Abstract
This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a selfforce that prevents the particle from moving on a geodesic of the background spacetime. The selfforce contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the selfforce matches the energy radiated away by the particle.
The field’s action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field’s singular part and show that it exerts no force on the particle — its only effect is to contribute to the particle’s inertia. What remains after subtraction is a regular field that is fully responsible for the selfforce. Because this field satisfies a homogeneous wave equation, it can be thought of as a free field that interacts with the particle; it is this interaction that gives rise to the selfforce.
The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors (Part I). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle’s word line (Part II). It continues with a thorough discussion of Green’s functions in curved spacetime (Part III). The review presents a detailed derivation of each of the three equations of motion (Part IV). Because the notion of a point mass is problematic in general relativity, the review concludes (Part V) with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.
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1 Introduction and summary
1.1 Invitation
The motion of a point electric charge in flat spacetime was the subject of active investigation since the early work of Lorentz, Abrahams, Poincaré, and Dirac [56], until Gralla, Harte, and Wald produced a definitive derivation of the equations motion [82] with all the rigour that one should demand, without recourse to postulates and renormalization procedures. (The field’s early history is well related in Ref. [154].) In 1960 DeWitt and Brehme [54] generalized Dirac’s result to curved spacetimes, and their calculation was corrected by Hobbs [95] several years later. In 1997 the motion of a point mass in a curved background spacetime was investigated by Mino, Sasaki, and Tanaka [130], who derived an expression for the particle’s acceleration (which is not zero unless the particle is a test mass); the same equations of motion were later obtained by Quinn and Wald [150] using an axiomatic approach. The case of a point scalar charge was finally considered by Quinn in 2000 [149], and this led to the realization that the mass of a scalar particle is not necessarily a constant of the motion.
This article reviews the achievements described in the preceding paragraph; it is concerned with the motion of a point scalar charge q, a point electric charge e, and a point mass m in a specified background spacetime with metric g_{ αβ }. These particles carry with them fields that behave as outgoing radiation in the wave zone. The radiation removes energy and angular momentum from the particle, which then undergoes a radiation reaction — its world line cannot be simply a geodesic of the background spacetime. The particle’s motion is affected by the nearzone field which acts directly on the particle and produces a selfforce. In curved spacetime the selfforce contains a radiationreaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angularmomentum transport. The selfforce is proportional to q^{2} in the case of a scalar charge, proportional to e^{2} in the case of an electric charge, and proportional to m^{2} in the case of a point mass.
In this review we derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely selfcontained, and all relevant materials are developed ab initio. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. One must first assimilate the basic theory of bitensors (Part I), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line (Part II). One must next formulate a theory of Green’s functions in curved spacetimes (Part III), and finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how they should act on the particle (Part IV). A dedicated reader, correctly skeptical that sense can be made of a point mass in general relativity, will also want to work through the last portion of the review (Part V), which provides a derivation of the equations of motion for a small, but physically extended, body; this reader will be reassured to find that the extended body follows the same motion as the point mass. The review is very long, but the satisfaction derived, we hope, will be commensurate.
In this introductory section we set the stage and present an impressionistic survey of what the review contains. This should help the reader get oriented and acquainted with some of the ideas and some of the notation. Enjoy!
1.2 Radiation reaction in flat spacetime
Let us first consider the relatively simple and wellunderstood case of a point electric charge e moving in flat spacetime [154, 101, 171]. The charge produces an electromagnetic vector potential A^{α} that satisfies the wave equation
together with the Lorenz gauge condition ∂_{ α }A^{α} = 0. (On page 294, Jackson [101] 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 fourdimensional 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 selfforce 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.
Difficult but not impossible. To find a way around this problem we note first that the situation considered here, in which the radiation is propagating outward and the charge is spiraling inward, breaks the timereversal invariance of Maxwell’s theory. A specific time direction was adopted when, among all possible solutions to the wave equation, we chose \(A^\alpha_{\rm{ret}}\), the retarded solution, as the physically relevant solution. Choosing instead the advanced solution \(A_{{\rm{adv}}}^\alpha\) would produce a timereversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition
would restore timereversal invariance: outgoing and incoming radiation would be present in equal amounts, there would be no net loss nor gain of energy by the system, and the charge would undergo no radiation reaction. In Eq. (1.2) the subscript ‘S’ stands for ‘symmetric’, as the vector potential depends symmetrically upon future and past.
Our second key observation is that while the potential of Eq. (1.2) does not exert a force on the charged particle, it is just as singular as the retarded potential in the vicinity of the world line. This follows from the fact that \(A_{{\rm{ret}}}^\alpha,\; A_{{\rm{adv}}}^\alpha\), and \(A_{\rm{S}}^\alpha\) all satisfy Eq. (1.1), whose source term is infinite on the world line. So while the wavezone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line — all three electromagnetic fields are dominated by the particle’s Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript ‘S’: it stands for ‘singular’ as well as ‘symmetric’.
Because \(A_{\rm{S}}^\alpha\) is just as singular as \(A_{{\rm{ret}}}^\alpha\), removing it from the retarded solution gives rise to a potential that is well behaved in a neighbourhood of the world line. And because \(A_{\rm{S}}^\alpha\) is known not to affect the motion of the charged particle, this new potential must be entirely responsible for the radiation reaction. We therefore introduce the new potential
and postulate that it, and it alone, exerts a force on the particle. The subscript ‘R’ stands for ‘regular’, because \(A_{\rm{R}}^\alpha\) is nonsingular on the world line. This property can be directly inferred from the fact that the regular potential satisfies the homogeneous version of Eq. (1.1), \(\square A^\alpha_{\rm{R}} = 0\); there is no singular source to produce a singular behaviour on the world line. Since \(A_{\rm{R}}^\alpha\) satisfies the homogeneous wave equation, it can be thought of as a free radiation field, and the subscript ‘R’ could also stand for ‘radiative’.
The selfaction of the charge’s own field is now clarified: a singular potential \(A_{\rm{S}}^\alpha\) can be removed from the retarded potential and shown not to affect the motion of the particle. What remains is a wellbehaved potential \(A_{\rm{R}}^\alpha\) that must be solely responsible for the radiation reaction. From the regular potential we form an electromagnetic field tensor \(F_{\alpha \beta}^{\rm{R}} = {\partial _\alpha}A_\beta ^{\rm{R}}  {\partial _\beta}A_\alpha ^{\rm{R}}\) and we take the particle’s equations of motion to be
where u^{μ} = dz^{μ}/dτ is the charge’s fourvelocity [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 regular field yields the more concrete expression
in which the second term is the selfforce that is responsible for the radiation reaction. We observe that the selfforce is proportional to e^{2}, it is orthogonal to the fourvelocity, and it depends on the rate of change of the external force. This is the result that was first derived by Dirac [56]. (Dirac’s original expression actually involved the rate of change of the acceleration vector on the righthand side. The resulting equation gives rise to the wellknown problem of runaway solutions. To avoid such unphysical behaviour we have submitted Dirac’s equation to a reductionoforder procedure whereby da^{ν}/dτ is replaced with \({m^{ 1}}df_{{\rm{ext}}}^\nu/d\tau\). This procedure is explained and justified, for example, in Refs. [112, 70], and further discussed in Section 24 below.)
To establish that the singular field exerts no force on the particle requires a careful analysis that is presented in the bulk of the paper. What really happens is that, because the particle is a monopole source for the electromagnetic field, the singular field is locally isotropic around the particle; it therefore exerts no force, but contributes to the particle’s inertia and renormalizes its mass. In fact, one could do without a decomposition of the field into singular and regular solutions, and instead construct the force by using the retarded field and averaging it over a small sphere around the particle, as was done by Quinn and Wald [150]. In the body of this review we will use both methods and emphasize the equivalence of the results. We will, however, give some emphasis to the decomposition because it provides a compelling physical interpretation of the selfforce as an interaction with a free electromagnetic field.
1.3 Green’s functions in flat spacetime
To see how Eq. (1.5) can eventually be generalized to curved spacetimes, we introduce a new layer of mathematical formalism and show that the decomposition of the retarded potential into singular and regular pieces can be performed at the level of the Green’s functions associated with Eq. (1.1). The retarded solution to the wave equation can be expressed as
in terms of the retarded Green’s function \(G_{+ \beta \prime}^{\alpha}(x,x{\prime}) = \delta _{\beta \prime}^\alpha \delta (t  t{\prime}  \vert x  x{\prime} \vert)/\vert x  x{\prime} \vert\). Here 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
in terms of the advanced Green’s function \(G_{+ \beta {\prime}}^{\alpha}(x,x{\prime}) = \delta _{\beta \prime}^\alpha \delta (t  t{\prime}  \vert x  x{\prime} \vert)/\vert x  x{\prime} \vert\). The retarded Green’s function is zero whenever 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
this states that the retarded Green’s function becomes the advanced Green’s function (and vice versa) when x and x′ are interchanged.
From the retarded and advanced Green’s functions we can define a singular Green’s function by
and a regular twopoint function by
By virtue of Eq. (1.8) the singular Green’s function is symmetric in its indices and arguments: \(G^{\rm{S}}_{\beta{\prime}\alpha}(x{\prime},x) = G^{\rm{S}}_{\alpha\beta{\prime}}(x,x{\prime})\). The regular twopoint function, on the other hand, is antisymmetric. The potential
satisfies the wave equation of Eq. (1.1) and is singular on the world line, while
satisfies the homogeneous equation □A^{α} = 0 and is well behaved on the world line.
Equation (1.6) implies that the retarded potential at x is generated by a single event in spacetime: the intersection of the world line and x’s past light cone (see Figure 1). We shall call this the retarded point associated with x and denote it z (u); u is the retarded time, the value of the propertime parameter at the retarded point. Similarly we find that the advanced potential of Eq. (1.7) is generated by the intersection of the world line and the future light cone of the field point x. We shall call this the advanced point associated with x and denote it z (v); v is the advanced time, the value of the propertime parameter at the advanced point.
1.4 Green’s functions in curved spacetime
In a curved spacetime with metric g_{ αβ } the wave equation for the vector potential becomes
where □ = 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 Eq. (1.13) take the same form as in Eqs. (1.6) and (1.7), except that dV′ now stands for \(\sqrt { g(x{\prime})} \,{d^4}x{\prime}\).
The causal structure of the Green’s functions is richer in curved spacetime: While in flat spacetime the retarded Green’s function has support only on the future light cone of 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 ≥ v.
The physically relevant solution to Eq. (1.13) is obviously the retarded potential \(A_{{\rm{ret}}}^{\alpha}(x)\), and as in flat spacetime, this diverges on the world line. The cause of this singular behaviour is still the pointlike nature of the source, and the presence of spacetime curvature does not change the fact that the potential diverges at the position of the particle. Once more this behaviour makes it difficult to figure out how the retarded field is supposed to act on the particle and determine its motion. As in flat spacetime we shall attempt to decompose the retarded solution into a singular part that exerts no force, and a regular part that produces the entire selfforce.
To decompose the retarded Green’s function into singular and regular parts is not a straightforward task in curved spacetime. The flatspacetime definition for the singular Green’s function, Eq. (1.9), cannot be adopted without modification: While the combination halfretarded plus halfadvanced Green’s functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Eq. (1.13), this candidate for the singular Green’s function would produce a selfforce with an unacceptable dependence on the particle’s future history. For suppose that we made this choice. Then the regular twopoint function would be given by the combination halfretarded minus halfadvanced Green’s functions, just as in flat spacetime. The resulting potential would satisfy the homogeneous wave equation, and it would be regular on the world line, but it would also depend on the particle’s entire history, both past (through the retarded Green’s function) and future (through the advanced Green’s function). More precisely stated, we would find that the regular potential at x depends on the particle’s state of motion at all times τ outside the interval u < τ < v; in the limit where x approaches the world line, this interval shrinks to nothing, and we would find that the regular potential is generated by the complete history of the particle. A selfforce constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and regular Green’s functions.
The proper definitions were identified by Detweiler and Whiting [53], who proposed the following generalization to Eq. (1.9):
The twopoint function \(H_{\;\beta {\prime}}^\alpha (x,x{\prime})\) is introduced specifically to cure the pathology described in the preceding paragraph. It is symmetric in its indices and arguments, so that \(G_{\alpha \beta{\prime}}^{\rm{S}}(x,x{\prime})\) will be also (since the retarded and advanced Green’s functions are still linked by a reciprocity relation); and it is a solution to the homogeneous wave equation, \(\square H^\alpha_{\ \beta{\prime}}(x,x{\prime})  R^\alpha_{\ \gamma}(x) H^\gamma_{\ \beta{\prime}}(x,x{\prime}) = 0\), so that the singular, retarded, and advanced Green’s functions will all satisfy the same wave equation. Furthermore, and this is its key property, the twopoint function is defined to agree with the advanced Green’s function when x is in the chronological past of \(x{\prime}:\,\,H_{\;\beta {\prime}}^\alpha (x,x{\prime}) = G_{ \beta {\prime}}^{\;\alpha}(x,x{\prime})\) 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 \(H_{\;\beta {\prime}}^\alpha (x,x{\prime})\) 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′.
The potential \(A_{\rm{S}}^\alpha (x)\) constructed from the singular Green’s function can now be seen to depend on the particle’s state of motion at times τ restricted to the interval u ≤ τ ≤ v (see Figure 3). Because this potential satisfies Eq. (1.13), 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.)
The DetweilerWhiting [53] definition for the regular twopoint function is then
The potential \(A^\alpha_{\rm{R}}(x)\) constructed from this depends on the particle’s state of motion at all times 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 \(A^\alpha_{\rm{S}}(x)\) can be shown to exert no force on the particle, we conclude that \(A^\alpha_{\rm{R}}(x)\) alone is responsible for the selfforce.
From the regular potential we form an electromagnetic field tensor \(F_{\alpha \beta}^{\rm{R}} = {\nabla _\alpha}A_\beta ^{\rm{R}}  {\nabla _\beta}A_\alpha ^{\rm{R}}\) and the curvedspacetime generalization to Eq. (1.4) is
where u^{μ} = dz^{μ}/dτ is again the charge’s fourvelocity, 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 subsection we add yet another layer of mathematical formalism and construct a convenient coordinate system to chart a neighbourhood of the particle’s world line. In the next subsection we will display explicit expressions for the retarded, singular, and regular fields of a point electric charge.
Let γ be the world line of a point particle in a curved spacetime. It is described by parametric relations z^{μ} (τ) in which τ is proper time. Its tangent vector is u^{μ} = dz^{μ}/dτ and its acceleration is a^{μ} = Du^{μ}/dτ; we shall also encounter ȧ^{μ}: = Da^{μ}/dτ.
On γ we erect an orthonormal basis that consists of the fourvelocity 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, \({g_{\mu \nu}}{u^\mu}{u^\nu} =  1,\,\,{g_{\mu \nu}}{u^\mu}e_a^\nu = 0\), and \({g_{\mu \nu}}e_a^\mu e_b^\nu = {\delta _{ab}}\). We take the spatial vectors to be FermiWalker transported on the world line: \(De_a^\mu/d\tau = {a_a}{u^\mu}\), where
are frame components of the acceleration vector; it is easy to show that FermiWalker transport preserves the orthonormality of the basis vectors. We shall use the tetrad to decompose various tensors evaluated on the world line. An example was already given in Eq. (1.17) but we shall also encounter frame components of the Riemann tensor,
as well as frame components of the Ricci tensor,
We shall use 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 twopoint function σ (x,z), known as Synge’s world function [169], is numerically equal to half the squared geodesic distance between z and x; it is positive if x and z are spacelike related, negative if they are timelike related, and σ (x, z) is zero if x and z are linked by a null geodesic. We denote its gradient ∂σ/∂z^{μ} by σ_{ μ } (x,z), and −σ^{μ} gives a meaningful notion of a separation vector (pointing from z tox).
To construct a coordinate system in this neighbourhood we locate the unique point x′:= z (u) on γ which is linked to x by a futuredirected null geodesic (this geodesic is directed from x′ to x); we shall refer to x′ as the retarded point associated with x, and u will be called the retarded time. To tensors at x′ we assign indices α′, β′, …; this will distinguish them from tensors at a generic point z (τ) on the world line, to which we have assigned indices μ, ν, …. We have σ (x, x′) = 0 and −σ^{α′} (x, x′) is a null vector that can be interpreted as the separation between x′ and x.
The 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
is an affine parameter on the null geodesic that links 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 we shall call r the retarded distance between x and the world line. The unit vector
is constant on the null geodesic that links x to x′. Because Ω^{a} is a different constant on each null geodesic that emanates from x′, keeping u fixed and varying Ω^{a} 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^{a}, 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 regular electromagnetic fields of a point electric charge
The retarded solution to Eq. (1.13) is
where the integration is over the world line of the point electric charge. Because the retarded solution is the physically relevant solution to the wave equation, it will not be necessary to put a label ‘ret’ on the vector potential.
From the vector potential we form the electromagnetic field tensor 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
where
are the frame components of the “tail part” of the field, which is given by
In these expressions, all tensors (or their frame components) are evaluated at the retarded point x′:= z (u) associated with x; for example, \({a_a} := {a_a}(u) := {a_{\alpha{\prime}}}e_a^{\alpha{\prime}}\). 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 Eqs. (1.23) and (1.24).
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 Fa0 diverges as r−^{2} when we approach the world line. But there are also terms that, though they stay bounded in the limit, possess a directional ambiguity at r = 0; for example F_{ ab } contains a term proportional to R_{a0bc}Ω^{c} whose limit depends on the direction of approach.
This singularity structure is perfectly reproduced by the singular field \(F^{\rm{S}}_{\alpha\beta}\) obtained from the potential
where \(G_{{\rm{S}}\,\mu}^{\;\alpha}(x,z)\) is the singular Green’s function of Eq. (1.14). Near the world line the singular field is given by
Comparison of these expressions with Eqs. (1.23) and (1.24) does indeed reveal that all singular terms are shared by both fields.
The difference between the retarded and singular fields defines the regular field \(F_{\alpha \beta}^{\rm{R}}(x)\). Its frame components are
and at x′ the regular field becomes
where \({\dot a^{\gamma{\prime}}} = D{a^{\gamma{\prime}}}/d\tau\) is the rate of change of the acceleration vector, and where the tail term was given by Eq. (1.26). We see that \(F_{\alpha \beta}^{\rm{R}}(x)\) is a regular tensor field, even on the world line.
1.7 Motion of an electric charge in curved spacetime
We have argued in Section 1.4 that the selfforce acting on a point electric charge is produced by the regular field, and that the charge’s equations of motion should take the form of \(m{a_\mu} = f_\mu ^{{\rm{ext}}} + eF_{\mu \nu}^{\rm{R}}{u^\nu}\), where \(f_\mu ^{{\rm{ext}}}\) is an external force also acting on the particle. Substituting Eq. (1.32) gives
in which 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 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 Eq. (1.33) we have reduced the order of the differential equation by replacing ȧ^{ν} with \({m^{ 1}}\dot f_{{\rm{ext}}}^\nu\) on the righthand side; this procedure was explained at the end of Section 1.2.
Equation (1.33) is the result that was first derived by DeWitt and Brehme [54] and later corrected by Hobbs [95]. (The original version of the equation did not include the Riccitensor term.) In flat spacetime the Ricci tensor is zero, the tail integral disappears (because the Green’s function vanishes everywhere within the domain of integration), and Eq. (1.33) reduces to Dirac’s result of Eq. (1.5). In curved spacetime the selfforce does not vanish even when the electric charge is moving freely, in the absence of an external force: it is then given by the tail integral, which represents radiation emitted earlier and coming back to the particle after interacting with the spacetime curvature. This delayed action implies that in general, the selfforce is nonlocal in time: it depends not only on the current state of motion of the particle, but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that the physical process that leads to Eq. (1.33) is simply an interaction between the charge and a free electromagnetic field \(F_{\alpha \beta}^{\rm{R}}\); it is this field that carries the information about the charge’s past.
1.8 Motion of a scalar charge in curved spacetime
The dynamics of a point scalar charge can be formulated in a way that stays fairly close to the electromagnetic theory. The particle’s charge q produces a scalar field Φ(x) which satisfies a wave equation
that is very similar to Eq. (1.13). Here, R is the spacetime’s Ricci scalar, and ξ is an arbitrary coupling constant; the scalar charge density μ (x) is given by a fourdimensional Dirac functional supported on the particle’s world line γ. The retarded solution to the wave equation is
where G_{+}(x, z) is the retarded Green’s function associated with Eq. (1.34). The field exerts a force on the particle, whose equations of motion are
where m is the particle’s mass; this equation is very similar to the Lorentzforce law. But the dynamics of a scalar charge comes with a twist: If Eqs. (1.34) and (1.36) 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
and in general 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.
The scalar field of Eq. (1.35) diverges on the world line and its singular part Φ_{S}(x) must be removed before Eqs. (1.36) and (1.37) can be evaluated. This procedure produces the regular field Φ_{R}(x), and it is this field (which satisfies the homogeneous wave equation) that gives rise to a selfforce. The gradient of the regular field takes the form of
when it is evaluated on the world line. The last term is the tail integral
and this brings the dependence on the particle’s past.
Substitution of Eq. (1.38) into Eqs. (1.36) and (1.37) gives the equations of motion of a point scalar charge. (At this stage we introduce an external force \(f_{{\rm{ext}}}^\mu\) and reduce the order of the differential equation.) The acceleration is given by
and the mass changes according to
These equations were first derived by Quinn [149]. (His analysis was restricted to a minimally coupled scalar field, so that ξ = 0 in his expressions. We extended Quinn’s results to an arbitrary coupling counstant for this review.)
In flat spacetime the Riccitensor term and the tail integral disappear and Eq. (1.40) takes the form of Eq. (1.5) with q^{2}/(3m) replacing the factor of 2e^{2}/(3m). In this simple case Eq. (1.41) reduces to dm/dτ = 0 and the mass is in fact a constant. This property remains true in a conformally flat spacetime when the wave equation is conformally invariant (ξ = 1/6): in this case the Green’s function possesses only a lightcone part and the righthand side of Eq. (1.41) vanishes. In generic situations the mass of a point scalar charge will vary with proper time.
1.9 Motion of a point mass, or a small body, in a background spacetime
The case of a point mass moving in a specified background spacetime presents itself with a serious conceptual challenge, as the fundamental equations of the theory are nonlinear and the very notion of a “point mass” is somewhat misguided. Nevertheless, to the extent that the perturbation h_{ aβ } (x) created by the point mass can be considered to be “small”, the problem can be formulated in close analogy with what was presented before.
We take the metric 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 tracereversed potentials γ_{ αβ } defined by
where h_{ aβ } is the difference between g_{ αβ }, the actual metric of the perturbed spacetime, and g_{ αβ }. The potentials satisfy the wave equation
together with the Lorenz gauge condition γ^{αβ}_{;β} = 0. Here and below, covariant differentiation refers to a connection that is compatible with the background metric, □ = g^{αβ} ∇_{ α } ∇_{ β } is the wave operator for the background spacetime, and T^{αβ} is the energymomentum tensor of the point mass; this is given by a Dirac distribution supported on the particle’s world line γ. The retarded solution is
where \(G_{+ \;\mu \nu}^{\;\alpha \beta}(x,z)\) is the retarded Green’s function associated with Eq. (1.43). The perturbation h_{ aβ } (x) can be recovered by inverting Eq. (1.42).
Equations of motion for the point mass can be obtained by formally demanding that the motion be geodesic in the perturbed spacetime with metric g_{ αβ } = g_{ αβ } + h_{ αβ }. After a mapping to the background spacetime, the equations of motion take the form of
The acceleration is thus proportional to m; in the testmass limit the world line of the particle is a geodesic of the background spacetime.
We now remove \(h_{\alpha \beta}^{\rm{S}}(x)\) from the retarded perturbation and postulate that it is the regular field \(h_{\alpha \beta}^{\rm{R}}(x)\) that should act on the particle. (Note that \(\gamma _{\alpha \beta}^{\rm{S}}\) satisfies the same wave equation as the retarded potentials, but that \(\gamma _{\alpha \beta}^{\rm{R}}\) is a free gravitational field that satisfies the homogeneous wave equation.) On the world line we have
where the tail term is given by
When Eq. (1.46) is substituted into Eq. (1.45) we find that the terms that involve the Riemann tensor cancel out, and we are left with
Only the tail integral appears in the final form of the equations of motion. It involves the current position 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 Eq. (1.48) were first derived by Mino, Sasaki, and Tanaka [130], and then reproduced with a different analysis by Quinn and Wald [150]. They are now known as the MiSaTaQuWa equations of motion. As noted by these authors, the MiSaTaQuWa equation has the appearance of the geodesic equation in a metric \({g_{\alpha \beta}} + h_{\alpha \beta}^{{\rm{tail}}}\). Detweiler and Whiting [53] have contributed the more compelling interpretation that the motion is actually geodesic in a spacetime with metric \({g_{\alpha \beta}} + h_{\alpha \beta}^{\rm{R}}\). The distinction is important: Unlike the first version of the metric, the DetweilerWhiting metric is regular on the world line and satisfies the Einstein field equations in vacuum; and because it is a solution to the field equations, it can be viewed as a physical metric — specifically, the metric of the background spacetime perturbed by a free field produced by the particle at an earlier stage of its history.
While Eq. (1.48) does indeed give the correct equations of motion for a small mass m moving in a background spacetime with metric g_{ αβ }, the derivation outlined here leaves much to be desired — to what extent should we trust an analysis based on the existence of a point mass? As a partial answer to this question, Mino, Sasaki, and Tanaka [130] produced an alternative derivation of their result, which involved a small nonrotating black hole instead of a point mass. In this alternative derivation, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: it must move according to Eq. (1.48). This alternative derivation (which was given a different implementation in Ref. [142]) is entirely free of singularities (except deep within the black hole), and it suggests that that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body’s internal structure can be ignored). This derivation, however, was limited to the case of a nonrotating black hole, and it relied on a number of unjustified and sometimes unstated assumptions [83, 144, 145]. The conclusion was made firm by the more rigorous analysis of Gralla and Wald [83] (as extended by Pound [144]), who showed that the MiSaTaQuWa equations apply to any sufficiently compact body of arbitrary internal structure.
It is important to understand that unlike Eqs. (1.33) and (1.40), which are true tensorial equations, Eq. (1.48) reflects a specific choice of coordinate system and its form would not be preserved under a coordinate transformation. In other words, the MiSaTaQuWa equations are not gauge invariant, and they depend upon the Lorenz gauge condition γ^{αβ}_{;β} = O (m^{2}). Barack and Ori [17] 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
is the “gauge acceleration”; D^{2}ξ^{ν}/dτ^{2} = (ξ^{ν}_{;μ}u^{μ})_{;ρ}u^{ρ} is the second covariant derivative of ξ^{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 wellposed physical question; to obtain such answers it is necessary to combine the equations of motion with the metric perturbation h_{ αβ } so as to form gaugeinvariant quantities that will correspond to direct observables. This point is very important and cannot be overemphasized.
The gravitational selfforce possesses a physical significance that is not shared by its scalar and electromagnetic analogues, because the motion of a small body in the strong gravitational field of a much larger body is a problem of direct relevance to gravitationalwave astronomy. Indeed, extrememassratio inspirals, involving solarmass compact objects moving around massive black holes of the sort found in galactic cores, have been identified as promising sources of lowfrequency gravitational waves for spacebased interferometric detectors such as the proposed Laser Interferometer Space Antenna (LISA [115]). These systems involve highly eccentric, nonequatorial, and relativistic orbits around rapidly rotating black holes, and the waves produced by such orbital motions are rich in information concerning the strongest gravitational fields in the Universe. This information will be extractable from the LISA data stream, but the extraction depends on sophisticated dataanalysis strategies that require a detailed and accurate modeling of the source. This modeling involves formulating the equations of motion for the small body in the field of the rotating black hole, as well as a consistent incorporation of the motion into a wavegeneration formalism. In short, the extraction of this wealth of information relies on a successful evaluation of the gravitational selfforce.
The finitemass corrections to the orbital motion are important. For concreteness, let us assume that the orbiting body is a black hole of mass m = 10 M_{⊙} and that the central black hole has a mass M = 10^{6} M_{⊙}. Let us also assume that the small black hole is in the deep field of the large hole, near the innermost stable circular orbit, so that its orbital period P is of the order of minutes. The gravitational waves produced by the orbital motion have frequencies f of the order of the mHz, which is well within LISA’s frequency band. The radiative losses drive the orbital motion toward a final plunge into the large black hole; this occurs over a radiationreaction timescale (M/m)P of the order of a year, during which the system will go through a number of wave cycles of the order of M/m = 10^{5}. The role of the gravitational selfforce is precisely to describe this orbital evolution toward the final plunge. While at any given time the selfforce provides fractional corrections of order m/M = 10^{−5} to the motion of the small black hole, these build up over a number of orbital cycles of order M/m = 10^{5} to produce a large cumulative effect. As will be discussed in some detail in Section 2.6, the gravitational selfforce is important, because it drives large secular changes in the orbital motion of an extrememassratio binary.
1.10 Case study: static electric charge in Schwarzschild spacetime
One of the first selfforce calculations ever performed for a curved spacetime was presented by Smith and Will [163]. They considered an electric charge e held in place at position r = r_{0} outside a Schwarzschild black hole of mass M. Such a static particle must be maintained in position with an external force that compensates for the black hole’s attraction. For a particle without electric charge this force is directed outward, and its radial component in Schwarzschild coordinates is given by \(f_{{\rm{ext}}}^r = {1 \over 2}mf{\prime}\), where m is the particle’s mass, f:= 1 − 2M/r_{0} is the usual metric factor, and a prime indicates differentiation with respect to r_{0}, so that \(f{\prime} = 2M/r_0^2\). Smith and Will found that for a particle of charge e, the external force is given instead by \(f_{{\rm{ext}}}^r = {1 \over 2}mf\prime  {e^2}M\,{f^{1/2}}/r_{\,0}^3\). The second term is contributed by the electromagnetic selfforce, and implies that the external force is smaller for a charged particle. This means that the electromagnetic selfforce acting on the particle is directed outward and given by
This is a repulsive force. It was shown by Zel’nikov and Frolov [186] that the same expression applies to a static charge outside a ReissnerNordström black hole of mass M and charge Q, provided that f is replaced by the more general expression \(f = 1  2M/{r_0} + {Q^2}/r_{\,\,0}^2\).
The repulsive nature of the electromagnetic selfforce acting on a static charge outside a black hole is unexpected. In an attempt to gain some intuition about this result, it is useful to recall that a blackhole horizon always acts as perfect conductor, because the electrostatic potential φ:= −A_{ t } is necessarily uniform across its surface. It is then tempting to imagine that the selfforce should result from a fictitious distribution of induced charge on the horizon, and that it could be estimated on the basis of an elementary model involving a spherical conductor. Let us, therefore, calculate the electric field produced by a point charge e situated outside a spherical conductor of radius R. The charge is placed at a distance r_{0} from the centre of the conductor, which is taken at first to be grounded. The electrostatic potential produced by the charge can easily be obtained with the method of images. It is found that an image charge e′ = −eR/r_{0} is situated at a distance r′_{0} = R^{2}/r_{0} from the centre of the conductor, and the potential is given by φ = e/s + e′/s′, where s is the distance to the charge, while s′ is the distance to the image charge. The first term can be identified with the singular potential φs, and the associated electric field exerts no force on the point charge. The second term is the regular potential φ_{ R }, and the associated field is entirely responsible for the selfforce. The regular electric field is \(E_{\rm{R}}^r =  {\partial _r}{\varphi _{\rm{R}}}\), and the selfforce is \(f_{{\rm{self}}}^r = eE_{\rm{R}}^r\). A simple computation returns
This is an attractive selfforce, because the total induced charge on the conducting surface is equal to e′, which is opposite in sign to e. With R identified with M up to a numerical factor, we find that our intuition has produced the expected factor of \({e^2}M/r_{\,0}^3\), but that it gives rise to the wrong sign for the selfforce. An attempt to refine this computation by removing the net charge e′ on the conductor (to mimic more closely the blackhole horizon, which cannot support a net charge) produces a wrong dependence on r_{0} in addition to the same wrong sign. In this case the conductor is maintained at a constant potential φ_{0} = −e′/R, and the situation involves a second image charge −e′ situated at r = 0. It is easy to see that in this case,
This is still an attractive force, which is weaker than the force of Eq. (1.51) by a factor of (R/r_{0})^{2}; the force is now exerted by an image dipole instead of a single image charge.
The computation of the selfforce in the blackhole case is almost as straightforward. The exact solution to Maxwell’s equations that describes a point charge e situated r = r_{0} and θ = 0 in the Schwarzschild spacetime is given by
where
is the solution first discovered by Copson in 1928 [43], while
is the monopole field that was added by Linet [114] to obtain the correct asymptotic behaviour φ ∼ e/r when r is much larger than r_{0}. It is easy to see that Copson’s potential behaves as e(1 − M/r_{0})/r at large distances, which reveals that in addition to e, φ^{s} comes with an additional (and unphysical) charge −eM/r_{0} situated at r = 0. This charge must be removed by adding to φ^{s} a potential that (i) is a solution to the vacuum Maxwell equations, (ii) is regular everywhere except at r = 0, and (iii) carries the opposite charge +eM/r_{0}; this potential must be a pure monopole, because higher multipoles would produce a singularity on the horizon, and it is given uniquely by φ^{R}. The Copson solution was generalized to ReissnerNordström spacetime by Léauté and Linet [113], who also showed that the regular potential of Eq. (1.55) requires no modification.
The identification of Copson’s potential with the singular potential φ^{s} is dictated by the fact that its associated electric field \(F_{\,tr}^{\rm{S}} = {\partial _r}{\varphi ^{\rm{S}}}\) is isotropic around the charge e and therefore exerts no force. The selfforce comes entirely from the monopole potential, which describes a (fictitious) charge +eM/r_{0} situated at r = 0. Because this charge is of the same sign as the original charge e, the selfforce is repulsive. More precisely stated, we find that the regular piece of the electric field is given by
and that it produces the selfforce of Eq. (1.50). The simple picture described here, in which the electromagnetic selfforce is produced by a fictitious charge eM/r_{0} situated at the centre of the black hole, is not easily extracted from the derivation presented originally by Smith and Will [163]. To the best of our knowledge, the monopolar origin of the selfforce was first noticed by Alan Wiseman [185]. (In his paper, Wiseman computed the scalar selfforce acting on a static particle in Schwarzschild spacetime, and found a zero answer. In this case, the analogue of the Copson solution for the scalar potential happens to satisfy the correct asymptotic conditions, and there is no need to add another solution to it. Because the scalar potential is precisely equal to the singular potential, the selfforce vanishes.)
We should remark that the identification of φ^{S} and φ^{R} with the DetweilerWhiting singular and regular fields, respectively, is a matter of conjecture. Although φ^{S} and φ^{R} satisfy the essential properties of the DetweilerWhiting decomposition — being, respectively, a regular homogenous solution and a singular solution sourced by the particle — one should accept the possibility that they may not be the actual DetweilerWhiting fields. It is a topic for future research to investigate the precise relation between the Copson field and the DetweilerWhiting singular field.
It is instructive to compare the electromagnetic selfforce produced by the presence of a grounded conductor to the selfforce produced by the presence of a black hole. In the case of a conductor, the total induced charge on the conducting surface is e′ = −eR/r_{0}, and it is this charge that is responsible for the attractive selfforce; the induced charge is supplied by the electrodes that keep the conductor grounded. In the case of a black hole, there is no external apparatus that can supply such a charge, and the total induced charge on the horizon necessarily vanishes. The origin of the selfforce is therefore very different in this case. As we have seen, the selfforce is produced by a fictitious charge eM/r_{0} situated at the centre of black hole; and because this charge is positive, the selfforce is repulsive.
1.11 Organization of this review
After a detailed review of the literature in Section 2, the main body of the review begins in Part I (Sections 3 to 7) with a description of the general theory of bitensors, the name designating tensorial functions of two points in spacetime. We introduce Synge’s world function σ (x, x′) and its derivatives in Section 3, the parallel propagator \(g_{\;\alpha {\prime}}^\alpha (x,x{\prime})\) in Section 5, and the van Vleck determinant Δ(x, x′) in Section 7. An important portion of the theory (covered in Sections 4 and 6) is concerned with the expansion of bitensors when x is very close to x′; expansions such as those displayed in Eqs. (1.23) and (1.24) are based on these techniques. The presentation in Part I borrows heavily from Synge’s book [169] and the article by DeWitt and Brehme [54]. These two sources use different conventions for the Riemann tensor, and we have adopted Synge’s conventions (which agree with those of Misner, Thorne, and Wheeler [131]). The reader is therefore warned that formulae derived in Part I may look superficially different from those found in De Witt and Brehme.
In Part II (Sections 8 to 11) we introduce a number of coordinate systems that play an important role in later parts of the review. As a warmup exercise we first construct (in Section 8) Riemann normal coordinates in a neighbourhood of a reference point x′. We then move on (in Section 9) to Fermi normal coordinates [122], which are defined in a neighbourhood of a world line γ. The retarded coordinates, which are also based at a world line and which were briefly introduced in Section 1.5, are covered systematically in Section 10. The relationship between Fermi and retarded coordinates is worked out in Section 11, which also locates the advanced point z (v) associated with a field point x. The presentation in Part II borrows heavily from Synge’s book [169]. In fact, we are much indebted to Synge for initiating the construction of retarded coordinates in a neighbourhood of a world line. We have implemented his program quite differently (Synge was interested in a large neighbourhood of the world line in a weakly curved spacetime, while we are interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally his.
In Part III (Sections 12 to 16) we review the theory of Green’s functions for (scalar, vectorial, and tensorial) wave equations in curved spacetime. We begin in Section 12 with a pedagogical introduction to the retarded and advanced Green’s functions for a massive scalar field in flat spacetime; in this simple context the allimportant Hadamard decomposition [88] of the Green’s function into “lightcone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in Section 13 along with its restrictions on the past and future null cones of a reference point x′. The retarded, advanced, singular, and regular Green’s functions for the scalar wave equation are introduced in Section 14. In Sections 15 and 16 we cover the vectorial and tensorial wave equations, respectively. The presentation in Part III is based partly on the paper by DeWitt and Brehme [54], but it is inspired mostly by Friedlander’s book [71]. The reader should be warned that in one important aspect, our notation differs from the notation of DeWitt and Brehme: While they denote the tail part of the Green’s function by −v (x, x′), we have taken the liberty of eliminating the silly minus sign and call it instead +V (x, x′). The reader should also note that all our Green’s functions are normalized in the same way, with a factor of −4π multiplying a fourdimensional Dirac functional of the righthand side of the wave equation. (The gravitational Green’s function is sometimes normalized with a −16π on the righthand side.)
In Part IV (Sections 17 to 19) we compute the retarded, singular, and regular fields associated with a point scalar charge (Section 17), a point electric charge (Section 18), and a point mass (Section 19). We provide two different derivations for each of the equations of motion. The first type of derivation was outlined previously: We follow Detweiler and Whiting [53] and postulate that only the regular field exerts a force on the particle. In the second type of derivation we take guidance from Quinn and Wald [150] and postulate that the net force exerted on a point particle is given by an average of the retarded field over a surface of constant proper distance orthogonal to the world line — this restframe average is easily carried out in Fermi normal coordinates. The averaged field is still infinite on the world line, but the divergence points in the direction of the acceleration vector and it can thus be removed by mass renormalization. Such calculations show that while the singular field does not affect the motion of the particle, it nonetheless contributes to its inertia.
In Part V (Sections 20 to 23), we show that at linear order in the body’s mass m, an extended body behaves just as a point mass, and except for the effects of the body’s spin, the world line representing its mean motion is governed by the MiSaTaQuWa equation. At this order, therefore, the picture of a point particle interacting with its own field, and the results obtained from this picture, is justified. Our derivation utilizes the method of matched asymptotic expansions, with an inner expansion accurate near the body and an outer expansion accurate everywhere else. The equation of motion of the body’s world line, suitably defined, is calculated by solving the Einstein equation in a buffer region around the body, where both expansions are accurate.
Concluding remarks are presented in Section 24, and technical developments that are required in Part V are relegated to Appendices. Throughout this review we use geometrized units and adopt the notations and conventions of Misner, Thorne, and Wheeler [131].
2 Computing the selfforce: a 2010 literature survey
Much progress has been achieved in the development of practical methods for computing the selfforce. We briefly summarize these efforts in this section, with the goal of introducing the main ideas and some key issues. A more detailed coverage of the various implementations can be found in Barack’s excellent review [9]. The 2005 collection of reviews published in Classical and Quantum Gravity [118] is also recommended for an introduction to the various aspects of selfforce theory and numerics. Among our favourites in this collection are the reviews by Detweiler [49] and Whiting [183].
An important point to bear in mind is that all the methods covered here mainly compute the selfforce on a particle moving on a fixed world line of the background spacetime. A few numerical codes based on the radiative approximation have allowed orbits to evolve according to energy and angularmomentum balance. As will be emphasized below, however, these calculations miss out on important conservative effects that are only accounted for by the full selfforce. Work is currently underway to develop methods to let the selfforce alter the motion of the particle in a selfconsistent manner.
2.1 Early work: DeWitt and DeWitt; Smith and Will
The first evaluation of the electromagnetic selfforce in curved spacetime was carried out by DeWitt and DeWitt [132] for a charge moving freely in a weakly curved spacetime characterized by a Newtonian potential Φ ≪ 1. In this context the righthand side of Eq. (1.33) reduces to the tail integral, because the particle moves in a vacuum region of the spacetime, and there is no external force acting on the charge. They found that the spatial components of the selfforce are given by
where M is the total mass contained in the spacetime, r = x is the distance from the centre of mass, \(\hat r = x/r\), and g = −∇Φ is the Newtonian gravitational field. (In these expressions the boldfaced symbols represent vectors in threedimensional flat space.) The first term on the righthand side of Eq. (2.1) is a conservative correction to the Newtonian force mg. The second term is the standard radiationreaction force; although it comes from the tail integral, this is the same result that would be obtained in flat spacetime if an external force mg were acting on the particle. This agreement is necessary, but remarkable!
A similar expression was obtained by Pfenning and Poisson [141] for the case of a scalar charge. Here
where 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 selfforce acting on a point mass moving in a weakly curved spacetime. The expression they obtained is in complete agreement (within its domain of validity) with the standard postNewtonian equations of motion.
The force required to hold an electric charge in place in a Schwarzschild spacetime was computed, without approximations, by Smith and Will [163]. As we reviewed previously in Section 1.10, the selfforce contribution to the total force is given by
where M is the blackhole mass, r the position of the charge (in Schwarzschild coordinates), and f:= 1 − 2M/r. When r ≫ M, this expression agrees with the conservative term in Eq. (2.1). This result was generalized to ReissnerNordström spacetime by Zel’nikov and Frolov [186]. Wiseman [185] calculated the selfforce acting on a static scalar charge in Schwarzschild spacetime. He found that in this case the selfforce vanishes. This result is not incompatible with Eq. (2.2), even for nonminimal coupling, because the computation of the weakfield selfforce requires the presence of matter, while Wiseman’s scalar charge lives in a purely vacuum spacetime.
2.2 Modesum method
Selfforce calculations involving a sum over modes were pioneered by Barack and Ori [16, 7], and the method was further developed by Barack, Ori, Mino, Nakano, and Sasaki [15, 8, 18, 20, 19, 127]; a somewhat related approach was also considered by Lousto [117]. It has now emerged as the method of choice for selfforce calculations in spacetimes such as Schwarzschild and Kerr. Our understanding of the method was greatly improved by the DetweilerWhiting decomposition [53] of the retarded field into singular and regular pieces, as outlined in Sections 1.4 and 1.8, and subsequent work by Detweiler, Whiting, and their collaborators [51].
2.2.1 DetweilerWhiting decomposition; mode decomposition; regularization parameters
For simplicity we consider the problem of computing the selfforce acting on a particle with a scalar charge q moving on a world line γ. (The electromagnetic and gravitational problems are conceptually similar, and they will be discussed below.) The potential Φ produced by the particle satisfies Eq. (1.34), which we rewrite schematically as
where □ is the wave operator in curved spacetime, and δ (x, z) represents a distributional source that depends on the world line γ through its coordinate representation z (τ). From the perspective of the DetweilerWhiting decomposition, the scalar selfforce is given by
where Φ, Φ_{S}, and Φ_{R} are the retarded, singular, and regular potentials, respectively. To evaluate the selfforce, then, is to compute the gradient of the regular potential.
From the point of view of Eq. (2.5), the task of computing the selfforce appears conceptually straightforward: Either (i) compute the retarded and singular potentials, subtract them, and take a gradient of the difference; or (ii) compute the gradients of the retarded and singular potentials, and then subtract the gradients. Indeed, this is the basic idea for most methods of selfforce computations. However, the apparent simplicity of this sequence of steps is complicated by the following facts: (i) except for a very limited number of cases, the retarded potential of a point particle cannot be computed analytically and must therefore be obtained by numerical means; and (ii) both the retarded and singular potential diverge at the particle’s position. Thus, any sort of subtraction will generally have to be performed numerically, and for this to be possible, one requires representations of the retarded and singular potentials (and/or their gradients) in terms of finite quantities.
In a modesum method, these difficulties are overcome with a decomposition of the potential in sphericalharmonic functions:
When the background spacetime is spherically symmetric, Eq. (2.4) gives rise to a fully decoupled set of reduced wave equations for the mode coefficients Φ^{lm} (t, r), and these are easily integrated with simple numerical methods. The dimensional reduction of the wave equation implies that each Φ^{lm} (t, r) is finite and continuous (though nondifferentiable) at the position of the particle. There is, therefore, no obstacle to evaluating each lmode of the field, defined by
The sum over modes, however, must reproduce the singular field evaluated at the particle’s position, and this is infinite; the mode sum, therefore, does not converge.
Fortunately, there is a piece of each lmode that does not contribute to the selfforce, and that can be subtracted out; this piece is the corresponding lmode of the singular field ∇_{ α } Φ_{S}. Because the retarded and singular fields share the same singularity structure near the particle’s world line (as described in Section 1.6), the subtraction produces a mode decomposition of the regular field ∇_{α}Φ_{R}. And because this field is regular at the particle’s position, the sum over all modes q (∇_{ a } Φ_{R},)_{ l } is guaranteed to converge to the correct value for the selfforce. The key to the modesum method, therefore, is the ability to express the singular field as a mode decomposition.
This can be done because the singular field, unlike the retarded field, can always be expressed as a local expansion in powers of the distance to the particle; such an expansion was displayed in Eqs. (1.28) and (1.29). (In a few special cases the singular field is actually known exactly [43, 114, 33, 86, 162].) This local expansion can then be turned into a multipole decomposition. Barack and Ori [18, 15, 20, 19, 9], and then Mino, Nakano, and Sasaki [127], were the first to show that this produces the following generic structure:
where A_{ α }, B_{ α }, C_{ α }, and so on are lindependent functions that depend on the choice of field (i.e., scalar, electromagnetic, or gravitational), the choice of spacetime, and the particle’s state of motion. These socalled regularization parameters are now ubiquitous in the selfforce literature, and they can all be determined from the local expansion for the singular field. The number of regularization parameters that can be obtained depends on the accuracy of the expansion. For example, expansions accurate through order r^{0} such as Eqs. (1.28) and (1.29) permit the determination of A_{ α }, B_{ α }, and C_{α}; to obtain D_{ α } one requires the terms of order r, and to get E_{ α } the expansion must be carried out through order r^{2}. The particular polynomials in l that accompany the regularization parameters were first identified by Detweiler and his collaborators [51]. Because the D_{ α } term is generated by terms of order r in the local expansion of the singular field, the sum of \({[(l  {1 \over 2})(l + {3 \over 2})]^{ 1}}\) from l = 0 to l = ∞ evaluates to zero. The sum of the polynomial in front of E_{ α } also evaluates to zero, and this property is shared by all remaining terms in Eq. (2.8).
2.2.2 Mode sum
With these elements in place, the selfforce is finally computed by implementing the modesum formula
where the infinite sum over l is truncated to a maximum mode number L. (This truncation is necessary in practice, because in general the modes must be determined numerically.) The remainder consists of the remaining terms in the sum, from l = L + 1 to l = ∞; it is easy to see that since the next regularization term would scale as l^{−6} for large l, the remainder scales as L^{−5}, and can be made negligible by summing to a suitably large value of l. This observation motivates the inclusion of the D_{ α } and E_{ α } terms within the mode sum, even though their complete sums evaluate to zero. These terms are useful because the sum must necessarily be truncated, and they permit a more rapid convergence of the mode sum. For example, exclusion of the D_{ α } and E_{ α } terms in Eq. (2.9) would produce a remainder that scales as L^{−1} instead of L^{−5}; while this is sufficient for convergence, the rate of convergence is too slow to permit highaccuracy computations. Rapid convergence therefore relies on a knowledge of as many regularization parameters as possible, but unfortunately these parameters are not easy to calculate. To date, only A_{ α }, B_{ α }, C_{ α }, and D_{ α } have been calculated for general orbits in Schwarzschild spacetime [51, 87], and only A_{ α }, B_{ α }, C_{ α } have been calculated for orbits in Kerr spacetime [19]. It is possible, however, to estimate a few additional regularization parameters by fitting numerical results to the structure of Eq. (2.8); this clever trick was first exploited by Detweiler and his collaborators [51] to achieve extremely high numerical accuracies. This trick is now applied routinely in modesum computations of the selfforce.
2.2.3 Case study: static electric charge in extreme ReissnerNordström spacetime
The practical use of the modesum method can be illustrated with the help of a specific example that can be worked out fully and exactly. We consider, as in Section 1.10, an electric charge e held in place at position r = r_{0} in the spacetime of an extreme ReissnerNordström black hole of mass M and charge Q = M. The reason for selecting this spacetime resides in the resulting simplicity of the sphericalharmonic modes for the electromagnetic field.
The metric of the extreme ReissnerNordström spacetime is given by
where f = (1 − M/r)^{2}. The only nonzero component of the electromagnetic field tensor is Ft_{ r } = −E_{ r }, and this is decomposed as
This field diverges at r = r_{0}, but the modes \(F_{tr}^{lm}(r)\) are finite, though discontinuous. The multipole coefficients of the field are defined to be
where the limit is taken in the direction of the particle’s position. The charge can be placed on the axis θ = 0, and this choice produces an axisymmetric field with contributions from m = 0 only. Because Y^{l0} = [(2l + 1)/4π ]^{1/2}Pl (cosθ) and Pl (1) = 1, we have
The sign of Δ is arbitrary, and (F_{ tr })_{ l } depends on the direction in which r_{0} is approached.
The charge density of a static particle can also be decomposed in spherical harmonics, and the mode coefficients are given by
where f_{0} = (1 − M/r_{0})^{2}. If we let
then Gauss’s law in the extreme ReissnerNordström spacetime can be shown to reduce to
in which a prime indicates differentiation with respect to r, and the index l on Φ is omitted to simplify the expressions. The solution to Eq. (2.16) can be expressed as Φ(r) = Φ_{>}(r)Θ(r − r_{0}) + Φ_{<}(r)Θ(r_{0} − r), where Φ_{>} and Φ_{<} are each required to satisfy the homogeneous equation (f Φ′)′ − l (l + 1)Φ/r^{2} = 0, as well as the junction conditions
with [Φ]:= Φ_{>}(r_{0}) − Φ_{<}(r_{0}) denoting the jump across r = r_{0}.
For l = 0 the general solution to the homogeneous equation is c_{1}r* + c_{2}, where c_{1} and c_{2} are constants and r * = ∫ f^{−1} dr. The solution for r < r_{ 0 } must be regular at r = M, and we select Φ_{<} = constant. The solution for r > r_{0} must produce a field that decays as r^{−2} at large r, and we again select Φ_{>} = constant. Since each constant is proportional to the total charge enclosed within a sphere of radius r, we arrive at
for l ≠ 0 the solutions to the homogeneous equation are
and
The constants c_{1} and c_{2} are determined by the junction conditions, and we get
The modes of the electromagnetic field are now completely determined.
According to the foregoing results, and recalling the definition of Eq. (2.13), the multipole coefficients of the electromagnetic field at r = r_{0} + 0^{+} are given by
for r = r_{0} + 0^{−} we have instead
We observe that the multipole coefficients lead to a diverging mode sum. We also observe, however, that the multipole structure is identical to the decomposition of the singular field displayed in Eq. (2.8). Comparison of the two expressions allows us to determine the regularization parameters for the given situation, and we obtain
Regularization of the mode sum via Eq. (2.9) reveals that the modes l ≠ 0 give rise to the singular field, while the regular field comes entirely from the mode l = 0. In this case, therefore, we can state that the exact expression for the regular field evaluated at the position of the particle is \(F_{tr}^{\rm{R}} = {({F_{tr}})_0}  {1 \over 2}A  B\), or \(F_{\,tr}^{\rm{R}}({r_0}) =  eM/r_{\,0}^3\). Because the regular field must be a solution to the vacuum Maxwell equations, its monopole structure guarantees that its value at any position is given by
This is the field of an image charge e′ = +eM/r_{0} situated at the centre of the black hole.
The selfforce acting on the static charge is then
This expression agrees with the SmithWill force of Eq. (1.50). The interpretation of the result in terms of an interaction between e and the image charge e′ was elaborated in Sec. 1.10.
2.2.4 Computations in Schwarzschild spacetime
The modesum method was successfully implemented in Schwarzschild spacetime to compute the scalar and electromagnetic selfforces on a static particle [31, 36]. It was used to calculate the scalar selfforce on a particle moving on a radial trajectory [10], circular orbit [30, 51, 87, 37], and a generic bound orbit [84]. It was also developed to compute the electromagnetic selfforce on a particle moving on a generic bound orbit [85], as well as the gravitational selfforce on a point mass moving on circular [21, 1] and eccentric orbits [22]. The modesum method was also used to compute unambiguous physical effects associated with the gravitational selfforce [50, 157, 11], and these results were involved in detailed comparisons with postNewtonian theory [50, 29, 28, 44, 11]. These achievements will be described in more detail in Section 2.6.
An issue that arises in computations of the electromagnetic and gravitational selfforces is the choice of gauge. While the selfforce formalism is solidly grounded in the Lorenz gauge (which allows the formulation of a wave equation for the potentials, the decomposition of the retarded field into singular and regular pieces, and the computation of regularization parameters), it is often convenient to carry out the numerical computations in other gauges, such as the popular ReggeWheeler gauge and the Chrzanowski radiation gauge described below. Compatibility of calculations carried out in different gauges has been debated in the literature. It is clear that the singular field is gauge invariant when the transformation between the Lorenz gauge and the adopted gauge is smooth on the particle’s world line; in such cases the regularization parameters also are gauge invariant [17], the transformation affects the regular field only, and the selfforce changes according to Eq. (1.49). The transformations between the Lorenz gauge and the ReggeWheeler and radiation gauges are not regular on the world line, however, and in such cases the regularization of the retarded field must be handled with extreme care.
2.2.5 Computations in Kerr spacetime; metric reconstruction
The reliance of the modesum method on a sphericalharmonic decomposition makes it generally impractical to apply to selfforce computations in Kerr spacetime. Wave equations in this spacetime are better analyzed in terms of a spheroidalharmonic decomposition, which simultaneously requires a Fourier decomposition of the field’s time dependence. (The eigenvalue equation for the angular functions depends on the mode’s frequency.) For a static particle, however, the situation simplifies, and Burko and Liu [35] were able to apply the method to calculate the selfforce on a static scalar charge in Kerr spacetime. More recently, Warburton and Barack [181] carried out a modesum calculations of the scalar selfforce on a particle moving on equatorial orbits of a Kerr black hole. They first solve for the spheroidal multipoles of the retarded potential, and then reexpress them in terms of sphericalharmonic multipoles. Fortunately, they find that a spheroidal multipole is well represented by summing over a limited number of spherical multipoles. The WarburtonBarack work represents the first successful computations of the selfforce in Kerr spacetime, and it reveals the interesting effect of the black hole’s spin on the behaviour of the selfforce.
The analysis of the scalar wave equation in terms of spheroidal functions and a Fourier decomposition permits a complete separation of the variables. For decoupling and separation to occur in the case of a gravitational perturbation, it is necessary to formulate the perturbation equations in terms of NewmanPenrose (NP) quantities [172], and to work with the Teukolsky equation that governs their behaviour. Several computer codes are now available that are capable of integrating the Teukolsky equation when the source is a point mass moving on an arbitrary geodesic of the Kerr spacetime. (A survey of these codes is given below.) Once a solution to the Teukolsky equation is at hand, however, there still remains the additional task of recovering the metric perturbation from this solution, a problem referred to as metric reconstruction.
Reconstruction of the metric perturbation from solutions to the Teukolsky equation was tackled in the past in the pioneering efforts of Chrzanowski [41], Cohen and Kegeles [42, 105], Stewart [166], and Wald [179]. These works have established a procedure, typically attributed to Chrzanowski, that returns the metric perturbation in a socalled radiation gauge. An important limitation of this method, however, is that it applies only to vacuum solutions to the Teukolsky equation. This makes the standard Chrzanowski procedure inapplicable in the selfforce context, because a point particle must necessarily act as a source of the perturbation. Some methods were devised to extend the Chrzanowski procedure to accommodate point sources in specific circumstances [121, 134], but these were not developed sufficiently to permit the computation of a selfforce. See Ref. [184] for a review of metric reconstruction from the perspective of selfforce calculations.
A remarkable breakthrough in the application of metricreconstruction methods in selfforce calculations was achieved by Keidl, Wiseman, and Friedman [107, 106, 108], who were able to compute a selfforce starting from a Teukolsky equation sourced by a point particle. They did it first for the case of an electric charge and a point mass held at a fixed position in a Schwarzschild spacetime [107], and then for the case of a point mass moving on a circular orbit around a Schwarzschild black hole [108]. The key conceptual advance is the realization that, according to the DetweilerWhiting perspective, the selfforce is produced by a regularized field that satisfies vacuum field equations in a neighbourhood of the particle. The regular field can therefore be submitted to the Chrzanowski procedure and reconstructed from a sourcefree solution to the Teukolsky equation.
More concretely, suppose that we have access to the Weyl scalar ψ_{0} produced by a point mass moving on a geodesic of a Kerr spacetime. To compute the selfforce from this, one first calculates the singular Weyl scalar \(\psi _0^{\rm{S}}\) from the DetweilerWhiting singular field \(h_{\alpha \beta}^{\rm{S}}\), and subtracts it from ψ_{0}. The result is a regularized Weyl scalar \(\psi _0^{\rm{R}}\), which is a solution to the homogeneous Teukolsky equation. This sets the stage for the metricreconstruction procedure, which returns (a piece of) the regular field \(h_{\alpha \beta}^{\rm{R}}\) in the radiation gauge selected by Chrzanowski. The computation must be completed by adding the pieces of the metric perturbation that are not contained in ψ_{0}; these are referred to either as the nonradiative degrees of freedom (since ψ_{0} is purely radiative), or as the l = 0 and l = 1 field multipoles (because the sum over multipoles that make up ψ_{0} begins at l = 2). A method to complete the Chrzanowski reconstruction of \(h_{\alpha \beta}^{\rm{R}}\) was devised by Keidl et al. [107, 108], and the end result leads directly to the gravitational selfforce. The relevance of the l = 0 and l = 1 modes to the gravitational selfforce was emphasized by Detweiler and Poisson [52].
2.2.6 Timedomain versus frequencydomain methods
When calculating the sphericalharmonic components Φ^{lm} (t, r) of the retarded potential Φ — refer back to Eq. (2.6) — one can choose to work either directly in the time domain, or perform a Fourier decomposition of the time dependence and work instead in the frequency domain. While the timedomain method requires the integration of a partial differential equation in t and r, the frequencydomain method gives rise to set of ordinary differential equations in r, one for each frequency ω. For particles moving on circular or slightly eccentric orbits in Schwarzschild spacetime, the frequency spectrum is limited to a small number of discrete frequencies, and a frequencydomain method is easy to implement and yields highly accurate results. As the orbital eccentricity increases, however, the frequency spectrum broadens, and the computational burden of summing over all frequency components becomes more significant. Frequencydomain methods are less efficient for large eccentricities, the case of most relevance for extrememassratio inspirals, and it becomes advantageous to replace them with timedomain methods. (See Ref. [25] for a quantitative study of this claim.) This observation has motivated the development of accurate evolution codes for wave equations in 1+1 dimensions.
Such codes must be able to accommodate pointparticle sources, and various strategies have been pursued to represent a Dirac distribution on a numerical grid, including the use of very narrow Gaussian pulses [116, 110, 34] and of “finite impulse representations” [168]. These methods do a good job with waveform and radiative flux calculations far away from the particle, but are of very limited accuracy when computing the potential in a neighborhood of the particle. A numerical method designed to provide an exact representation of a Dirac distribution in a timedomain computation was devised by Lousto and Price [120] (see also Ref. [123]). It was implemented by Haas [84, 85] for the specific purpose of evaluating Φ^{lm} (t, r) at the position of particle and computing the selfforce. Similar codes were developed by other workers for scalar [176] and gravitational [21, 22] selfforce calculations.
Most extant timedomain codes are based on finitedifference techniques, but codes based on pseudospectral methods have also been developed [67, 68, 37, 38]. Spectral codes are a powerful alternative to finitedifference codes, especially when dealing with smooth functions, because they produce much faster convergence. The fact that selfforce calculations deal with point sources and field modes that are not differentiable might suggest that spectral convergence should not be expected in this case. This objection can be countered, however, by placing the particle at the boundary between two spectral domains. Functions are then smooth in each domain, and discontinuities are handled by formulating appropriate boundary conditions; spectral convergence is thereby achieved.
2.3 Effectivesource method
The modesum methods reviewed in the preceding subsection have been developed and applied extensively, but they do not exhaust the range of approaches that may be exploited to compute a selfforce. Another set of methods, devised by Barack and his collaborators [12, 13, 60] as well as Vega and his collaborators [176, 177, 175], begin by recognizing that an approximation to the exact singular potential can be used to regularize the deltafunction source term of the original field equation. We shall explain this idea in the simple context of a scalar potential Φ.
We continue to write the wave equation for the retarded potential Φ in the schematic form
where □ is the wave operator in curved spacetime, and δ (x, z) is a distributional source term that depends on the particle’s world line γ through its coordinate representation z (τ). By construction, the exact singular potential Φ_{S} satisfies the same equation, and an approximation to the singular potential, denoted \({\tilde \Phi _{\rm{S}}}\), will generally satisfy an equation of the form
for some integer n > 0, where r is a measure of distance to the world line. A “better” approximation to the singular potential is one with a higher value of n. From the approximated singular potential we form an approximation to the regular potential by writing
where W is a window function whose properties will be specified below. The approximated regular potential is governed by the wave equation
and the righthand side of this equation defines the effective source term S (x, z). This equation is much less singular than Eq. (2.27), and it can be integrated using numerical methods designed to handle smooth functions.
To see this, we write the effective source more specifically as
With the window function W designed to approach unity as x → z, we find that the delta function that arises from the third term on the righthand side precisely cancels out the fourth term. To keep the other terms in S well behaved on the world line, we further restrict the window function to satisfy ∇_{α}W = O (r^{p}) with p ≥ 2; this ensures that multiplication by \({\nabla _\alpha}{\tilde \Phi _{\rm{S}}} = O({r^{ 2}})\) leaves behind a bounded quantity. In addition, we demand that □W = O (r^{q}) with q ≥ 1, so that multiplication by \({\tilde \Phi _{\rm{S}}} = O({r^{ 1}})\) again produces a bounded quantity. It is also useful to require that W (x) have compact (spatial) support, to ensure that the effective source term S (x, z) does not extend beyond a reasonably small neighbourhood of the world line; this property also has the virtue of making \({\tilde \Phi _{\rm{R}}}\) precisely equal to the retarded potential Φ outside the support of the window function. This implies, in particular, that \({\tilde \Phi _{\rm{R}}}\) can be used directly to compute radiative fluxes at infinity. Another considerable virtue of these specifications for the window function is that they guarantee that the gradient of \({\tilde \Phi _{\rm{R}}}\) is directly tied to the selfforce. We indeed see that
with the second line following by virtue of the imposed conditions on W, and the third line from the properties of the approximated singular field.
The effectivesource method therefore consists of integrating the wave equation
for the approximated regular potential \({\tilde \Phi _{\rm{R}}}\), with a source term S (x, z) that has become a regular function (of limited differentiability) of the spacetime coordinates x. The method is also known as a “puncture approach,” in reference to a similar regularization strategy employed in numerical relativity. It is well suited to a 3+1 integration of the wave equation, which can be implemented on mature codes already in circulation within the numericalrelativity community. An important advantage of a 3+1 implementation is that it is largely indifferent to the choice of background spacetime, and largely insensitive to the symmetries possessed by this spacetime; a selfforce in Kerr spacetime is in principle just as easy to obtain as a selfforce in Schwarzschild spacetime.
The method is also well suited to a selfconsistent implementation of the selfforce, in which the motion of the particle is not fixed in advance, but determined by the action of the computed selfforce. This amounts to combining Eq. (2.33) with the selfforce equation
in which the field is evaluated on the dynamically determined world line. The system of equations is integrated jointly, and selfconsistently. The 3+1 version of the effectivesource approach presents a unique opportunity for the numericalrelativity community to get involved in selfforce computations, with only a minimal amount of infrastructure development. This was advocated by Vega and Detweiler [176], who first demonstrated the viability of the approach with a 1+1 timedomain code for a scalar charge on a circular orbit around a Schwarzschild black hole. An implementation with two separate 3+1 codes imported from numerical relativity was also accomplished [177].
The work of Barack and collaborators [12, 13] is a particular implementation of the effectivesource approach in a 2+1 numerical calculation of the scalar selfforce in Kerr spacetime. (See also the independent implementation by Lousto and Nakano [119].) Instead of starting with Eq. (2.27), they first decompose Φ according to
and formulate reduced wave equations for the Fourier coefficients Φ^{m}. Each coefficient is then regularized with an appropriate singular field \(\tilde \Phi _{\rm{S}}^m\), which eliminates the deltafunction from Eq. (2.27). This gives rise to regularized source terms for the reduced wave equations, which can then be integrated with a 2+1 evolution code. In the final stage of the computation, the selfforce is recovered by summing over the regularized Fourier coefficients. This strategy, known as the mmode regularization scheme, is currently under active development. Recently it was successfully applied by Dolan and Barack [60] to compute the selfforce on a scalar charge in circular orbit around a Schwarzschild black hole.
2.4 Quasilocal approach with “matched expansions”
As was seen in Eqs. (1.33), (1.40), and (1.47), the selfforce can be expressed as an integral over the past world line of the particle, the integrand involving the Green’s function for the appropriate wave equation. Attempts have been made to compute the Green’s function directly [132, 141, 33, 86], and to evaluate the worldline integral. The quasilocal approach, first introduced by Anderson and his collaborators [4, 3, 6, 5], is based on the expectation that the worldline integral might be dominated by the particle’s recent past, so that the Green’s function can be represented by its Hadamard expansion, which is restricted to the normal convex neighbourhood of the particle’s current position. To help with this enterprise, Ottewill and his collaborators [136, 182, 137, 39] have pushed the Hadamard expansion to a very high order of accuracy, building on earlier work by Décanini and Folacci [48].
The weakfield calculations performed by DeWitt and DeWitt [132] and Pfenning and Poisson [141] suggest that the worldline integral is not, in fact, dominated by the recent past. Instead, most of the selfforce is produced by signals that leave the particle at some time in the past, scatter off the central mass, and reconnect with the particle at the current time; such signals mark the boundary of the normal convex neighbourhood. The quasilocal evaluation of the worldline integral must therefore be supplemented with contributions from the distant past, and this requires a representation of the Green’s function that is not limited to the normal convex neighbourhood. In some spacetimes it is possible to express the Green’s function as an expansion in quasinormal modes, as was demonstrated by Casals and his collaborators for a static scalar charge in the Nariai spacetime [40]. Their study provided significant insights into the geometrical structure of Green’s functions in curved spacetime, and increased our understanding of the nonlocal character of the selfforce.
2.5 Adiabatic approximations
The accurate computation of longterm waveforms from extrememassratio inspirals (EMRIs) involves a lengthy sequence of calculations that include the calculation of the selfforce. One can already imagine the difficulty of numerically integrating the coupled linearized Einstein equation for durations much longer than has ever been attempted by existing numerical codes. While doing so, the code would also have to evaluate the selfforce reasonably often (if not at each time step) in order to remain close to the true dynamics of the point mass. Moreover, gravitationalwave data analysis via matched filtering require full evolutions of the sort just described for a large sample of systems parameters. All these considerations underlie the desire for simplified approximations to fully selfconsistent selfforce EMRI models.
The adiabatic approximation refers to one such class of potentially useful approximations. The basic assumption is that the secular effects of the selfforce occur on a timescale that is much longer than the orbital period. In an extrememassratio binary, this assumption is valid during the early stage of inspiral; it breaks down in the final moments, when the orbit transitions to a quasiradial infall called the plunge. From the adiabaticity assumption, numerous approximations have been formulated: For example, (i) since the particle’s orbit deviates only slowly from geodesic motion, the selfforce can be calculated from a field sourced by a geodesic; (ii) since the radiationreaction timescale t_{ rr }, over which a significant shrinking of the orbit occurs due to the selfforce, is much longer than the orbital period, periodic effects of the selfforce can be neglected; and (iii) conservative effects of the selfforce can be neglected (the radiative approximation).
A seminal example of an adiabatic approximation is the PetersMathews formalism [140, 139], which determines the longterm evolution of a binary orbit by equating the timeaveraged rate of change of the orbital energy E and angular momentum L to, respectively, the flux of gravitationalwave energy and angular momentum at infinity. This formalism was used to successfully predict the decreasing orbital period of the HulseTaylor pulsar, before more sophisticated methods, based on postNewtonian equations of motion expanded to 2.5pn order, were incorporated in timesofarrival formulae.
In the hope of achieving similar success in the context of the selfforce, considerable work has been done to formulate a similar approximation for the case of an extrememassratio inspiral [124, 125, 126, 98, 61, 62, 159, 158, 78, 128, 94]. Bound geodesics in Kerr spacetime are specified by three constants of motion — the energy E, angular momentum L, and Carter constant C. If one could easily calculate the rates of change of these quantities, using a method analogous to the PetersMathews formalism, then one could determine an approximation to the longterm orbital evolution of the small body in an EMRI, avoiding the lengthy process of regularization involved in the direct integration of the selfforced equation of motion. In the early 1980s, Gal’tsov [77] showed that the average rates of change of E and L, as calculated from balance equations that assume geodesic source motion, agree with the averaged rates of change induced by a selfforce constructed from a radiative Green’s function defined as \({G_{{\rm{rad}}}} := {{1 \over 2}}({G_ }  {G_ +})\). As discussed in Section 1.4, this is equal to the regular twopoint function Gr in flat spacetime, but G_{rad} ≠ G_{R} in curved spacetime; because of its timeasymmetry, it is purely dissipative. Mino [124], building on the work of Gal’tsov, was able to show that the true selfforce and the dissipative force constructed from G_{rad} cause the same averaged rates of change of all three constants of motion, lending credence to the radiative approximation. Since then, the radiative Green’s function was used to derive explicit expressions for the rates of change of E, L, and C in terms of the particle’s orbit and wave amplitudes at infinity [159, 158, 78], and radiative approximations based on such expressions have been concretely implemented by Drasco, Hughes and their collaborators [99, 61, 62].
The relevance of the conservative part of the selfforce — the part left out when using G_{rad} — was analyzed in numerous recent publications [32, 148, 146, 147, 94, 97]. As was shown by Pound et al. [148, 146, 147], neglect of the conservative effects of the selfforce generically leads to longterm errors in the phase of an orbit and the gravitational wave it produces. These phasing errors are due to orbital precession and a direct shift in orbital frequency. This shift can be understood by considering a conservative force acting on a circular orbit: the force is radial, it alters the centripetal acceleration, and the frequency associated with a given orbital radius is affected. Despite these errors, a radiative approximation may still suffice for gravitationalwave detection [94]; for circular orbits, which have minimal conservative effects, radiative approximations may suffice even for parameterestimation [97]. However, at this point in time, these analyses remain inconclusive because they all rely on extrapolations from postNewtonian results for the conservative part of the selfforce. For a more comprehensive discussion of these issues, the reader is referred to Ref. [94, 143].
Hinderer and Flanagan performed the most comprehensive study of these issues [69], utilizing a twotimescale expansion [109, 145] of the field equations and selfforced equations of motion in an EMRI. In this method, all dynamical variables are written in terms of two time coordinates: a fast time t and a slow time \(\tilde t := (m/M)t\). In the case of an EMRI, the dynamical variables are the metric and the phasespace variables of the world line. The fasttime dependence captures evolution on the orbital timescale ∼ M, while the slowtime dependence captures evolution on the radiationreaction timescale ∼ M^{2}/m. One obtains a sequence of fasttime and slowtime equations by expanding the full equations in the limit of small m while treating the two time coordinates as independent. Solving the leadingorder fasttime equation, in which \(\tilde {t}\) is held fixed, yields a metric perturbation sourced by a geodesic, as one would expect from the linearized field equations for a point particle. The leadingorder effects of the selfforce are incorporated by solving the slowtime equation and letting \(\tilde {t}\) vary. (Solving the nexthigherorder slowtime equation determines similar effects, but also the backreaction that causes the parameters of the large black hole to change slowly.)
Using this method, Hinderer and Flanagan identified the effects of the various pieces of the selfforce. To describe this we write the selfforce as
where ‘rr’ denotes a radiationreaction, or dissipative, piece of the force, and ‘c’ denotes a conservative piece. Hinderer and Flanagan’s principal result is a formula for the orbital phase (which directly determines the phase of the emitted gravitational waves) in terms of these quantities:
where ϕ^{(0)} depends on an averaged piece of \(f_{(1){\rm{rr}}}^\mu\), while ϕ^{(1)} depends on \(f_{(1){\rm{c}}}^\mu\), the oscillatory piece of \(f_{(1){\rm{rr}}}^\mu\), and the averaged piece of \(f_{(2){\rm{rr}}}^\mu\). From this result, we see that the radiative approximation yields the leadingorder phase, but fails to determine the first subleading correction. We also see that the approximations (i)–(iii) mentioned above are consistent (so long as the parameters of the ‘geodesic’ source are allowed to vary slowly) at leading order in the twotimescale expansion, but diverge from one another beyond that order. Hence, we see that an adiabatic approximation is generically insufficient to extract parameters from a waveform, since doing so requires a description of the inspiral accurate up to small (i.e., smaller than order1) errors. But we also see that an adiabatic approximation based on the radiative Green’s function may be an excellent approximation for other purposes, such as detection.
To understand this result, consider the following naive analysis of a quasicircular EMRI — that is, an orbit that would be circular were it not for the action of the selfforce, and which is slowly spiraling into the large central body. We write the orbital frequency as \({\omega ^{(0)}}(E) + (m/M)\omega _1^{(1)}(E) + \cdots\), where ω^{(0)}(E) is the frequency as a function of energy on a circular geodesic, and \((m/M)\omega _1^{(1)}(E)\) is the correction to this due to the conservative part of the firstorder selfforce (part of the correction also arises due to oscillatory zerothorder effects combining with oscillatory firstorder effects, but for simplicity we ignore this contribution). Neglecting oscillatory effects, we write the energy in terms only of its slowtime dependence: \(E = {E^{(0)}}(\tilde t) + (m/M){E^{(1)}}(\tilde t) + \cdots\). The leadingorder term E^{(0)} is determined by the dissipative part of firstorder selfforce, while E^{(1)} is determined by both the dissipative part of the secondorder force and a combination of conservative and dissipative parts of the firstorder force. Substituting this into the frequency, we arrive at
where \(\omega _2^{(1)} = {E^{(1)}}\partial {\omega ^{(0)}}/\partial E\), in which the partial derivative is evaluated at E = E^{(0)} Integrating this over a radiationreaction time, we arrive at the orbital phase of Eq. (2.37). (In a complete description, E (t) will have oscillatory pieces, which are functions of t rather than \(\tilde t\), and one must know these in order to correctly determine ϕ^{(1)}.) Such a result remains valid even for generic orbits, where, for example, orbital precession due to the conservative force contributes to the analogue of \(\omega _1^{(1)}\)
2.6 Physical consequences of the selfforce
To be of relevance to gravitationalwave astronomy, the paramount goal of the selfforce community remains the computation of waveforms that properly encode the longterm dynamical evolution of an extrememassratio binary. This requires a fully consistent orbital evolution fed to a wavegeneration formalism, and to this day the completion of this program remains as a future challenge. In the meantime, a somewhat less ambitious, though no less compelling, undertaking is that of probing the physical consequences of the selfforce on the motion of point particles.
2.6.1 Scalar charge in cosmological spacetimes
The intriguing phenomenon of a scalar charge changing its rest mass because of an interaction with its selffield was studied by Burko, Harte, and Poisson [33] and Haas and Poisson [86] in the simple context of a particle at rest in an expanding universe. The scalar Green’s function could be computed explicitly for a wide class of cosmological spacetimes, and the action of the field on the particle determined without approximations. It is found that for certain cosmological models, the mass decreases and then increases back to its original value. For other models, the mass is restored only to a fraction of its original value. For de Sitter spacetime, the particle radiates all of its rest mass into monopole scalar waves.
2.6.2 Physical consequences of the gravitational selfforce
The earliest calculation of a gravitational selfforce was performed by Barack and Lousto for the case of a point mass plunging radially into a Schwarzschild black hole [14]. The calculation, however, depended on a specific choice of gauge and did not identify unambiguous physical consequences of the selfforce. To obtain such consequences, it is necessary to combine the selfforce (computed in whatever gauge) with the metric perturbation (computed in the same gauge) in the calculation of a welldefined observable that could in principle be measured. For example, the conservative pieces of the selfforce and metric perturbation can be combined to calculate the shifts in orbital frequencies that originate from the gravitational effects of the small body; an application of such a calculation would be to determine the shift (as measured by frequency) in the innermost stable circular orbit of an extrememassratio binary, or the shift in the rate of periastron advance for eccentric orbits. Such calculations, however, must exclude all dissipative aspects of the selfforce, because these introduce an inherent ambiguity in the determination of orbital frequencies.
A calculation of this kind was recently achieved by Barack and Sago [22, 23], who computed the shift in the innermost stable circular orbit of a Schwarzschild black hole caused by the conservative piece of the gravitational selfforce. The shift in orbital radius is gauge dependent (and was reported in the Lorenz gauge by Barack and Sago), but the shift in orbital frequency is measurable and therefore gauge invariant. Their main result — a genuine milestone in selfforce computations — is that the fractional shift in frequency is equal to 0.4870m/M; the frequency is driven upward by the gravitational selfforce. Barack and Sago compare this shift to the ambiguity created by the dissipative piece of the selfforce, which was previously investigated by Ori and Thorne [135] and Sundararajan [167]; they find that the conservative shift is very small compared with the dissipative ambiguity. In a followup analysis, Barack, Damour, and Sago [11] computed the conservative shift in the rate of periastron advance of slightly eccentric orbits in Schwarzschild spacetime.
Conservative shifts in the innermost stable circular orbit of a Schwarzschild black hole were first obtained in the context of the scalar selfforce by DiazRivera et al. [55]; in this case they obtain a fractional shift of 0.0291657q^{2}/(mM), and here also the frequency is driven upward.
2.6.3 Detweiler’s redshift factor
In another effort to extract physical consequences from the gravitational selfforce on a particle in circular motion in Schwarzschild spacetime, Detweiler discovered [50] that u^{t}, the time component of the velocity vector in Schwarzschild coordinates, is invariant with respect to a class of gauge transformations that preserve the helical symmetry of the perturbed spacetime. Detweiler further showed that 1/ut is an observable: it is the redshift that a photon suffers when it propagates from the orbiting body to an observer situated at a large distance on the orbital axis. This gaugeinvariant quantity can be calculated together with the orbital frequency Ω, which is a second gaugeinvariant quantity that can be constructed for circular orbits in Schwarzschild spacetime. Both u^{t} and Ω acquire corrections of fractional order m/M from the selfforce and the metric perturbation. While the functions u^{t} (r) and Ω(r) are still gauge dependent, because of the dependence on the radial coordinate r, elimination of r from these relations permits the construction of u^{t} (Ω), which is gauge invariant. A plot of u^{t} as a function of Ω therefore contains physically unambiguous information about the gravitational selfforce.
The computation of the gaugeinvariant relation ut(Ω) opened the door to a detailed comparison between the predictions of the selfforce formalism to those of postNewtonian theory. This was first pursued by Detweiler [50], who compared u^{t} (Ω) as determined accurately through second postNewtonian order, to selfforce results obtained numerically; he reported full consistency at the expected level of accuracy. This comparison was pushed to the third postNewtonian order [29, 28, 44, 11]. Agreement is remarkable, and it conveys a rather deep point about the methods of calculation. The computation of ut(Ω), in the context of both the selfforce and postNewtonian theory, requires regularization of the metric perturbation created by the point mass. In the selfforce calculation, removal of the singular field is achieved with the DetweilerWhiting prescription, while in postNewtonian theory it is performed with a very different prescription based on dimensional regularization. Each prescription could have returned a different regularized field, and therefore a different expression for ut(Ω). This, remarkably, does not happen; the singular fields are “physically the same” in the selfforce and postNewtonian calculations.
A generalization of Detweiler’s redshift invariant to eccentric orbits was recently proposed and computed by Barack and Sago [24], who report consistency with corresponding postNewtonian results in the weakfield regime. They also computed the influence of the conservative gravitational selfforce on the periastron advance of slightly eccentric orbits, and compared their results with full numerical relativity simulations for modest massratio binaries. Thus, in spite of the unavailability of selfconsistent waveforms, it is becoming clear that selfforce calculations are already proving to be of value: they inform postNewtonian calculations and serve as benchmarks for numerical relativity.
3 Part I: General Theory of Bitensors
4 Synge’s world function
4.1 Definition
In this and the following sections we will construct a number of bitensors, tensorial functions of two points in spacetime. The first is x′, which we call the “base point”, and to which we assign indices α′, β′, etc. The second is x, which we call the “field point”, and to which we assign indices α, β, etc. We assume that x belongs to N (x′), the normal convex neighbourhood of x′; this is the set of points that are linked to x′ by a unique geodesic. The geodesic segment 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}/dλ is tangent to the geodesic, and it obeys the geodesic equation Dt^{μ}/dλ = 0. The situation is illustrated in Figure 5.
Synge’s world function is a scalar function of the base point x′ and the field point x. It is defined by
and the integral is evaluated on the geodesic 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 \({1 \over 2}\varepsilon {({\lambda _1}  {\lambda _0})^2}\). If the geodesic is timelike, then λ can be set equal to the proper time τ, which implies that ε = −1 and \(\sigma =  {1 \over 2}{(\Delta \tau)^2}\). If the geodesic is spacelike, then λ can be set equal to the proper distance s, which implies that ε = 1 and \(\sigma = {1 \over 2}{(\Delta s)^2}\). If the geodesic is null, then σ = 0. Quite generally, therefore, the world function is half the squared geodesic distance between the points x′ and x.
In flat spacetime, the geodesic linking x to x^{′} is a straight line, and \(\sigma = {1 \over 2}{\eta _{\alpha \beta}}{(x  x{\prime})^\alpha}{(x  x{\prime})^\beta}\) in Lorentzian coordinates.
4.2 Differentiation of the world function
The world function σ (x, x′) can be differentiated with respect to either argument. We let σ_{ α } = ∂σ/∂x^{α} be its partial derivative with respect to x, and σ_{α′} = ∂σ/∂x^{α′} its partial derivative with respect to x′. It is clear that σ_{ α } behaves as a dual vector with respect to tensorial operations carried out at x, but as a scalar with respect to operations carried out x′. Similarly, σ_{α′} is a scalar at x but a dual vector at x′.
We let σ_{ αβ }:= ∇_{ β }σ_{ α } be the covariant derivative of σ_{ α } with respect to x; this is a rank2 tensor at x and a scalar at x′. Because σ is a scalar at x, we have that this tensor is symmetric: σ_{ βα } = σ_{ αβ }. Similarly, we let σ_{αβ′} := ∂_{β′}σ_{ α } = ∂^{2}σ/∂x^{β′} ∂x^{α} be the partial derivative of σ_{ α } with respect to x′; this is a dual vector both at x and x′. We can also define σ_{α′β} := ∂_{ β }σ_{ α } _{′} = ∂^{2}σ/∂x^{β}∂x^{α′} to be the partial derivative of σ_{α′} with respect to x. Because partial derivatives commute, these bitensors are equal: σ_{β′α} = σ_{ αβ } _{′}. Finally, we let σ_{ σα′β } := ∇_{β′}σ_{α′} be the covariant derivative of σ_{α′} with respect to x′; this is a symmetric rank2 tensor at x′ and a scalar at x.
The notation is easily extended to any number of derivatives. For example, we let σ_{σαβγ′}:= ∇_{δ′}∇_{γ}∇_{ β } ∇_{ α }σ, which is a rank3 tensor at x and a dual vector at x′. This bitensor is symmetric in the pair of indices α and β, but not in the pairs α and γ, nor β and γ. Because ∇_{δ′} is here an ordinary partial derivative with respect to x′, the bitensor is symmetric in any pair of indices involving δ′. The ordering of the primed index relative to the unprimed indices is therefore irrelevant: the same bitensor can be written as σ_{δ′αβγ} or σ_{αδ′βγ} or σ_{αβδ′γ}, making sure that the ordering of the unprimed indices is not altered.
More generally, we can show that derivatives of any bitensor Ω…(x,x′) satisfy the property
in which “⋯” stands for any combination of primed and unprimed indices. We start by establishing the symmetry of Ω…_{;αβ′} with respect to the pair α and β′. This is most easily done by adopting Fermi normal coordinates (see Section 9) 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 (3.2) then follows by further differentiation with respect to either x or x′.
The message of Eq. (3.2), when applied to derivatives of the world function, is that while the ordering of the primed and unprimed indices relative to themselves is important, their ordering with respect to each other is arbitrary. For example, σ_{α′β′γδ′δ} = σ_{α′β′δ′γδ} = σ_{γϵα′β′δ′}.
4.3 Evaluation of first derivatives
We can compute σ_{ α } by examining how σ varies when the field point x moves. We let the new field point be x + δx, and δσ:= σ (x + δx, x′) − σ (x, x′) is the corresponding variation of the world function. We let β + δβ be the unique geodesic segment that links x + δx to x′; it is described by relations z^{μ} (λ) + δz^{μ} (λ), in which the affine parameter is scaled in such a way that it runs from λ_{0} to λ_{1} also on the new geodesic. We note that δz (λ_{0}) = δx′ = 0 and δz (λ_{1}) = δx.
Working to first order in the variations, Eq. (3.1) implies
where Δλ = λ_{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
The integral vanishes because 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 = Δλg_{ αβ }t^{α} δx^{β}, or
in which the metric and the tangent vector are both evaluated at 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 Eq. (3.3) 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.
A virtually identical calculation reveals how 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 δσ = −Δλg_{α′β′}t^{α}′ δx^{β}′. This shows that
in which the metric and the tangent vector are both evaluated at x′. Apart from a factor Δλ, we see that σ^{α′} (x, x′) is minus the geodesic’s tangent vector at x′.
It is interesting to compute the norm of σ_{ α }. According to Eq. (3.3) we have \({g_{\alpha \beta}}{\sigma ^\alpha}{\sigma ^\beta} = {(\Delta \lambda)^2}{g_{\alpha \beta}}{t^\alpha}{t^\beta} = {(\Delta \lambda)^2}\varepsilon\). According to Eq. (3.1), this is equal to 2σ. We have obtained
and similarly,
These important relations will be the starting point of many computations to be described below.
We note that in flat spacetime, σ_{ α } = η_{ αβ } (x − x^{′})^{β} and σα′ = −η_{ αβ } (x − x′)^{β} in Lorentzian coordinates. From this it follows that σ_{ αβ } = σ_{α′β′} = −σ_{αβ′} = −σ_{ α′β } = η_{ αβ }, and finally, g^{αβ}σ_{ αβ } = 4 = g^{a′β′}σ_{a′β′}.
4.4 Congruence of geodesics emanating from x′
If the base point x′ is kept fixed, σ can be considered to be an ordinary scalar function of x. According to Eq. (3.5), this function is a solution to the nonlinear differential equation \({1 \over 2}{g^{\alpha \beta}}{\sigma _\alpha}{\sigma _\beta} = \sigma\). Suppose that we are presented with such a scalar field. What can we say about it?
An additional differentiation of the defining equation reveals that the vector σ^{α} := σ^{;α} satisfies
which is the geodesic equation in a nonaffine parameterization. The vector field is therefore tangent to a congruence of geodesics. The geodesics are timelike where 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.
The vector
is a normalized tangent vector that satisfies the geodesic equation in affineparameter form: u^{α} _{;β}u^{β} = 0. The parameter λ is then proper time τ. If λ* denotes the original parameterization of the geodesics, we have that dλ*/dτ = 2σ ^{−1/2}, and we see that the original parameterization is singular at σ = 0.
In the affine parameterization, the expansion of the congruence is calculated to be
where θ* = (δV)^{−1}(d/dλ*)(δV) is the expansion in the original parameterization (δV is the congruence’s crosssectional volume). While θ* is well behaved in the limit σ → 0 (we shall see below that θ* → 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 \({1 \over 2}{g^{\alpha \beta}}{\sigma _\alpha}{\sigma _\beta} = \sigma\), are clearly compatible with our preceding explicit construction of the world function.
5 Coincidence limits
It is useful to determine the limiting behaviour of the bitensors σ… as x approaches x′. We introduce the notation
to designate the limit of any bitensor Ω…(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.
5.1 Computation of coincidence limits
From Eqs. (3.1), (3.3), and (3.4) we already have
Additional results are obtained by repeated differentiation of the relations (3.5) and (3.6). For example, Eq. (3.5) implies σ_{γ} = g^{αβ}σ_{ α }σ_{ βγ } = σ^{β}σ_{ βγ }, or (g_{ βγ } − σ_{ βγ })t^{β} = 0 after using Eq. (3.3). 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_{ α }β] = g_{α′β′}. By very similar calculations we obtain all other coincidence limits for the second derivatives of the world function. The results are
From these relations we infer that [\([\sigma _{\,\,\,\alpha}^\alpha ] = 4\) 4, so that [θ* ] = 3, where θ* was defined in Eq. (3.9).
To generate coincidence limits of bitensors involving primed indices, it is efficient to invoke Synge’s rule,
in which “⋯” designates any combination of primed and unprimed indices; this rule will be established below. For example, according to Synge’s rule we have [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 Eq. (4.2). Similarly, [σ_{α′β′}] = [σ_{α′}]_{;β′}, − [σ_{ α′β }] = − [σ_{βα′}] = g_{a′β′}. The results of Eq. (4.2) can thus all be generated from the known result for [σ_{ αβ }].
The coincidence limits of Eq. (4.2) were derived from the relation \({\sigma _\alpha} = \sigma _{\;\alpha}^\delta {\sigma _\delta}\). We now differentiate this twice more and obtain \({\sigma _{\alpha \beta \gamma}} = \sigma _{\;\alpha \beta \gamma}^\delta {\sigma _\delta} + \sigma _{\;\alpha \beta}^\delta {\sigma _{\delta \gamma}} + \sigma _{\;\alpha \gamma}^\delta {\sigma _{\delta \beta}} + \sigma _{\;\alpha}^\delta {\sigma _{\delta \beta \gamma}}\). At coincidence we have
or [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 \([{\sigma _{\alpha \gamma \beta}}] + [{\sigma _{\alpha \beta \gamma}}] =0\), or \(2[{\sigma _{\alpha \beta \gamma}}]  [R_{\;\alpha \beta \gamma}^\delta {\sigma _\delta}] = 0\), or \(2[{\sigma _{\alpha \beta \gamma}}] = R_{\;\alpha{\prime}\beta{\prime}\gamma{\prime}}^{\delta{\prime}}[{\sigma _{\delta{\prime}}}]\). Since the last factor is zero, we arrive at
The last three results were derived from [σ_{ αβγ }] = 0 by employing Synge’s rule. We now differentiate the relation \({\sigma _\alpha} = {\sigma ^\delta}_\alpha {\sigma _\delta}\) three times and obtain
At coincidence this reduces to [σ_{ σαβγ } ] + [σ_{ σαδβ } ] + [σ_{ σαγβ } ] =0. To simplify the third term we differentiate Ricci’s identity \({\sigma _{\alpha \gamma \beta}} = {\sigma _{\alpha \beta \gamma}}  R_{\;\alpha \beta \gamma}^{\epsilon}{\sigma _{\epsilon}}\) with respect to x^{δ} and then take the coincidence limit. This gives us [σ_{ σαγβ } ] = [σ_{ σαβγ } ] + R_{α′δ′β′γ′}. The same manipulations on the second term give [σ_{ αδβγ }] = [σ_{ αβδγ }]+ R_{α′γ′β′δ′}. Using the identity \({\sigma _{\alpha \beta \delta \gamma}} = {\sigma _{\alpha \beta \gamma \delta}}  R_{\;\alpha \gamma \delta}^{\epsilon}{\sigma _{{\epsilon}\beta}}  R_{\;\beta \gamma \delta}^{\epsilon}{\sigma _{\alpha {\epsilon}}}\) and the symmetries of the Riemann tensor, it is then easy to show that [σ_{ σαβδ } ] = [σ_{ σαβγ } ]. Gathering the results, we obtain 3[σ_{ σαβγ } ] + R_{α′γ′β′δ′} + R_{α′δ′β′γ′} = 0, and Synge’s rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are
5.2 Derivation of Synge’s rule
We begin with any bitensor Ω_{AB′} (x,x′) in which A = α ⋯ β is a multiindex that represents any number of unprimed indices, and B′ = γ′ ⋯ δ′ a multiindex 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 multiindex 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 \(P_{\;\;;\alpha}^A{t^\alpha} = 0\). 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 \(Q_{\;\;;\alpha{\prime}}^{B{\prime}}{t^{\alpha{\prime}}} = 0\). With Ω, P, and Q we form a biscalar H (x,x′) defined by
Having specified the geodesic that links 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
Alternatively,
and these two expressions give
because the lefthand side is the limit of [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 Ω_{AB′;α′}t^{α′} P^{A} Q^{B′}, and in the limit this becomes [Ω_{AB′;α′}]t^{α′} P^{A′} Q^{B′}. 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 \([{\Omega _{AB{\prime} ;\alpha}}]{t^{\alpha {\prime}}}{P^{A{\prime}}}{Q^{B{\prime}}}\). Finally, \(H({\lambda _0},{\lambda _0}) = [{\Omega _{AB{\prime}}}]{P^{A{\prime}}}{Q^{B{\prime}}}\), and its derivative with respect to λ_{0} is \({[{\Omega _{AB{\prime}}}]_{;\alpha {\prime}}}{t^{\alpha {\prime}}}{P^{A{\prime}}}{Q^{B{\prime}}}\). Gathering the results we find that
and the final statement of Synge’s rule,
follows from the fact that the tensors P^{M} and Q^{N}, and the direction of the selected geodesic β, are all arbitrary. Equation (4.6) reduces to Eq. (4.3) when σ… is substituted in place of Ω_{AB′}.
6 Parallel propagator
6.1 Tetrad on β
On the geodesic segment 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 a, b,…, run from 0 to 3 and the basis vectors satisfy
where 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
and we define a dual tetrad \(e_\mu ^{\rm{a}}(z)\) by
this is also parallel transported on β. In terms of the dual tetrad the completeness relations take the form
and it is easy to show that the tetrad and its dual satisfy \(e_\mu ^{\rm{a}}e_{\rm{b}}^\mu = \delta _{\;{\rm{b}}}^{\rm{a}}\) and \(e_\nu ^{\rm{a}}e_{\rm{a}}^\mu = \delta _{\;\nu}^\mu\). Equations (5.1) — (5.4) hold everywhere on β. In particular, with an appropriate change of notation they hold at x′ and x; for example, \({g_{\alpha \beta}} = {\eta _{{\rm{ab}}}}\,e_\alpha ^{\rm{a}}e_\beta ^{\rm{b}}\) zz is the metric at x.
(You will have noticed that we use sansserif symbols for the frame indices. This is to distinguish them from another set of frame indices that will appear below. The frame indices introduced here run from 0 to 3; those to be introduced later will run from 1 to 3.)
6.2 Definition and properties of the parallel propagator
Any vector field 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 \({A^\alpha} = ({A^{\alpha {\prime}}}\,{\rm e}_{\alpha {\prime}}^{\rm{a}})e_{\rm{a}}^\alpha\), or
The object \(g_{\,\,\,\alpha \prime}^\alpha = e_{\rm{a}}^\alpha e_{\alpha \prime}^{\rm{a}}\) is the parallel propagator: it takes a vector at x′ and paralleltransports it to x along the unique geodesic that links these points.
Similarly, we find that
and we see that \(g_{\,\,\alpha}^{\alpha{\prime}} = e_{\rm{a}}^{\alpha \prime}e_\alpha ^{\rm{a}}\) performs the inverse operation: it takes a vector at x and paralleltransports it back to x′. Clearly,
and these relations formally express the fact that \(g_{\;\alpha}^{\alpha {\prime}}\) is the inverse of \(g_{\;\alpha {\prime}}^\alpha\).
The relation \(g_{\;\alpha {\prime}}^\alpha = e_{\rm{a}}^\alpha e_{\alpha {\prime}}^{\rm{a}}\) can also be expressed as \(g_\alpha ^{\;\alpha \prime} = e_\alpha ^{\rm{a}}e_{\rm{a}}^{\alpha \prime}\), and this reveals that
The ordering of the indices, and the ordering of the arguments, are arbitrary.
The action of the parallel propagator on tensors of arbitrary rank is easy to figure out. For example, suppose that the dual vector \({p_\mu} = {p_a}\,e_\mu ^a\) is parallel transported on 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
It is therefore the inverse propagator \(g_{\;\;\alpha}^{\alpha {\prime}}\) that takes a dual vector at x′ and paralleltransports it to x. As another example, it is easy to show that a tensor A^{αβ} at x obtained by parallel transport from x′ must be given by
Here we need two occurrences of the parallel propagator, one for each tensorial index. Because the metric tensor is covariantly constant, it is automatically parallel transported on β, and a special case of Eq. (5.10) is \({g_{\alpha \beta}} = g_{\;\alpha}^{\alpha {\prime}}g_{\;\beta}^{\beta {\prime}}\,{g_{\alpha {\prime}\beta {\prime}}}\).
Because the basis vectors are parallel transported on 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
Another useful property of the parallel propagator follows from the fact that if t^{μ} = dz^{μ}/dλ is tangent to the geodesic connecting x to x′, then \({t^\alpha} = g_{\;\alpha {\prime}}^\alpha {t^{\alpha {\prime}}}\). Using Eqs. (3.3) and (3.4), this observation gives us the relations
6.3 Coincidence limits
Eq. (5.5) and the completeness relations of Eqs. (5.2) or (5.4) imply that
Other coincidence limits are obtained by differentiation of Eqs. (5.11). For example, the relation \(g_{\;\beta {\prime};\gamma}^\alpha {\sigma ^\gamma} = 0\) implies \(g_{\;\beta {\prime};\gamma \delta}^\alpha {\sigma ^\gamma} + g_{\;\beta {\prime};\gamma}^\alpha \sigma _{\;\delta}^\gamma = 0\), and at coincidence we have
the second result was obtained by applying Synge’s rule on the first result. Further differentiation gives
and at coincidence we have \([g_{\;\beta {\prime};\gamma \delta}^\alpha ] + [g_{\;\beta {\prime};\delta \gamma}^\alpha ] = 0\), or \(2[g_{\;\beta {\prime};\gamma \delta}^\alpha ] + R_{\;\beta {\prime}\gamma {\prime}\delta {\prime}}^{\alpha {\prime}} = 0\). The coincidence limit for \(g_{\;\beta {\prime};\gamma \delta {\prime}}^\alpha = g_{\;\beta {\prime};\delta {\prime}\gamma}^\alpha\) can then be obtained from Synge’s rule, and an additional application of the rule gives \([g_{\;\beta {\prime};\gamma {\prime}\delta {\prime}}^\alpha ]\). Our results are
7 Expansion of bitensors near coincidence
7.1 General method
We would like to express a bitensor Ω_{α′β′} (x, x′) near coincidence as an expansion in powers of −σ^{α′} (x, x′), the closest analogue in curved spacetime to the flatspacetime quantity (x − x′)^{α}. For concreteness we shall consider the case of rank2 bitensor, and for the moment we will assume that the tensorial indices all refer to the base point x′.
The expansion we seek is of the form
in which the “expansion coefficients” 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}′.
To find the expansion coefficients we differentiate Eq. (6.1) repeatedly and take coincidence limits. Equation (6.1) immediately implies \([{\Omega _{\alpha {\prime}\beta {\prime}}}] = {A_{\alpha {\prime}\beta {\prime}}}\). After one differentiation we obtain \(\Omega_{\alpha{\prime}\beta{\prime};\gamma{\prime}} = A_{\alpha{\prime}\beta{\prime};\gamma{\prime}} + A_{\alpha{\prime}\beta{\prime}\epsilon{\prime};\gamma{\prime}} \sigma^{\epsilon{\prime}} + A_{\alpha{\prime}\beta{\prime}\epsilon{\prime}} \sigma^{\epsilon{\prime}}_{\ \gamma{\prime}} + \frac{1}{2}\, A_{\alpha{\prime}\beta{\prime}\epsilon{\prime}\iota{\prime};\gamma{\prime}} \sigma^{\epsilon{\prime}} \sigma^{\iota{\prime}} + A_{\alpha{\prime}\beta{\prime}\epsilon{\prime}\iota{\prime}} \sigma^{\epsilon{\prime}} \sigma^{\iota{\prime}}_{\ \gamma{\prime}} + O(\epsilon^2)\), and at coincidence this reduces to \([{\Omega _{\alpha {\prime}\beta {\prime};\gamma {\prime}}}] = {A_{\alpha {\prime}\beta {\prime};\gamma {\prime}}} + {A_{\alpha {\prime}\beta {\prime}\gamma {\prime}}}\). Taking the coincidence limit after two differentiations yields \([{\Omega _{\alpha {\prime}\beta {\prime};\gamma {\prime}\delta {\prime}}}] = {A_{\alpha {\prime}\beta {\prime};\gamma {\prime}\delta {\prime}}} + {A_{\alpha {\prime}\beta {\prime}\gamma {\prime};\delta {\prime}}} + {A_{\alpha {\prime}\beta {\prime}\delta {\prime};\gamma {\prime}}} + {A_{\alpha {\prime}\beta {\prime}\gamma {\prime}\delta {\prime}}}\). The expansion coefficients are therefore
These results are to be substituted into Eq. (6.1), and this gives us Ωα′β′ (x,x′) to second order in ϵ.
Suppose now that the bitensor is Ω_{ 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 \({\tilde \Omega _{\alpha {\prime}\beta {\prime}}} := g_{\;\beta {\prime}}^\beta {\Omega _{\alpha {\prime}\beta}}\) whose indices all refer to the point x′. Then \({\tilde \Omega _{\alpha {\prime}\beta {\prime}}}\) can be expanded as in Eq. (6.1), and the original bitensor is reconstructed as \({\Omega _{\alpha {\prime}\beta}} = g_{\;\beta}^{\beta {\prime}}{\tilde \Omega _{\alpha {\prime}\beta {\prime}}}\), or
The expansion coefficients can be obtained from the coincidence limits of \({\tilde \Omega _{\alpha {\prime}\beta {\prime}}}\) and its derivatives. It is convenient, however, to express them directly in terms of the original bitensor Ω_{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 Eq. (5.13) — (5.15) we find
The only difference with respect to Eq. (6.3) is the presence of a Riemanntensor term in B_{α′β′γ′δ′}.
Suppose finally that the bitensor to be expanded is Ω_{ αβ }, whose indices all refer to x. Much as we did before, we introduce an auxiliary bitensor \({\tilde \Omega _{\alpha {\prime}\beta {\prime}}} = g_{\;\alpha {\prime}}^\alpha g_{\;\beta {\prime}}^\beta {\Omega _{\alpha \beta}}\) whose indices all refer to x′, we expand \({\tilde \Omega _{\alpha {\prime}\beta {\prime}}}\) as in Eq. (6.1), and we then reconstruct the original bitensor. This gives us
and the expansion coefficients are now
This differs from Eq. (6.4) by the presence of an additional Riemanntensor term in C_{α′β′γ′δ′}.
7.2 Special cases
We now apply the general expansion method developed in the preceding subsection to the bitensors 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_{α′β′} = g_{α′β′}, C_{α′β′γ′} = 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
Taking the trace of the last equation returns \(\sigma _{\;\;\alpha}^\alpha = 4  {1 \over 3}{R_{\gamma {\prime}\delta {\prime}}}\,{\sigma ^{\gamma {\prime}}}{\sigma ^{\delta {\prime}}} + O({\epsilon^3})\), or
where \({\theta ^{\ast}} := \sigma _{\;\,\alpha}^\alpha  1\) was shown in Section 3.4 to describe the expansion of the congruence of geodesics that emanate from x′. Equation (6.10) reveals that timelike geodesics are focused if the Ricci tensor is nonzero and the strong energy condition holds: when \({R_{\alpha {\prime}\beta {\prime}}}\,{\sigma ^{\alpha {\prime}}}{\sigma ^{\beta {\prime}}} > 0\) we see that θ* is smaller than 3, the value it would take in flat spacetime
The expansion method can easily be extended to bitensors of other tensorial ranks. In particular, it can be adapted to give expansions of the first derivatives of the parallel propagator. The expansions
and thus easy to establish, and they will be needed in part III of this review.
7.3 Expansion of tensors
The expansion method can also be applied to ordinary tensor fields. For concreteness, suppose that we wish to express a rank2 tensor 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
if the tensor field is parallel transported on the geodesic β that links x to x′, then Eq. (6.12) reduces to Eq. (5.10). The extension of this formula to tensors of other ranks is obvious.
To derive this result we express 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 5.1 that is an orthonormal basis that is parallel transported on β; recall also that the affine parameter γ ranges from γ_{0} (its value at x′) to γ_{1} (its value at x). We have \({A_{\alpha{\prime} \beta{\prime}}}(x{\prime}) = {A_{{\rm{ab}}}}({\lambda _0}){\rm e}_{\alpha{\prime}}^{\rm{a}}e_{\beta{\prime}}^{\rm{b}},\; \;{A_{\alpha \beta}}(x) = {A_{{\rm{ab}}}}({\lambda _1})e_\alpha ^{\rm{a}}e_\beta ^{\rm{b}}\), and A_{ab}(γ_{1}) can be expressed in terms of quantities at γ = γ_{0} by straightforward Taylor expansion. Since, for example,
where we have used Eq. (3.4), we arrive at Eq. (6.12) after involving Eq. (5.6).
8 van Vleck determinant
8.1 Definition and properties
The van Vleck biscalar Δ(x, x′) is defined by
As we shall show below, it can also be expressed as
where g is the metric determinant at x and g′ the metric determinant at x′.
Eqs. 4.2) and (5.13) imply that at coincidence, \([\Delta _{\;\beta {\prime}}^{\alpha {\prime}}] = \delta _{\;\beta {\prime}}^{\alpha {\prime}}\) and [Δ] = 1. Equation (6.8), on the other hand, implies that near coincidence,
so that
This last result follows from the fact that for a “small” matrix a, det(1 + a) = 1 + tr(a) + O (a^{2}).
We shall prove below that the van Vleck determinant satisfies the differential equation
which can also be written as (ln Δ), \({(\ln \Delta)_{,\alpha}}{\sigma ^\alpha} = 4  \sigma _{\;\,\alpha}^\alpha\), or
in the notation introduced in Section 3.4. Equation (7.6) reveals that the behaviour of the van Vleck determinant is governed by the expansion of the congruence of geodesics that emanate from x′. If θ* < 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* > 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 Eq. (7.4): the sign of Δ − 1 near x′ is determined by the sign of \({R_{\alpha {\prime}\beta {\prime}}}\,{\sigma ^{\alpha {\prime}}}{\sigma ^{\beta {\prime}}}\).
8.2 Derivations
To show that Eq. (7.2) follows from Eq. (7.1) we rewrite the completeness relations at x, \({g^{\alpha \beta}} = {\eta ^{{\rm{ab}}}}e_{\rm{a}}^\alpha e_{\rm{b}}^\beta\), in the matrix form g^{−1} = E_{ η }E^{T}, where E denotes the 4 × 4 matrix whose entries correspond to \(e_{\rm{a}}^\alpha\). (In this translation we put tensor and frame indices on an equal footing.) With 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 \(x{\prime},\,\,{g^{\alpha {\prime}\,\beta {\prime}}} = {\eta ^{{\rm{ab}}}}e_{\rm{a}}^{\alpha {\prime}}e_{\rm{b}}^{\beta {\prime}}\), in the matrix form g′^{−1} = E′_{ η }E^{′}^{T}, where E′ is the matrix corresponding to \(e_{\rm{a}}^{\alpha {\prime}}\). With e′ denoting its determinant, we have 1/g′ = − e′^{2}, or \(e{\prime} = 1/\sqrt { g{\prime}}\). Now, the parallel propagator is defined by \(g_{\;\alpha {\prime}}^\alpha = {\eta ^{{\rm{ab}}}}{g_{\alpha {\prime}\beta {\prime}}}e_{\rm{a}}^\alpha e_{\rm{b}}^{\beta {\prime}}\), and the matrix form of this equation is \(\hat g = E\eta {E{\prime}^T}{g{\prime}^T}\). The determinant of the parallel propagator is therefore \(\hat g =  ee{\prime}g{\prime} = \sqrt { g{\prime}}/\sqrt { g}\). So we have
and Eq. (7.2) follows from the fact that the matrix form of Eq. (7.1) is \(\Delta =  {\hat g^{ 1}}{g^{ 1}}\sigma\), where σ is the matrix corresponding to σ_{ββ′}.
To establish Eq. (7.5) we differentiate the relation \(\sigma = {1 \over 2}{\sigma ^\gamma}{\sigma _\gamma}\) twice and obtain \({\sigma _{\alpha \beta {\prime}}} = \sigma _{\;\alpha}^\gamma {\sigma _{\gamma \beta {\prime}}} + {\sigma ^\gamma}{\sigma _{\gamma \alpha \beta {\prime}}}\). If we replace the last factor by σ_{σαβ′} and multiply both sides by − g^{α′α} we find
In this expression we make the substitution \({\sigma _{\alpha \beta {\prime}}} =  {g_{\alpha \alpha {\prime}}}\,\Delta _{\;\beta {\prime}}^{\alpha {\prime}}\), which follows directly from Eq. (7.1). This gives us
where we have used Eq. (5.11). At this stage we introduce an inverse \((\Delta^{1})_{\ \beta{\prime}}^{\alpha{\prime}}\), to the van Vleck bitensor, defined by \(\Delta _{\;\;\beta {\prime}}^{\alpha {\prime}}({\Delta ^{ 1}})_{\;\;\gamma {\prime}}^{\beta {\prime}} = \delta _{\;\;\gamma {\prime}}^{\alpha {\prime}}\). After multiplying both sides of Eq. (7.8) by \((\Delta^{1})^{\beta{\prime}}_{\ \gamma{\prime}}\), we find
and taking the trace of this equation yields
We now recall the identity δ ln detM = Tr(M^{−1}δM), 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
9 Part II: Coordinate Systems
10 Riemann normal coordinates
10.1 Definition and coordinate transformation
Given a fixed base point x′ and a tetrad \(e_{\rm{a}}^{\alpha {\prime}}(x{\prime})\) (x′), we assign to a neighbouring point x the four coordinates
where \(e_{\alpha \prime}^{\rm{a}} = {\eta ^{{\rm{ab}}}}{g_{\alpha \prime \beta \prime}}e_{\rm{b}}^{\beta \prime}\) is the dual tetrad attached to 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
Thus, \({\eta _{{\rm{ab}}}}{\hat {x}^{\rm{a}}}{\hat {x}^{\rm{b}}}\) is the squared geodesic distance between x and the base point x′. It is obvious that x′ is at the origin of the RNC, where \({\hat {x}^{\rm{a}}} = 0\).
If we move the point x to x + δx, the new coordinates change to \({\hat x^{\rm{a}}} + \delta {\hat x^{\rm{a}}} =  e_{\alpha \prime}^{\rm{a}}{\sigma ^{\alpha \prime}}(x + \delta x,x\prime) = {\hat x^{\rm{a}}}  {\rm{e}}_{\alpha \prime}^{\rm{a}}\sigma _{\;\beta}^{\alpha \prime}\,\delta {x^\beta}\), so that
The coordinate transformation is therefore determined by \(\partial {\hat x^{\rm{a}}}/\partial {x^\beta} =  e_{\alpha \prime}^{\rm{a}}\sigma _{\;\beta}^{\alpha \prime}\), and at coincidence we have
the second result follows from the identities \(e_{\alpha \prime}^{\rm{a}}e_{\rm{b}}^{\alpha \prime} = \delta _{\;\,{\rm{b}}}^{\rm{a}}\) and \(e_{\rm{a}}^{\alpha {\prime}}e_{\beta {\prime}}^{\rm{a}} = \delta _{\;\,\beta {\prime}}^{\alpha {\prime}}\).
It is interesting to note that the Jacobian of the transformation of Eq. (8.3), \(J := {\rm{det}}(\partial {\hat x^{\rm{a}}}/\partial {x^\beta})\), is given by \(J = \sqrt { g} \Delta (x,\;x\prime)\), where g is the determinant of the metric in the original coordinates, and Δ(x,x′) is the Van Vleck determinant of Eq. (7.2). 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 RNC, the determinant of the metric is given by
It is easy to show that the geodesics emanating from x′ are straight coordinate lines in 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 flatspacetime value of \(d^4 \hat{x}\) if Δ > 1, that is, if the geodesics are focused by the spacetime curvature.
10.2 Metric near x′
We now would like to invert Eq. (8.3) in order to express the line element \(d{s^2} = {g_{\alpha \beta}}\,d{x^\alpha}d{x^\beta}\) in terms of the displacements \(d{\hat x^{\rm{a}}}\). We shall do this approximately, by working in a small neighbourhood of x′. We recall the expansion of Eq. (6.8),
and in this we substitute the frame decomposition of the Riemann tensor, \(R_{\;\;\gamma \prime \beta \prime \delta \prime}^{\alpha \prime} = R_{\;\;\;{\rm{cbd}}}^{\rm{a}}\,e_{\rm{a}}^{\alpha \prime}e_{\gamma \prime}^{\rm{c}}e_{\beta \prime}^{\rm{b}}e_{\delta \prime}^{\rm{d}}\) and the tetrad decomposition of the parallel propagator, \(g_{\;\beta}^{\beta \prime} = e_{\rm{b}}^{\beta \prime}e_\beta ^{\rm{b}}\), where \(e_\beta ^{\rm{b}}(x)\)(x) is the dual tetrad at x obtained by parallel transport of \(e_{\beta {\prime}}^{\rm{b}}(x{\prime})\) (x′). After some algebra we obtain
where we have used Eq. (8.1). Substituting this into Eq. (8.3) yields
and this is easily inverted to give
This is the desired approximate inversion of Eq. (8.3). It is useful to note that Eq. (8.7), when specialized from the arbitrary coordinates x^{α} to \({\hat x^{\rm{a}}}\), gives us the components of the dual tetrad at x in RNC. And since \(e_{\rm{a}}^{\alpha \prime} = \delta _{\;{\rm{a}}}^{\alpha \prime}\) in RNC, we immediately obtain the components of the parallel propagator: \(g_{\;{\rm{b}}}^{{\rm{a}}\prime} = \delta _{\;{\rm{b}}}^{\rm{a}}  {1 \over 6}R_{\;\,{\rm{cbd}}}^{\rm{a}}{\hat x^{\rm{c}}}{\hat x^{\rm{d}}} + O({x^3})\).
We are now in a position to calculate the metric in the new coordinates. We have \(d{s^2} = {g_{\alpha \beta}}\,d{x^\alpha}d{x^\beta} = ({\eta _{{\rm{ab}}}}e_\alpha ^{\rm{a}}e_\beta ^{\rm{b}})d{x^\alpha}d{x^\beta} = {\eta _{{\rm{ab}}}}(e_\alpha ^{\rm{a}}d{x^\alpha})(e_\beta ^{\rm{b}}d{x^\beta})\), and in this we substitute Eq. (8.7). The final result is \(d{s^2} = {g_{{\rm{ab}}}}\,d{\hat x^{\rm{a}}}d{\hat x^{\rm{b}}}\), with
The quantities R_{acbd} appearing in Eq. (8.8) are the frame components of the Riemann tensor evaluated at the base point x′,
and these are independent of \(\hat{x}^{\rm a}\). They are also, by virtue of Eq. (8.4), the components of the (basepoint) Riemann tensor in RNC, because Eq. (8.9) can also be expressed as
which is the standard transformation law for tensor components.
It is obvious from Eq. (8.8) that \({g_{{\rm{ab}}}}(x\prime) = {\eta _{{\rm{ab}}}}\) and \(\Gamma _{\;{\rm{bc}}}^{\rm{a}}(x {\prime}) = 0\), where \(\Gamma _{\;{\rm{bc}}}^{\rm{a}} =  {1 \over 3}(R_{\;{\rm{bcd}}}^{\rm{a}} + R_{\;{\rm{cbd}}}^{\rm{a}}){\hat x^{\rm{d}}} + O({x^2})\) is the connection compatible with the metric g_{ab}. The Riemann normal coordinates therefore provide a constructive proof of the local flatness theorem.
11 Fermi normal coordinates
11.1 FermiWalker transport
Let γ be a timelike curve described by parametric relations z^{μ} (τ) in which τ is proper time. Let u^{μ} = dz^{μ}/dτ be the curve’s normalized tangent vector, and let a^{μ} = Du^{μ}/dτ be its acceleration vector.
A vector field v^{μ} is said to be FermiWalker transported on γ if it is a solution to the differential equation
Notice that this reduces to parallel transport when a^{μ} = 0 and γ is a geodesic.
The operation of FermiWalker (FW) transport satisfies two important properties. The first is that u^{μ} is automatically FW transported along γ; this follows at once from Eq. (9.1) and the fact that u^{μ} is orthogonal to a^{μ}. The second is that if the vectors v^{μ} and w^{μ} are both FW transported along γ, then their inner product v_{ μ }w^{μ} is constant on γ: D (v_{ μ }w^{μ})/dτ = 0; this also follows immediately from Eq. (9.1).
11.2 Tetrad and dual tetrad on γ
Let \(\bar{z}\) be an arbitrary reference point on γ. At this point we erect an orthonormal tetrad \(({u^{\bar \mu}},\,e_a^{\bar \mu})\) where, as a modification 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
From the tetrad on γ we define a dual tetrad \((e_{\mu} ^{0},e_{\mu} ^{a})\) by the relations
this also is FW transported on γ. The tetrad and its dual give rise to the completeness relations
11.3 Fermi normal coordinates
To construct the Fermi normal coordinates (FNC) of a point 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 \(\bar{x} := z(t)\) := z (t), with t denoting the value of the propertime 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
the last statement determines \(\bar{x}\) from the requirement that \(\sigma^{\bar{\alpha}}\), the vector tangent to 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 Eq. (9.4) it follows that
so 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.
Suppose that x is moved to x + δx. This typically induces a change in the spacelike geodesic β, which moves to β + δδβ, and a corresponding change in the intersection point \(\bar{x}\), which moves to \(x{\prime}{\prime} := \bar{x} + \delta \bar{x}\), 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 \({\sigma _{\alpha \prime\prime}}(x + \delta x,x\prime\prime){u^{\alpha \prime\prime}}(x\prime\prime) = 0\). Expanding these relations to first order in the displacements, and simplifying using Eqs. (9.2), yields
where μ is determined by \(\mu^{1} =  (\sigma_{\bar{\alpha}\bar{\beta}} u^{\bar{\alpha}} u^{\bar{\beta}} + \sigma_{\bar{\alpha}} a^{\bar{\alpha}})\).
11.4 Coordinate displacements near γ
The relations of Eq. (9.7) can be expressed as expansions in powers of s, the spatial distance from \(\bar{x}\) to x. For this we use the expansions of Eqs. (6.7) and (6.8), in which we substitute \({\sigma ^{\bar {\alpha}}} =  e_a^{\bar {\alpha}}{\hat {x}^{a}}\) and \(g_{\;\,\alpha}^{\bar {\alpha}} = {u^{\bar {\alpha}}}\bar{e}_{\alpha}^0 + e_{a}^{\bar \alpha}\bar {e}_{\alpha}^{a}\), 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
where \({a_a}(t) := {a_{\bar \alpha}}e_a^{\bar \alpha}\) are frame components of the acceleration vector, and \({R_{0c0d}}(t) := {R_{\bar \alpha \bar \gamma \bar \beta \bar \delta}}{u^{\bar \alpha}}e_c^{\bar \gamma}{u^{\bar \beta}}e_d^{\bar \delta}\) are frame components of the Riemann tensor evaluated on γ. This last result is easily inverted to give
Proceeding similarly for the other relations of Eq. (9.7), we obtain
and
where \({R_{ac0d}}(t) := {R_{\bar \alpha \bar \gamma \bar \beta \bar \delta}}e_a^{\bar \alpha}e_c^{\bar \gamma}{u^{\bar \beta}}e_d^{\bar \delta}\) and \({R_{acbd}}(t) := {R_{\bar {\alpha} \bar {\gamma} \bar {\beta} \bar {\delta}}}e_{a}^{\bar {\alpha}}e_{c}^{\bar {\gamma}}e_{b}^{\bar {\beta}}e_{d}^{\bar {\delta}}\) are additional frame components of the Riemann tensor evaluated on γ. (Note that frame indices are raised with δ^{ab}.)
As a special case of Eqs. (9.8) and (9.9) we find that
because in the limit \(x \rightarrow \bar{x}\), the dual tetrad \((\bar{e}_{\alpha} ^0,\, \bar{e}_{\alpha} ^{a})\) at 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 FermiWalker transported tetrad:
This implies that the frame components of the acceleration vector, a_{ a } (t), are also the components of the acceleration vector in FNC; orthogonality between \({u^{\bar {\alpha}}}\) and \({a^{\bar {\alpha}}}\) means that a_{0} = 0. Similarly, R_{0c 0d}(t), R_{0cbd}(t), and R_{ acbd } (t) are the components of the Riemann tensor (evaluated on γ) in Fermi normal coordinates.
11.5 Metric near γ
Inversion of Eqs. (9.8) and (9.9) gives
and
These relations, when specialized to the FNC, give the components of the dual tetrad at 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
This is the metric near 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_{0c 0d}(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 Eqs. (9.14) — (9.16) reduces to g_{ tt } = −1 and g_{ ab } = δ_{ ab }. On the other hand, the nonvanishing Christoffel symbols (on γ) are \(\Gamma _{\;ta}^t = \Gamma _{\;tt}^a = {a_a}\); these are zero (and the FNC enforce local flatness on the entire curve) when γ is a geodesic.
11.6 Thorne—Hartle—Zhang coordinates
The form of the metric can be simplified when the Ricci tensor vanishes on the world line:
In such circumstances, a transformation from the Fermi normal coordinates \((t,\hat{x}^a)\) to the ThorneHartleZhang (THZ) coordinates \((t,{\hat y^a})\) brings the metric to the form
We see that the transformation leaves g_{ tt } and g_{ ta } unchanged, but that it diagonalizes g_{ ab }. This metric was first displayed in Ref. [174] and the coordinate transformation was later produced by Zhang [187].
The key to the simplification comes from Eq. (9.17), which dramatically reduces the number of independent components of the Riemann tensor. In particular, Eq. (9.17) implies that the frame components R_{ acbd } of the Riemann tensor are completely determined by \({{\mathcal E}_{ab}} := {R_{0a0b}}\), which in this special case is a symmetrictracefree tensor. To prove this we invoke the completeness relations of Eq. (9.4) and take frame components of Eq. (9.17). This produces the three independent equations
the last of which stating that \({{\mathcal E}_{ab}}\) has a vanishing trace. Taking the trace of the first equation gives \({\delta ^{ab}}{\delta ^{cd}}{R_{acbd}} = 0\), 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
and \({\mathcal E}_{ab} := R_{0a0b}\) satisfies \(\delta^{ab} {\mathcal E}_{ab} = 0\).
We may also note that the relation 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 symmetrictracefree tensor \({{\mathcal B}_{ab}}\). We take the independent components to be \({R_{0112}} :=  {{\mathcal B}_{13}},\,\,{R_{0113}} := {{\mathcal B}_{12}},\,\,\,{R_{0123}} :=  {{\mathcal B}_{11}},\,\,\,{R_{0212}} :=  {{\mathcal B}_{23}}\), and \(R_{0213} := {\mathcal B}_{22}\), 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
where ε_{ abc } is the threedimensional permutation symbol. The inverse relation is \({\mathcal B}_{\;b}^a =  {1 \over 2}{\varepsilon ^{acd}}{R_{0bcd}}\).
Substitution of Eq. (9.21) into Eq. (9.16) gives
and we have not yet achieved the simple form of Eq. (9.20). The missing step is the transformation from the FNC \(\hat{x}^a\) to the THZ coordinates \(\hat{y}^a\). This is given by
It is easy to see that this transformation does not affect g_{ tt } nor g_{ ta } at orders s and s^{2}. The remaining components of the metric, however, transform according to g_{ ab } (THZ) = g_{ ab } (FNC) − ξ_{ a;b } − ξ_{b;a}, where
It follows that \(g_{ab}^{{\rm{THZ}}} = {\delta _{ab}}(1  {{\mathcal E}_{cd}}{\hat {y}^{c}}{\hat {y}^{d}}) + O({\hat {y}^3})\), which is just the same statement as in Eq. (9.20).
Alternative expressions for the components of the THZ metric are
12 Retarded coordinates
12.1 Geometrical elements
We introduce the same geometrical elements as in Section 9: we have a timelike curve 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 FW transported on the world line according to
where \({a_a}(\tau) = {a_\mu}e_a^\mu\) are the frame components of the acceleration vector. Finally, we have a dual tetrad \((e_\mu ^0,e_\mu ^a)\), with \(e_\mu ^0 =  {u_\mu}\) and \(e_\mu ^a = {\delta ^{ab}}{g_{\mu \nu}}e_b^\nu\). The tetrad and its dual give rise to the completeness relations
which are the same as in Eq. (9.4).
The Fermi normal coordinates of Section 9 were constructed on the basis of a spacelike geodesic connecting a field point x to the world line. The retarded coordinates are based instead on a null geodesic going from the world line to the field point. We thus let x be within the normal convex neighbourhood of γ, β be the unique futuredirected null geodesic that goes from the world line to x, and x′ := z (u) be the point at which β intersects the world line, with u denoting the value of the propertime parameter at this point.
From the tetrad at 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
and the parallel propagator from x′ to x is given by
12.2 Definition of the retarded coordinates
The quasiCartesian version of the retarded coordinates are defined by
the last statement indicates that x′ and x are linked by a null geodesic. From the fact that σ^{β′} is a null vector we obtain
and r is a positive quantity by virtue of the fact that β is a futuredirected null geodesic — this makes σ^{β′} pastdirected. In flat spacetime, σ^{β′} = − (x − x′)^{β}, 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 \(e_\alpha ^a = {\delta ^{ab}}{g_{\alpha \beta}}e_b^\beta\) can still be called the retarded distance between the point x and the world line. Another consequence of Eq. (10.5) is that
where \({\Omega ^a} := {\hat x^a}/r\) is a unit spatial vector that satisfies δ_{ ab } Ω^{a} Ω^{b} = 1.
A straightforward calculation reveals that under a displacement of the point x, the retarded coordinates change according to
where k_{ α } = σ_{ β }/r is a futuredirected 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 Eq. (10.8) follows from the fact that a displacement along the world line is described by δx^{α′} = uα′ δu.
12.3 The scalar field r (x) and the vector field k^{α}(x)
If we keep x′ linked to x by the relation σ (x, x′) = 0, then the quantity
can be viewed as an ordinary scalar field defined in a neighbourhood of 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
Similarly, we can view
as an ordinary vector field, which is tangent to the congruence of null geodesics that emanate from x′. It is easy to check that this vector satisfies the identities
from which we also obtain \({\sigma _{\alpha \prime\beta}}{u^{\alpha \prime}}{k^\beta} = 1\). From this last result and Eq. (10.10) we deduce the important relation
In addition, combining the general statement \({\sigma ^\alpha} =  g_{\;\alpha \prime}^\alpha {\sigma ^{\alpha \prime}}\) cf. Eq. (5.12) — with Eq. (10.7) gives
the vector at x is therefore obtained by parallel transport of \({u^{\alpha \prime}} + {\Omega ^a}e_a^{\alpha \prime}\) on α. From this and Eq. (10.4) we get the alternative expression
which confirms that k^{α} is a futuredirected null vector field (recall that \({\Omega ^a} = {\hat x^a}/r\) is a unit vector).
The covariant derivative of k_{ α } can be computed by finding how the vector changes under a displacement of x. (It is in fact easier to calculate first how rk_{ α } changes, and then substitute our previous expression for ∂_{ α }r.) The result is
From this we infer that k^{α} satisfies the geodesic equation in affineparameter form, \(k_{\;;\beta}^\alpha {k^\beta} = 0\), and Eq. (10.13) 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 Eqs. (10.8) and (10.12), we find that this statement becomes du = 0 and \(d{\hat x^a} = ({\hat x^a}/r)\,dr\), which integrate to u = constant 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
in the retarded coordinates. Here, the two angles θ^{A} (A = 1, 2) serve to parameterize the unit vector Ω^{a}, which is independent of r.
Eq. (10.16) also implies that the expansion of the congruence is given by
Using Eq. (6.10), we find that this becomes \(r\theta = 2  {1 \over 3}{R_{\alpha \prime \beta \prime}}{\sigma ^{\alpha \prime}}{\sigma ^{\beta \prime}} + O({r^3})\), or
after using Eq. (10.7). Here, \({R_{00}} = {R_{\alpha \prime \beta \prime}}{u^{\alpha \prime}}{u^{\beta \prime}},\;\; {R_{0a}} = {R_{\alpha \prime \beta \prime}}{u^{\alpha \prime}}e_a^{\beta \prime}\), and \({R_{ab}} = {R_{\alpha \prime\beta \prime}}e_a^{\alpha \prime}e_b^{\beta \prime}\) are the frame components of the Ricci tensor evaluated at 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′.
Finally, we infer from Eq. (10.16) that k^{α} is hypersurface orthogonal. This, together with the property that k^{α} satisfies the geodesic equation in affineparameter form, implies that there exists a scalar field u (x) such that
This scalar field was already identified in Eq. (10.8): it is numerically equal to the propertime parameter of the world line at x′. We conclude that the geodesics to which k^{α} is tangent are the generators of the null cone u = constant. As Eq. (10.17) indicates, a specific generator is selected by choosing a direction Ω^{a} (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.
12.4 Frame components of tensor fields on the world line
The metric at 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,
are the frame components of the acceleration at proper time u.
Similarly,
are the frame components of the Riemann tensor evaluated on γ. From these we form the useful combinations
in which the quantities \({\Omega ^a} := {\hat x^a}/r\) depend on the angles θ^{A} only — they are independent of u and r.
We have previously introduced the frame components of the Ricci tensor in Eq. (10.19). The identity
follows easily from Eqs. (10.23) — (10.25) and the definition of the Ricci tensor.
In Section 9 we saw that the frame components of a given tensor were also the components of this tensor (evaluated on the world line) in the Fermi normal coordinates. We should not expect this property to be true also in the case of the retarded coordinates: the frame components of a tensor are not to be identified with the components of this tensor in the retarded coordinates. The reason is that the retarded coordinates are in fact singular on the world line. As we shall see, they give rise to a metric that possesses a directional ambiguity at r = 0. (This can easily be seen in Minkowski spacetime by performing the coordinate transformation \(u = t  \sqrt {{x^2} + {y^2} + {z^2}}\).) Components of tensors are therefore not defined on the world line, although they are perfectly well defined for r ≠ 0. Frame components, on the other hand, are well defined both off and on the world line, and working with them will eliminate any difficulty associated with the singular nature of the retarded coordinates.
12.5 Coordinate displacements near γ
The changes in the quasiCartesian retarded coordinates under a displacement of x are given by Eq. (10.8). In these we substitute the standard expansions for σ_{α′β′} and σ_{ α′β }, as given by Eqs. (6.7) and (6.8), as well as Eqs. (10.7) and (10.14). After a straightforward (but fairly lengthy) calculation, we obtain the following expressions for the coordinate displacements:
Notice that the result for du is exact, but that \(d{\hat x^a}\) is expressed as an expansion in powers of r
These results can also be expressed in the form of gradients of the retarded coordinates:
Notice that Eq. (10.29) follows immediately from Eqs. (10.15) and (10.20). From Eq. (10.30) and the identity \({\partial _\alpha}r = {\Omega _a}{\partial _\alpha}{\hat x^a}\) we also infer
where we have used the facts that S_{ a } = S_{ ab } Ω^{b} and S = S_{ a }Ω^{a}; these last results were derived in Eqs. (10.24) and (10.25). It may be checked that Eq. (10.31) agrees with Eq. (10.10).
12.6 Metric near γ
It is straightforward (but fairly tedious) to invert the relations of Eqs. (10.27) and (10.28) and solve for \(e_\alpha ^0\,d{x^\alpha}\) and \(e_\alpha ^a\,d{x^\alpha}\). The results are
These relations, when specialized to the retarded coordinates, give us the components of the dual tetrad \((e_\alpha ^0,e_\alpha ^a)\) at x. The metric is then computed by using the completeness relations of Eq. (10.3). We find
where a^{2} := δ_{ ab }a^{a}a^{b}. We see (as was pointed out in Section 10.4) that the metric possesses a directional ambiguity on the world line: the metric at 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.
By setting S_{ ab } = S_{ a } = S = 0 in Eqs. (10.34)–(10.36) we obtain the metric of flat spacetime in the retarded coordinates. This we express as
In spite of the directional ambiguity, the metric of flat spacetime has a unit determinant everywhere, and it is easily inverted:
The inverse metric also is ambiguous on the world line.
To invert the curvedspacetime metric of Eqs. (10.34) — (10.36) we express it as g_{ αβ } = η_{ αβ } + h_{ αβ } + O (r^{3}) and treat h_{ αβ } = O (r^{2}) as a perturbation. The inverse metric is then \({g^{\alpha \beta}} = {\eta ^{\alpha \beta}}  {\eta ^{\alpha \gamma}}{\eta ^{\beta \delta}}{h_{\gamma \delta}} + O({r^3})\), or
The results for g^{uu} and g^{ua} are exact, and they follow from the general relations g^{αβ} (∂_{ αu })(∂_{ βu }) = 0 and g^{αβ} (∂_{ αu })(∂_{ βr }) = −1 that are derived from Eqs. (10.13) and (10.20).
The metric determinant is computed from \(\sqrt { g} = 1 + {1 \over 2}{\eta ^{\alpha \beta}}{h_{\alpha \beta}} + O({r^3})\), which gives
where we have substituted the identity of Eq. (10.26). Comparison with Eq. (10.19) gives us the interesting relation \(\sqrt { g} = {1 \over 2}r\theta + O({r^3})\), where θ is the expansion of the generators of the null cones u = constant.
12.7 Transformation to angular coordinates
Because the vector \({\Omega ^a} = {\hat x^a}/r\) satisfies 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 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 10.3 that the angles θ^{A} are constant on the generators of the null cones u = constant, 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
where the transformation matrix
satisfies the identity \({\Omega _a}\Omega _A^a = 0\).
We introduce the quantities
which act as a (nonphysical) metric in the subspace spanned by the angular coordinates. In the canonical parameterization, Ω_{ ab } = diag(1, sin^{2} θ). We use the inverse of Ω_{ ab }, denoted Ω^{AB}, to raise uppercase Latin indices. We then define the new object
which satisfies the identities
The second result follows from the fact that both sides are simultaneously symmetric in a and b, orthogonal to Ω_{ a } and Ω^{b}, and have the same trace.
From the preceding results we establish that the transformation from \({\hat x^a}\) to (r, θ^{A}) is accomplished by
while the transformation from (r, θ^{A}) to \(\hat{x}^a\) is accomplished by
With these transformation rules it is easy to show that in the angular coordinates, the metric is given by
The results 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 }.
The nonvanishing components of the inverse metric are
The results g^{uu} = 0, g^{ur} = −1, and g^{uA} = 0 are exact, and they follow from the same reasoning as before.
Finally, we note that in the angular coordinates, the metric determinant is given by
where Ω is the determinant of Ω_{ ab }; in the canonical parameterization, \(\sqrt \Omega = \sin \,\theta\).
12.8 Specialization to a^{μ} = 0 = R_{ μν }
In this subsection we specialize our previous results to a situation where γ is a geodesic on which the Ricci tensor vanishes. We therefore set a^{μ} = 0 = R_{ μν } everywhere on γ.
We have seen in Section 9.6 that when the Ricci tensor vanishes on γ, all frame components of the Riemann tensor can be expressed in terms of the symmetrictracefree tensors \({\mathcal E}_{ab}(u)\) and \({\mathcal B}_{ab}(u)\). The relations are \({R_{a0b0}} = {{\mathcal E}_{ab}},\;\;{R_{a0bc}} = {\varepsilon _{bcd}}{\mathcal B}_{\;\;a}^d\), 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 Eqs. (10.23) — (10.25) to give
In these expressions the dependence on retarded time u is contained in \({{\mathcal E}_{ab}}\) and \({{\mathcal B}_{ab}}\), while the angular dependence is encoded in the unit vector Ω^{a}.
It is convenient to introduce the irreducible quantities
These are all orthogonal to \({\Omega ^a}:\,\,\,{\mathcal E}_a^\ast{\Omega ^a} = {\mathcal B}_a^\ast{\Omega ^a} = 0\) and \({\mathcal E}_{ab}^{\ast}{\Omega ^b} = {\mathcal B}_{ab}^{\ast}{\Omega ^b} = 0\). In terms of these Eqs. (10.59) — (10.61) become
When Eqs. (10.67) — (10.69) are substituted into the metric tensor of Eqs. (10.34) — (10.36) — in which a_{ a } is set equal to zero — we obtain the compact expressions
The metric becomes
after transforming to angular coordinates using the rules of Eq. (10.48). Here we have introduced the projections
It may be noted that the inverse relations are \({\mathcal E}_a^\ast = {\mathcal E}_A^\ast\Omega _a^A,\,\, {\mathcal B}_a^\ast = {\mathcal B}_A^\ast\Omega _a^A,\,\, {\mathcal E}_{ab}^\ast = {\mathcal E}_{AB}^\ast\Omega _a^A\Omega _b^B\), and \({\mathcal B}_{ab}^* = {\mathcal B}_{AB}^*\Omega _a^A\Omega _b^B\), where \(\Omega _{a}^{A}\) was introduced in Eq. (10.46).
13 Transformation between Fermi and retarded coordinates; advanced point
A point x in the normal convex neighbourhood of a world line γ can be assigned a set of Fermi normal coordinates (as in Section 9), or it can be assigned a set of retarded coordinates (Section 10). These coordinate systems can obviously be related to one another, and our first task in this section (which will occupy us in Sections 11.1–11.3) will be to derive the transformation rules. We begin by refining our notation so as to eliminate any danger of ambiguity.
The Fermi normal coordinates of x refer to a point \(\bar {x} := z(t)\) 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 propertime 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 \(s{\omega ^a} =  e_{\bar \alpha}^a{\sigma ^{\bar \alpha}}\) 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 \(({u^{\bar \alpha}},\;\; e_a^{\bar \alpha})\) 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 futuredirected null geodesic; see Figure 8. We refer to this point as x’s retarded point, and to tensors at x′ we assign indices α′, β′, etc. We let (u, r Ω^{a}) be the retarded coordinates of x, with u denoting the value of γ’s propertime parameter at x″ representing the affineparameter distance from x′ to x along the null geodesic, and Ω^{a} denoting a unit vector (δ_{ ab } Ω^{a} Ω^{b} = 1) that determines the direction of the geodesic. The retarded coordinates are defined by \(r{\Omega ^a} =  e_{\alpha \prime}^a{\sigma ^{\alpha \prime}}\) and σ (x, x′) = 0. Finally, we denote by \((e_0^\alpha ,\; e_a^\alpha)\) the tetrad at x that is obtained by parallel transport of \(({u^{\alpha \prime}},\; e_a^{\alpha \prime})\) on the null geodesic.
The reader who does not wish to follow the details of this discussion can be informed that: (i) our results concerning the transformation from the retarded coordinates (u, r, Ω^{a}) to the Fermi normal coordinates (t, s, ω^{a}) are contained in Eqs. (11.1) — (11.3) below; (ii) our results concerning the transformation from the Fermi normal coordinates (t, s, ω^{a}) to the retarded coordinates (u, r, Ω^{a}) are contained in Eqs. (11.4) — (11.6); (iii) the decomposition of each member of \((\bar e_0^\alpha ,\bar e_a^\alpha)\) in the tetrad \((e_0^\alpha ,\; e_a^\alpha)\) is given in retarded coordinates by Eqs. (11.7) and (11.8); and (iv) the decomposition of each member of \((e_0^\alpha ,\; e_a^\alpha)\) in the tetrad \((\bar e_0^\alpha ,\bar e_a^\alpha)\) is given in Fermi normal coordinates by Eqs. (11.9) and (11.10).
Our final task will be to define, along with the retarded and simultaneous points, an advanced point x″ on the world line γ; see Figure 8. This is taken on in Section 11.4.
13.1 From retarded to Fermi coordinates
Quantities at \(\bar{x} := z(t)\) can be related to quantities at x′ := z (u) by Taylor expansion along the world line γ. To implement this strategy we must first find an expression for Δ := t − u. (Although we use the same notation, this should not be confused with the van Vleck determinant introduced in Section 7.)
Consider the function p (τ) of the propertime parameter τ defined by
in which 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,
where overdots (or a number within brackets) indicate repeated differentiation with respect to τ. We have
where a^{μ} = Du^{μ}/dτ, ȧ^{μ} = Da^{μ}/dτ, and ä^{μ} = Dȧ^{μ}/dτ.
We now express all of this in retarded coordinates by invoking the expansion of Eq. (6.7) for σ_{ σα′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 Eq. (10.7). With a degree of accuracy sufficient for our purposes we obtain
where S = R_{a0b0}Ω^{a} Ω^{b} was first introduced in Eq. (10.25), and where \({\dot a_0} := {\dot a_{\alpha \prime}}{u^{\alpha \prime}},\,\,{\dot a_a} := {\dot a_{\alpha \prime}}e_a^{\alpha \prime}\) are the frame components of the covariant derivative of the acceleration vector. To arrive at these results we made use of the identity \({a_{\alpha \prime}}{a^{\alpha \prime}} + {\dot a_{\alpha \prime}}{u^{\alpha \prime}} = 0\) 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 \({\dot a_a} := {\dot a_\mu}e_a^\mu\) 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 FermiWalker transported on the world line) into the identity \(d{a_a}/d\tau = D({a_\nu}e_a^\nu)/d\tau\).
Collecting our results we obtain
which can readily be solved for Δ := t − u expressed as an expansion in powers of r. The final result is
where we show explicitly that all frame components are evaluated at the retarded point z (u).
To obtain relations between the spatial coordinates we consider the functions
in which 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
with
To arrive at these results we have used the same expansions as before and reintroduced S_{ a } = R_{a0b0}Ω^{b} − R_{ab0c}Ω^{b} Ω^{c}, as it was first defined in Eq. (10.24).
Collecting our results we obtain
which becomes
after substituting Eq. (11.1) for Δ := t − u. From squaring Eq. (11.2) and using the identity δ_{ ab }ω^{a} ω^{b} = 1 we can also deduce
for the spatial distance between x and z (t).
13.2 From Fermi to retarded coordinates
The techniques developed in the preceding subsection can easily be adapted to the task of relating the retarded coordinates of x to its Fermi normal coordinates. Here we use \(\bar{x} := z(t)\) as the reference point and express all quantities at x^{′} := z (u) as Taylor expansions about τ = t.
We begin by considering the function
of the propertime parameter τ on γ. We have that \(\sigma (t) = {1 \over 2}{s^2}\) 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
Expressions for the derivatives of p (τ) evaluated at τ = t can be constructed from results derived previously in Section 11.1: it suffices to replace all primed indices by barred indices and then substitute the relation \({\sigma ^{\bar \alpha}} =  s{\omega ^a}e_a^{\bar \alpha}\) that follows immediately from Eq. (9.5). This gives
and then
after recalling that p (t) = 0. Solving for Δ as an expansion in powers of s returns
in which we emphasize that all frame components are evaluated at the simultaneous point z (t).
An expression for r = p (u) can be obtained by expanding p (t − Δ) in powers of Δ. We have
and substitution of our previous results gives
for the retarded distance between x and z (u).
Finally, the retarded coordinates 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
Results from the preceding subsection can again be imported with mild alterations, and we find
This, together with Eq. (11.4), gives
It may be checked that squaring this equation and using the identity δ_{ ab } Ω^{a} Ω^{b} = 1 returns the same result as Eq. (11.5).
13.3 Transformation of the tetrads at x
Recall that we have constructed two sets of basis vectors at 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 \(\bar x := z(t)\). The second set is 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 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 subsection. For this purpose we shall consider the functions
in which x is a fixed point in a neighbourhood of γ, z (τ) is an arbitrary point on the world line, and \(g_{\;\mu}^\alpha (x,z)\) 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 \(e_a^\alpha = p_a^\alpha (u)\).
We begin with the decomposition of \((\bar e_0^\alpha ,\bar e_a^\alpha)\) in the tetrad \((e_0^\alpha ,e_a^\alpha)\) associated with the retarded point z (u). This decomposition will be expressed in the retarded coordinates as an expansion in powers of r. As in Section 9.1 we express quantities at z (t) in terms of quantities at z (u) by expanding in powers of Δ := t − u. We have
with
where we have used the expansions of Eq. (6.11) as well as the decompositions of Eq. (10.4). Collecting these results and substituting Eq. (11.1) for Δ yields
Similarly, we have
with
and all this gives
We now turn to the decomposition of \((e_0^\alpha ,e_a^\alpha)\) in the tetrad \((\bar e_0^\alpha ,\bar e_a^\alpha)\) associated with the simultaneous point z (t). This decomposition will be expressed in the Fermi normal coordinates as an expansion in powers of. Here, as in Section 9.2, we shall express quantities at z (u) in terms of quantities at z (t). We begin with
and we evaluate the derivatives of 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 Eq. (6.11), and on the decomposition of \(g_{\;\;\bar \alpha}^\alpha (x,\bar x)\) in the tetrads at x and \(\bar{x}\). This gives
and we finally obtain
Similarly, we write
in which we substitute
as well as Eq. (11.4) for Δ := t − u. Our final result is
13.4 Advanced point
It will prove convenient to introduce on the world line, along with the retarded and simultaneous points, an advanced point associated with the field point x. The advanced point will be denoted x ″ := z (v), with v denoting the value of the propertime parameter at x ″; to tensors at this point we assign indices α ″, β ″, etc. The advanced point is linked to by a pastdirected 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 futuredirected null vector. The affineparameter distance between x and x″ along the null geodesic is given by
and we shall call this the advanced distance between x and the world line. Notice that r_{adv} is a positive quantity.
We wish first to find an expression for v in terms of the retarded coordinates of x. For this purpose we define Δ″ := v − u and reintroduce the function σ (τ) := σ (x, z (τ)) first considered in Section 11.2. 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
Using the expressions for the derivatives of p (τ) that were first obtained in Section 11.1, we write this as
Solving for Δ′ as an expansion in powers of r, we obtain
in which all frame components are evaluated at the retarded point z (u).
Our next task is to derive an expression for the advanced distance 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
which then becomes
After substituting Eq. (11.12) for Δ′ and witnessing a number of cancellations, we arrive at the simple expression
From Eqs. (10.29), (10.30), and (11.12) we deduce that the gradient of the advanced time is given by
where the expansion in powers of r was truncated to a sufficient number of terms. Similarly, Eqs. (10.30), (10.31), and (11.13) imply that the gradient of the advanced distance is given by
where S_{ a } and S were first introduced in Eqs. (10.24) and (10.25), respectively. We emphasize that in Eqs. (11.14) and (11.15), all frame components are evaluated at the retarded point z (u).
14 Part III: Green’s Functions
15 Scalar Green’s functions in flat spacetime
15.1 Green’s equation for a massive scalar field
To prepare the way for our discussion of Green’s functions in curved spacetime, we consider first the slightly nontrivial case of a massive scalar field Φ(x) in flat spacetime. This field satisfies the wave equation
where □ = η^{αβ}∂_{ α }∂_{ β } is the wave operator, μ (x) a prescribed source, and where the parameter k has a dimension of inverse length. We seek a Green’s function G (x, x′) such that a solution to Eq. (12.1) can be expressed as
where the integration is over all of Minkowski spacetime. The relevant wave equation for the Green’s function is
where δ_{4}(x − x′) = δ (t − t′)δ (x − x′)δ (y − y′)δ (z − z′) is a fourdimensional Dirac distribution in flat spacetime. Two types of Green’s functions will be of particular interest: the retarded Green’s function, a solution to Eq. (12.3) with the property that it vanishes when is in the past of x′, and the advanced Green’s function, which vanishes when x is in the future of x′.
To solve Eq. (12.3) we appeal to Lorentz invariance and the fact that the spacetime is homogeneous to argue that the retarded and advanced Green’s functions must be given by expressions of the form
where \(\sigma = {1 \over 2}{\eta _{\alpha \beta}}{(x  x\prime)^\alpha}{(x  x\prime)^\beta}\) 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.
15.2 Integration over the source
The Dirac functional on the righthand side of Eq. (12.3) is a highly singular quantity, and we can avoid dealing with it by integrating the equation over a small fourvolume V that contains x′ = 0. This volume is bounded by a closed hypersurface ∂V. After using Gauss’ theorem on the first term of Eq. (12.3), we obtain ∮_{ ∂V }G^{α}d∑_{ α } −k^{2} ∫_{ V } G dV = −4π, where dσ_{ 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
It should be emphasized that the fourvolume V must contain the point x^{′}.
To examine Eq. (12.5) we introduce coordinates (w, χ, θ, θ) defined by
and we let ∂V be a surface of constant w. The metric of flat spacetime is given by
in the new coordinates, where d Ω^{2} = dθ^{2} + sin^{2} θ dϕ^{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 cos2_{ χ } < 0. Straightforward computations reveal that in these coordinates, \(\sigma =  {1 \over 2}{w^2}\cos 2\chi ,\,\,\sqrt { g} = {w^3}{\sin ^2}\chi \sin \theta ,\,\,{g^{ww}} =  \cos 2\chi ,\,\,{g^{w\chi}} = {w^{ 1}}\sin 2\chi ,\,\,{g^{\chi \chi}} = {w^{ 2}}\cos 2\chi\), and the only nonvanishing component of the surface element is dΣ_{ w } = w^{3} sin^{2} _{ χ }d_{ χ }d Ω, where d Ω = sin θ dθdϕ. To calculate the gradient of the Green’s function we express it as \(G = \theta (\pm t)g(\sigma) = \theta (\pm w\cos \chi)g( {1 \over 2}{w^2}\cos 2\chi)\), with the upper (lower) sign belonging to the retarded (advanced) Green’s function. Calculation gives G;^{α} dΣ_{ α } = θ (± cos _{ χ })w^{4} sin^{2} χg′(σ) d_{ χ }d Ω, 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.
Integration of G^{;α}dΣ_{ α } with respect to d Ω is immediate, and we find that Eq. (12.5) reduces to
for the retarded Green’s function, the step function restricts the domain of integration to 0 < χ < π/2, in which σ increases from \( {1 \over 2}{w^2}\) to \({1 \over 2}{w^2}\). Changing the variable of integration from χ to σ transforms Eq. (12.6) into
where \(\epsilon := {1 \over 2}{w^2}\). For the advanced Green’s function, the domain of integration is π/2 < χ < π, in which σ decreases from \({1 \over 2}{w^2}\) to \( {1 \over 2}{w^2}\). Changing the variable of integration from χ to σ also produces Eq. (12.7).
15.3 Singular part of g (σ)
We have seen that Eq. (12.7) properly encodes the influence of the singular source term on both the retarded and advanced Green’s function. The function g (σ) that enters into the expressions of Eq. (12.4) must therefore be such that Eq. (12.7) is satisfied. It follows immediately that g (σ) must be a singular function, because for a smooth function the integral of Eq. (12.7) would be of order ϵ and the lefthand side of Eq. (12.7) could never be made equal to −1. The singularity, however, must be integrable, and this leads us to assume that g′(σ) must be made out of Dirac δfunctions and derivatives.
We make the ansatz
where 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 δfunctions to be constants. To see this we invoke the distributional identities
from which it follows that σ^{2}δ′(σ) = σ^{3}δ ″(σ) = … =0. A term like f (σ)δ (σ) is then distributionally equal to f (0)δ (σ), while a term like f (σ)δ′(σ) is distributionally equal to f (σ)δ′(σ) − f′(0)δ (σ), and a term like f (σ)δ ″(σ) is distributionally equal to f (0)δ ″(σ) − 2f′(0)δ′ (σ) + 2f ″(0)δ (σ); here f (σ) is an arbitrary test function. Summing over such terms, we recover an expression of the form of Eq. (12.9), and there is no need to make A, B, C, … functions of σ.
Differentiation of Eq. (12.8) and substitution into Eq. (12.7) yields
where overdots (or a number within brackets) indicate repeated differentiation with respect to 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
with V (σ) a smooth function that cannot be determined from Eq. (12.7) alone.
15.4 Smooth part of g (σ)
To determine V (σ) we must go back to the differential equation of Eq. (12.3). 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 (□ − ^{2})g (σ) = 0, the homogeneous version of Eq. (12.3). We have ∇_{ α }g = g′σ_{ α }, ∇_{ α } ∇_{ β }g = g ″σ_{ α }σ_{ β } + g′σ_{ αβ }, □g = 2σg ″ + 4g′, so that Green’s equation reduces to the ordinary differential equation
if we substitute Eq. (12.10) into this we get
where we have used the identities of Eq. (12.9). The lefthand side will vanish as a distribution if we set
These equations determine V (σ) uniquely, even in the absence of a second boundary condition at σ = 0, because the differential equation is singular at σ = 0 while V is known to be smooth.
To solve Eq. (12.12) we let V = F (z)/z, with \(z := k\sqrt { 2\sigma}\). This gives rise to Bessel’s equation for the new function F:
The solution that is well behaved near z = 0 is F = aJ_{1}(z), where a is a constant to be determined. We have that \({J_1}(z)\sim {1 \over 2}z\) for small values of z, and it follows that V ∼ a/2. From Eq. (12.12) we see that a = − k^{2}. So we have found that the only acceptable solution to Eq. (12.12) is
To summarize, the retarded and advanced solutions to Eq. (12.3) are given by Eq. (12.4) with g (σ) given by Eq. (12.10) and V (σ) given by Eq. (12.13).
15.5 Advanced distributional methods
The techniques developed previously to find Green’s functions for the scalar wave equation are limited to flat spacetime, and they would not be very useful for curved spacetimes. To pursue this generalization we must introduce more powerful distributional methods. We do so in this subsection, and in the next we shall use them to recover our previous results.
Let θ_{+} (x, Σ) be a generalized step function, defined to be one when x is in the future of the spacelike hypersurface Σ and zero otherwise. Similarly, define θ_{−}(x, Σ) := 1 − θ_{+} (x, Σ) to be one when x is in the past of the spacelike hypersurface Σ and zero otherwise. Then define the lightcone step functions
so that θ_{+} (−σ) is one if x is within I^{+} (x′), the chronological future of x′, and zero otherwise, and θ_{−}(−σ) is one if x is within 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_{−}(−σ) = θ (−σ). Define also the lightcone Dirac functionals
so that 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 Eqs. (12.14) and (12.15), σ is the world function for flat spacetime; it is negative when x and x′ are timelike related, and positive when 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 θ_{±}(σ + ϵ) in some sensitive computations, and then take the limit ϵ → 0^{+}. Notice that the equation σ + ϵ = 0 describes a twobranch hyperboloid that is located just within the light cone of the reference point x′. The hyperboloid does not include x′, and θ_{+} (x, Σ) is one everywhere on its future branch, while θ −(x, Σ) is one everywhere on its past branch. These factors, therefore, become invisible to differential operators. For example, θ′+ (−σ − ϵ) = θ_{+} (x, Σ)θ′(−σ − ϵ) = −θ_{+}(x, Σ)δ (σ + ϵ) = − δ_{+} (σ + ϵ). This manipulation shows that after the shift from σ to σ + ϵ, the distributions of Eqs. (12.14) and (12.15) can be straightforwardly differentiated with respect to σ.
In the next paragraphs we shall establish the distributional identities
in fourdimensional flat spacetime. These will be used in the next subsection to recover the Green’s functions for the scalar wave equation, and they will be generalized to curved spacetime in Section 13.
The derivation of Eqs. (12.16) — (12.18) relies on a “master” distributional identity, formulated in threedimensional flat space:
with \(r := \vert x \vert := \sqrt {{x^2} + {y^2} + {z^2}}\). This follows from yet another identity, ∇^{2}r^{−1} = − 4πδ_{ 3 }(x), in which we write the lefthand side as lim_{ ϵ→0 }+ ∇^{2}R^{−1}; since R^{−1} is nonsingular at x = 0 it can be straightforwardly differentiated, and the result is ∇^{2}R −^{1} = −6ϵ/R^{5}, from which Eq. (12.19) follows.
To prove Eq. (12.16) we must show that ϵδ_{±}(σ + ϵ) vanishes as a distribution in the limit ϵ − 0^{+}. For this we must prove that a functional of the form
where f (x) = f (t, x) is a smooth test function, vanishes for all such functions f. Our first task will be to find a more convenient expression for θ_{±}(σ + ϵ). Once more we set x′ = 0 (without loss of generality) and we note that 2(σ + ϵ) = −t^{2} + r^{2} + 2ϵ = −(t − R)(t + R), where we have used Eq. (12.19). It follows that
and from this we find
which establishes Eq. (12.16).
The validity of Eq. (12.17) is established by a similar computation. Here we must show that a functional of the form
vanishes for all test functions f. We have
and the identity of Eq. (12.17) is proved. In these manipulations we have let an overdot indicate partial differentiation with respect to t, and we have used ∂R/de = 1/R.
To establish Eq. (12.18) we consider the functional
and show that it evaluates to 2πf (0, 0). We have
as required. This proves that Eq. (12.18) holds as a distributional identity in fourdimensional flat spacetime.
15.6 Alternative computation of the Green’s functions
The retarded and advanced Green’s functions for the scalar wave equation are now defined as the limit of the functions \(G_ \pm ^\epsilon (x,x\prime)\) when ϵ → 0^{+}. For these we make the ansatz
and we shall prove that \(G_ \pm ^\epsilon (x,x\prime)\) satisfies Eq. (12.3) in the limit. We recall that the distributions θ_{±} and δ_{±} were defined in the preceding subsection, and we assume that V (σ) is a smooth function of \(\sigma (x,x\prime) = {1 \over 2}{\eta _{\alpha \beta}}{(x  x\prime)^\alpha}{(x  x\prime)^\beta}\); because this function is smooth, it is not necessary to evaluate V at σ + ϵ in Eq. (12.21). 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
and
where Σ is a spacelike hypersurface that contains x′. We will also determine the form of the function V (σ).
The functions that appear in Eq. (12.21) can be straightforwardly differentiated. The manipulations are similar to what was done in Section 12.4, and dropping all labels, we obtain (□ − k^{2})G = 2σ G″ + 4G′ − k^{2}G, with a prime indicating differentiation with respect to σ. From Eq. (12.21) we obtain G′ = δ′ − Vδ + V′θ and G″ = δ″ − Vδ′ − 2V′δ + V″θ. The identities of Eq. (12.9) can be expressed as (σ + ϵ)δ′(σ + ϵ) = − δ (σ + ϵ) and (σ + ϵ)δ″ (σ + ϵ) = −2δ′(σ + ϵ), and combining this with our previous results gives
According to Eq. (12.16) — (12.18), the last two terms on the righthand side disappear in the limit ϵ → 0^{+}, and the third term becomes − 4πδ_{4}(x − x′). Provided that the first two terms vanish also, we recover (□ − k^{2})G (x, x′) = − 4πδ_{4}(x − x′) in the limit, as required. Thus, the limit of \(G^\epsilon_{\pm}(x,x\prime)\) when ϵ − 0^{+} will indeed satisfy Green’s equation provided that V (σ) is a solution to
these are the same statements as in Eq. (12.12). The solution to these equations was produced in Eq. (12.13):
and this completely determines the Green’s functions of Eqs. (12.22) and (12.23).
16 Distributions in curved spacetime
The distributions introduced in Section 12.5 can also be defined in a fourdimensional spacetime with metric g_{ αβ }. Here we produce the relevant generalizations of the results derived in that section.
16.1 Invariant Dirac distribution
We first introduce δ_{4}(x,x′), an invariant Dirac functional in a fourdimensional curved spacetime. This is defined by the relations
where f (x) is a smooth test function, V any fourdimensional region that contains x′, and V′ any fourdimensional region that contains x. These relations imply that δ_{4}(x,x′) is symmetric in its arguments, and it is easy to see that
where 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) fourdimensional Dirac functional. The relations of Eq. (13.2) 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′.
The invariant Dirac distribution satisfies the identities
where Ω…(x, x′) is any bitensor and \(g_{\;\alpha \prime}^\alpha (x,x\prime),\,\,g_{\;\alpha}^{\alpha \prime}(x,x\prime)\) are parallel propagators. The first identity follows immediately from the definition of the δfunction. 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 Eqs. (13.1) implies
and on the other hand,
which establishes the second identity of Eq. (13.3). Notice that in these manipulations, the integrations involve 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 Eq. (13.3) is proved in a similar way.
16.2 Lightcone distributions
For the remainder of Section 13 we assume that \(x \in {\mathcal N}(x\prime)\), so that a unique geodesic β links these two points. We then let σ (x,x′) be the curved spacetime world function, and we define lightcone step functions by
where θ_{+} (x, Σ) is one when 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 Eq. (12.14). We have that θ_{+} (−σ) is one when x is within I^{+}(x′), the chronological future of x′, and zero otherwise, and θ −(−σ) is one when x is within I^{−}(x′), the chronological past of x′, and zero otherwise. We also have θ+ (−σ) + θ_{−}(−σ) = θ (−σ).
We define the curvedspacetime version of the lightcone Dirac functionals by
an immediate generalization of Eq. (12.15). We have that 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 σ is negative when x and x′ are timelike related, and positive when they are spacelike related.
For the same reasons as those mentioned in Section 12.5, it is sometimes convenient to shift the argument of the step and δfunctions from σ to σ + ϵ, where ϵ is a small positive quantity. With this shift, the lightcone distributions can be straightforwardly differentiated with respect to σ. For example, \(\delta_{\pm}(\sigma + \epsilon) = \theta\prime_{\pm}(\sigma\epsilon)\), with a prime indicating differentiation with respect to σ.
We now prove that the identities of Eq. (12.16)–(12.18) generalize to
in a fourdimensional curved spacetime; the only differences lie with the definition of the world function and the fact that it is the invariant Dirac functional that appears in Eq. (13.8). To establish these identities in curved spacetime we use the fact that they hold in flat spacetime — as was shown in Section 12.5 — and that they are scalar relations that must be valid in any coordinate system if they are found to hold in one. Let us then examine Eqs. (13.6) — (13.7) in the Riemann normal coordinates of Section 8; these are denoted \({\hat x^\alpha}\) and are based at x′. We have that \(\sigma (x,x\prime) = {1 \over 2}{\eta _{\alpha \beta}}{{\hat x}^\alpha}{{\hat x}^\beta}\) and δ_{4}(x,x′) = Δ(x,x′)δ_{4}(x − x′) = δ_{4}(x − x′), where Δ(x,x′) is the van Vleck determinant, whose coincidence limit is unity. In Riemann normal coordinates, therefore, Eqs. (13.6) — (13.8) take exactly the same form as Eqs. (12.16) — (12.18). 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.
17 Scalar Green’s functions in curved spacetime
17.1 Green’s equation for a massless scalar field in curved spacetime
We consider a massless scalar field Φ(x) in a curved spacetime with metric g_{ αβ }. The field satisfies the wave equation
where □ = 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 Eq. (14.1) can be expressed as
where the integration is over the entire spacetime. The wave equation for the Green’s function is
where δ_{4}(x, x′) is the invariant Dirac functional introduced in Section 13.1. It is easy to verify that the field defined by Eq. (14.2) is truly a solution to Eq. (14.1).
We let G +(x,x′) be the retarded solution to Eq. (14.3), and G_{ − } (x,x′) is the advanced solution; when viewed as functions of x, G_{+}(x,x′) is nonzero in the causal future of x′, while G_{−}(x,x′) is nonzero in its causal past. We assume that the retarded and advanced Green’s functions exist as distributions and can be defined globally in the entire spacetime.
17.2 Hadamard construction of the Green’s functions
Assuming throughout this subsection that x is restricted to the normal convex neighbourhood of x′, we make the ansatz
where 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.
Before we substitute the Green’s functions of Eq. (14.4) into the differential equation of Eq. (14.3) we proceed as in Section 12.6 and shift σ by the small positive quantity ϵ. We shall therefore consider the distributions
and later recover the Green’s functions by taking the limit ϵ → 0^{+}. Differentiation of these objects is straightforward, and in the following manipulations we will repeatedly use the relation σ^{α}σ_{ α } = 2σ satisfied by the world function. We will also use the distributional identities \(\sigma {\delta _ \pm}(\sigma + \epsilon) =  \epsilon {\delta _ \pm}(\sigma + \epsilon),\,\sigma {\delta \prime_ \pm}(\sigma + \epsilon) =  {\delta _ \pm}(\sigma + \epsilon)  \epsilon {\delta \prime_ \pm}(\sigma + \epsilon)\), and \(\sigma {\delta \prime\prime_ \pm}(\sigma + \epsilon) =  2\delta \prime(\sigma + \epsilon)  \epsilon\delta \prime\prime(\sigma + \epsilon)\). After a routine calculation we obtain
which becomes
in the limit ϵ → 0^{+}, after using the identities of Eqs. (13.6) — (13.8).
According to Eq. (14.3), the righthand side of Eq. (14.5) should be equal to − 4πδ_{4}(x, x′). This immediately gives us the coincidence condition
for the biscalar U (x,x′). To eliminate the δ′_{±} term we make its coefficient vanish:
As we shall now prove, these two equations determine U (x,x′) uniquely.
Recall from Section 3.3 that 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 Eq. (14.7) therefore represents the logarithmic rate of change of U (x,x′) along β, and this can be expressed as 2λdU/d λ. For the second term we recall from Section 7.1 the differential equation Δ^{−1}(Δσα)_{;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 Eq. (14.7) becomes
It follows that U^{2}/Δ is constant on β, and this must therefore be equal to its value at the starting point x′: U^{2}/Δ = [U^{2}/Δ] = 1, by virtue of Eq. (14.6) and the property [Δ] = 1 of the van Vleck determinant. Because this statement must be true for all geodesics β that emanate from x′, we have found that the unique solution to Eqs. (14.6) and (14.7) is
We must still consider the remaining terms in Eq. (14.5). The 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
where we indicate that the righthand side also must be restricted to the light cone. The first term of Eq. (14.9) can be expressed as \(\lambda d\check{V}/d\lambda\) and this equation can be integrated along any null geodesic that generates the null cone 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 Eq. (14.9) and the fact that V (x, x′) must be smooth at coincidence.
Eqs. (7.4) and (14.8) imply that near coincidence, U (x,x′) admits the expansion
where R_{α′β′} is the Ricci tensor at x′ and λ is the affineparameter distance to x (which can be either on or off the light cone). Differentiation of this relation gives
and eventually,
Using also \([\sigma _{\;\,\alpha}^\alpha ] = 4\), we find that the coincidence limit of Eq. (14.9) gives
and this provides the initial values required for the integration of Eq. (14.9) on the null cone.
Eqs. (14.9) and (14.13) give us a means to construct \(\check{V}(x,x\prime)\), the restriction of V (x, x′) on the null cone σ(x, x′) = 0. These values can then be used as characteristic data for the wave equation
which is obtained by elimination of the θ_{±} term in Eq. (14.5). 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 Eq. (14.8) and V (x,x′) determined uniquely by the wave equation of Eq. (14.14) and the characteristic data constructed with Eqs. (14.9) and (14.13), the retarded and advanced Green’s functions of Eq. (14.4) do indeed satisfy Eq. (14.3). It should be emphasized that the construction provided in this subsection is restricted to \({\mathcal N}(x \prime)\), the normal convex neighbourhood of the reference point x′.
17.3 Reciprocity
We shall now establish the following reciprocity relation between the (globally defined) retarded and advanced Green’s functions:
Before we get to the proof we observe that by virtue of Eq. (14.15), the biscalar V (x,x′) must be symmetric in its arguments:
To go from Eq. (14.15) to Eq. (14.16) we simply note that when \(x \in {\mathcal N}(x\prime)\) and belongs to I^{+} (x′), then G_{+} (x,x′) = V (x,x′) and G_{−}(x′,x) = V (x′,x).
To prove the reciprocity relation we invoke the identities
and
and take their difference. On the lefthand side we have
while the righthand side gives
. Integrating both sides over a large fourdimensional region that contains both x′ and x ″, we obtain
where ∂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 lefthand side of the equation evaluates to zero in the limit. This gives us the statement G_{+} (x ″,x′) = G_{−}(x′,x ″), which is just Eq. (14.15) with x ″ replacing x.
17.4 Kirchhoff representation
Suppose that the values for a scalar field Φ(x′) and its normal derivative \({n^{\alpha \prime}}{\nabla _{\alpha \prime}}\Phi (x\prime)\) are known on a spacelike hypersurface Σ. Suppose also that the scalar field satisfies the homogeneous wave equation
then the value of the field at a point x in the future of Σ is given by Kirchhoff’s formula,
where \(d{\Sigma _{\alpha \prime}}\) is the surface element on Σ. If nx′ is the futuredirected unit normal, then \(d{\Sigma _{\alpha \prime}} =  {n_{\alpha \prime}}dV\), with dV denoting the invariant volume element on Σ; notice that \(d{\Sigma _{\alpha \prime}}\) is past directed.
To establish this result we start with the equations
in which x and x′ refer to arbitrary points in spacetime. Taking their difference gives
and this we integrate over a fourdimensional region V that is bounded in the past by the hypersurface Σ. We suppose that V contains x and we obtain
where \(d{\Sigma _{\alpha \prime}}\) is the outwarddirected 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 outwarddirected surface element on ∂V, we must change the sign of the lefthand side to be consistent with the convention adopted previously. With this change we have
which is the same statement as Eq. (14.18) if we take into account the reciprocity relation of Eq. (14.15).
17.5 Singular and regular Green’s functions
In Part IV of this review we will compute the retarded field of a moving scalar charge, and we will analyze its singularity structure near the world line; this will be part of our effort to understand the effect of the field on the particle’s motion. The retarded solution to the scalar wave equation is the physically relevant solution because it properly incorporates outgoingwave boundary conditions at infinity — the advanced solution would come instead with incomingwave boundary conditions. The retarded field is singular on the world line because a point particle produces a Coulomb field that diverges at the particle’s position. In view of this singular behaviour, it is a subtle matter to describe the field’s action on the particle, and to formulate meaningful equations of motion.
When facing this problem in flat spacetime (recall the discussion of Section 1.3) it is convenient to decompose the retarded Green’s function G_{+} (x, x′) into a singular Green’s function \({G_{\rm{S}}}(x,x\prime) := {1 \over 2}[{G_ +}(x,x\prime) + {G_ }(x,x\prime)]\) and a regular twopoint function \({G_{\rm{R}}}(x,x\prime) := {1 \over 2}[{G_ +}(x,x\prime)  {G_ }(x,x\prime)]\). The singular Green’s function takes its name from the fact that it produces a field with the same singularity structure as the retarded solution: the diverging field near the particle is insensitive to the boundary conditions imposed at infinity. We note also that G_{S}(x,x′) satisfies the same wave equation as the retarded Green’s function (with a Dirac functional as a source), and that by virtue of the reciprocity relations, it is symmetric in its arguments. The regular twopoint function, on the other hand, takes its name from the fact that it satisfies the homogeneous wave equation, without the Dirac functional on the righthand side; it produces a field that is regular on the world line of the moving scalar charge. (We reserve the term “Green’s function” to a twopoint function that satisfies the wave equation with a Dirac distribution on the righthand side; when the source term is absent, the object is called a “twopoint function”.)
Because the singular Green’s function is symmetric in its argument, it does not distinguish between past and future, and it produces a field that contains equal amounts of outgoing and incoming radiation — the singular solution describes a standing wave at infinity. Removing G_{S}(x, x′) from the retarded Green’s function will have the effect of removing the singular behaviour of the field without affecting the motion of the particle. The motion is not affected because it is intimately tied to the boundary conditions: If the waves are outgoing, the particle loses energy to the radiation and its motion is affected; if the waves are incoming, the particle gains energy from the radiation and its motion is affected differently. With equal amounts of outgoing and incoming radiation, the particle neither loses nor gains energy and its interaction with the scalar field cannot affect its motion. Thus, subtracting G_{S}(x,x′) from the retarded Green’s function eliminates the singular part of the field without affecting the motion of the scalar charge. The subtraction leaves behind the regular twopoint function, which produces a field that is regular on the world line; it is this field that will govern the motion of the particle. The action of this field is well defined, and it properly encodes the outgoingwave boundary conditions: the particle will lose energy to the radiation.
In this subsection we attempt a decomposition of the curvedspacetime retarded Green’s function into singular and regular pieces. The flatspacetime relations will have to be amended, however, because of the fact that in a curved spacetime, the advanced Green’s function is generally nonzero when x′ is in the chronological future of x. This implies that the value of the advanced field at x depends on events x′ that will unfold in the future; this dependence would be inherited by the regular field (which acts on the particle and determines its motion) if the naive definition \({G_{\rm{R}}}(x,x\prime) := {1 \over 2}[{G_ +}(x,x\prime)  {G_ }(x,x\prime)]\) were to be adopted.
We shall not adopt this definition. Instead, we shall follow Detweiler and Whiting [53] and introduce a singular Green’s function with the properties

S1: G_{S}(x,x′) satisfies the inhomogeneous scalar wave equation,
$$(\square \,  \xi R){G_{\rm{S}}}(x,x\prime) =  4\pi {\delta _4}(x,x\prime);$$(14.19) 
S2: G_{S}(x,x′) is symmetric in its arguments,
$${G_{\rm{S}}}(x\prime ,x) = {G_{\rm{S}}}(x,x\prime);$$(14.20) 
S3: G_{S}(x,x′) vanishes if x is in the chronological past or future of x′,
$${G_{\rm{S}}}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ \pm}(x\prime).$$(14.21)
Properties S1 and S2 ensure that the singular Green’s function will properly reproduce the singular behaviour of the retarded solution without distinguishing between past and future; and as we shall see, property S3 ensures that the support of the regular twopoint function will not include the chronological future of x.
The regular twopoint function is then defined by
where G_{+} (x,x′) is the retarded Green’s function. This comes with the properties

R1: G_{R}(x,x′) satisfies the homogeneous wave equation,
$$(\square\,  \xi R){G_{\rm{R}}}(x,x\prime) = 0;$$(14.23) 
R2: G_{R}(x,x′) agrees with the retarded Green’s function if x is in the chronological future of
$${G_{\rm{R}}}(x,x\prime) = {G_ +}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(14.24) 
R3: G_{R}(x,x′) vanishes if x is in the chronological past of x′,
$${G_{\rm{R}}}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ }(x\prime).$$(14.25)
Property R1 follows directly from Eq. (14.22) and property S1 of the singular Green’s function. Properties R2 and R3 follow from S3 and the fact that the retarded Green’s function vanishes if x is in past of x′. The properties of the regular twopoint function ensure that the corresponding regular field will be nonsingular at the world line, and will depend only on the past history of the scalar charge.
We must still show that such singular and regular Green’s functions can be constructed. This relies on the existence of a twopoint function H (x,x′) that would possess the properties

H1: H (x,x′) satisfies the homogeneous wave equation,
$$(\square\,  \xi R)H(x,x\prime) = 0;$$(14.26) 
H2: H (x,x′) is symmetric in its arguments,
$$H(x\prime ,x) = H(x,x\prime);$$(14.27) 
H3: H (x,x′) agrees with the retarded Green’s function if x is in the chronological future of
$$H(x,x\prime) = {G_ +}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(14.28) 
H4: H (x,x′) agrees with the advanced Green’s function if x is in the chronological past of
$$H(x,x\prime) = {G_ }(x,x\prime)\qquad {\rm{when}}\;x \in {I^ }(x\prime).$$(14.29)
With a biscalar H (x, x′) satisfying these relations, a singular Green’s function defined by
will satisfy all the properties listed previously: S1 comes as a consequence of H1 and the fact that both the advanced and the retarded Green’s functions are solutions to the inhomogeneous wave equation, S2 follows directly from H2 and the definition of Eq. (14.30), and S3 comes as a consequence of H3, H4 and the properties of the retarded and advanced Green’s functions.
The question is now: does such a function H (x, x′) exist? We will present a plausibility argument for an affirmative answer. Later in this section we will see that H (x,x′) is guaranteed to exist in the local convex neighbourhood of x′, where it is equal to V (x,x′). And in Section 14.6 we will see that there exist particular spacetimes for which H (x, x′) can be defined globally.
To satisfy all of H1–H4 might seem a tall order, but it should be possible. We first note that property H4 is not independent from the rest: it follows from H2, H3, and the reciprocity relation (14.15) satisfied by the retarded and advanced Green’s functions. Let x ∈ I^{−}(x′), so that x′ ∈ I^{+} (x). Then H (x,x′) = H (x′,x) by H2, and by H3 this is equal to G_{+} (x′,x). But by the reciprocity relation this is also equal to G_{−}(x,x′), and we have obtained H4. Alternatively, and this shall be our point of view in the next paragraph, we can think of H3 as following from H2 and H4.
Because H (x,x′) satisfies the homogeneous wave equation (property H1), it can be given the Kirkhoff representation of Eq. (14.18): if Σ is a spacelike hypersurface in the past of both x and x′, then
where \(d\Sigma_{\alpha\prime\prime}\) 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 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 the entire 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 H2 is enforced and we conclude that the properties H1, H2, H3, and H4 can all be satisfied.
When x is restricted to the normal convex neighbourhood of x′, properties H1–H4 imply that
it should be stressed here that while 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 Eqs. (14.4) and (14.30) we find that the singular Green’s function is given explicitly by
in the normal convex neighbourhood. Equation (14.32) shows very clearly that the singular Green’s function does not distinguish between past and future (property S2), and that its support excludes I^{±}(x′), in which θ (σ) = 0 (property S3). From Eq. (14.22) we get an analogous expression for the regular twopoint function:
This reveals directly that the regular twopoint function coincides with G_{+}(x,x′) in I^{+}(x′), in which θ (σ) = 0 and θ_{+}(−σ) = 1 (property R2), and that its support does not include I^{−}(x′), in which θ (σ) = θ_{+}(−σ) = 0 (property R3).
17.6 Example: Cosmological Green’s functions
To illustrate the general theory outlined in the previous subsections we consider here the specific case of a minimally coupled (ξ = 0) scalar field in a cosmological spacetime with metric
where 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 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 [33]; it can be extended to other cosmologies [86].
To solve Green’s equation □G (x,x′) = −4πδ_{4}(x,x′) we first introduce a reduced Green’s function g (x,x′) defined by
Substitution yields
where x = (x, y, z) is a vector in threedimensional flat space, and ∇^{2} is the Laplacian operator in this space. We next expand g (x, x′) in terms of planewave solutions to Laplace’s equation,
and we substitute this back into Eq. (14.36). The result, after also Fourier transforming δ_{3}(x − x′), is an ordinary differential equation for \(\tilde g(\eta ,\eta \prime;k)\):
where k^{2} = k · k. To generate the retarded Green’s function we set
in which we indicate that \(\hat{g}\) depends only on the modulus of the vector k. 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.
Substitution of Eq. (14.39) into Eq. (14.38) reveals that \(\hat{g}\) must satisfy the homogeneous equation
together with the boundary conditions
Inserting Eq. (14.39) into Eq. (14.37) and integrating over the angular variables associated with the vector k yields
where Δη:= η − η′ and R:= x − x′.
Eq. (14.40) has cos(k Δη) − (kη)^{−1} sin(k Δη) and sin(k Δη) + (kη)^{−1} cos(k Δη) as linearly independent solutions, and \(\hat g(\eta ,\eta \prime;k)\) must be given by a linear superposition. The coefficients can be functions of η′, and after imposing Eqs. (14.41) we find that the appropriate combination is
Substituting this into Eq. (14.42) and using the identity \((2/\pi)\int\nolimits_0^\infty {\sin} (\omega x)\sin (\omega x\prime)\,d\omega = \delta (x  x\prime)  \delta (x + x\prime)\) yields
after integration by parts. The integral evaluates to θ (Δη − R).
We have arrived at
for our final expression for the retarded Green’s function. The advanced Green’s function is given instead by
The distributions g_{±}(x,x′) are solutions to the reduced Green’s equation of Eq. (14.36). The actual Green’s functions are obtained by substituting Eqs. (14.44) and (14.45) into Eq. (14.35). 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′.
It may be verified that the symmetric twopoint function
satisfies all of the properties H1–H4 listed in Section 14.5; it may thus be used to define singular and regular Green’s functions. According to Eq. (14.30) the singular Green’s function is given by
and its support is limited to the interval − x − x ′ ≤ η − η′ ≤ x −x′ According to Eq. (14.22) the regular twopoint function is given by
its support is given by η − η′ ≥ − x − x′ and for η − η′ ≥ x − x′ the regular twopoint function agrees with the retarded Green’s function.
As a final observation we note that for this cosmological spacetime, the normal convex neighbourhood of any point x consists of the whole spacetime manifold (which excludes the cosmological singularity at a = 0). The Hadamard construction of the Green’s functions is therefore valid globally, a fact that is immediately revealed by Eqs. (14.44) and (14.45).
18 Electromagnetic Green’s functions
18.1 Equations of electromagnetism
The electromagnetic field tensor 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
where □ = g^{αβ} ∇_{ α } ∇_{ β } is the wave operator, \(R_{\;\beta}^\alpha\) the Ricci tensor, and j^{α} a prescribed current density. The wave equation enforces the condition ∇_{ α }j^{α} = 0, which expresses charge conservation.
The solution to the wave equation is written as
in terms of a Green’s function \(G_{\;\beta \prime}^\alpha (x,x\prime)\) that satisfies
where \(g_{\;\beta \prime}^\alpha (x,x\prime)\) is a parallel propagator and η_{4}(x,x′) an invariant Dirac distribution. The parallel propagator is inserted on the righthand side of Eq. (15.3) to keep the index structure of the equation consistent from side to side; because \(g_{\;\beta \prime}^\alpha (x,x\prime){\delta _4}(x,x\prime)\) is distributionally equal to \([g_{\;\beta \prime}^\alpha ]{\delta _4}(x,x\prime) = \delta _{\;\beta \prime}^{\alpha \prime}{\delta _4}(x,x\prime)\), it could have been replaced by either \(\delta _{\;\beta \prime}^{\alpha \prime}\). It is easy to check that by virtue of Eq. (15.3), the vector potential of Eq. (15.2) satisfies the wave equation of Eq. (15.1).
We will assume that the retarded Green’s function \(G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\), which is nonzero if x is in the causal future of x′, and the advanced Green’s function \(G_{ \beta \prime}^{\;\alpha}(x,x\prime)\), which is nonzero if x is in the causal past of x′, exist as distributions and can be defined globally in the entire spacetime.
18.2 Hadamard construction of the Green’s functions
Assuming throughout this subsection that x is in the normal convex neighbourhood of x′, we make the ansatz
where θ_{±}(−σ), δ_{±}(σ) are the lightcone distributions introduced in Section 13.2, and where \(U_{\;\beta \prime}^\alpha (x,x\prime),\,\,V_{\;\beta \prime}^\alpha (x,x\prime)\) are smooth bitensors.
To conveniently manipulate the Green’s functions we shift σ by a small positive quantity ϵ. The Green’s functions are then recovered by the taking the limit of
as ϵ → 0^{+}. When we substitute this into the lefthand side of Eq. (15.3) and then take the limit, we obtain
after a routine computation similar to the one presented at the beginning of Section 14.2. Comparison with Eq. (15.3) returns: (i) the equations
and
that determine \(U_{\;\beta \prime}^\alpha (x,x\prime)\) (ii) the equation
that determines \(\check V{\,^\alpha}_{\beta \prime}(x,x\prime)\), the restriction of \(V_{\;\beta \prime}^\alpha (x,x\prime)\) on the light cone σ (x, x′) = 0; and (iii) the wave equation
that determines \(V_{\;\beta \prime}^\alpha (x,x\prime)\) inside the light cone.
Eq. (15.6) can be integrated along the unique geodesic β that links x′ to x. The initial conditions are provided by Eq. (15.5), and if we set \(U_{\;\beta \prime}^\alpha (x,x\prime) = g_{\;\beta \prime}^\alpha (x,x\prime)U(x,x\prime)\), we find that these equations reduce to Eqs. (14.7) and (14.6), respectively. According to Eq. (14.8), then, we have
which reduces to
near coincidence, with λ denoting the affineparameter distance between x′ and x. Differentiation of this relation gives
and eventually,
Similarly, Eq. (15.7) can be integrated along each null geodesic that generates the null cone σ (x, x′) = 0. The initial values are obtained by taking the coincidence limit of this equation, using Eqs. (15.5), (15.13), and the additional relation \([\sigma^\gamma_{\ \gamma}] = 4\). We arrive at
With the characteristic data obtained by integrating Eq. (15.7), the wave equation of Eq. (15.8) admits a unique solution.
To summarize, the retarded and advanced electromagnetic Green’s functions are given by Eq. (15.4) with \(U_{\;\beta \prime}^\alpha (x,x\prime)\) given by Eq. (15.9) and \(V_{\;\beta \prime}^\alpha (x,x\prime)\) determined by Eq. (15.8) and the characteristic data constructed with Eqs. (15.7) and (15.14). It should be emphasized that the construction provided in this subsection is restricted to \({\mathcal N}(x\prime)\), the normal convex neighbourhood of the reference point x′.
18.3 Reciprocity and Kirchhoff representation
Like their scalar counterparts, the (globally defined) electromagnetic Green’s functions satisfy a reciprocity relation, the statement of which is
The derivation of Eq. (15.15) is virtually identical to what was presented in Section 14.3, and we shall not present the details. It suffices to mention that it is based on the identities
and
A direct consequence of the reciprocity relation is
the statement that the bitensor V_{αβ′} (x,x′) is symmetric in its indices and arguments.
The Kirchhoff representation for the electromagnetic vector potential is formulated as follows. Suppose that A^{α} (x) satisfies the homogeneous version of Eq. (15.1) and that initial values A^{α′} (x′), n^{β′} ∇_{β′} A^{α′} (x′) 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
where d Σ_{γ′} = −n_{γ′}dV is a surface element on Σ; n_{γ′} is the futuredirected unit normal and dV is the invariant volume element on the hypersurface. The derivation of Eq. (15.17) is virtually identical to what was presented in Section 14.4.
18.4 Relation with scalar Green’s functions
In a spacetime that satisfies the Einstein field equations in vacuum, so that R_{ αβ } = 0 everywhere in the spacetime, the (retarded and advanced) electromagnetic Green’s functions satisfy the identities [54]
where G_{±} are the corresponding scalar Green’s functions.
To prove this we differentiate Eq. (15.3) covariantly with respect to x^{α} and use Eq. (13.3) to express the righthand side as + 4π∂_{β′}δ_{4}(x,x′). After repeated use of Ricci’s identity to permute the ordering of the covariant derivatives on the lefthand side, we arrive at the equation
all terms involving the Riemann tensor disappear by virtue of the fact that the spacetime is Ricciflat. Because Eq. (15.19) is also the differential equation satisfied by G_{;β′}, and because the solutions are chosen to satisfy the same boundary conditions, we have established the validity of Eq. (15.18).
18.5 Singular and regular Green’s functions
We shall now construct singular and regular Green’s functions for the electromagnetic field. The treatment here parallels closely what was presented in Section 14.5, and the reader is referred to that section for a more complete discussion.
We begin by introducing the bitensor \(H_{\;\beta \prime}^\alpha (x,x\prime)\) with properties

H1: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) satisfies the homogeneous wave equation,
$$\square H_{\;\,\beta \prime}^\alpha (x,x\prime)  R_{\;\,\beta}^\alpha (x)H_{\;\,\beta \prime}^\beta (x,x\prime) = 0;$$(15.20) 
H2: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) is symmetric in its indices and arguments,
$${H_{\beta \prime \alpha}}(x\prime ,x) = {H_{\alpha \beta \prime}}(x,x\prime);$$(15.21) 
H3: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,
$$H_{\;\,\beta \prime}^\alpha (x,x\prime) = G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(15.22) 
H4: \(H^\alpha_{\ \beta\prime}(x,x\prime)\) agrees with the advanced Green’s function if x is in the chronological past of x′,
$$H_{\;\,\beta \prime}^\alpha (x,x\prime) = G_{ \beta \prime}^{\;\alpha}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ }(x\prime).$$(15.23)
It is easy to prove that property H4 follows from H2, H3, and the reciprocity relation (15.15) satisfied by the retarded and advanced Green’s functions. That such a bitensor exists can be argued along the same lines as those presented in Section 14.5.
Equipped with the bitensor \(H^\alpha_{\ \beta\prime}(x,x\prime)\) we define the singular Green’s function to be
This comes with the properties

S1: \(G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime)\) satisfies the inhomogeneous wave equation,
$$\square G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime)  R_{\;\,\beta}^\alpha (x)G_{{\rm{S}}\,\beta \prime}^{\;\beta}(x,x\prime) =  4\pi g_{\;\,\beta \prime}^\alpha (x,x\prime){\delta _4}(x,x\prime);$$(15.25) 
S2: \(G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime)\) is symmetric in its indices and arguments,
$$G_{\beta \prime \alpha}^{\rm{S}}(x\prime ,x) = G_{\alpha \beta \prime}^{\rm{S}}(x,x\prime);$$(15.26) 
S3: \(G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime)\) vanishes if x is in the chronological past or future of x′,
$$G_{{\rm{S}}\,\beta \prime}^{\;\alpha}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ \pm}(x\prime).$$(15.27)
These can be established as consequences of H1–H4 and the properties of the retarded and advanced Green’s functions.
The regular twopoint function is then defined by
and it comes with the properties

R1: \(G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime)\) satisfies the homogeneous wave equation,
$$\square G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime)  R_{\;\,\beta}^\alpha (x)G_{{\rm{R}}\,\beta \prime}^{\;\,\beta}(x,x\prime) = 0;$$(15.29) 
R2: \(G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,
$$G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime) = G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(15.30) 
R3: \(G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime)\) vanishes if x is in the chronological past of x′,
$$G_{{\rm{R}}\,\beta \prime}^{\;\,\alpha}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ }(x\prime).$$(15.31)
Those follow immediately from S1—S3 and the properties of the retarded Green’s function.
When x is restricted to the normal convex neighbourhood of x′, we have the explicit relations
From these we see clearly that the singular Green’s function does not distinguish between past and future (property S2), and that its support excludes I^{±}(x′) (property S3). We see also that the regular twopoint function coincides with \(G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\) in I^{+} (x′) (property R2), and that its support does not include I^{−}(x′) (property R3).
19 Gravitational Green’s functions
19.1 Equations of linearized gravity
We are given a background spacetime for which the metric g_{ αβ } satisfies the Einstein field equations in vacuum. We then perturb the metric from g_{ αβ } to
The metric perturbation 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 tracereversed potentials γ_{ αβ } defined by
and we impose the Lorenz gauge condition,
In this equation, and in all others below, indices are raised and lowered with the background metric g_{ αβ }. Similarly, the connection involved in Eq. (16.3), and in all other equations below, is the one that is compatible with the background metric. If is the perturbing energymomentum tensor, then by virtue of the linearized Einstein field equations the perturbation field obeys the wave equation
in which □ = g^{αβ} ∇_{ α } ∇_{ β } is the wave operator and R_{ γαδβ } the Riemann tensor. In firstorder perturbation theory, the energymomentum tensor must be conserved in the background spacetime: \(T_{\;\;\;;\beta}^{\alpha \beta} = 0\).
The solution to the wave equation is written as
in terms of a Green’s function \(G_{\;\;\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) that satisfies [161]
where \(g_{\;\gamma \prime}^\alpha (x,x\prime)\) is a parallel propagator and δ_{4}(x,x′) an invariant Dirac functional. The parallel propagators are inserted on the righthand side of Eq. (16.6) 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 Eq. (16.6), the perturbation field of Eq. (16.5) satisfies the wave equation of Eq. (16.4). 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 \(G_{+ \;\,\gamma \prime\delta \prime}^{\;\alpha \beta}(x,x\prime)\), which is nonzero if x is in the causal future of x′, and the advanced Green’s function \(G_{ \;\,\gamma \prime\delta \prime}^{\;\alpha \beta}(x,x\prime)\), which is nonzero if x is in the causal past of x′, exist as distributions and can be defined globally in the entire background spacetime.
19.2 Hadamard construction of the Green’s functions
Assuming throughout this subsection that is in the normal convex neighbourhood of x′, we make the ansatz
where θ_{±}(−σ), δ_{±}(σ) are the lightcone distributions introduced in Section 13.2, and where \(U_{\;\,\;\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime),\ \;V_{\,\,\;\;\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime)\) are smooth bitensors.
To conveniently manipulate the Green’s functions we shift σ by a small positive quantity ϵ. The Green’s functions are then recovered by the taking the limit of
as ϵ → 0^{+}. When we substitute this into the lefthand side of Eq. (16.6) and then take the limit, we obtain
after a routine computation similar to the one presented at the beginning of Section 14.2. Comparison with Eq. (16.6) returns: (i) the equations
and
that determine \(U_{\;\;\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) (ii) the equation
that determine \(\check{V}^{\alpha\beta}_{\ \ \gamma\prime\delta\prime}(x,x\prime)\), the restriction of \(V_{\;\;\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime)\) on the light cone σ (x, x′) = 0; and (iii) the wave equation
that determines \(V_{\;\beta \prime}^\alpha (x,x\prime)\) inside the light cone.
Eq. (16.9) can be integrated along the unique geodesic β that links x′ to x. The initial conditions are provided by Eq. (16.8), and if we set \(U_{\;\beta \prime}^\alpha (x,x\prime) = g_{\;\beta \prime}^\alpha (x,x\prime)U(x,x\prime)\), we find that these equations reduce to Eqs. (14.7) and (14.6), respectively. According to Eq. (14.8), then, we have
which reduces to
near coincidence, with λ denoting the affineparameter 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
and eventually,
this last result follows from the fact that \([U_{\;\;\gamma \prime\delta \prime;\epsilon \iota}^{\alpha \beta}]\) is antisymmetric in the last pair of indices.
Similarly, Eq. (16.10) can be integrated along each null geodesic that generates the null cone σ (x, x′) = 0. The initial values are obtained by taking the coincidence limit of this equation, using Eqs. (16.8), (16.16), and the additional relation \([\sigma _{\;\gamma}^\gamma ] = 4\). We arrive at
With the characteristic data obtained by integrating Eq. (16.10), the wave equation of Eq. (16.11) admits a unique solution.
To summarize, the retarded and advanced gravitational Green’s functions are given by Eq. (16.7) with \(U_{\;\;\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) given by Eq. (16.12) and \(V_{\;\;\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) determined by Eq. (16.11) and the characteristic data constructed with Eqs. (16.10) and (16.17). It should be emphasized that the construction provided in this subsection is restricted to \({\mathcal N}(x\prime)\), the normal convex neighbourhood of the reference point x′.
19.3 Reciprocity and Kirchhoff representation
The (globally defined) gravitational Green’s functions satisfy the reciprocity relation
The derivation of this result is virtually identical to what was presented in Sections 14.3 and 15.3. A direct consequence of the reciprocity relation is the statement
The Kirchhoff representation for the tracereversed gravitational perturbation γ_{ αβ } is formulated as follows. Suppose that γ^{αβ} (x) satisfies the homogeneous version of Eq. (16.4) 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
where dΣ_{ϵ′}= − n_{ϵ′}dV is a surface element on Σ; n_{ϵ′} is the futuredirected unit normal and dV is the invariant volume element on the hypersurface. The derivation of Eq. (16.20) is virtually identical to what was presented in Sections 14.4 and 15.3.
19.4 Relation with electromagnetic and scalar Green’s functions
In a spacetime that satisfies the Einstein field equations in vacuum, so that R_{ αβ } = 0 everywhere in the spacetime, the (retarded and advanced) gravitational Green’s functions satisfy the identities [144]
and
where \(d{\Sigma _{\epsilon\prime}} =  {n_{\epsilon\prime}}dV\) are the corresponding electromagnetic Green’s functions, and G_{±} the corresponding scalar Green’s functions.
To prove Eq. (16.21) we differentiate Eq. (16.6) covariantly with respect to x^{β}, use Eq. (13.3) to work on the righthand side, and invoke Ricci’s identity to permute the ordering of the covariant derivatives on the lefthand side. After simplification and involvement of the Ricciflat condition (which, together with the Bianchi identities, implies that \(R_{\alpha \gamma \beta \delta}^{\;\;\;\;\;\,\,\,\,;\beta} = 0\), we arrive at the equation
Because this is also the differential equation satisfied by \(G_{\;\,(\beta \prime;\gamma \prime)}^\alpha\), and because the solutions are chosen to satisfy the same boundary conditions, we have established the validity of Eq. (16.21).
The identity of Eq. (16.22) follows simply from the fact that \({g^{\gamma \prime\delta \prime}}G_{\,\,\,\,\gamma \prime\delta \prime}^{\alpha \beta}\) and g^{αβ}G satisfy the same tensorial wave equation in a Ricciflat spacetime.
19.5 Singular and regular Green’s functions
We shall now construct singular and regular Green’s functions for the linearized gravitational field. The treatment here parallels closely what was presented in Sections 14.5 and 15.5.
We begin by introducing the bitensor \(H_{\;\;\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) with properties

H1: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) satisfies the homogeneous wave equation,
$$\square H_{\;\;\,\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime) + 2R_{\gamma \;\,\delta}^{\;\alpha \;\,\beta}(x)H_{\;\;\gamma \prime \delta \prime}^{\gamma \delta}(x,x\prime) = 0;$$(16.24) 
H2: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) is symmetric in its indices and arguments,
$${H_{\gamma \prime \delta \prime \alpha \beta}}(x\prime ,x) = {H_{\alpha \beta \gamma \prime \delta \prime}}(x,x\prime);$$(16.25) 
H3: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,
$$H_{\;\;\,\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime) = G_{+ \;\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(16.26) 
H4: \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) agrees with the advanced Green’s function if x is in the chronological past of x′,
$$H_{\;\;\,\gamma \prime \delta \prime}^{\alpha \beta}(x,x\prime) = G_{ \;\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ }(x\prime)$$(16.27)
It is easy to prove that property H4 follows from H2, H3, and the reciprocity relation (16.18) satisfied by the retarded and advanced Green’s functions. That such a bitensor exists can be argued along the same lines as those presented in Section 14.5.
Equipped with \(H_{\;\;\,\gamma \prime\delta \prime}^{\alpha \beta}(x,x\prime)\) we define the singular Green’s function to be
This comes with the properties

S1: \(G_{{\rm{S}}\,\,\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\) satisfies the inhomogeneous wave equation,
$$\square G_{{\rm{S}}\;\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime) + 2R_{\gamma \;\,\delta}^{\;\alpha \;\beta}(x)G_{{\rm{S}}\;\,\gamma \prime \delta \prime}^{\;\gamma \delta}(x,x\prime) =  4\pi g_{\;\gamma \prime}^{(\alpha}(x,x\prime)g_{\;\delta \prime}^{\beta)}(x,x\prime){\delta _4}(x,x\prime);$$(16.29) 
S2: \(G_{{\rm{S}}\,\,\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\) is symmetric in its indices and arguments,
$$G_{\gamma \prime \delta \prime \alpha \beta}^{\rm{S}}(x\prime ,x) = G_{\alpha \beta \gamma \prime \delta \prime}^{\rm{S}}(x,x\prime);$$(16.30) 
S3: \(G_{{\rm{S}}\,\,\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\) vanishes if x is in the chronological past or future of x′,
$$G_{{\rm{S}}\;\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ \pm}(x\prime)$$(16.31)
These can be established as consequences of H1–H4 and the properties of the retarded and advanced Green’s functions.
The regular twopoint function is then defined by
and it comes with the properties

R1: \(G_{{\rm{R}}\;\,\gamma \prime\delta \prime}^{\;\,\alpha \beta}(x,x\prime)\) satisfies the homogeneous wave equation,
$$\square G_{{\rm{R}}\;\,\,\gamma \prime \delta \prime}^{\;\,\alpha \beta}(x,x\prime) + 2R_{\gamma \;\,\delta}^{\;\alpha \;\,\beta}(x)G_{{\rm{R}}\;\,\gamma \prime \delta \prime}^{\;\,\gamma \delta}(x,x\prime) = 0;$$(16.33) 
R2: \(G_{{\rm{R}}\;\,\gamma \prime\delta \prime}^{\;\,\alpha \beta}(x,x\prime)\) agrees with the retarded Green’s function if x is in the chronological future of x′,
$$G_{{\rm{R}}\;\,\gamma \prime \delta \prime}^{\;\,\alpha \beta}(x,x\prime) = G_{+ \;\,\,\gamma \prime \delta \prime}^{\;\alpha \beta}(x,x\prime)\qquad {\rm{when}}\;x \in {I^ +}(x\prime);$$(16.34) 
R3: \(G_{{\rm{R}}\;\,\gamma \prime\delta \prime}^{\;\,\alpha \beta}(x,x\prime)\) vanishes if x is in the chronological past of x′,
$$G_{{\rm{R}}\;\,\,\gamma \prime \delta \prime}^{\;\,\alpha \beta}(x,x\prime) = 0\qquad {\rm{when}}\;x \in {I^ }(x\prime)$$(16.35)
Those follow immediately from S1–S3 and the properties of the retarded Green’s function.
When x is restricted to the normal convex neighbourhood of x′, we have the explicit relations
From these we see clearly that the singular Green’s function does not distinguish between past and future (property S2), and that its support excludes I^{±}(x′) (property S3). We see also that the regular twopoint function coincides with \(G_{\, + \;\,\,\gamma \prime\delta \prime}^{\;\alpha \beta}(x,x\prime)\) in I^{+}(x′) (property R2), and that its support does not include I^{−}(x′) (property R3).
20 Part IV: Motion of Point Particles
21 Motion of a scalar charge
21.1 Dynamics of a point scalar charge
A point particle carries a 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
where S_{field} is an action functional for a free scalar field in a spacetime with metric g_{ αβ }, 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.
The field action is given by
where the integration is over all of spacetime; the field is coupled to the Ricci scalar R by an arbitrary constant ξ. The particle action is
where 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
Notice that both S_{particle} and S_{interaction} are invariant under a reparameterization λ → λ′(λ) of the world line.
Demanding that the total action be stationary under a variation δΦ(x) of the field configuration yields the wave equation
for the scalar potential, with a charge density μ(x) defined by
These equations determine the field Φ_{ 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. We have here adopted τ as the parameter on the world line, and introduced the fourvelocity u^{μ}(τ) := dz^{μ}/dτ. The dynamical mass that appears in Eq. (17.7) is defined by m(τ) := m_{0} − qΦ(z), which can also be expressed in differential form as
It should be clear that Eqs. (17.7) and (17.8) are valid only in a formal sense, because the scalar potential obtained from Eqs. (17.5) and (17.6) diverges on the world line. Before we can make sense of these equations we have to analyze the field’s singularity structure near the world line.
21.2 Retarded potential near the world line
The retarded solution to Eq. (17.5) is \(\Phi (x) = \int {{G_ +}} (x,x\prime)\mu (x\prime)\sqrt {g\prime} \,{d^4}x\prime\), where G_{+}(x, x′) is the retarded Green’s function introduced in Section 14. After substitution of Eq. (17.6) we obtain
in which z(τ) gives the description of the world line γ. Because the retarded Green’s function is defined globally in the entire spacetime, Eq. (17.9) applies to any field point x.
We now specialize Eq. (17.9) to a point 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 propertime 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 Eq. (17.9) can be broken up into the three integrals
The third integration vanishes because 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 14.2. This gives
and to evaluate this we refer back to Section 10 and let x′ := z(u) be the retarded point associated with x; these points are related by σ(x, x′) = 0 and \(r := {\sigma _{\alpha \prime}}{u^{\alpha \prime}}\) is the retarded distance between x and the world line. We resume the index convention of Section 10: 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.
To perform the first integration we change variables from τ to σ, noticing that σ increases as z(τ) passes through x′. The change of σ on the world line is given by dσ:= σ(x, z + dz) − σ(x, z) = σ_{ μ }u^{μ} dτ, and we find that the first integral evaluates to U(x, z)/(σ_{ μ }u^{μ}) 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:
This expression applies to a point x sufficiently close to the world line that there exists a nonempty intersection between \({\mathcal N}(x)\) and γ.
21.3 Field of a scalar charge in retarded coordinates
When we differentiate the potential of Eq. (17.10) we must keep in mind that a variation in x induces a variation in x′ because the new points x + δx and x′ + δx′ must also be linked by a null geodesic — you may refer back to Section 10.2 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
where the “tail integral” is defined by
In the second form of the definition we integrate ∇_{ 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 up into contributions that refer to \({\mathcal N}(x)\) and its complement, respectively.
We shall now expand Φ_{ α }(x) in powers of r, and express the results in terms of the retarded coordinates (u, r, Ω^{a}) introduced in Section 10. 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 10. The expansion relies on Eq. (10.29) for ∂_{ α }u, Eq. (10.31) for ∂_{ α }r, and we shall need
which follows from Eq. (14.10) and the relation \({\sigma ^{\alpha \prime}} = \,  r({u^{\alpha \prime}} + {\Omega ^a}e_a^{\alpha \prime})\) first encountered in Eq. (10.7); recall that
are frame components of the Ricci tensor evaluated at x′. We shall also need the expansions
and
which follow from Eqs. (14.11); recall from Eq. (10.4) that the parallel propagator can be expressed as \(g_{\;\alpha}^{\alpha \prime} = {u^{\alpha \prime}}e_\alpha ^0 + e_a^{\alpha \prime}e_\alpha ^a\). And finally, we shall need
a relation that was first established in Eq. (14.13); here R:= R(u) is the Ricci scalar evaluated at x′.
Collecting all these results gives
where \({a_a} = {a_{\alpha \prime}}e_a^{\alpha \prime}\) are the frame components of the acceleration vector,
are frame components of the Riemann tensor evaluated at x′, and
are the frame components of the tail integral evaluated at x′. Equations (17.17) and (17.18) 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.
21.4 Field of a scalar charge in Fermi normal coordinates
The gradient of the scalar potential can also be expressed in the Fermi normal coordinates of Section 9. To effect this translation we make x:= z(t) the new reference point on the world line. We resume here the notation of Section 11 and assign indices \(\bar \alpha ,\,\,\bar \beta ,\,\, \ldots\) to tensors at \(\bar{x}\). The Fermi normal coordinates are denoted (t, s, ω^{a}), and we let \((\bar{e}^\alpha_0, \bar{e}^\alpha_a)\) be the tetrad at x that is obtained by parallel transport of \(({u^{\bar \alpha}},e_a^{\bar \alpha})\) on the spacelike geodesic that links x to \(\bar{x}\).
Our first task is to decompose Φ_{ α }(x) in the tetrad \((\bar e_0^\alpha ,\bar e_a^\alpha)\), thereby defining \({\bar \Phi _0} := {\Phi _\alpha}\bar e_0^\alpha\) and \({\bar \Phi _a} := {\Phi _\alpha}\bar e_a^\alpha\). For this purpose we use Eqs. (11.7), (11.8), (17.17), and (17.18) to obtain
and
where all frame components are still evaluated at x′, except for \(\bar \Phi _0^{{\rm{tail}}}\) and \(\bar \Phi _{a}^{{\rm{tail}}}\) which are evaluated at \(\bar{x}\).
We must still translate these results into the Fermi normal coordinates (t, s, ω^{a}). For this we involve Eqs. (11.4), (11.5), and (11.6), from which we deduce, for example,
and
in which all frame components (on the righthand side of these relations) are now evaluated at \(\bar{x}\); to obtain the second relation we expressed a_{ a }(u) as a_{ a }(t) − sȧ_{ a }(t) + O(s^{2}) since according to Eq. (11.4), u = t − s + O(s^{2}).
Collecting these results yields
In these expressions, \({a_a}(t) = {a_{\bar \alpha}}e_a^{\bar \alpha}\) are the frame components of the acceleration vector evaluated at \(\bar x,\,\,{\dot a_0}(t) = {\dot a_{\bar \alpha}}{u^{\bar \alpha}}\) and \({\dot a_a}(t) = {\dot a_{\bar \alpha}}e_a^{\bar \alpha}\) are frame components of its covariant derivative, \({R_{a0b0}}(t) = {R_{\bar \alpha \bar \gamma \bar \beta \bar \delta}}e_a^{\bar \alpha}{u^{\bar \gamma}}e_b^{\bar \beta}{u^{\bar \delta}}\) are frame components of the Riemann tensor evaluated at \(\bar{x}\),
are frame components of the Ricci tensor, and R(t) is the Ricci scalar evaluated at \(\bar{x}\). Finally, we have that
are the frame components of the tail integral — see Eq. (17.12) — evaluated at \(\bar x := z(t)\).
We shall now compute the averages of \({\bar \Phi _0}\) and \({\bar \Phi _{a}}\) over S(t, s), a twosurface 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 twosurface is charted by angles θ^{A} (A = 1, 2) and it is described, in the Fermi normal coordinates, by the parametric relations \({\hat x^a} = s{\omega ^a}({\theta ^A})\); a canonical choice of parameterization is ω^{a} = (sin θ cos ϕ, sin θ sin ϕ, cos θ). Introducing the transformation matrices \(\omega _A^a := \partial {\omega ^a}/\partial {\theta ^A}\), we find from Eq. (9.16) that the induced metric on S(t, s) is given by
where \({\omega _{AB}}: = {\delta _{ab}}\omega _A^a\omega _B^b\) is the metric of the unit twosphere, and where \({R_{AB}}: = {R_{acbd}}\omega _A^a{\omega ^c}\omega _B^b{\omega ^d}\) depends on t and the angles θ^{A}. From this we infer that the element of surface area is given by
where \(d\omega = \sqrt {{\rm{det}}[{\omega _{AB}}]} \,{d^2}\theta\) is an element of solid angle — in the canonical parameterization, dω = sin θ dθdϕ. Integration of Eq. (17.24) produces the total surface area of S(t, s), and \({\mathcal A} = 4\pi {s^2}[1  {1 \over {18}}{s^2}R_{\;\;ab}^{ab} + O({s^3})]\).
The averaged fields are defined by
where the quantities to be integrated are scalar functions of the Fermi normal coordinates. The results
are easy to establish, and we obtain
The averaged field is still singular on the world line. Regardless, we shall take the formal limit s → 0 of the expressions displayed in Eqs. (17.27) and (17.28). In the limit the tetrad \((\bar e_0^\alpha ,\bar e_a^\alpha)\) reduces to \(({u^{\bar \alpha}},e_a^{\bar \alpha})\)), and we can reconstruct the field at \(\bar{x}\) by invoking the completeness relations \(\delta _{\;\bar \beta}^{\bar \alpha} =  {u^{\bar \alpha}}{u_{\bar \beta}} + e_a^{\bar \alpha}e_{\bar \beta}^a\). We thus obtain
where the tail integral can be copied from Eq. (17.12),
The tensors appearing in Eq. (17.29) all refer to \(\bar x := z(t)\), which now stands for an arbitrary point on the world line γ.
21.5 Singular and regular fields
The singular potential
is the (unphysical) solution to Eqs. (17.5) and (17.6) that is obtained by adopting the singular Green’s function of Eq. (14.30) instead of the retarded Green’s function. As we shall see, the resulting singular field \(\Phi _\alpha ^{\rm{S}}(x)\) reproduces the singular behaviour of the retarded solution; the difference, \(\Phi _\alpha ^{\rm{R}}(x) = {\Phi _\alpha}(x)  \Phi _\alpha ^{\rm{S}}(x)\), is smooth on the world line.
To evaluate the integral of Eq. (17.31) we assume once more that 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 propertime parameter at which γ enters and leaves \({\mathcal N}(x)\), respectively. Then Eq. (17.31) can be broken up into the three integrals
The first integration vanishes because x is then in the chronological future of z(τ), and G_{s}(x, z) = 0 by Eq. (14.21). 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 Eq. (14.32), and we have
To evaluate these we reintroduce the retarded point x′ := z(u) and let x″:= z(v) be the advanced point associated with x; we recall from Section 11.4 that these points are related by σ(x, x″) = 0 and that \({r_{{\rm{adv}}}}: =  {\sigma _{\alpha \prime\prime}}{u^{\alpha \prime\prime}}\) is the advanced 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(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″)/r_{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:
We observe that Φ^{S} (x) depends on the state of motion of the scalar charge between the retarded time u and the advanced time v; contrary to what was found in Section 17.2 for the retarded potential, there is no dependence on the particle’s remote past.
We use the techniques of Section 17.3 to differentiate the potential of Eq. (17.32). We find
and we would like to express this as an expansion in powers of r. For this we shall rely on results already established in Section 17.3, 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 Eq. (11.12), that an expression for the advanced distance was displayed in Eq. (11.13), and that Eqs. (11.14) and (11.15) give expansions for ∂_{ α }v and ∂_{ α }r_{adv}, respectively.
To derive an expansion for U(x, x″) we follow the general method of Section 11.4 and define a function U(τ) := U(x, z(τ)) of the propertime parameter on γ. We have that
where overdots indicate differentiation with respect to τ, and where Δ′ := v − u. The leading term U(u) := U(x, x′) was worked out in Eq. (17.13), and the derivatives of U(τ) are given by
and
according to Eqs. (17.15) and (14.11). Combining these results together with Eq. (11.12) for Δ′ gives
which should be compared with Eq. (17.13). It should be emphasized that in Eq. (17.34) and all equations below, the frame components of the Ricci tensor are evaluated at the retarded point x′ := z(u), and not at the advanced point. The preceding computation gives us also an expansion for \(U_{;\alpha\prime\prime} u^{\alpha\prime\prime} := \dot{U}(v) = \dot{U}(u) + \ddot{U}(u) \Delta^{\!\prime} + O(\Delta^{\!\prime 2})\). This becomes
which should be compared with Eq. (17.15).
We proceed similarly to derive an expansion for U_{;α}(x, x″). Here we introduce the functions U_{ α }(τ) := U_{;α}(x, z(τ)) and express U_{;α}(x, x″) as \(U_\alpha(v) = U_\alpha(u) + \dot{U}_\alpha(u) \Delta{\prime} + O(\Delta\prime^{2})\). The leading term U_{ α }(u) := U_{;α}(x, x′) was computed in Eq. (17.14), and
follows from Eq. (14.11). Combining these results together with Eq. (11.12) for Δ′ gives
and this should be compared with Eq. (17.14).
The last expansion we shall need is
which follows at once from Eq. (17.16) and the fact that V(x, x″) − V(x, x′) = O(r); the Ricci scalar is evaluated at the retarded point x′.
It is now a straightforward (but tedious) matter to substitute these expansions (all of them!) into Eq. (17.33) and obtain the projections of the singular field \(\Phi^{\mathrm{S}}_\alpha(x)\) in the same tetrad \((e_0^\alpha, e_a^\alpha)\) that was employed in Section 17.3. This gives
in which all frame components are evaluated at the retarded point x′ := z(u). Comparison of these expressions with Eqs. (17.17) and (17.18) reveals that the retarded and singular fields share the same singularity structure.
The difference between the retarded field of Eqs. (17.17), (17.18) and the singular field of Eqs. (17.38), (17.39) defines the regular field \(\Phi _\alpha ^{\rm{R}}(x)\). Its frame components are
and we see that \(\Phi _\alpha ^{\rm{R}}(x)\) is a regular vector field on the world line. There is therefore no obstacle in evaluating the regular field directly at 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
where the tail term can be copied from Eq. (17.12),
The tensors appearing in Eq. (17.42) all refer to the retarded point x′ := z(u), which now stands for an arbitrary point on the world line γ.
21.6 Equations of motion
The retarded field Φ_{ α }(x) of a point scalar charge is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and affect its motion. The field’s singularity structure was analyzed in Sections 17.3 and 17.4, and in Section 17.5 it was shown to originate from the singular field \(\Phi _\alpha ^{\rm{S}}(x)\); the regular field \(\Phi _\alpha ^{\rm{R}}(x) = {\Phi _\alpha}(x)  \Phi _\alpha ^{\rm{S}}(x)\) was then shown to be regular on the world line.
To make sense of the retarded field’s action on the particle we temporarily model the scalar charge not as a point particle, but as a small hollow shell that appears spherical when observed in a reference frame that is momentarily comoving with the particle; the shell’s radius is 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 qΦ_{ α }(τ, s_{0}, ω^{a}) over the surface of the shell. Assuming that the field on the shell is equal to the field of a point particle evaluated at s = s_{0}, and ignoring terms that disappear in the limit s_{0} → 0, we obtain from Eq. (17.29)
where
is formally a divergent quantity and
is the tail part of the force; all tensors in Eq. (17.44) are evaluated at an arbitrary point z(τ) on the world line.
Substituting Eqs. (17.44) and (17.46) into Eq. (17.7) gives rise to the equations of motion
for the scalar charge, with m:= m_{0} − qΦ(z) denoting the (also formally divergent) dynamical mass of the particle. We see that m and δm combine in Eq. (17.47) 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 Eqs. (17.44) and (17.46) into Eq. (17.8), in which we replace m by m_{obs} = m + δm, returns an expression for the rate of change of the observed mass,
That the observed mass is 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 timedependent metric, a scalar charge radiates monopole waves and the radiated energy comes at the expense of the particle’s inertial mass.
We must confess that the derivation of the equations of motion outlined above returns the wrong expression for the selfenergy of a spherical shell of scalar charge. We obtained δm = q^{2}/(3s_{0}), while the correct expression is δm = q^{2}/(2s_{0}); we are wrong by a factor of 2/3. We believe that this discrepancy originates in a previously stated assumption, that the field on the shell (as produced by the shell itself) is equal to the field of a point particle evaluated at s = s_{0}. We believe that this assumption is in fact wrong, and that a calculation of the field actually produced by a spherical shell would return the correct expression for δm. We also believe, however, that except for the diverging terms that determine δm, the difference between the shell’s field and the particle’s field should vanish in the limit s_{0} → 0. Our conclusion is therefore that while our expression for δm is admittedly incorrect, the statement of the equations of motion is reliable.
Apart from the term proportional to δm, the averaged field of Eq. (17.44) has exactly the same form as the regular field of Eq. (17.42), which we reexpress as
The force acting on the point particle can therefore be thought of as originating from the regular field, while the singular field simply contributes to the particle’s inertia. After mass renormalization, Eqs. (17.47) and (17.48) are equivalent to the statements
where we have dropped the superfluous label “obs” on the particle’s observed mass. Another argument in support of the claim that the motion of the particle should be affected by the regular field only was presented in Section 14.5.
The equations of motion displayed in Eqs. (17.47) and (17.48) are thirdorder differential equations for the functions z^{μ}(τ). It is well known that such a system of equations admits many unphysical solutions, such as runaway situations in which the particle’s acceleration increases exponentially with τ, even in the absence of any external force [56, 101]. And indeed, our equations of motion do not yet incorporate an external force which presumably is mostly responsible for the particle’s acceleration. Both defects can be cured in one stroke. We shall take the point of view, the only admissible one in a classical treatment, that a point particle is merely an idealization for an extended object whose internal structure — the details of its charge distribution — can be considered to be irrelevant. This view automatically implies that our equations are meant to provide only an approximate description of the object’s motion. It can then be shown [112, 70] that within the context of this approximation, it is consistent to replace, on the righthand side of the equations of motion, any occurrence of the acceleration vector by \(f_{{\rm{ext}}}^\mu/m\), where \(f_{{\rm{ext}}}^\mu\) is the external force acting on the particle. Because \(f_{{\rm{ext}}}^\mu\) is a prescribed quantity, differentiation of the external force does not produce higher derivatives of the functions z^{μ}(τ), and the equations of motion are properly of the second order.
We shall strengthen this conclusion in Part V of the review, when we consider the motion of an extended body in a curved external spacetime. While the discussion there will concern the gravitational selfforce, many of the lessons learned in Part V apply just as well to the case of a scalar (or electric) charge. And the main lesson is this: It is natural — indeed it is an imperative — to view an equation of motion such as Eq. (17.47) as an expansion of the acceleration in powers of q^{2}, and it is therefore appropriate — indeed imperative — to insert the zerothorder expression for ȧ^{ν} within the term of order q^{2}. The resulting expression for the acceleration is then valid up to correction terms of order q^{4}. Omitting these error terms, we shall write, in final analysis, the equations of motion in the form
and
where m denotes the observed inertial mass of the scalar charge, and where all tensors are evaluated at 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 17.3. Equations (17.51) and (17.52) were first derived by Theodore C. Quinn in 2000 [149]. In his paper Quinn also establishes that the total work done by the scalar selfforce matches the amount of energy radiated away by the particle.
22 Motion of an electric charge
22.1 Dynamics of a point electric charge
A point particle carries an 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
where S_{field} is an action functional for a free electromagnetic field in a spacetime with metric g_{ αβ }, 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.
The field action is given by
where the integration is over all of spacetime. The particle action is
where 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
Notice that both S_{particle} and S_{interaction} are invariant under a reparameterization λ → λ′(λ) of the world line.
Demanding that the total action be stationary under a variation δA^{α}(x) of the vector potential yields Maxwell’s equations
with a current density j^{α}(x) defined by
These equations determine the electromagnetic field 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. We have adopted τ as the parameter on the world line, and introduced the fourvelocity u^{μ}(τ) := dz^{μ}/dτ.
The electromagnetic field 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,
Under this condition the Maxwell equations of Eq. (18.5) reduce to a wave equation for the vector potential,
where D = g_{ αβ }∇_{ α }∇_{ β } is the wave operator and \(R_{\;\beta}^\alpha\) is the Ricci tensor. Having adopted τ as the parameter on the world line, we can reexpress the current density of Eq. (18.6) as
and we shall use Eqs. (18.9) and (18.10) to determine the electromagnetic field of a point electric charge. The motion of the particle is in principle determined by Eq. (18.7), but because the vector potential obtained from Eq. (18.9) is singular on the world line, these equations have only formal validity. Before we can make sense of them we will have to analyze the field’s singularity structure near the world line. The calculations to be carried out parallel closely those presented in Section 17 for the case of a scalar charge; the details will therefore be kept to a minimum and the reader is referred to Section 17 for additional information.
22.2 Retarded potential near the world line
The retarded solution to Eq. (18.9) is \({A^\alpha}(x) = \int {G_{+ \beta \prime}^{\;\alpha}} (x,x\prime){j^{\beta \prime}}(x\prime)\sqrt {g\prime} \,{d^4}x\prime\), where \(G_{+ \beta \prime}^{\;\alpha}(x,x\prime)\) is the retarded Green’s function introduced in Section 15. After substitution of Eq. (18.10) we obtain
in which 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, Eq. (18.11) applies to any field point x.
We now specialize Eq. (18.11) to a point 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 17.2 we let τ_{<} and τ_{>} be the values of the propertime parameter at which γ enters and leaves \({\mathcal N}(x)\), respectively. Then Eq. (18.11) can be expressed as
The third integration vanishes because 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 15.2. This gives
and to evaluate this we let x′ := z(u) be the retarded point associated with x; these points are related by σ(x, x′) = 0 and \(r := {\sigma _{\alpha \prime}}{u^{\alpha \prime}}\) 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 \(U_{\;\beta \prime}^\alpha {u^{\beta \prime}}/r\). 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:
This expression applies to a point x sufficiently close to the world line that there exists a nonempty intersection between \({\mathcal N}(x)\) and γ.
22.3 Electromagnetic field in retarded coordinates
When we differentiate the vector potential of Eq. (18.12) we must keep in mind that a variation in x induces a variation in 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
where the “tail integral” is defined by
The second form of the definition, in which we integrate the gradient of the retarded Green’s function from 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.
We shall now expand F_{ αβ } = ∇_{ α }A_{ β } − ∇_{}