In this section we consider the motion of a point particle of mass subjected to its own gravitational field in addition to an external field. The particle moves on a world line in a curved spacetime whose background metric is assumed to be a vacuum solution to the Einstein field equations. We shall suppose that is small, so that the perturbation created by the particle can also be considered to be small. In the final analysis we shall find that obeys a linear wave equation in the background spacetime, and this linearization of the field equations will allow us to fit the problem of determining the motion of a point mass within the general framework developed in Sections 17 and 18. We shall find that is not a geodesic of the background spacetime because acts on the particle and produces an acceleration proportional to ; the motion is geodesic in the test-mass limit only.

While we can make the problem fit within the general framework, it is important to understand that the problem of motion in gravitation is conceptually very different from the versions encountered previously in the case of a scalar or electromagnetic field. In these cases, the field equations satisfied by the scalar potential or the vector potential are fundamentally linear; in general relativity the field equations satisfied by are fundamentally nonlinear, and this makes a major impact on the formulation of the problem. (In all cases the coupled problem of determining the field and the motion of the particle is nonlinear.) Another difference resides with the fact that in the previous cases, the field equations and the law of motion could be formulated independently of each other (because the action functional could be varied independently with respect to the field and the world line); in general relativity the law of motion follows from energy-momentum conservation, which is itself a consequence of the field equations.

The dynamics of a point mass in general relativity must therefore be formulated with care. We shall describe a formal approach to this problem, based on the fiction that the spacetime of a point particle can be constructed exactly in general relativity. (This is indeed a fiction, because it is known [80] that the metric of a point particle, as described by a Dirac distribution on a world line, is much too singular to be defined as a distribution in spacetime. The construction, however, makes distributional sense at the level of the linearized theory.) The outcome of this approach will be an approximate formulation of the equations of motion that relies on a linearization of the field equations, and which turns out to be closely analogous to the scalar and electromagnetic cases encountered previously. We shall put the motion of a small mass on a much sounder foundation in Part V, where we take to be a (small) extended body instead of a point particle.

Let a point particle of mass move on a world line in a curved spacetime with metric . This is the exact metric of the perturbed spacetime, and it depends on as well as all other relevant parameters. At a later stage of the discussion will be expressed as sum of a “background” part that is independent of , and a “perturbation” part that contains the dependence on . The world line is described by relations in which is an arbitrary parameter – this will later be identified with proper time in the background spacetime. We use sans-serif symbols to denote tensors that refer to the perturbed spacetime; tensors in the background spacetime will be denoted, as usual, by italic symbols.

The particle’s action functional is

where is tangent to the world line and the metric is evaluated at . We assume that the particle provides the only source of matter in the spacetime – an explanation will be provided below – so that the Einstein field equations take the form of where is the Einstein tensor constructed from and is the particle’s energy-momentum tensor, obtained by functional differentiation of with respect to ; the parallel propagators appear naturally by expressing as .On a formal level the metric is obtained by solving the Einstein field equations, and the world line is determined by the equations of energy-momentum conservation, which follow from the field equations. From Eqs. (5.14), (13.3), and (19.3) we obtain

and additional manipulations reduce this to

where is the covariant acceleration and is a scalar field on the world line. Energy-momentum conservation therefore produces the geodesic equation and measures the failure of to be an affine parameter on the geodesic .

At this stage we begin treating as a small quantity, and we write

with denoting the limit of the metric , and containing the dependence on . We shall refer to as the “metric of the background spacetime” and to as the “perturbation” produced by the particle. We insist, however, that no approximation is introduced at this stage; the perturbation is the exact difference between the exact metric and the background metric . Below we shall use the background metric to lower and raise indices.We introduce the tensor field

as the exact difference between , the connection compatible with the exact metric , and , the connection compatible with the background metric . A covariant differentiation indicated by will refer to , while a covariant differentiation indicated by will continue to refer to .We express the exact Einstein tensor as

where is the Einstein tensor of the background spacetime, which is assumed to vanish. The second term is the linearized Einstein operator defined by where is the wave operator in the background spacetime, and is the “trace-reversed” metric perturbation (with all indices raised with the background metric). The third term contains the remaining nonlinear pieces that are excluded from .

The exact Einstein field equations can be expressed as

where the effective energy-momentum tensor is defined by Because satisfies the Bianchi-like identities , the effective energy-momentum tensor is conserved in the background spacetime: This statement is equivalent to , as can be inferred from the equations , , and the definition of . Equation (19.14), in turn, is equivalent to Eq. (19.5), which states that the motion of the point particle is geodesic in the perturbed spacetime.

Eq. (19.12) expresses the full and exact content of Einstein’s field equations. It is written in such a way that the left-hand side is linear in the perturbation , while the right-hand side contains all nonlinear terms. It may be viewed formally as a set of linear differential equations for with a specified source term . This equation is of mixed hyperbolic-elliptic type, and as such it is a poor starting point for the selection of retarded solutions that enforce a strict causal link between the source and the field. This inadequacy, however, can be remedied by imposing the Lorenz gauge condition

which converts into a strictly hyperbolic differential operator. In this gauge the field equations become This is a tensorial wave equation formulated in the background spacetime, and while the left-hand side is manifestly linear in , the right-hand side continues to incorporate all nonlinear terms. Equations (19.15) and (19.16) still express the full content of the exact field equations.A formal solution to Eq. (19.16) is

where is the retarded Green’s function introduced in Section 16. With the help of Eq. (16.21), it is easy to show that follows directly from Eq. (19.17); is the electromagnetic Green’s function introduced in Section 15. This equation indicates that the Lorenz gauge condition is automatically enforced when the conservation equation is imposed. Conversely, Eq. (19.18) implies that , which indicates that imposition of automatically enforces the conservation equation. There is a one-to-one correspondence between the conservation equation and the Lorenz gauge condition.The split of the Einstein field equations into a wave equation and a gauge condition directly tied to the conservation of the effective energy-momentum tensor is a most powerful tool, because it allows us to disentangle the problems of obtaining and determining the motion of the particle. This comes about because the wave equation can be solved first, independently of the gauge condition, for a particle moving on an arbitrary world line ; the world line is determined next, by imposing the Lorenz gauge condition on the solution to the wave equation. More precisely stated, the source term for the wave equation can be evaluated for any world line , without demanding that the effective energy-momentum tensor be conserved, and without demanding that be a geodesic of the perturbed spacetime. Solving the wave equation then returns as a functional of the arbitrary world line, and the metric is not yet fully specified. Because imposing the Lorenz gauge condition is equivalent to imposing conservation of the effective energy-momentum tensor, inserting within Eq. (19.15) finally determines , and forces it to be a geodesic of the perturbed spacetime. At this stage the full set of Einstein field equations is accounted for, and the metric is fully specified as a tensor field in spacetime. The split of the field equations into a wave equation and a gauge condition is key to the formulation of the gravitational self-force; in this specific context the Lorenz gauge is conferred a preferred status among all choices of gauge.

An important question to be addressed is how the wave equation is to be integrated. A method of principle, based on the assumed smallness of and , is suggested by post-Minkowskian theory [180, 26]. One proceeds by iterations. In the first iterative stage, one fixes and substitutes within ; evaluation of the integral in Eq. (19.17) returns the first-order approximation for the perturbation. In the second stage is inserted within and Eq. (19.17) returns the second-order approximation for the perturbation. Assuming that this procedure can be repeated at will and produces an adequate asymptotic series for the exact perturbation, the iterations are stopped when the -order approximation is deemed to be sufficiently accurate. The world line is then determined, to order , by subjecting the approximated field to the Lorenz gauge condition. It is to be noted that the procedure necessarily produces an approximation of the field, and an approximation of the motion, because the number of iterations is necessarily finite. This is the only source of approximation in our formulation of the dynamics of a point mass.

Conservation of energy-momentum implies Eq. (19.5), which states that the motion of the point mass is geodesic in the perturbed spacetime. The equation is expressed in terms of the exact connection , and with the help of Eq. (19.8) it can be re-written in terms of the background connection . We get , where the left-hand side is the covariant acceleration in the background spacetime, and is given by Eq. (19.6). At this stage the arbitrary parameter can be identified with proper time in the background spacetime. With this choice the equations of motion become

where is the velocity vector in the background spacetime, the covariant acceleration, and Eq. (19.19) is an exact statement of the equations of motion. It expresses the fact that while the motion is geodesic in the perturbed spacetime, it may be viewed as accelerated motion in the background spacetime. Because is calculated as an expansion in powers of , the acceleration also is eventually obtained as an expansion in powers of . Here, in keeping with the preceding sections, we will use order-reduction to make that expansion well-behaved; in Part V of the review, we will formulate the expansion more clearly as part of more systematic approach.

While our formulation of the dynamics of a point mass is in principle exact, any practical implementation will rely on an approximation method. As we saw previously, the most immediate source of approximation concerns the number of iterations involved in the integration of the wave equation. Here we perform a single iteration and obtain the perturbation and the equations of motion to first order in the mass .

In a first iteration of the wave equation we fix and set , , where

is the particle’s energy-momentum tensor in the background spacetime. This implies that , and Eq. (19.16) becomes Its solution is and from this we obtain . Equation (19.8) gives rise to , and from Eq. (19.20) we obtain ; we can discard the second term because it is clear that the acceleration will be of order . Inserting these results within Eq. (19.19), we obtain We express this in the equivalent form to emphasize the fact that the acceleration is orthogonal to the velocity vector.It should be clear that Eq. (19.25) is valid only in a formal sense, because the potentials obtained from Eqs. (19.23) diverge on the world line. To make sense of these equations we will proceed as in Sections 17 and 18 with a careful analysis of the field’s singularity structure; regularization will produce a well-defined version of Eq. (19.25). Our formulation of the dynamics of a point mass makes it clear that a proper implementation requires that the wave equation of Eq. (19.22) and the equations of motion of Eq. (19.25) be integrated simultaneously, in a self-consistent manner.

In the preceding discussion we started off with an exact formulation of the problem of motion for a small mass in a background spacetime with metric , but eventually boiled it down to an implementation accurate to first order in . Would it not be simpler and more expedient to formulate the problem directly to first order? The answer is a resounding no: By doing so we would be driven toward a grave inconsistency; the nonlinear formulation is absolutely necessary if one wishes to contemplate a self-consistent integration of Eqs. (19.22) and (19.25).

A strictly linearized formulation of the problem would be based on the field equations , where is the energy-momentum tensor of Eq. (19.21). The Bianchi-like identities dictate that must be conserved in the background spacetime, and a calculation identical to the one leading to Eq. (19.5) would reveal that the particle’s motion must be geodesic in the background spacetime. In the strictly linearized formulation, therefore, the gravitational potentials of Eq. (19.23) must be sourced by a particle moving on a geodesic, and there is no opportunity for these potentials to exert a self-force. To get the self-force, one must provide a formulation that extends beyond linear order. To be sure, one could persist in adopting the linearized formulation and “save the phenomenon” by relaxing the conservation equation. In practice this could be done by adopting the solutions of Eq. (19.23), demanding that the motion be geodesic in the perturbed spacetime, and relaxing the linearized gauge condition to . While this prescription would produce the correct answer, it is largely ad hoc and does not come with a clear justification. Our exact formulation provides much more control, at least in a formal sense. We shall do even better in Part V.

An alternative formulation of the problem provided by Gralla and Wald [83] avoids the inconsistency by refraining from performing a self-consistent integration of Eqs. (19.22) and (19.25). Instead of an expansion of the acceleration in powers of , their approach is based on an expansion of the world line itself, and it returns the equations of motion for a deviation vector which describes the offset of the true world line relative to a reference geodesic. While this approach is mathematically sound, it eventually breaks down as the deviation vector becomes large, and it does not provide a justification of the self-consistent treatment of the equations.

The difference between the Gralla–Wald approach and a self-consistent one is the difference between a regular expansion and a general one. In a regular expansion, all dependence on a small quantity is expanded in powers:

In a general expansion, on the other hand, the functions are allowed to retain some dependence on the small quantity: Put simply, the goal of a general expansion is to expand only part of a function’s dependence on a small quantity, while holding fixed some specific dependence that captures one or more of the function’s essential features. In the self-consistent expansion that we advocate here, our iterative solution returns in which the functional dependence on the world line incorporates a dependence on the expansion parameter . We deliberately introduce this functional dependence on a mass-dependent world line in order to maintain a meaningful and accurate description of the particle’s motion. Although the regular expansion can be retrieved by further expanding the dependence within , the reverse statement does not hold: the general expansion cannot be justified on the basis of the regular one. The notion of a general expansion is at the core of singular perturbation theory [63, 96, 109, 111, 178, 145]. We shall return to these issues in our treatment of asymptotically small bodies, and in particular, in Section 22.5 below.

To conclude this subsection we should explain why it is desirable to restrict our discussion to spacetimes that contain no matter except for the point particle. Suppose, in contradiction with this assumption, that the background spacetime contains a distribution of matter around which the particle is moving. (The corresponding vacuum situation has the particle moving around a black hole. Notice that we are still assuming that the particle moves in a region of spacetime in which there is no matter; the issue is whether we can allow for a distribution of matter somewhere else.) Suppose also that the matter distribution is described by a collection of matter fields . Then the field equations satisfied by the matter have the schematic form , and the background metric is determined by the Einstein field equations , in which stands for the matter’s energy-momentum tensor. We now insert the point particle in the spacetime, and recognize that this displaces the background solution to a new solution (. The perturbations are determined by the coupled set of equations and . After linearization these take the form of

where , , , and are suitable differential operators acting on the perturbations. This is a coupled set of partial differential equations for the perturbations and . These equations are linear, but they are much more difficult to deal with than the single equation for that was obtained in the vacuum case. And although it is still possible to solve the coupled set of equations via a Green’s function technique, the degree of difficulty is such that we will not attempt this here. We shall, therefore, continue to restrict our attention to the case of a point particle moving in a vacuum (globally Ricci-flat) background spacetime.

Going back to Eq. (19.23), we have that the gravitational potentials associated with a point particle of mass moving on world line are given by

up to corrections of order ; here gives the description of the world line, is the velocity vector, and is the retarded Green’s function introduced in Section 16. Because the retarded Green’s function is defined globally in the entire background spacetime, Eq. (19.29) describes the gravitational perturbation created by the particle at any point in that spacetime.For a more concrete expression we take to be in a neighbourhood of the world line. The manipulations that follow are very close to those performed in Section 17.2 for the case of a scalar charge, and in Section 18.2 for the case of an electric charge. Because these manipulations are by now familiar, it will be sufficient here to present only the main steps. There are two important simplifications that occur in the case of a massive particle. First, it is clear that

and we will take the liberty of performing a pre-emptive order-reduction by dropping all terms involving the acceleration vector when computing and to first order in ; otherwise we would arrive at an equation for the acceleration that would include an antidamping term [92, 93, 150]. Second, because we take to be a solution to the vacuum field equations, we are also allowed to set in our computations.With the understanding that is close to the world line (refer back to Figure 9), we substitute the Hadamard construction of Eq. (16.7) into Eq. (19.29) and integrate over the portion of that is contained in . The result is

in which primed indices refer to the retarded point associated with , is the retarded distance from to , and is the proper time at which enters from the past.In the following subsections we shall refer to as the gravitational potentials at produced by a particle of mass moving on the world line , and to as the gravitational field at . To compute this is our next task.

Keeping in mind that and are related by , a straightforward computation reveals that the covariant derivatives of the gravitational potentials are given by

where the “tail integral” is defined by The second form of the definition, in which the integration is cut short at to avoid the singular behaviour of the retarded Green’s function at , is equivalent to the first form.We wish to express in the retarded coordinates of Section 10, as an expansion in powers of . For this purpose we decompose the field in the tetrad that is obtained by parallel transport of on the null geodesic that links to ; this construction is detailed in Section 10. We recall from Eq. (10.4) that the parallel propagator can be expressed as . The expansion relies on Eq. (10.29) for and Eq. (10.31) for , both simplified by setting . We shall also need

which follows from Eq. (16.13), which follow from Eqs. (16.14) and (16.15), respectively, as well as the relation first encountered in Eq. (10.7). And finally, we shall need which follows from Eq. (16.17).Making these substitutions in Eq. (19.3) and projecting against various members of the tetrad gives

where, for example, are frame components of the Riemann tensor evaluated at . We have also introduced the frame components of the tail part of the gravitational field, which are obtained from Eq. (19.34) evaluated at instead of ; for example, . We may note here that while is the only component of the gravitational field that diverges when , the other components are nevertheless singular because of their dependence on the unit vector ; the only exception is , which is regular.

The translation of the results contained in Eqs. (19.39) – (19.44) into the Fermi normal coordinates of Section 9 proceeds as in Sections 17.4 and 18.4, but is simplified by setting in Eqs. (11.7), (11.8), (11.4), (11.5), and (11.6) that relate the Fermi normal coordinates to the retarded coordinates. We recall that the Fermi normal coordinates refer to a point on the world line that is linked to by a spacelike geodesic that intersects orthogonally.

The translated results are

where all frame components are now evaluated at instead of .It is then a simple matter to average these results over a two-surface of constant and . Using the area element of Eq. (17.24) and definitions analogous to those of Eq. (17.25), we obtain

The averaged gravitational field is regular in the limit , in which the tetrad coincides with . Reconstructing the field at from its frame components gives where the tail term can be copied from Eq. (19.34), The tensors that appear in Eq. (19.57) all refer to the simultaneous point , which can now be treated as an arbitrary point on the world line .

The singular gravitational potentials

are solutions to the wave equation of Eq. (19.22); the singular Green’s function was introduced in Section 16.5. We will see that the singular field reproduces the singular behaviour of the retarded solution near the world line, and that the difference, , is smooth on the world line.To evaluate the integral of Eq. (19.59) we take to be close to the world line (see Figure 9), and we invoke Eq. (16.31) as well as the Hadamard construction of Eq. (16.37). This gives

where primed indices refer to the retarded point , double-primed indices refer to the advanced point , and where is the advanced distance between and the world line.Differentiation of Eq. (19.60) yields

and we would like to express this as an expansion in powers of . For this we will rely on results already established in Section 19.3, as well as additional expansions that will involve the advanced point . We recall 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 and , respectively; these results can be simplified by setting , which is appropriate in this computation.To derive an expansion for we follow the general method of Section 11.4 and introduce the functions . We have that

where overdots indicate differentiation with respect to and . The leading term was worked out in Eq. (19.35), and the derivatives of are given by

and

according to Eqs. (19.37) and (16.15). Combining these results together with Eq. (11.12) for gives

which should be compared with Eq. (19.35). It should be emphasized that in Eq. (19.62) and all equations below, all frame components are evaluated at the retarded point , and not at the advanced point. The preceding computation gives us also an expansion forwhich becomes

and which is identical to Eq. (19.37).We proceed similarly to obtain an expansion for . Here we introduce the functions and express as . The leading term was computed in Eq. (19.36), and

follows from Eq. (16.14). Combining these results together with Eq. (11.12) for gives

and this should be compared with Eq. (19.36). The last expansion we shall need is which is identical to Eq. (19.38).We obtain the frame components of the singular gravitational field by substituting these expansions into Eq. (19.61) and projecting against the tetrad . After some algebra we arrive at

in which all frame components are evaluated at the retarded point . Comparison of these expressions with Eqs. (19.39) – (19.44) reveals identical singularity structures for the retarded and singular gravitational fields.The difference between the retarded field of Eqs. (19.39) – (19.44) and the singular field of Eqs. (19.66) – (19.71) defines the regular gravitational field . Its frame components are

and we see that is regular in the limit . We may therefore evaluate the regular field directly at , where the tetrad coincides with . After reconstructing the field at from its frame components, we obtain where the tail term can be copied from Eq. (19.34), The tensors that appear in Eq. (19.79) all refer to the retarded point , which can now be treated as an arbitrary point on the world line .

The retarded gravitational field of a point particle is singular on the world line, and this behaviour makes it difficult to understand how the field is supposed to act on the particle and influence its motion. The field’s singularity structure was analyzed in Sections 19.3 and 19.4, and in Section 19.5 it was shown to originate from the singular field ; the regular field was then shown to be regular on the world line.

To make sense of the retarded field’s action on the particle we can follow the discussions of Section 17.6 and 18.6 and postulate that the self gravitational field of the point particle is either , as worked out in Eq. (19.57), or , as worked out in Eq. (19.78). These regularized fields are both given by

and in which all tensors are now evaluated at an arbitrary point on the world line .The actual gravitational perturbation is obtained by inverting Eq. (19.10), which leads to . Substituting Eq. (19.80) yields

where the tail term is given by the trace-reversed counterpart to Eq. (19.81): When this regularized field is substituted into Eq. (19.13), we find that the terms that depend on the Riemann tensor cancel out, and we are left with We see that only the tail term is involved in the final form of the equations of motion. The tail integral of Eq. (19.83) involves the current position of the particle, at which all tensors with unprimed indices are evaluated, as well as all prior positions , at which all tensors with primed indices are evaluated. The tail integral is cut short at to avoid the singular behaviour of the retarded Green’s function at coincidence; this limiting procedure was justified at the beginning of Section 19.3.Eq. (19.84) was first derived by Yasushi Mino, Misao Sasaki, and Takahiro Tanaka in 1997 [130]. (An incomplete treatment had been given previously by Morette-DeWitt and Ging [133].) An alternative derivation was then produced, also in 1997, by Theodore C. Quinn and Robert M. Wald [150]. These equations are now known as the MiSaTaQuWa equations of motion, and other derivations [83, 144], based on an extended-body approach, will be reviewed below in Part V. It should be noted that Eq. (19.84) is formally equivalent to the statement that the point particle moves on a geodesic in a spacetime with metric , where is the regular metric perturbation obtained by trace-reversal of the potentials ; this perturbed metric is regular on the world line, and it is a solution to the vacuum field equations. This elegant interpretation of the MiSaTaQuWa equations was proposed in 2003 by Steven Detweiler and Bernard F. Whiting [53]. Quinn and Wald [151] have shown that under some conditions, the total work done by the gravitational self-force is equal to the energy radiated (in gravitational waves) by the particle.

Living Rev. Relativity 14, (2011), 7
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