i,. ,,lar, remainder, which is the difference between the retarded field and singular
field, is believed to be smooth and well behaved on the world-line of the particle,
and is summed over all modes to provide the self-force or the radiation reaction
effect. The mode-decomposed solution of the retarded field is obtained numerically
while the mode-decomposed piece of the singular field is determined in a purely
analytic manner. The latter is generically referred to as BR.l, ,lri, .: -i.:..n Parameters.
In this ('!h plter we deal with the self-force of a scalar charge orbiting a
Schwarzschild black hole. Section 4.1 introduces a recent method to split the
retarded scalar field in curved spacetime -i '-.- -1. 1 by Detweiler and Whiting
[16]. This follows Dirac's idea in his flat spacetime problem [2], and gives good
interpretations not only in the singular behavior of the retarded field, but also in
its differentiability. In Section 4.2 we give the overview of the mode-sum method
originated by Barack and Ori [9], and present the analytic results for the singular
field, i.e. the regularization parameters, which were obtained by Kim [17]. The
regularization parameters are locally well defined and should describe the singular
behavior and the differentiability of the field precisely. Higher order expansions of
the singular field will generate higher order regularization parameters, and their use
in the mode sums for the self-force calculation will result in more rapid convergence
and more accurate final results. To facilitate the computations of the regularization
parameters, an in-depth analysis of the local spacetime would be demanded, and in
Section 4.3 we develop an elaborate perturbation analysis of the local geometry for
this purpose. Then, Section 4.4 is devoted to the calculations of the regularization
parameters. These results are then combined with the numerical computations of
the retarded field to provide the self-force ultimately. This final task is done in the
Section 4.5.