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