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