4.5 An Example: Self-force on Circular Orbits about a Schwarzschild
Black Hole
In this Section we present an example of self-force calculation using the
formalism developed above. We assume that the particle's orbit is a circle with the
radius ro 10M about a Schwarzschild black hole of mass M. In the numerical
work we put M = 1 and also the scalar charge is assumed to be q = 1. For a
circular orbit problem, only the r-component of the self-force is non-vanishing.
From Eq. (4-32) we then have
self lim [r (r) f,(r)] Z (ro), (4-259)
t t
via which we compute our self-force.
Ffr is determined by numerically integrating the scalar wave equation (3-
10) along with Eq. (3-11). The practical details for this task are described well
in Appendix E of Ref. [18]. And from Eq. (4-33) we have a description for the
singular part JFS, in the coincidence limit r -- ro,
( s 1 22]0Dr
lim Fs, + A, + B, 1)(2 3)
rro 2 (2f- 1)(2+ 3)
)(-1)k+12k+3/2 [(2k + 1)!!]2
S(2 2k 1)(2- 2k + 1)... (2 + 2k + 1)(2 + 2k + 3)
+O(-%2(k+2)), (4-260)
where the regularization parameters are simplified from those for a general orbit
Eqs. (4-34)-(4-44) due to the condition that r = 0:
[ro(ro 3M)1/2
Ar, -sgn(A) ((r (4-26)
(o- 2M)
B ro 3M 11/2[ (o _23M)F /2] (4-262)
B,, L(ro -- 2-M) /12- 2(r0 -- 2M) ] '