Suppose we take our buffer-region expansion of to be valid everywhere in the interior of (in ), rather than just in the buffer region. This is a meaningful supposition in a distributional sense, since the singularity in is locally integrable even at . Note that the extension of the buffer-region expansion is not intended to provide an accurate or meaningful approximation in the interior; it is used only as a means of determining the field in the exterior. We can do this because the field values in are entirely determined by the field values on , so using the buffer-region expansion in the interior of leaves the field values in unaltered. Now, given the extension of the buffer-region expansion, it follows from Stokes’ law that the integral over in Eq. (21.8) can be replaced by a volume integral over the interior of the tube, plus two surface integrals over the “caps” and , which fill the “holes” in and , respectively, where they intersect . Schematically, we can write Stokes’ law as , where is the interior of . This is now valid as a distributional identity. (Note that the “interior” here means the region bounded by ; does not refer to the set of interior points in the point-set defined by .) The minus sign in front of the integral over accounts for the fact that the directed surface element in Eq. (21.8) points into the tube. Because does not lie in the past of any point in , it does not contribute to the perturbation at . Hence, we can rewrite Eq. (21.8) as
Gralla and Wald  provided an alternative derivation of the same result, using distributional methods to prove that the distributional source for the linearized Einstein equation must be that of a point particle in order for the solution to diverge as . One can understand this by considering that the most divergent term in the linearized Einstein tensor is a Laplacian acting on the perturbation, and the Laplacian of is a flat-space delta function; the less divergent corrections are due to the curvature of the background, which distorts the flat-space distribution into a covariant curved-spacetime distribution.
We have just seen that the solution to the wave equation with a point-mass source is given by
We expand the direct term in in powers of using the following: the near-coincidence expansion ; the relationship between and , given by Eq. (11.5); and the coordinate expansion of the parallel-propagators, obtained from the formula , where the retarded tetrad can be expanded in terms of using Eqs. (11.9), (11.10), (9.12), and (9.13). We expand the tail integral similarly: noting that , we expand about as ; each term is then expanded using the near-coincidence expansions and , where barred indices correspond to the point , and is given by
Combining the expansions of the direct and tail parts of the perturbation, we arrive at the expansion in Fermi coordinates:
The Detweiler–Whiting singular field is given by
The regular field could be calculated from the regular Green’s function. But it is more straightforwardly calculated using . The result is
With the metric perturbation fully determined, we can now express the self-force in terms of tail integrals. Reading off the components of from Table 2 and inserting the results into Eq. (22.84), we arrive at
Living Rev. Relativity 14, (2011), 7
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