such that the integral over the surface E can be computed explicitly in terms of an
integral over proper time and an integral over solid angle. By letting r and T2 both
approach r, Eq. (2-144) becomes
0 mo dr + lim ga'TabdEb.
e->0 J4,
(2-146)
One shall focus on the evaluation of the second term of this equation to derive the
equations of motion of the electric charge.
First, the retarded and advanced field strength tensors of Eq. (2-129) must
be expressed in the form of expansions. After a very tedious algebra involving a
number of perturbations one finds
F _g 1 -1 2Xa'Qb' + 1 -3.a' b' 1 5 '2
2ega[a'9lbb'] -2 a5 a/ + 2
L 2
-1 1-3* a''b2' _2 -4-a'b' + lnah'2c 'R
2 3 12
+-1/a RbK-I c, s if ~-- a' b' c' ,c + -1la'Qb' Pc'd' c' d'
2 12
1 1 b' d ,c' da -lb cL ,c' d
2 12
t --K 3 a c'd'e' ,c'd'Q 1 -2a Z bc',
6 3
Tadv/ret
where the tail term has been written in terms of the Green's function G /ret (x, z(r))
rather than the Hadamard expansion term Vaa' (x, z(r)) for later convenience.
From Eq. (2-147) it follows that the field Fradab is everywhere finite. At the
location of the particle, it is described as
Fret a'b' adv a'b'
14, (a *' + 42b' K-2[a')R b]cc
3 3
+2e I V[b'Gret a'] Cc" (Tr")dr" d [badva'] c,"
7-0o J ,Tadv
(2-148)
F adv/ret /
Iab r )
Frad a'b'