which is shown to vanish [9]. Then the self-force is ultimately calculated as
.Fself l- [iimi _ret(P) (C) A (4-30)
e (p)- + A B D (4-30)
o 2 -P' a + -
Our approach closely follows Barack and Ori [9], but there is some difference in
the regularization scheme due to our different split of Qret as described in
Eq. (4-24). From our perspective, via Eq. (4-1) the self-force can be explicitly
evaluated from
pelf li [. et(p)- _F(p)] =-T(pI)
q lim Va('ret- s) qVaR, (4-31)
where similarly to the above, we expand both et(p) and FS(p) into multiple f-
modes e-,T Pt (p) and e-e^S,(p), respectively, with rt(p) determined numerically
and F~a(p) determined analytically. This implies that our self-force is
r
.Fself lir [Pm(p)-_ Sa(P)] yj t
q alim V (,, l q liVa (4-32)
p____tp/ ,, 7
m m
evaluated at the location of the particle. Here the individual fm components
'rt and are finite at the location of the particle even though their sums
are both singular. The f-mode derivatives F =- qVa Zm(,' itm) and Fsa
qVa (' "' YAm) are also finite at the point of the particle, and we take the
difference between the two, which is T, = qVa Em QmYm, then take the sum
of this quantity over which produces a convergent value for the self-force.
Our computation of the retarded field part is identical to that of Barack and
Ori, but our f mode-decomposition of the singular field part, i.e. Fsa is slightly
different from their J7r. We describe JTsa in the coincidence limit p -i p' via the
regularization parameters