to be responsible for the self-force. Descriptions of the two fields are given in the
following Subsections.
5.3.1 Singular Field hsb
A description of the metric perturbation hab near a point mass p moving
along a geodesic world-line F is most convenient with the coordinates in which
the background geometry looks as flat as possible in the vicinity of the particle.
As it did for the scalar field problem, the Thorne-Hartle-Zhang normal coordinate
system introduced in Section 4.3 of C'! lpter 4 will serve this purpose best also in
our gravitational self-force problem. With THZ coordinates the background metric
is
gAB = AB + 2HAB + O(p3/73), (5-18)
where 2HAB is defined by Eq. (4-50) and p (X2 + y2 + Z2)1/2 with X, Y,
Z being the spatial THZ coordinates and R represents a length scale of the
background geometry.
To avoid the singularity in hab in our perturbation analysis of Section 5.2,
we replace the point particle model by a small Schwarzschild black hole. Then
the difficulty caused by the formal singularity is replaced by the requirement of
boundary conditions at the event horizon. When a small Schwarzschild black hole
of mass p moves through a background spacetime, the metric of the small black
hole is perturbed by tidal forces arising from 2HAB in Eq. (5-18), and the actual
metric near the black hole is written in THZ coordinates as [10]
gA = g"A + 2hAB + O(p3/3), (5-19)
where the quadrupole metric perturbation 2hAB is a solution of the perturbed
Einstein equations (5-6) with the appropriate boundary conditions that the
perturbation be well behaved on the future event horizon and that 2hAB 2HAB
in the buffer region [20], where p < p < R. For p < 7?, 2/AB is governed by