by combining both analytical and numerical computations. In their analysis, the
self-force may be considered to be calculated from
self lim [Fet() di(p)], (4-25)
p-p'
where p' is the event on F where the self-force is to be determined and p is an
event in the neighborhood of p', and Fa(p) is related to Q(p) via Eq. (4-1). For
use of this equation, both Fret(p) and F"ir(p) would be expanded into multiple
f-modes, i.e y:, F7 (p) and Z:, F r' (p), respectively, where ire (p) is determined
numerically and Ff (p) determined analytically. In order to determine ~j (p), we
solve Eq. (4-3) using spherical harmonic expansions. The source Q in the equation
is expanded into spherical harmonics, and similarly the field Qret is expanded
', r ')(r, t), (4-26)
where (r, t) is found numerically. The individual components '(r, t) in this
expansion are finite at the location of the particle even though their sum, the right
hand side of Eq. (4-26) is singular. Then we have
qret q V iretv ) (4-27)
rn
which is also finite. The remaining part ff(p) is determined by a local analysis
of the Green's function in the neighborhood of the particle's world-line. Ref. [9]
provides
lim j-- (p) (i A B + + + O(f-2), (4-28)
P P' \ 2 / +
where Aa, Ba and Ca are constants and are generically referred to as R il., rization
Parameters. The remainder is defined as
D [- lim f (p) + 2)A B --e- (4-29)
So-P' 20 f -
=0