The method of matched asymptotic expansions relies on the fact that the inner and outer expansion agree term by term when re-expanded in the buffer region, where . To illustrate this idea of matching, consider the forms of the two expansions in the buffer region. The inner expansion holds constant (since ) while expanding for small . But if is replaced with its value , the inner expansion takes the form , where each term has a dependence on that can be expanded in the limit to arrive at the schematic forms and , where signifies “plus terms of the form” and the expanded quantities can be taken to be components in Fermi coordinates. Here we have preemptively restricted the form of the expansions, since terms such as must vanish because they would have no corresponding terms in the outer expansion. Putting these two expansions together, we arrive at
On the other hand, the outer expansion holds constant (since is formally of the order of the global external coordinates) while expanding for small , leading to the form . But very near the world line, each term in this expansion can be expanded for small , leading to and . (Again, we have restricted this form because terms such as cannot arise in the inner expansion.) Putting these two expansions together, we arrive at
One can make use of this fact by first determining the inner and outer expansions as fully as possible, then fixing any unknown functions in them by matching them term by term in the buffer region; this was the route taken in, e.g., Refs. [130, 142, 49, 170]. However, such an approach is complicated by the subtleties of matching in a diffeomorphism-invariant theory, where the inner and outer expansions are generically in different coordinate systems. See Ref.  for an analysis of the limitations of this approach as it has typically been implemented. Alternatively, one can take the opposite approach, working in the buffer region first, constraining the forms of the two expansions by making use of their matching, then using the buffer-region information to construct a global solution; this was the route taken in, e.g., Refs. [102, 83, 144]. In general, some mixture of these two approaches can be taken. Our calculation follows Ref. . The only information we take from the inner expansion is its general form, which is characterized by the multipole moments of the body. From this information, we determine the external expansion, and thence the equation of motion of the world line.
Over the course of our calculation, we will find that the external metric perturbation in the buffer region is expressed as the sum of two solutions: one that formally diverges at and is entirely determined from a combination of (i) the multipole moments of the internal background metric , (ii) the Riemann tensor of the external background , and (iii) the acceleration of the world line ; and a second solution that is formally regular at and depends on the past history of the body and the initial conditions of the field. At leading order, these two solutions are identified as the Detweiler–Whiting singular and regular fields and , respectively, and the self-force is determined entirely by . Along with the self-force, the acceleration of the world line includes the Papapetrou spin force . This calculation leaves us with the self-force in terms of the the metric perturbation in the neighbourhood of the body. In Section 23, we use the local information from the buffer region to construct a global solution for the metric perturbation, completing the solution of the problem.
Before proceeding, we define some notation. We use the multi-index notation . Angular brackets denote the STF combination of the enclosed indices, and a tensor bearing a hat is an STF tensor. To accommodate this, we now write the Fermi spatial coordinates as , instead of as they were written in previous sections. Finally, we define the one-forms and .
One should note that the coordinate transformation between Fermi coordinates and the global coordinates is -dependent, since Fermi coordinates are tethered to an -dependent world line. If one were using a regular expansion, then this coordinate transformation would devolve into a background coordinate transformation to a Fermi coordinate system centered on a geodesic world line, combined with a gauge transformation to account for the -dependence. But in the self-consistent expansion, the transformation is purely a background transformation, because the -dependence in it is reducible to that of the fixed world line.
Because the dependence on in the coordinate transformation cannot be reduced to a gauge transformation, in Fermi coordinates the components of the background metric become -dependent. This dependence takes the explicit form of factors of the acceleration and its derivatives, for which we have assumed the expansion . There is also an implicit dependence on in that the proper time on the world line depends on if written as a function of the global coordinates; but this dependence can be ignored so long as we work consistently with Fermi coordinates.
Of course, even in these -dependent coordinates, remains the background metric of the outer expansion, and is an exact solution to the wave equation (21.7). At first order we will, therefore, obtain exactly in Fermi coordinates, for arbitrary . However, for some purposes an approximate solution of the wave equation may suffice, in which case we may utilize the expansion of . Substituting that expansion into and yields the buffer-region expansions
Now, we merely assume that in the buffer region there exists a smooth coordinate transformation between the local coordinates and the Fermi coordinates such that , , and . The buffer region corresponds to asymptotic infinity (or ) in the internal spacetime. So after re-expressing as , the internal background metric can be expanded as
We assume that the perturbation can be similarly expanded in powers of at fixed ,
The expansion of may or may not hold the acceleration fixed. Regardless of this choice, the general form of the expansion remains valid: incorporating the expansion of the acceleration would merely shuffle terms from one coefficient to another. And since the internal metric must equal the external metric , the general form of the above expansions of and completely determines the general form of the external perturbations:
To obtain a general solution to the Einstein equation, we write each as an expansion in terms of irreducible symmetric trace-free pieces:
Now, since the wave equations (21.4) and (21.5) are covariant, they must still hold in the new coordinate system, despite the additional -dependence. Thus, both equations could be solved for arbitrary acceleration in the buffer region. However, due to the length of the calculations involved, we will instead solve the equations
Unlike the wave equations, the gauge conditions (21.10) and (21.11) already incorporate the expansion of the acceleration. As such, they are unmodified by the replacement of the second-order wave equation (21.5) with its approximation (22.17). So we can write
In what follows, the reader may safely assume that all calculations are lengthy unless noted otherwise.
In principle, solving the first-order Einstein equation in the buffer region is straightforward. One need simply substitute the expansion of , given in Eq. (22.11), into the linearized wave equation (22.16) and the gauge condition (22.18). Equating powers of in the resulting expansions then yields a sequence of equations that can be solved for successively higher-order terms in . Solving these equations consists primarily of expressing each quantity in its irreducible STF form, using the decompositions (B.3) and (B.7); since the terms in this STF decomposition are linearly independent, we can solve each equation term by term. This calculation is aided by the fact that , so that, for example, the wave operator consists of a flat-space Laplacian plus corrections of order . Appendix B also lists many useful identities, particularly , , and the fact that is an eigenvector of the flat-space Laplacian: .
Before proceeding with the calculation, which consists mostly of tedious and lengthy algebra, we summarize the results. The first-order perturbation consists of two pieces, which we will eventually identify with the Detweiler–Whiting regular and singular fields. In the buffer-region expansion, the regular field consists entirely of unknowns, which is to be expected since as a free radiation field, it must be provided by boundary data. Only when we consider the global solution, in Section 23, will we express it in terms of a tail integral. On the other hand, the singular field is locally determined, and it is characterized by the body’s monopole moment . More precisely, it is fully determined by the tidal fields of the external background spacetime and the Arnowitt–Deser–Misner mass of the internal background spacetime . By itself the wave equation does not restrict the behaviour of this monopole moment, but imposing the gauge condition produces the evolution equations
We now proceed to the details of the calculation. We begin with the most divergent term in the wave equation: the order , flat-space Laplacian term
At the next order, , along with the acceleration of the world line and the time-derivative of the mass, first appears in the Einstein equation. The order term in the wave equation is
At the next order, , along with squares and derivatives of the acceleration, first appears in the Einstein equation, and the tidal fields of the external background couple to . The order term in the wave equation becomes
Substituting this into the order terms in the gauge condition, we find
Thus, the order component of is
To summarize the results of this section, we have , where is given in Eq. (22.27), is given in Eq. (22.29), and is given in Eq. (22.38). In addition, we have determined that the ADM mass of the internal background spacetime is time-independent, and that the acceleration of the body’s world line vanishes at leading order.
Though the calculations are much lengthier, solving the second-order Einstein equation in the buffer region is essentially no different from solving the first. We seek to solve the approximate wave equation (22.17), along with the gauge condition (22.19), for the second-order perturbation ; doing so will also, more importantly, determine the acceleration . In this calculation, the acceleration is set to everywhere except in the left-hand side of the gauge condition, , which is linear in .
We first summarize the results. As at first order, the metric perturbation contains a regular, free radiation field and a singular, bound field; but in addition to these pieces, it also contains terms sourced by the first-order perturbation. Again, the regular field requires boundary data to be fully determined. And again, the singular field is characterized by the multipole moments of the body: the mass dipole of the internal background metric , which measures the shift of the body’s centre of mass relative to the world line; the spin dipole of , which measures the spin of the body about the world line; and an effective correction to the body’s mass. The wave equation by itself imposes no restriction on these quantities, but by imposing the gauge condition we find the evolution equations – and (iii) the free radiation field created by the body – this is the self-force. We identify the world line as the body’s by the condition . If we start with initial conditions , then the mass dipole remains zero for all times if and only if the world line satisfies the equation
We now proceed to the details of the calculation. Substituting the expansion
To begin, the most divergent, order term in the wave equation reads
The most divergent, order terms in the gauge condition similarly involve only ; they read
The metric perturbation in this form depends on five free functions of time. However, from calculations in flat spacetime, we know that order terms in the metric perturbation can be written in terms of two free functions: a mass dipole and a spin dipole. We transform the perturbation into this “canonical” form by performing a gauge transformation (cf. Ref. ). The transformation is generated by , the effect of which is to remove and from the metric. This transformation is a refinement of the Lorenz gauge. (Effects at higher order in and will be automatically incorporated into the higher-order perturbations.) The condition then becomes . The remaining two functions are related to the ADM momenta of the internal spacetime:
At the next order, , because the acceleration is set to zero, does not contribute to , and does not contribute to . The wave equation hence reads
We next substitute and into the order terms in the gauge condition. The -component becomes
Thus, the order term in is given bybackground spacetime, which is based on the inner limit that holds fixed. A term of the form appears as a perturbation of this background, even when, as in this case, it is part of the mass monopole of the body. This is equivalent to the ambiguity in any expansion in one’s choice of small parameter: one could expand in powers of , or one could expand in powers of , and so on. It is also equivalent to the ambiguity in defining the mass of a non-isolated body; whether the “mass” of the body is taken to be or is a matter of taste. As we shall discover, the time-dependent part of is constructed from the tail terms in the first-order metric perturbation. Hence, the ambiguity in the definition of the mass is, at least in part, equivalent to whether or not one chooses to include the free gravitational field induced by the body in what one calls its mass. (In fact, any order incoming radiation, not just that originally produced by the body, will contribute to this effective mass.) In any case, we will define the “correction” to the mass as .
We next move to the order terms in the wave equation, and the order terms in the gauge condition, which read
Finally, we arrive at the order terms in the wave equation. At this order, the body’s tidal moments become coupled to those of the external background. The equation reads
Foregoing the details, after some algebra we can read off the solution
In solving Eq. (22.64), we also find that the logarithmic term in the expansion becomes uniquely determined:
We now move to the final equation in the buffer region: the order gauge condition. This condition will determine the acceleration . At this order, first contributes to Eq. (22.19):
We have now completed our calculation in the buffer region. In summary, the second-order perturbation in the buffer region is given by , where is given in Eq. (22.50), in Eq. (22.60), in Eq. (22.65), and in Eq. (22.68). In addition, we have found evolution equations for an effective correction to the body’s mass, given by Eq. (22.39), and mass and spin dipoles, given by Eqs. (22.59) and (22.71).
Eq. (22.71) is the principal result of our calculation. After simplification, it reads
Eq. (22.72) is a type of master equation of motion, describing the position of the body relative to a world line of unspecified (though small) acceleration, in terms of the metric perturbation on the world line, the tidal fields of the spacetime it lies in, and the spin of the body. It contains two types of accelerations: and . The first type is the second time derivative of the body’s mass dipole moment (or the first derivative of its ADM linear momentum), as measured in a frame centered on the world line . The second type is the covariant acceleration of the world line through the external spacetime. In other words, measures the acceleration of the body’s centre of mass relative to the centre of the coordinate system, while measures the acceleration of the coordinate system itself. As discussed in Section 21, our aim is to identify the world line as that of the body, and we do so via the condition that the mass dipole vanishes for all times, meaning that the body is centered on the world line for all times. If we start with initial conditions , then the mass dipole remains zero for all times if and only if the world line satisfies the equation[92, 93, 150] if we had not assumed that the acceleration possesses an expansion of the form given in Eq. (21.9).
In our self-consistent approach, we began with the aim of identifying by the condition that the body must be centered about it for all time. However, we could have begun with a regular expansion, in which the world line is taken to be the remnant of the body in the outer limit of with only fixed. In that case the acceleration of the world line would necessarily be -independent, so would be the full acceleration of . Hence, when we found , we would have identified the world line as a geodesic, and there would be no corrections for . We would then have arrived at the equation of motion (although they phrased their expansion in terms of an explicit expansion of the world line, with a deviation vector on , rather than the mass dipole, measuring the correction to the motion). It describes the drift of the body away from the reference geodesic . This drift is driven partially by the local curvature of the background, as seen in the geodesic-deviation term , and by the coupling between the body’s spin and the local curvature. It is also driven by the self-force, as seen in the terms containing and , but unlike in the self-consistent equation, the fields that produce the self-force are generated by a geodesic past history (plus free propagation from initial data) rather than by the corrected motion.
Although perfectly valid, such an equation is of limited use. If the external background is curved, then has meaning only if the body is “close” to the world line. Thus, is a meaningful acceleration only for a short time, since will generically grow large as the body drifts away from the reference world line. On that short timescale of validity, the deviation vector defined by accurately points from to a “corrected” world line ; that world line, the approximate equation of motion of which is given in Eq. (22.73), accurately tracks the motion of the body. After a short time, when the mass dipole grows large and the regular expansion scheme begins to break down, the deviation vector will no longer correctly point to the corrected world line. Errors will also accumulate in the field itself, because it is being sourced by the geodesic, rather than corrected, motion.
The self-consistent equation of motion appears to be more robust, and offers a much wider range of validity. Furthermore, even beyond the above step, where we had the option to choose to set either or to zero, the self-consistent expansion continues to contain within it the regular expansion. Starting from the solution in the self-consistent expansion, one can recover the regular expansion, and its equation of motion (22.74), simply by assuming an expansion for the world line and following the usual steps of deriving the geodesic deviation equation.
Regardless of which equation of motion we opt to use, we have now completed the derivation of the gravitational self-force, in the sense that, given the metric perturbation in the neighbourhood of the body, the self-force is uniquely determined by irreducible pieces of that perturbation. Explicitly, the terms that appear in the self-force are given by
However, before doing so, we emphasize some important features of the self-force and the field near the body. First, note that the first-order external field splits into two distinct pieces. There is the singular piece , given by
Next, there is the Detweiler–Whiting regular field , given by
Now, the acceleration of the body is given by[89, 90, 91] that exerts a direct force on the body, and a nonhomogeneous field that exerts only an indirect force by altering the body’s multipole moments. His results should be generalizable to the gravitational case as well.
We now turn to the question of how the world line transforms under a gauge transformation. We begin with the equation of motion (22.41), presented again here:
Suppose that we had not chosen a world line for which the mass dipole vanishes, but instead had chosen some “nearby” world line. Then Eq. (22.85) provides the relationship between the acceleration of that world line, the mass dipole relative to it, and the first-order metric perturbations (we again neglect spin for simplicity). The mass dipole is given by , which has the covariant form
Now consider a gauge transformation generated by , where is bounded as , and diverges as . More specifically, we assume the expansions and . (The dependence on appears in the form of dependence on proper time . Other dependences could appear, but it would not affect the result.) This transformation preserves the presumed form of the outer expansion, both in powers of and in powers of . The metric perturbations transform as
From these results, we find that the left- and right-hand sides of Eq. (22.88) transform in the same way:equation is valid in any gauge, not that the value of the acceleration is the same in every gauge. Under a gauge transformation, a new fixed world line is adopted, and the value of the acceleration on it is related to that on the old world line according to Eq. (22.93). In the particular case that has no angle-dependence on the world line, this relationship reduces to . (Here we’ve replaced the tidal field with its expression in terms of the Riemann tensor to more transparently agree with equations in the literature.) An argument of this form was first presented by Gralla  for the case of a regular expansion, and was extended to the case of a self-consistent expansion in Ref. .
Living Rev. Relativity 14, (2011), 7
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