Taking its derivative with respect to x', the coordinates for p, one obtains [6]
qv [p, p'(rt)] Varet = q [vT1 Vaa
qR(p) X
R (p) xa ) + O(p/3, p (4-14)
12p
where Eq. (4-9) was used for v [p,p'(Qret)] near F. The spatial part of the right
hand side of Eq. (4-14) is not defined when p is on F, thus the differentiability is
not guaranteed in general on the world-line if the Ricci scalar of the background
is not zero-similarly, the electromagnetic potential At1il and the gravitational
metric perturbation h1ai are not differentiable at the point of the particle unless
(Rab 'gabR) Ub and R ,,, R ,i', respectively, are zero in the background [16].
Therefore, in order to obtain a well defined contribution to the self-force out of the
tail part, one first averages Va', "' over a small, spatial two-sphere surrounding the
particle, thus removing the spatial part of Eq. (4-14), then takes the limit of this
average as the radius of the two-sphere tends to zero [3, 4, 5, 6].
4.1.2 New Method of Splitting the Retarded Field
According to Detweiler and Whiting [16], a new symmetric Green's function
can be constructed by adding to the first in Eq. (4-7) any bi-scalar which is
a homogeneous solution of Eq. (4-5). Dewitt and Brehme [3] show that the
symmetric bi-scalar v(p,p') is a solution of the homogeneous wave equation,
V2(p,p') = 0. (4-15)
Then, using this we generate a new symmetric Green's function
Gs(p,p') GY(p,p') + -v(p,p')
87
S[u(p,p') () + v (p,p')e(a)]. (4-16)
87
This new symmetric Green's function has support on the null cone of p just as
Gsym does, and has support outside the null cone, but not within the null cone,