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 , a point electric charge , and a point mass in a specified background spacetime with metric . These particles carry with them fields that behave as outgoing radiation in the wave zone. The radiation removes energy and angular momentum from the particle, which then undergoes a radiation reaction – its world line cannot be simply a geodesic of the background spacetime. The particle’s motion is affected by the near-zone field which acts directly on the particle and produces a self-force. In curved spacetime the self-force contains a radiation-reaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angular-momentum transport. The self-force is proportional to in the case of a scalar charge, proportional to in the case of an electric charge, and proportional to in the case of a point mass.

In this review we derive the equations that govern the motion of a point particle in a curved background spacetime. The presentation is entirely self-contained, and all relevant materials are developed ab initio. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. One must first assimilate the basic theory of bitensors (Part I), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line (Part II). One must next formulate a theory of Green’s functions in curved spacetimes (Part III), and finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how they should act on the particle (Part IV). A dedicated reader, correctly skeptical that sense can be made of a point mass in general relativity, will also want to work through the last portion of the review (Part V), which provides a derivation of the equations of motion for a small, but physically extended, body; this reader will be reassured to find that the extended body follows the same motion as the point mass. The review is very long, but the satisfaction derived, we hope, will be commensurate.

In this introductory section we set the stage and present an impressionistic survey of what the review contains. This should help the reader get oriented and acquainted with some of the ideas and some of the notation. Enjoy!

Let us first consider the relatively simple and well-understood case of a point electric charge moving in flat spacetime [154, 101, 171]. The charge produces an electromagnetic vector potential that satisfies the wave equation

together with the Lorenz gauge condition . (On page 294, Jackson [101] explains why the term “Lorenz gauge” is preferable to “Lorentz gauge”.) The vector is the charge’s current density, which is formally written in terms of a four-dimensional Dirac functional supported on the charge’s world line: the density is zero everywhere, except at the particle’s position where it is infinite. For concreteness we will imagine that the particle moves around a centre (perhaps another charge, which is taken to be fixed) and that it emits outgoing radiation. We expect that the charge will undergo a radiation reaction and that it will spiral down toward the centre. This effect must be accounted for by the equations of motion, and these must therefore include the action of the charge’s own field, which is the only available agent that could be responsible for the radiation reaction. We seek to determine this self-force acting on the particle.An immediate difficulty presents itself: the vector potential, and also the electromagnetic field tensor, diverge on the particle’s world line, because the field of a point charge is necessarily infinite at the charge’s position. This behaviour makes it most difficult to decide how the field is supposed to act on the particle.

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 time-reversal invariance of Maxwell’s theory. A specific time direction was adopted when, among all possible solutions to the wave equation, we chose , the retarded solution, as the physically relevant solution. Choosing instead the advanced solution would produce a time-reversed picture in which the radiation is propagating inward and the charge is spiraling outward. Alternatively, choosing the linear superposition

would restore time-reversal 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 , , and all satisfy Eq. (1.1), whose source term is infinite on the world line. So while the wave-zone behaviours of these solutions are very different (with the retarded solution describing outgoing waves, the advanced solution describing incoming waves, and the symmetric solution describing standing waves), the three vector potentials share the same singular behaviour near the world line – all three electromagnetic fields are dominated by the particle’s Coulomb field and the different asymptotic conditions make no difference close to the particle. This observation gives us an alternative interpretation for the subscript ‘S’: it stands for ‘singular’ as well as ‘symmetric’.

Because is just as singular as , removing it from the retarded solution gives rise to a potential that is well behaved in a neighbourhood of the world line. And because 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 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), ; there is no singular source to produce a singular behaviour on the world line. Since 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 self-action of the charge’s own field is now clarified: a singular potential can be removed from the retarded potential and shown not to affect the motion of the particle. What remains is a well-behaved potential that must be solely responsible for the radiation reaction. From the regular potential we form an electromagnetic field tensor and we take the particle’s equations of motion to be

where is the charge’s four-velocity [ gives the description of the world line and is proper time], its acceleration, its (renormalized) mass, and 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 self-force that is responsible for the radiation reaction. We observe that the self-force is proportional to , it is orthogonal to the four-velocity, and it depends on the rate of change of the external force. This is the result that was first derived by Dirac [56]. (Dirac’s original expression actually involved the rate of change of the acceleration vector on the right-hand side. The resulting equation gives rise to the well-known problem of runaway solutions. To avoid such unphysical behaviour we have submitted Dirac’s equation to a reduction-of-order procedure whereby is replaced with . 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 self-force as an interaction with a free electromagnetic field.

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 . Here is an arbitrary field point, is a source point, and ; tensors at are identified with unprimed indices, while primed indices refer to tensors at . Similarly, the advanced solution can be expressed as in terms of the advanced Green’s function . The retarded Green’s function is zero whenever lies outside of the future light cone of , and is infinite at these points. On the other hand, the advanced Green’s function is zero whenever lies outside of the past light cone of , and 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 and are interchanged.From the retarded and advanced Green’s functions we can define a singular Green’s function by

and a regular two-point function by By virtue of Eq. (1.8) the singular Green’s function is symmetric in its indices and arguments: . The regular two-point 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 and is well behaved on the world line.Equation (1.6) implies that the retarded potential at is generated by a single event in spacetime: the intersection of the world line and ’s past light cone (see Figure 1). We shall call this the retarded point associated with and denote it ; is the retarded time, the value of the proper-time 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 . We shall call this the advanced point associated with and denote it ; is the advanced time, the value of the proper-time parameter at the advanced point.

In a curved spacetime with metric the wave equation for the vector potential becomes

where is the covariant wave operator and is the spacetime’s Ricci tensor; the Lorenz gauge conditions becomes , and 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 now stands for .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 , in curved spacetime its support extends inside the light cone as well; is therefore nonzero when , which denotes the chronological future of . 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 is now generated by the point charge during its entire history prior to the retarded time associated with : the potential depends on the particle’s state of motion for all times (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 , in curved spacetime its support extends inside the light cone, and is nonzero when , which denotes the chronological past of . This implies that the advanced potential at is generated by the point charge during its entire future history following the advanced time associated with : the potential depends on the particle’s state of motion for all times .

The physically relevant solution to Eq. (1.13) is obviously the retarded potential , and as in flat spacetime, this diverges on the world line. The cause of this singular behaviour is still the pointlike nature of the source, and the presence of spacetime curvature does not change the fact that the potential diverges at the position of the particle. Once more this behaviour makes it difficult to figure out how the retarded field is supposed to act on the particle and determine its motion. As in flat spacetime we shall attempt to decompose the retarded solution into a singular part that exerts no force, and a regular part that produces the entire self-force.

To decompose the retarded Green’s function into singular and regular parts is not a straightforward task in curved spacetime. The flat-spacetime definition for the singular Green’s function, Eq. (1.9), cannot be adopted without modification: While the combination half-retarded plus half-advanced Green’s functions does have the property of being symmetric, and while the resulting vector potential would be a solution to Eq. (1.13), this candidate for the singular Green’s function would produce a self-force with an unacceptable dependence on the particle’s future history. For suppose that we made this choice. Then the regular two-point function would be given by the combination half-retarded minus half-advanced Green’s functions, just as in flat spacetime. The resulting potential would satisfy the homogeneous wave equation, and it would be regular on the world line, but it would also depend on the particle’s entire history, both past (through the retarded Green’s function) and future (through the advanced Green’s function). More precisely stated, we would find that the regular potential at depends on the particle’s state of motion at all times outside the interval ; in the limit where approaches the world line, this interval shrinks to nothing, and we would find that the regular potential is generated by the complete history of the particle. A self-force constructed from this potential would be highly noncausal, and we are compelled to reject these definitions for the singular and regular Green’s functions.

The proper definitions were identified by Detweiler and Whiting [53], who proposed the following generalization to Eq. (1.9):

The two-point function is introduced specifically to cure the pathology described in the preceding paragraph. It is symmetric in its indices and arguments, so that 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, , 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 two-point function is defined to agree with the advanced Green’s function when is in the chronological past of : when . This ensures that vanishes when is in the chronological past of . In fact, reciprocity implies that will also agree with the retarded Green’s function when is in the chronological future of , and it follows that the symmetric Green’s function vanishes also when is in the chronological future of .The potential constructed from the singular Green’s function can now be seen to depend on the particle’s state of motion at times restricted to the interval (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 Detweiler–Whiting [53] definition for the regular two-point function is then

The potential constructed from this depends on the particle’s state of motion at all times prior to the advanced time : . 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 can be shown to exert no force on the particle, we conclude that alone is responsible for the self-force.From the regular potential we form an electromagnetic field tensor and the curved-spacetime generalization to Eq. (1.4) is

where is again the charge’s four-velocity, but is now its covariant acceleration.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 be the world line of a point particle in a curved spacetime. It is described by parametric relations in which is proper time. Its tangent vector is and its acceleration is ; we shall also encounter .

On we erect an orthonormal basis that consists of the four-velocity and three spatial vectors labelled by a frame index . These vectors satisfy the relations , , and . We take the spatial vectors to be Fermi–Walker transported on the world line: , where

are frame components of the acceleration vector; it is easy to show that Fermi–Walker 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 and its inverse to lower and raise frame indices, respectively.Consider a point in a neighbourhood of the world line . We assume that is sufficiently close to the world line that a unique geodesic links to any neighbouring point on . The two-point function , known as Synge’s world function [169], is numerically equal to half the squared geodesic distance between and ; it is positive if and are spacelike related, negative if they are timelike related, and is zero if and are linked by a null geodesic. We denote its gradient by , and gives a meaningful notion of a separation vector (pointing from to ).

To construct a coordinate system in this neighbourhood we locate the unique point on which is linked to by a future-directed null geodesic (this geodesic is directed from to ); we shall refer to as the retarded point associated with , and will be called the retarded time. To tensors at we assign indices , , …; this will distinguish them from tensors at a generic point on the world line, to which we have assigned indices , , …. We have and is a null vector that can be interpreted as the separation between and .

The retarded coordinates of the point are , where 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 to ; it can be loosely interpreted as the time delay between and as measured by an observer moving with the particle. This therefore gives a meaningful notion of distance between and the retarded point, and we shall call the retarded distance between and the world line. The unit vector is constant on the null geodesic that links to . Because is a different constant on each null geodesic that emanates from , keeping fixed and varying produces a congruence of null geodesics that generate the future light cone of the point (the congruence is hypersurface orthogonal). Each light cone can thus be labelled by its retarded time , each generator on a given light cone can be labelled by its direction vector , and each point on a given generator can be labelled by its retarded distance . We therefore have a good coordinate system in a neighbourhood of .To tensors at we assign indices , , …. These tensors will be decomposed in a tetrad that is constructed as follows: Given we locate its associated retarded point on the world line, as well as the null geodesic that links these two points; we then take the tetrad at and parallel transport it to along the null geodesic to obtain .

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 , which we decompose in the tetrad 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 . 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 associated with ; for example, . 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 ; this represents the past history of the particle. The integral is cut short at to avoid the singular behaviour of the retarded Green’s function when coincides with ; 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 near the world line does indeed reveal many singular terms. We first recognize terms that diverge when ; for example the Coulomb field diverges as when we approach the world line. But there are also terms that, though they stay bounded in the limit, possess a directional ambiguity at ; for example contains a term proportional to whose limit depends on the direction of approach.

This singularity structure is perfectly reproduced by the singular field obtained from the potential

where 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 . Its frame components are

and at the regular field becomes where is the rate of change of the acceleration vector, and where the tail term was given by Eq. (1.26). We see that is a regular tensor field, even on the world line.

We have argued in Section 1.4 that the self-force 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 , where is an external force also acting on the particle. Substituting Eq. (1.32) gives

in which all tensors are evaluated at , the current position of the particle on the world line. The primed indices in the tail integral refer to a point which represents a prior position; the integration is cut short at 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 with on the right-hand 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 Ricci-tensor term.) In flat spacetime the Ricci tensor is zero, the tail integral disappears (because the Green’s function vanishes everywhere within the domain of integration), and Eq. (1.33) reduces to Dirac’s result of Eq. (1.5). In curved spacetime the self-force does not vanish even when the electric charge is moving freely, in the absence of an external force: it is then given by the tail integral, which represents radiation emitted earlier and coming back to the particle after interacting with the spacetime curvature. This delayed action implies that in general, the self-force is nonlocal in time: it depends not only on the current state of motion of the particle, but also on its past history. Lest this behaviour should seem mysterious, it may help to keep in mind that the physical process that leads to Eq. (1.33) is simply an interaction between the charge and a free electromagnetic field ; it is this field that carries the information about the charge’s past.

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 produces a scalar field which satisfies a wave equation

that is very similar to Eq. (1.13). Here, is the spacetime’s Ricci scalar, and is an arbitrary coupling constant; the scalar charge density is given by a four-dimensional Dirac functional supported on the particle’s world line . The retarded solution to the wave equation is where 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 is the particle’s mass; this equation is very similar to the Lorentz-force law. But the dynamics of a scalar charge comes with a twist: If 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 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 must be removed before Eqs. (1.36) and (1.37) can be evaluated. This procedure produces the regular field , and it is this field (which satisfies the homogeneous wave equation) that gives rise to a self-force. 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 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 in his expressions. We extended Quinn’s results to an arbitrary coupling counstant for this review.)In flat spacetime the Ricci-tensor term and the tail integral disappear and Eq. (1.40) takes the form of Eq. (1.5) with replacing the factor of . In this simple case Eq. (1.41) reduces to and the mass is in fact a constant. This property remains true in a conformally flat spacetime when the wave equation is conformally invariant (): in this case the Green’s function possesses only a light-cone part and the right-hand side of Eq. (1.41) vanishes. In generic situations the mass of a point scalar charge will vary with proper time.

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 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 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 in terms of trace-reversed potentials defined by

where is the difference between , the actual metric of the perturbed spacetime, and . The potentials satisfy the wave equation together with the Lorenz gauge condition . Here and below, covariant differentiation refers to a connection that is compatible with the background metric, is the wave operator for the background spacetime, and is the energy-momentum tensor of the point mass; this is given by a Dirac distribution supported on the particle’s world line . The retarded solution is where is the retarded Green’s function associated with Eq. (1.43). The perturbation 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 . After a mapping to the background spacetime, the equations of motion take the form of

The acceleration is thus proportional to ; in the test-mass limit the world line of the particle is a geodesic of the background spacetime.We now remove from the retarded perturbation and postulate that it is the regular field that should act on the particle. (Note that satisfies the same wave equation as the retarded potentials, but that 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 of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions , at which tensors with primed indices are evaluated. As before the integral is cut short at 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 . Detweiler and Whiting [53] have contributed the more compelling interpretation that the motion is actually geodesic in a spacetime with metric . The distinction is important: Unlike the first version of the metric, the Detweiler-Whiting metric is regular on the world line and satisfies the Einstein field equations in vacuum; and because it is a solution to the field equations, it can be viewed as a physical metric — specifically, the metric of the background spacetime perturbed by a free field produced by the particle at an earlier stage of its history.

While Eq. (1.48) does indeed give the correct equations of motion for a small mass moving in a background spacetime with metric , the derivation outlined here leaves much to be desired – to what extent should we trust an analysis based on the existence of a point mass? As a partial answer to this question, Mino, Sasaki, and Tanaka [130] produced an alternative derivation of their result, which involved a small nonrotating black hole instead of a point mass. In this alternative derivation, the metric of the black hole perturbed by the tidal gravitational field of the external universe is matched to the metric of the background spacetime perturbed by the moving black hole. Demanding that this metric be a solution to the vacuum field equations determines the motion of the black hole: it must move according to Eq. (1.48). This alternative derivation (which was given a different implementation in Ref. [142]) is entirely free of singularities (except deep within the black hole), and it suggests that that the MiSaTaQuWa equations can be trusted to describe the motion of any gravitating body in a curved background spacetime (so long as the body’s internal structure can be ignored). This derivation, however, was limited to the case of a non-rotating black hole, and it relied on a number of unjustified and sometimes unstated assumptions [83, 144, 145]. The conclusion was made firm by the more rigorous analysis of Gralla and Wald [83] (as extended by Pound [144]), who showed that the MiSaTaQuWa equations apply to any sufficiently compact body of arbitrary internal structure.

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 . Barack and Ori [17] have shown that under a coordinate transformation of the form , where are the coordinates of the background spacetime and is a smooth vector field of order , the particle’s acceleration changes according to , where

is the “gauge acceleration”; is the second covariant derivative of in the direction of the world line. This implies that the particle’s acceleration can be altered at will by a gauge transformation; could even be chosen so as to produce , making the motion geodesic after all. This observation provides a dramatic illustration of the following point: The MiSaTaQuWa equations of motion are not gauge invariant and they cannot by themselves produce a meaningful answer to a well-posed physical question; to obtain such answers it is necessary to combine the equations of motion with the metric perturbation so as to form gauge-invariant quantities that will correspond to direct observables. This point is very important and cannot be over-emphasized.The gravitational self-force possesses a physical significance that is not shared by its scalar and electromagnetic analogues, because the motion of a small body in the strong gravitational field of a much larger body is a problem of direct relevance to gravitational-wave astronomy. Indeed, extreme-mass-ratio inspirals, involving solar-mass compact objects moving around massive black holes of the sort found in galactic cores, have been identified as promising sources of low-frequency gravitational waves for space-based interferometric detectors such as the proposed Laser Interferometer Space Antenna (LISA [115]). These systems involve highly eccentric, nonequatorial, and relativistic orbits around rapidly rotating black holes, and the waves produced by such orbital motions are rich in information concerning the strongest gravitational fields in the Universe. This information will be extractable from the LISA data stream, but the extraction depends on sophisticated data-analysis strategies that require a detailed and accurate modeling of the source. This modeling involves formulating the equations of motion for the small body in the field of the rotating black hole, as well as a consistent incorporation of the motion into a wave-generation formalism. In short, the extraction of this wealth of information relies on a successful evaluation of the gravitational self-force.

The finite-mass corrections to the orbital motion are important. For concreteness, let us assume that the orbiting body is a black hole of mass and that the central black hole has a mass . 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 is of the order of minutes. The gravitational waves produced by the orbital motion have frequencies of the order of the mHz, which is well within LISA’s frequency band. The radiative losses drive the orbital motion toward a final plunge into the large black hole; this occurs over a radiation-reaction timescale of the order of a year, during which the system will go through a number of wave cycles of the order of . The role of the gravitational self-force is precisely to describe this orbital evolution toward the final plunge. While at any given time the self-force provides fractional corrections of order to the motion of the small black hole, these build up over a number of orbital cycles of order to produce a large cumulative effect. As will be discussed in some detail in Section 2.6, the gravitational self-force is important, because it drives large secular changes in the orbital motion of an extreme-mass-ratio binary.

One of the first self-force calculations ever performed for a curved spacetime was presented by Smith and Will [163]. They considered an electric charge held in place at position outside a Schwarzschild black hole of mass . 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 , where is the particle’s mass, is the usual metric factor, and a prime indicates differentiation with respect to , so that . Smith and Will found that for a particle of charge , the external force is given instead by . The second term is contributed by the electromagnetic self-force, and implies that the external force is smaller for a charged particle. This means that the electromagnetic self-force 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 Reissner–Nordström black hole of mass and charge , provided that is replaced by the more general expression .The repulsive nature of the electromagnetic self-force 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 black-hole horizon always acts as perfect conductor, because the electrostatic potential is necessarily uniform across its surface. It is then tempting to imagine that the self-force 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 situated outside a spherical conductor of radius . The charge is placed at a distance 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 is situated at a distance from the centre of the conductor, and the potential is given by , where is the distance to the charge, while is the distance to the image charge. The first term can be identified with the singular potential , and the associated electric field exerts no force on the point charge. The second term is the regular potential , and the associated field is entirely responsible for the self-force. The regular electric field is , and the self-force is . A simple computation returns

This is an attractive self-force, because the total induced charge on the conducting surface is equal to , which is opposite in sign to . With identified with up to a numerical factor, we find that our intuition has produced the expected factor of , but that it gives rise to the wrong sign for the self-force. An attempt to refine this computation by removing the net charge on the conductor (to mimic more closely the black-hole horizon, which cannot support a net charge) produces a wrong dependence on in addition to the same wrong sign. In this case the conductor is maintained at a constant potential , and the situation involves a second image charge situated at . 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 ; the force is now exerted by an image dipole instead of a single image charge.The computation of the self-force in the black-hole case is almost as straightforward. The exact solution to Maxwell’s equations that describes a point charge situated and 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 when is much larger than . It is easy to see that Copson’s potential behaves as at large distances, which reveals that in addition to , comes with an additional (and unphysical) charge situated at . This charge must be removed by adding to a potential that (i) is a solution to the vacuum Maxwell equations, (ii) is regular everywhere except at , and (iii) carries the opposite charge ; this potential must be a pure monopole, because higher multipoles would produce a singularity on the horizon, and it is given uniquely by . The Copson solution was generalized to Reissner–Nordström 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 is dictated by the fact that its associated electric field is isotropic around the charge and therefore exerts no force. The self-force comes entirely from the monopole potential, which describes a (fictitious) charge situated at . Because this charge is of the same sign as the original charge , the self-force is repulsive. More precisely stated, we find that the regular piece of the electric field is given by

and that it produces the self-force of Eq. (1.50). The simple picture described here, in which the electromagnetic self-force is produced by a fictitious charge 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 self-force was first noticed by Alan Wiseman [185]. (In his paper, Wiseman computed the scalar self-force 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 self-force vanishes.)We should remark that the identification of and with the Detweiler–Whiting singular and regular fields, respectively, is a matter of conjecture. Although and satisfy the essential properties of the Detweiler–Whiting decomposition – being, respectively, a regular homogenous solution and a singular solution sourced by the particle – one should accept the possibility that they may not be the actual Detweiler–Whiting fields. It is a topic for future research to investigate the precise relation between the Copson field and the Detweiler–Whiting singular field.

It is instructive to compare the electromagnetic self-force produced by the presence of a grounded conductor to the self-force produced by the presence of a black hole. In the case of a conductor, the total induced charge on the conducting surface is , and it is this charge that is responsible for the attractive self-force; the induced charge is supplied by the electrodes that keep the conductor grounded. In the case of a black hole, there is no external apparatus that can supply such a charge, and the total induced charge on the horizon necessarily vanishes. The origin of the self-force is therefore very different in this case. As we have seen, the self-force is produced by a fictitious charge situated at the centre of black hole; and because this charge is positive, the self-force is repulsive.

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 and its derivatives in Section 3, the parallel propagator in Section 5, and the van Vleck determinant in Section 7. An important portion of the theory (covered in Sections 4 and 6) is concerned with the expansion of bitensors when is very close to ; 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 DeWitt 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 . We then move on (in Section 9) to Fermi normal coordinates [122], which are defined in a neighbourhood of a world line . 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 associated with a field point . The presentation in Part II borrows heavily from Synge’s book [169]. In fact, we are much indebted to Synge for initiating the construction of retarded coordinates in a neighbourhood of a world line. We have implemented his program quite differently (Synge was interested in a large neighbourhood of the world line in a weakly curved spacetime, while we are interested in a small neighbourhood in a strongly curved spacetime), but the idea is originally his.

In Part III (Sections 12 to 16) we review the theory of Green’s functions for (scalar, vectorial, and tensorial) wave equations in curved spacetime. We begin in Section 12 with a pedagogical introduction to the retarded and advanced Green’s functions for a massive scalar field in flat spacetime; in this simple context the all-important Hadamard decomposition [88] of the Green’s function into “light-cone” and “tail” parts can be displayed explicitly. The invariant Dirac functional is defined in Section 13 along with its restrictions on the past and future null cones of a reference point . 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 , we have taken the liberty of eliminating the silly minus sign and call it instead . The reader should also note that all our Green’s functions are normalized in the same way, with a factor of multiplying a four-dimensional Dirac functional of the right-hand side of the wave equation. (The gravitational Green’s function is sometimes normalized with a on the right-hand side.)

In Part IV (Sections 17 to 19) we compute the retarded, singular, and regular fields associated with a point scalar charge (Section 17), a point electric charge (Section 18), and a point mass (Section 19). We provide two different derivations for each of the equations of motion. The first type of derivation was outlined previously: We follow Detweiler and Whiting [53] and postulate that only the regular field exerts a force on the particle. In the second type of derivation we take guidance from Quinn and Wald [150] and postulate that the net force exerted on a point particle is given by an average of the retarded field over a surface of constant proper distance orthogonal to the world line — this rest-frame average is easily carried out in Fermi normal coordinates. The averaged field is still infinite on the world line, but the divergence points in the direction of the acceleration vector and it can thus be removed by mass renormalization. Such calculations show that while the singular field does not affect the motion of the particle, it nonetheless contributes to its inertia.

In Part V (Sections 20 to 23), we show that at linear order in the body’s mass , 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].

Living Rev. Relativity 14, (2011), 7
http://www.livingreviews.org/lrr-2011-7 |
This work is licensed under a Creative Commons License. E-mail us: |