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

Since this expansion relies on both an expansion at fixed and an expansion at fixed , it can be expected to be accurate if and – that is, in the buffer region .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

Since this expansion relies on both an expansion at fixed and an expansion for small , it can be expected to be accurate in the buffer region . As we can see, the two buffer-region expansions have an identical form; and because they are expansions of the same exact metric , they must agree term by term.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. [145] 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. [144]. 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 [138]. 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

where indices refer to Fermi coordinates, is linear in and its derivatives, and for future compactness of notation we define , where the subscript ‘B’ stands for ‘buffer’. In the case that , these expansions will significantly reduce the complexity of calculations in the buffer region. For that reason, we shall use them in solving the second-order wave equation, but we stress that they are simply a means of economizing calculations in Fermi coordinates; they do not play a fundamental role in the formalism, and one could readily do without them.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

As mentioned above, since the outer expansion has no negative powers of , we exclude them from the inner expansion. Furthermore, since , we must have , since these are the only terms independent of both and . Thus, noting that , we can write implying that the internal background spacetime is asymptotically flat.We assume that the perturbation can be similarly expanded in powers of at fixed ,

and that each coefficient can be expanded in powers of to yield where , the coefficient of and , is a function of and (and potentially a functional of ). Again, the form of this expansion is constrained by the fact that no negative powers of can appear in the buffer region. (One might think that terms with negative powers of could be allowed in the expansion of if they are exactly canceled by terms in the expansion of , but the differing powers of in the two expansions makes this impossible.) Note that explicit powers of appear because . Also note that we allow for a logarithmic term at second order in ; this term arises because the retarded time in the internal background includes a logarithmic correction of the form (e.g., in Schwarzschild coordinates). Since we seek solutions to a wave equation, this correction to the characteristic curves induces a corresponding correction to the first-order perturbations.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:

where each depends only on and , along with an implicit functional dependence on . If the internal expansion is performed with held fixed, then the internal and external quantities are related order by order: e.g., , , and . Since we are not concerned with determining the internal perturbations, the only such relationship of interest is . This equality tells us that the most divergent, piece of the th-order perturbation is defined entirely by the th-order piece of the internal background metric , which is the metric of the body if it were isolated.To obtain a general solution to the Einstein equation, we write each as an expansion in terms of irreducible symmetric trace-free pieces:

Here a hat indicates that a tensor is STF with respect to , angular brackets indicate the STF combination of enclosed indices, parentheses indicate the symmetric combination of enclosed indices, and symbols such as are functions of time (and potentially functionals of ) and are STF in all their indices. Each term in this expansion is linearly independent of all the other terms. All the quantities on the right-hand side are flat-space Cartesian tensors; their indices can be raised or lowered with . Refer to Appendix B for more details about this expansion.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

where and . In analogy with the notation used for , and would be linear in , and would be linear in and quadratic in , and so on. For a function , , , and correspond to the coefficients of in expansions in powers of . Equation (22.16) is identical to Eq. (21.4). Equation (22.17) follows directly from substituting Eqs. (22.3) and (22.4) into Eq. (21.5); in the buffer region, it captures the dominant behaviour of , represented by the approximation , but it does not capture its full dependence on acceleration. If one desired a global second-order solution, one might need to solve Eq. (21.5), but for our purpose, which is to determine the first-order acceleration , Eq. (22.17) will suffice.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

where the first equation is identical to Eq. (21.10), and the second to Eq. (21.11). (The second identity holds because , since differs from by and higher acceleration terms, which are set to zero in .) We remind the reader that while this gauge choice is important for finding the external perturbations globally, any other choice would suffice in the buffer region calculation.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

Hence, at leading order, the body behaves as a test particle, with constant mass and vanishing acceleration.

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

The -component of this equation is from which we read off that is arbitrary and must vanish for all . The -component is from which we read off that is arbitrary and all other coefficients must vanish. Lastly, the -component is from which we read off that and are arbitrary and all other coefficients must vanish. Thus, we find that the wave equation constrains to be This is further constrained by the most divergent, term in the gauge condition, which reads From the -component of this equation, we read off ; from the -component, and . Thus, depends only on a single function of time, . By the definition of the ADM mass, this function (times ) must be twice the mass of the internal background spacetime. Thus, is fully determined to be where is defined to be the mass at time divided by the initial mass . (Because the mass will be found to be a constant, is merely a placeholder; it is identically unity. We could instead set equal to unity at the end of the calculation, in which case would simply be the mass at time . Obviously, the difference between the two approaches is immaterial.)

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

where the terms on the right arise from the wave operator acting on . This equation constrains to be Substituting this result into the order term in the gauge condition, we find Thus, both the leading-order part of the acceleration and the rate of change of the mass of the body vanish:

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

From the -component, we read off that is arbitrary, , and ; from the -component, , , and are arbitrary, , and ; from the component, , , , and are arbitrary, and , , , and .Substituting this into the order terms in the gauge condition, we find

where the equation is to be evaluated at . From the -component, we read off From the -component, It is understood that both these equations hold only when evaluated at .Thus, the order component of is

where and are constrained to satisfy Eqs. (22.36) and (22.37).

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

The first of these tells us that the free radiation field created by the body creates a time-varying shift in the body’s mass. We can immediately integrate it to find We note that this mass correction is entirely gauge dependent; it could be removed by redefining the time coordinate on the world line. In addition, one could choose to incorporate into the leading-order mass . The second of the equations tells us that the body’s spin is constant at this order; at higher orders, time-dependent corrections to the spin dipole would arise. The last of the equations is the principal result of this section. It tells us that the relationship between the acceleration of the world line and the drift of the body away from it is governed by (i) the local curvature of the background spacetime, as characterized by – this is the same term that appears in the geodesic deviation equation – (ii) the coupling of the body’s spin to the local curvature – this is the Papapetrou spin force [138] – 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 This is the equation of motion we sought. It, along with the more general equation containing , will be discussed further in the following section.

We now proceed to the details of the calculation. Substituting the expansion

and the results for from the previous section into the wave equation and the gauge condition again yields a sequence of equations that can be solved for coefficients of successively higher-order powers (and logarithms) of . Due to its length, the expansion of the second-order Ricci tensor is given in Appendix A. Note that since the approximate wave equation (22.17) contains an explicit correction, will be determined only up to corrections. For simplicity, we omit these symbols from the equations in this section; note, however, that these corrections do not effect the gauge condition, as discussed above.To begin, the most divergent, order term in the wave equation reads

where the right-hand side is the most divergent part of the second-order Ricci tensor, as given in Eq. (A.3). From the -component of this equation, we read off and that is arbitrary. From the -component, , , and are arbitrary. From the -component, , , and , , , and are arbitrary.The most divergent, order terms in the gauge condition similarly involve only ; they read

After substituting the results from the wave equation, the -component of this equation determines that . The -component determines that , , and Thus, the order part of is given by where is given by Eq. (22.47).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. [45]). 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:

where is such that is proportional to the ADM linear momentum of the internal spacetime, and is the ADM angular momentum. is a mass dipole term; it is what would result from a transformation applied to the term in . is a spin dipole term. Thus, the order part of reads

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

where is given in Eqs. (A.4)–(A.6). The -component of this equation implies , from which we read off that is arbitrary and . The -component implies , from which we read off and that is arbitrary. The -component implies from which we read off that is arbitrary, , , , and is arbitrary. This restricts to the formWe next substitute and into the order terms in the gauge condition. The -component becomes

from which we read off And the -component becomes from which we read off and that the angular momentum of the internal background is constant at leading order:Thus, the order term in is given by

Note that the undetermined function appears in precisely the form of a mass monopole. The value of this function will never be determined (though its time-dependence will be). This ambiguity arises because the mass that we have defined is the mass of the internal background 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

From this we determineFinally, 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

where comprises the contributions from and , given in Eqs. (A.10), (A.15), and (A.21). The contribution from the second-order Ricci tensor is given in Eqs. (A.7) – (A.9).Foregoing the details, after some algebra we can read off the solution

where each one of the STF tensors is listed in Table 1.In solving Eq. (22.64), we also find that the logarithmic term in the expansion becomes uniquely determined:

This term arises because the sources in the wave equation (22.64) contain a term , which cannot be equated to any term in . Thus, the wave equation cannot be satisfied without including a logarithmic term.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):

The contribution from is most easily calculated by making use of Eqs. (B.24) and (B.25). After some algebra, we find that the -component of the gauge condition reduces to and the -component reduces to After removing common factors, these equations become Eqs. (22.39) and (22.41). We remind the reader that these equations are valid only when evaluated at , except in the term that arose from .

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

We recall that is the body’s mass dipole moment, and are components of the Riemann tensor of the background spacetime evaluated on the world line, is the first-order acceleration of the world line, is the body’s spin angular momentum, and , are vector fields on the world line that have yet to be determined. The equation is formulated in Fermi normal coordinates.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

This equation of motion contains two types of terms: a Papapetrou spin force, given by , which arises due to the coupling of the body’s spin to the local magnetic-type tidal field of the external spacetime; and a self-force, arising from homogenous terms in the wave equation. Note that the right-hand side of this equation is to be evaluated at , and that it would contain an antidamping term [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

This equation of motion was first derived by Gralla and Wald [83] (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

This is all that is needed to incorporate the motion of the body into a dynamical system that can be numerically evolved; at each timestep, one simply needs to calculate the field near the world line and decompose it into irreducible pieces in order to determine the acceleration of the body. The remaining difficulty is to actually determine the field at each timestep. In the next section, we will use the formal integral representation of the solution to determine the metric perturbation at the location of the body in terms of a tail integral.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

This field is a solution to the homogenous wave equation for , but it is divergent at . It is the generalization of the Newtonian field of the body, as perturbed by the tidal fields of the external spacetime . Comparing with results to be derived below in Section 23.2, we find that it is precisely the Detweiler–Whiting singular field for a point mass.Next, there is the Detweiler–Whiting regular field , given by

This field is a solution to the homogeneous wave equation even at . It is a free radiation field in the neighbourhood of the body. And it contains all the free functions in the buffer-region expansion.Now, the acceleration of the body is given by

which we can rewrite as where . In other words, up to order errors, a body with order or smaller spin (i.e., one for which ), moves on a geodesic of a spacetime , where is a free radiation field in the neighbourhood of the body; a local observer would measure the “background spacetime,” in which the body is in free fall, to have the metric instead of . If we performed a transformation into Fermi coordinates in , the metric would contain no acceleration term, and it would take the simple form of a smooth background plus a singular perturbation. Hence, the Detweiler–Whiting axiom is a consequence, rather than an assumption, of our derivation, and we have recovered precisely the picture it provides in the point particle case. In the electromagnetic and scalar cases, Harte has shown that this result is quite general: even for a finite extended body, the field it produces can be split into a homogeneous field [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:

Setting , we derive the first-order acceleration of , given in Eq. (22.73). If, for simplicity, we neglect the Papapetrou spin term, then that acceleration is given by where it is understood that explicit appearances of the acceleration are to be set to zero on the right-hand side. The first equality follows directly from Eq. (22.73) and the definitions of and . The second equality follows from the STF decomposition of and the integral identities (B.26) – (B.28). We could also readily derive the form of the force given by the Quinn–Wald method of regularization: . However, in order to derive a gauge-invariant equation of motion, we shall use the form in Eq. (22.86).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

where a primed index corresponds to a point on the world line. Note that the parallel propagator does not interfere with the angle-averaging, because in Fermi coordinates, . One can also rewrite the first-order-metric-perturbation terms in Eq. (22.85) using the form given in Eq. (22.86). We then have Eq. (22.85) in the covariant formNow 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

Using the results for , the effect of this transformation on is given by The order term arises from in the gauge transformation. On the right-hand side of Eq. (22.88), the metric-perturbation terms transform as The only remaining term in the equation is . If we extend the acceleration off the world line in any smooth manner, then it defines a vector field that transforms as . Since , this means that – it is invariant under a gauge transformation. It is important to note that this statement applies to the acceleration on the original world line; it does not imply that the acceleration of the body itself is gauge-invariant.From these results, we find that the left- and right-hand sides of Eq. (22.88) transform in the same way:

Therefore, Eq. (22.88) provides a gauge-invariant relationship between the acceleration of a chosen fixed world line, the mass dipole of the body relative to that world line, and the first-order metric perturbations. So suppose that we begin in the Lorenz gauge, and we choose the fixed world line such that the mass dipole vanishes relative to it. Then in some other gauge, the mass dipole will no longer vanish relative to , and we must adopt a different, nearby fixed world line . If the mass dipole is to vanish relative to , then the acceleration of that new world line must be given by , where Hence, this is a covariant and gauge-invariant form of the first-order acceleration. By that we mean the 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 as first derived by Barack and Ori [17]. (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 [81] for the case of a regular expansion, and was extended to the case of a self-consistent expansion in Ref. [145].
Living Rev. Relativity 14, (2011), 7
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