G (p p') [u(p,p')(a) v(p,p')(-a)] (4-7)
87
where u(p,p') and v(p,p') are bi-scalars described by Dewitt and Brehme, and their
expansions are known to be convergent within a finite neighborhood of F if the
geometry is analytic. In the vicinity of F, Dewitt and Brehme show that
u(p,p') = 1 + RabVaVba + O(p3/3), (4-8)
12
and that
v(p,p') = R(p') + O(p/R3), (4-9)
12
where p is the proper distance from p to F measured along the spatial geodesic
which is orthogonal to F, and R represents a length scale of the background
geometry (the the smallest of the radius of curvature, the scale of inhomogeneities
and time scale for changes in curvature along F). The ((-o) guarantees that only
when p and p' are timelike-related is there a contribution from v(p,p'). In any
Green's function the terms containing u and v are frequently referred to as the
"direct" and I i!" parts, respectively. Also, the retarded and advanced Green's
functions are expressed in terms of Gsym(p,p') as,
Gret(p,p/) 20 [E(p),p'] Gsym(p,p/),
Gadv (p p') = 20 [p', (p)] Gym(p,p'), (4-10)
respectively, where 0 [E(p),p'] = 1- 0 [p', (p)] equals 1 if p' is in the past
of a spacelike hypersurface E(p) that intersects p, and is 0 otherwise. As Dirac
decomposed the retarded electromagnetic field Fa'" into two parts as in Eq. (3-2),
we may try to decompose our scalar field 'ret into