G (p p') [u(p,p')(a) v(p,p')(-a)] (4-7)
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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