*x*to

*x*′; this construction is detailed in Section 3.3. Note that throughout this section we set

*ω*

_{ab}= 0, where

*ω*

_{ab}is the rotation tensor defined by Equation (138): The tetrad vectors \(e_a^{{\alpha \prime}}\) are taken to be parallel transported on

*γ*. We recall from Equation (141) that the parallel propagator can be expressed as \({g^{{\alpha \prime}}}_\alpha = {u^{{\alpha \prime}}}e_\alpha ^0 + e_a^{{\alpha \prime}}e_\alpha ^a\). The expansion relies on Equation (166) for

*∂*

_{γ}

*u*and Equation (168) for

*∂*

_{γ}

*r*, both specialized to the case of geodesic motio,

*a*

_{a}= 0. We shall also need

*x*′ ≡

*z*(

*u*). We have also introduced the frame components of the tail part of the gravitational field, which are obtained from Equation (500) evaluated at

*x*′ instead of

*x*; for example, \(\gamma _{000}^{{\rm{tail}}} = {u^{{\alpha \prime}}}{u^{{\beta \prime}}}{u^{{\gamma \prime}}}\gamma _{{\alpha \prime}{\beta \prime}{\gamma \prime}}^{{\rm{tail}}}({x\prime})\). We may note here that while

*γ*

_{00c}is the only component of the gravitational field that diverges when

*r*→ 0, the other components are nevertheless singular because of their dependence on the unit vector Ω

^{a}; the only exception is

*γ*

_{ab0}, which is smooth.

#### 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.

*x*′.

*t*and

*s*. Using the area element of Equation (404) and definitions analogous to those of Equation (405), we obtain

*s*→ 0, in which the tetrad \((\bar e_0^\alpha, \bar e_a^\alpha)\) coincides with \(({u^{\bar \alpha}},e_a^{\bar \alpha})\) Reconstructing the field at \({\bar x}\) from its frame components gives

*γ*.

#### 5.3.5 Singular and radiative fields

*x*to be close to the world line (see Figure 9), and we invoke Equation (373) as well as the Hadamard construction of Equation (379). This gives

*x*′ ≡

*z*(

*u*), double-primed indices refer to the advanced point

*x*″ ≡

*z*(

*v*), and where \({r_{{\rm{adv}}}} \equiv - {\sigma _{{\alpha ^{\prime \prime}}}}{u^\alpha}^{\prime \prime}\) is the advanced distance between

*x*and the world line.

*r*. For this we will rely on results already established in Section 5.3.3, as well as additional expansions that will involve the advanced point

*x*″. We recall 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

*∂*

_{γ}

*v*and

*∂*

_{γ}

*r*

_{adv}, respectively; these results can be simplified by setting \({a_a} = {{\dot a}_0} = {{\dot a}_a} = 0\) which is appropriate in this computation.

*τ*and Δ′ ≡

*v*−

*u*. The leading term \({U_{\alpha \beta}}(u) \equiv {U_{\alpha \beta {\alpha \prime}{\beta \prime}}}{u^{{\alpha \prime}}}{u^{{\beta \prime}}}\) was worked out in Equation (501), and the derivatives of

*U*

_{αβ}(

*τ*) are given by

*x*′, and not at the advanced point. The preceding computation gives us also an expansion for

*x*′. Comparison of these expressions with Equations (505, 506, 507, 508, 509, 510) reveals identical singularity structures for the retarded and singular gravitational fields.

*r*→ 0. We may therefore evaluate the radiative field directly at

*x*=

*x*′, where the tetrad \((e_0^\alpha, e_a^\alpha)\) coincides with \(({u^{{\alpha \prime}}},e_a^{{\alpha \prime}})\). After reconstructing the field at

*x*′ from its frame components, we obtain

*x*′ ≡

*z*(

*u*) which can now be treated as an arbitrary point on the world line

*γ*.

#### 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.

*γ*

_{μν;λ}〉 as worked out in Equation (523), or \(\gamma _{\mu \nu; \lambda}^{\rm{R}}\) as worked out in Equation (544). These regularized fields are both given by

*z*(

*τ*) on the world line

*γ*.

*h*

_{αβ}is obtained by inverting Equation (491), which leads to \({h_{\mu \nu; \lambda}} = {\gamma _{\mu \nu; \gamma}} - {1 \over 2}{g_{\mu \nu}}{\gamma ^\rho}_{\rho; \lambda}\). Substituting Equation (546) yields

*z*(

*τ*) of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions

*z*(

*τ′*), at which all tensors with primed indices are evaluated. The tail integral is cut short at

*τ*′ =

*τ*

^{−}≡

*τ*− 0

^{+}to avoid the singular behaviour of the retarded Green’s function at coincidence; this limiting procedure was justified at the beginning of Section 5.3.3.

Equation (550) was first derived by Yasushi Mino, Misao Sasaki, and Takahiro Tanaka in 1997 [39]. (An incomplete treatment had been given previously by Morette-DeWitt 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 trace-reversal 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 self-force 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].

*x*

^{α}are the coordinates of the background spacetime, and

*ξ*

^{α}is a vector field that we take to be of order m. We assume that

*ξ*

^{α}is smooth in a neighbourhood of the world line

*γ*. The coordinate transformation changes the background metric according to

*a*

^{μ}is the acceleration of Equation (550) and

*a*[

*ξ*]

^{μ}is the “gauge acceleration” generated by the vector field

*ξ*

^{α}.

*ξ*

_{λ;νρ}−

*ξ*

_{λ;ρν}=

*R*

_{νρωλ}

*ξ*

^{ω}, and the fact that \({a^\mu} = {\mathcal O}(m)\). The final expression is

*ξ*

^{ν}in the direction of the world line. The expression within the large brackets is familiar from the equation of geodesic deviation, which states that this quantity vanishes if

*ξ*

^{μ}is a deviation vector between two neighbouring geodesics. Equation (553), with

*a*[

*ξ*]

^{μ}given by Equation (554), is therefore a generalized version of this statement.

### 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 well-behaved 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 strong-field, fast-motion situation.)

*r*be a meaningful measure of distance from the small black hole, and let us consider a region of spacetime defined by

*r*<

*r*

_{i}, where

*r*

_{i}is a constant that is much smaller than \({\mathcal R}\). This inequality defines a narrow world tube that surrounds the small black hole, and we shall call this region the

*internal zone*(see Figure 10). In the internal zone the gravitational field is dominated by the black hole, and the metric can be expressed as

*g*(black hole) is the metric of a nonrotating black hole in isolation (as given by the unperturbed Schwarzschild solution), while

*H*

_{1}and

*H*

_{2}are corrections associated with the conditions in the external universe. The metric of Equation (555) represents a black hole that is distorted by the tidal gravitational field of the external universe, and

*H*

_{1},

*H*

_{2}are functions of m and the spacetime coordinates that can be obtained by solving the Einstein field equations. They must be such that the spacetime possesses a regular event horizon near

*r*= 2

*m*, and such that g(internal zone) agrees with the metric of the external universe — the metric of the background spacetime in the absence of the black hole — when

*r*≫

*m*. As we shall see in Section 5.4.2,

*H*

_{1}actually vanishes and the small correction \({H_2}/{{\mathcal R}^2}\) can be obtained by employing the well-developed tools of black-hole perturbation theory [51, 59, 63].

*r*>

*r*

_{e}, where

*r*

_{e}is a constant that is much larger than m; this region will be called the

*external zone*(see Figure 10). In the external zone the gravitational field is dominated by the conditions in the external universe, and the metric can be expressed as

*g*(background spacetime) is the unperturbed metric of the background spacetime in which the black hole is moving, while

*h*

_{1}and

*h*

_{2}are corrections associated with the hole’s presence; these are functions of \({\mathcal R}\) and the spacetime coordinates that can be obtained by solving the Einstein field equations. We shall truncate Equation (556) to first order in

*m*, and

*mh*

_{1}will be calculated in Section 5.4.3 by linearizing the field equations about the metric of the background spacetime. In the external zone the perturbation associated with the presence of a black hole cannot be distinguished from the perturbation produced by a point particle of the same mass, and

*mh*

_{1}will therefore be obtained by solving Equation (493) in the background spacetime.

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 external-zone 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 self-gravitating bodies is determined by the Einstein field equations.

*r*≫

*m*in the internal zone, the metric of the unperturbed black hole can be expanded as

*g*(black hole) =

*η*⊕

*m/r*⊕

*m*

^{2}/

*r*

^{2}⊕ …, where

*η*is the metric of flat spacetime (in asymptotically inertial coordinates) and the symbol ⊕ means “and a term of the form… ”. On the other hand, dimensional analysis dictates that \({H_1}/{\mathcal R}\) be of the form \(r/{\mathcal R} \oplus m/{\mathcal R} \oplus {m^2}/\left({r{\mathcal R}} \right) \oplus \ldots\) while \({H_2}/{{\mathcal R}^2}\) should be expressed as \({r^2}/{{\mathcal R}^2} \oplus mr/{{\mathcal R}^2} \oplus {m^2}/{{\mathcal R}^2} \oplus \ldots\). Altogether we obtain

*r*be much smaller than \({\mathcal R}\) in the external zone, we have that the metric of the background spacetime can be expressed as \(g\left({{\rm{background}}\,{\rm{spacetime}}} \right) = \eta \oplus r/{\mathcal R} \oplus {r^2}/{{\mathcal R}^2} \oplus \ldots\), where the expansion now uses world-line based coordinates such as the Fermi normal coordinates of Section 3.2 or the retarded coordinates of Section 3.3. On dimensional grounds we also have \(m{h_1} = m/r \oplus m/{\mathcal R} \oplus mr/{{\mathcal R}^2} \oplus \ldots\) and \({m^2}{h_2} = {m^2}/{r^2} \oplus {m^2}/\left({r{\mathcal R}} \right) \oplus {m^2}/{{\mathcal R}^2} \oplus \ldots\). Altogether this gives

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 internal-zone 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.

*g*(black hole), the Schwarzschild metric of an isolated black hole of mass

*m*, in terms of retarded Eddington-Finkelstein coordinates \(\left({\bar u,\,\bar r,\,{{\bar \theta}^A}} \right)\) where

*ū*is retarded time, \(\bar r\) the usual areal radius, and \({\bar \theta ^A} = \left({\bar \theta, \, \bar \phi} \right)\) are two angles spanning the two-spheres of constant

*α*and \(\bar r\). The metric is given by

*r*≫

*m*this metric achieves the asymptotic values

*H*

_{1}and

*H*

_{2}in Equation (555) encode the information that our black hole is not isolated but in fact immersed in an external universe whose metric becomes

*g*(background spacetime) asymptotically. In the internal zone the metric of the background spacetime can be expanded in powers of \(\bar r/{\mathcal R}\) and expressed in a form that can be directly imported from Section 3.3. If we assume for the moment that the “world line” \(\bar r = 0\) has no acceleration in the background spacetime (a statement that will be justified shortly), then the asymptotic values of g(internal zone) must be given by Equations (210, 211, 212, 213):

*ū*through the frame components \({\mathcal E}ab \equiv {R_{a0b0}} = {\mathcal O}\left({1/{{\mathcal R}^2}} \right)\) and \({{\mathcal B}^a}_b \equiv {1 \over 2}{\varepsilon ^{acd}}{R_{0bcd}} = {\mathcal O}\left({1/{{\mathcal R}^2}} \right)\) of the Riemann tensor. (This is the Riemann tensor of the background spacetime evaluated at \(\bar r = 0\).) Notice that we have incorporated the fact that the Ricci tensor vanishes at \(\bar r = 0:\) The black hole moves in a vacuum spacetime.

*e*

_{1},

*e*

_{2},

*e*

_{3},

*b*

_{2}, and

*b*

_{3}can all be determined by solving the Einstein field equations; they must all approach unity when

*r*≫

*m*and be well-behaved at

*r*= 2

*m*(so that the tidally distorted black hole will have a nonsingular event horizon). It is clear from Equations (562, 563, 564, 565) that the assumed deviation of g(internal zone) with respect to

*g*(black hole) scales as \(1/{{\mathcal R}^2}\). It is therefore of the form of Equation (555) with

*H*

_{1}= 0. The fact that

*H*

_{1}vanishes comes as a consequence of our previous assumption that the “world line” \(\bar r = 0\) has a zero acceleration in the background spacetime; a nonzero acceleration of order \(1/{\mathcal R}\) would bring terms of order \(1/{\mathcal R}\) to the metric, and

*H*

_{1}would then be nonzero.

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 odd-parity perturbations of degree *ℓ* =1 correspond to a shift in the black hole’s angular-momentum 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 odd-parity, *ℓ* =1 terms in *H*_{2}. Finally, Zerilli has shown that in a vacuum spacetime, even-parity 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 even-parity, *ℓ* = 1 terms from the perturbed metric. This leads us to Equations (562, 563, 564, 565), which contain only *ℓ* = 2 perturbation modes; the even-parity modes are contained in those terms that involve \({{\mathcal E}_{ab}}\), while the odd-parity 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.

*e*

_{1},

*e*

_{2},

*e*

_{3},

*b*

_{2}, and

*b*

_{3}. For this it is sufficient to take, say, \({{\mathcal E}_{12}} = {{\mathcal E}_{21}}\) and \({{\mathcal B}_{12}} = {{\mathcal B}_{21}}\) as the only nonvanishing components of the tidal fields

*ε*

_{ab}and \({{\mathcal B}_{ab}}\). And since the equations for even-parity and odd-parity perturbations decouple, each case can be considered separately. Including only even-parity perturbations, Equations (562)–(565) become

_{αβ}= 0. Calculating the Einstein tensor is simplified by linearizing with respect to \({{\mathcal E}_{12}}\) and discarding its derivatives with respect to

*ū*: Since the time scale over which \({{\mathcal E}_{ab}}\) changes is of order \({\mathcal R}\), the ratio between temporal and spatial derivatives is of order \(\bar r/{\mathcal R}\) and therefore small in the internal zone; the temporal derivatives can be consistently neglected. The field equations produce ordinary differential equations to be satisfied by the functions

*e*

_{1},

*e*

_{2}, and

*e*

_{3}. Those are easily decoupled, and demanding that the functions all approach unity as

*r*→ ∞ and be well-behaved at

*r*= 2

*m*yields the unique solutions

#### 5.4.3 Metric in the external zone

*m*≡

*r*

_{e}<

*r*. As was pointed out in Section 5.4.1, in the external zone the gravitational perturbation associated with the presence of a black hole cannot be distinguished from the perturbation produced by a point particle of the same mass; it can therefore be obtained by solving Equation (493) in a background spacetime with metric

*g*(background spacetime). The external-zone metric is decomposed as

*g*

_{αβ}is the metric of the background spacetime and \({h_{\alpha \beta}} = {\mathcal O}\left(m \right)\) is the perturbation; we shall work consistently to first order in

*m*and systematically discard all terms of higher order. We relate

*h*

_{αβ}to trace-reversed potentials

*γ*

_{αβ},

*z*

^{μ}(

*τ*) gives the description of the world line

*γ*,

*τ*is proper time in the background spacetime,

*u*

^{μ}=

*dz*

^{μ}/

*dτ*is the four-velocity, and \(G_{+ \,\mu \nu}^{\,\alpha \beta}\left({x,\,z} \right)\) is the retarded Green’s function associated with Equation (493); the potentials of Equation (578) satisfy the Lorenz-gauge condition \({\gamma ^{\alpha \beta}}_{;\beta} = 0\). As was pointed out in Section 5.4.1,

*γ*

_{αβ}(and therefore

*h*

_{αβ}) are functionals of a world line

*γ*that will be determined by matching g(external zone) to g(internal 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*, *r*Ω^{a}) of Section 3.3 and rely on the machinery developed there.

*g*

_{αβ}, the metric of the background spacetime. We have seen in Section 3.3.8 that if the world line

*γ*is a geodesic, if the vectors \(e_a^\mu\) are parallel transported on the world line, and if the Ricci tensor vanishes on

*γ*, then the metric takes the form given by Equations (207, 208, 209). This form, however, is too restrictive for our purposes: We must allow

*γ*to have an acceleration, and allow the basis vectors to be transported in the most general way compatible with their orthonormality property; this transport law is given by Equation (138),

*a*

^{μ}=

*Du*

^{μ}/

*dτ*, and

*ω*

_{ab}(

*τ*) =

^{−}

*ω*

_{ba}(

*τ*) is a rotation tensor to be determined. Anticipating that

*a*

_{a}and

*ω*

_{ab}will both be proportional to m, we express the metric of the background spacetime as

*a*

_{a}and

*ω*

_{ab}and specializing to a zero Ricci tensor.

*h*

_{αβ}as an expansion in powers of \(r/{\mathcal R}\) we first go back to Equation (498) and rewrite it in the form

*x*′ ≡

*z*(

*u*) associated with

*x*, and

*τ*

_{<}is the proper time at which

*γ*enters

*x*’s normal convex neighbourhood from the past). We next expand the first term on the right-hand side of Equation (583) with the help of Equation (501), and the tail term is expanded using Equation (93) in which we substitute Equation (504) and the familiar relation \({\sigma ^{\alpha {\prime}}} = - r\left({{u^{\alpha {\prime}}} + {\Omega ^a}e_a^{\alpha {\prime}}} \right)\). This gives

*x*′, and where

*u*,

*r*Ω

^{a}).

*h*

_{αβ}in the tetrad \(\left({e_0^\alpha, \, e_a^\alpha} \right)\) that is obtained by parallel transport of \(\left({{u^{\alpha {\prime}}},\,e_a^{\alpha {\prime}}} \right)\) on the null geodesic that links

*x*to its corresponding retarded point

*x*′ ≡

*z*(

*u*) on the world line. (The vectors are parallel transported in the background spacetime.) The projections are

*u*only.

*a*

_{a}and

*ω*

_{ab}can both be set equal to zero in these equations (because they would produce negligible terms of order

*m*

^{2}in

*h*

_{αβ}), and that

*S*

_{ab},

*S*

_{a}, and

*S*can all be expressed in terms of the tidal fields \({{\mathcal E}^{\ast}},\,{\mathcal E}_a^{\ast},\,{\mathcal E}_{ab}^{\ast},\,{\mathcal B}_a^{\ast}\) and \({\mathcal B}_{ab}^{\ast}\) using Equations (204, 205, 206), we arrive at

*coordinate components*of the metric perturbation

*h*

_{αβ}in the retarded coordinates (

*u*,

*rΩ*

^{a}), expressed in terms of

*frame components*of the tail fields \(h_{\alpha {\prime}\beta {\prime}}^{{\rm{tails}}}\) and \(h_{\alpha {\prime}\beta {\prime}\gamma {\prime}}^{{\rm{tails}}}\). The perturbation is expanded in powers of \(r/{\mathcal R}\) and it also involves the tidal gravitational fields of the background spacetime.

*g*

_{αβ}as given by Equations (580, 581, 582) to

*h*

_{αβ}as given by Equations (593, 594, 595). The final result is

*a*

_{a}and

*ω*

_{ab}are both of order \(m/{{\mathcal R}^2}\); this will be confirmed in Section 5.4.5.

#### 5.4.4 Transformation from external to internal coordinates

Comparison of Equations (568, 569, 570) and Equations (596, 597, 598) reveals that the internal-zone and external-zone 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.

*m*/

*r*and \(mr/{{\mathcal R}^2}\). This transformation redefines the radial coordinate —

*r*→

*r*′ =

*r*+

*m*— and incorporates in

*u*− the gravitational time delay contributed by the small mass

*m*. After performing the coordinate transformation the metric becomes

*m/r*′, but there is still a mismatch at orders \(m/{\mathcal R}\) and \(mr{\prime}/{{\mathcal R}^2}\).

*u*. The metric now matches g(internal zone) at orders \(1,\,{{r{\prime\prime}}^2}/{{\mathcal R}^2},\,m/r{\prime\prime}\), and \(m/{\mathcal R}\), but there is still a mismatch at order \(mr{\prime\prime}/{{\mathcal R}^2}\).

*a*

_{a}and

*ω*

_{ab}, this metric is equal to g(internal zone) as given by Equations (568, 569, 570) linearized with respect to

*m*.

#### 5.4.5 Motion of the black hole in the background spacetime

*γ*; the rotation tensor is nonzero and given by Equation (617).

### 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 self-forces, most especially in the gravitational case.

#### 5.5.1 Conservation of energy-momentum

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 energy-momentum conservation, in spite of the fact that historically, it is one of the most important.

Conservation of energy-momentum 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 self-force. 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.)

*T*

^{αβ}includes a contribution from the particle and a contribution from the field; the particle’s contribution is a Dirac functional on the world line, and the field’s contribution diverges as 1/

*r*

^{4}near the world line. (I am using retarded coordinates in this discussion.) While in flat spacetime the differential statement of energy-momentum conservation can immediately be turned into an integral statement, the same is not true in a curved spacetime (unless the spacetime possesses at least one Killing vector). To proceed it is necessary to rewrite the conservation equation as

*g*

^{μα}(

*z*,

*x*) is a parallel propagator from

*x*to an arbitrary point

*z*on the world line. Integrating this equation over the interior of a world-tube segment that consists of a “wall” of constant

*r*and two “caps” of constant

*u*, we obtain

*dV*an invariant, four-dimensional volume element.

There is no obstacle in evaluating the wall integral, for which *T*^{αβ} reduces to the field’s stress-energy 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 stress-energy 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

*F*

_{αβ}over the shell’s surface. (The averaging is performed in the shell’s rest frame, and the shell is spherical in the sense that its proper distance from the world line is the same in all directions.) The averaged field is next evaluated on the world line, in the limit of a zero-radius shell. Because the radiative field is smooth on the world line, this yields

*ma*

_{μ}=

*e*〈

*F*

_{μν}〉

*u*

^{ν}, where

*m*is the particle’s bare mass. With the preceding results we arrive at \({m_{{\rm{obs}}}}{a_\mu} = eF_{\mu \nu}^{\rm{R}}{u^\nu}\), where

*m*

_{obs}≡

*m*+

*δm*is the particle’s observed (renormalized) inertial mass.

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 near-source 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 Detweiler-Whiting 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 Detweiler-Whiting axiom produces the statement that the point massthe singular field exerts no force on the particle (it merely contributes to the particle’s inertia); the entire self-force arises from the action of the radiative field.

*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.

#### 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 point-particle 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 point-mass description than to perform the matching of two metrics in two coordinate systems.

#### 5.5.5 Evaluation of the gravitational self-force

The concrete evaluation of the scalar, electromagnetic, and gravitational self-forces 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 self-force 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 low-frequency gravitational waves for the Laser Interferometer Space Antenna (LISA) [32]; to calculate these waves requires an accurate description of the motion, beyond the test-mass approximation which ignores the object’s radiation reaction.

*g*

_{αβ}is the Kerr metric, and we wish to calculate

*a*

^{μ}[

*h*

^{R}] for a body moving in the Kerr spacetime. This calculation is challenging and it involves a large number of steps.

*h*

_{αβ}produced by a point particle moving on a specified geodesic of the Kerr spacetime. A method for doing this was elaborated by Lousto and Whiting [34] and Ori [44], building on the pioneering work of Teukolsky [57], Chrzanowski [18], and Wald [61]. The procedure consists of

solving the Teukolsky equation for one of the Newman-Penrose 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.

*ψ*

_{0}or

*ψ*

_{4}is expressed as a multipole expansion, summing over modes with (spheroidal-harmonic) indices

*l*and

*m*. In fact, the procedure outlined above relies heavily on this mode decomposition, and the metric perturbation returned at the end of the procedure is also expressed as a sum over modes \(h_{\alpha \beta}^l\). (For each

*l*,

*m*ranges from −

*l*to

*l*, and summation of

*m*over this range is henceforth understood.) From these, mode contributions to the self-acceleration can be computed:

*aμ*[

*h*

_{l}] is obtained from our preceding expression for the self-acceleration by substituting

*h*

_{αβ}in place of \(h_{\alpha \beta}^{\rm{R}}\). These mode contributions do not diverge on the world line, but

*a*

^{μ}[

*h*

_{l}] is discontinuous at the radial position of the particle. The sum over modes, on the other hand, does not converge, because the “bare” acceleration (constructed from the retarded field

*h*

_{αβ}) is formally infinite.

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 self-acceleration; 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 self-acceleration 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 self-acceleration 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 mode-decomposing 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 well-developed 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 Newman-Penrose 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 angular-momentum 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 self-force

The successful computation of the gravitational self-force 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 gauge-invariant information associated with the body’s corrected motion. Because the MiSaTaQuWa equations of motion are not by themselves gauge-invariant, this step will necessitate going beyond the self-force.

To see how this might be done, imagine that the small body is a pulsar, and that it emits light pulses at regular proper-time 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 times-of-arrival of the pulses. Because these can be measured directly by a distant observer, they clearly constitute gauge-invariant information. But the times-of-arrival 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 gauge-invariant 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 gauge-invariant 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 first-order problem. Schematically, one introduces a perturbation h that satisfies a wave equation □*h* = *T*[*z*] in the background spacetime, where *T*[*z*] is the stress-energy tensor of the moving body, which is a functional of the world line *z*(*τ*). In first-order perturbation theory, the stress-energy tensor must be conserved in the background spacetime, and *z*(*τ*) must describe a geodesic. It follows that in first-order perturbation theory, the waveforms constructed from the perturbation *h* contain no information about the body’s corrected motion.

The first-order 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 self-force, one would now resolve the wave equation with a corrected stress-energy 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 gauge-invariant. In fact, to be consistent one would have to include *all* second-order 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 stress-energy tensor). But provided one can find a way to make this second-order problem well posed, and provided one can solve it (or at least the relevant part of it), the waveforms constructed from the second-order 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 right-hand side. The resulting equation gives rise to the well-known problem of runaway solutions. To avoid such unphysical behaviour I have submitted Dirac’s equation to a reduction-of-order 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 minimally-coupled 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 sans-serif symbols for the frame indices. This is to distinguish them from another set of frame indices that will appear below. The frame indices introduced here run from 0 to 3; those to be introduced later will run from 1 to 3.

## 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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