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 [83] 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
One can also obtain this result from Eq. (19.29 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:
As the final step, each of these terms is decomposed into irreducible STF pieces using the formulas (B.1![]() |
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The Detweiler–Whiting singular field is given by
Using the Hadamard decomposition 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
http://www.livingreviews.org/lrr-2011-7 |
Living Rev. Relativity 14, (2011), 7
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