In this approach, the self-force is considered to result via
Fa qV= V (4-1)
from the interaction of the charge with the field
.1 = ret direct. (4-2)
Now is investigated in the following manner. We have the scalar field equation
V2 -47Q, (4-3)
where
Q(P) = q (g)-1/2(4) (- p'()) d (4-4)
is the source function for a scalar charge q moving along a world-line F, described
by p'(r), with 7 representing the proper time along the world-line. This field
equation is solved in terms of a Green's function,
V2G(p,p') (-g)-1/ (4)(p- p'). (4-5)
The scalar field of this charge is then
b(p) 47 fG(p, p') Q(p')dp'
(p)
S 4q G [p, p'()] dr. (4-6)
Jr
Dewitt and Brehme analyze the scalar field in curved spacetime using the
Hadamard expansions of the Green's function near F. A bi-scalar quantity a(p,p'),
termed Synge's .- i Id function" [13] is defined as half of the square of the distance
measured along a geodesic from p to p', and a < 0 for a timelike geodesic, a = 0 on
the past and future null cones of p, and a > 0 for a spacelike geodesic. The usual
symmetric scalar field Green's function is derived from the Hadamard form to be