unlike Gsym. We consider GS(p,p') only in a local neighborhood of the particle,
thus the use of Gs(p,p') is not complicated by the need for knowledge of the entire
past history of the source and is amenable to local analysis. By Eqs. (4-6) and
(4-16), the corresponding field is
s(p)= -q ,() + q ) + v [p, p'(r)] dr, (4-17)
2 o t| 2 I L r2jt
which is an inhomogeneous solution of Eq. (4-3) just as i, is, and is analogous
to Dirac's singular field (Fb + F'dv). Following Dirac's pioneering idea, one can
define
GR(p,p') Gret(p,p) GS(p,p'). (4-18)
It is remarkable that like Gret(p,p'), GR(p,p') has no support inside the future null
cone. Corresponding to GR(p,p'), we construct
rR ret- S
qu [, p'(r)] Tadv (Tret 1 /Tadv)
q v [p, p'(T)] dT, (4-19)
2( T t 2 rett
1 b (F Fab >
which is analogous to Dirac's radiation field (Fe Fadv)
As both ', 1' and s are inhomogeneous solutions of the same differential
equation, Eq. (4-3), consequently, QR, as defined by the first line of Eq. (4-19)
is a homogeneous solution and therefore expected to be differentiable on F. The
relation between QR and is
q Tadv v [p, p'(Q)] d. (4-20)
VTret
Here is observed a result of great significance: qR can replace ', for an ex-
plicit computation of the self-force, since the integral term in Eq. (4-20) gives
no contribution to a self-force. For a field point p near F, via Eq. (4-9) and