CHAPTER 4
 PRACTICAL SCHEMES FOR CALCULATIONS OF SELF-FORCE (A):
 SCALAR FIELD

 In C!I ipter 3 we reviewed the analytic approaches to the self-force calculations

by Dewitt and Dewitt [7] and Pfenning and Poisson [8]. From the reviews, it was

meaningful to see that their calculations show desired correspondence limit to

the flat spacetime theory or the agreement with the low-order post-Newtonian

approximations. In their calculations of the self-force, however, special conditions

such as non-relativistic velocities of particles and a weak gravitational field had

to be imposed to enable the entire calculations to be treated fully analytically.

More realistic self-force problems having more general conditions would require

completely different approaches for the calculations, in which we combine both

analytical and numerical methods.

 Here, we introduce an alternative approach to the self-force calculations,

known as the "mode-- iin method, which was originally devised by Barack and

Ori [9]. Employing both analytical and numerical techniques, this method does not

limit the particle's velocities and the field's strength, and should practically work

for all kinds of fields under consideration, whether it is scalar, electromagnetic,

or gravitational. It is particularly powerful for the problems in a spherically

symmetric spacetime, such as Schwarzschild. In principle, we take advantage of the

spherical symmetry of the background geometry to decompose the retarded Green's

function in the I il!" term into spherical-harmonic modes which can be computed

individually. Then, from the mode-decomposition of the retarded Green's function

we obtain a mode-decomposition of the retarded field, and from this subtract a

mode-decomposition of the -.:,lu;,lar field, which is locally well described. The