where


 2HO -2S1jXIXJ
 2HOK 2 10 JIJ K 2K I 2]
 2t = PKpQBQIXpXI+ 21
 3 21 5

 2HIJ = X(;' J)PQQKXpXK PQ(IJ) Qxp2 (4-59)

and


 3Hoo0 lIJKXIXJXK
 3
 3HOK KPQBQIjXp XJ
 3
 3HIJ = O(p4/4). (4-60)

 The metric perturbation HAB is the trace reversed version of HAB at linear

order,
 1
 HAB HAB- 9ABHCc, (4-61)
 2
and the expansion shown in Eqs. (4-49)-(4-51) precisely corresponds to Zhang's

[21], the first leading terms of which are expressed in Eqs. (4-58)-(4-60).
4.3.2 Approximation for the Singular Field in THZ Coordinates

 It was seen at the beginning of the previous Subsection that the motivation

for the use of THZ coordinates is to obtain a simpler approximation of the singular

source field, as represented by Eq. (4-45). The result in this equation can be

derived via Eq. (4-17). We develop local expansions in the THZ coordinates for the

elements u(p,p'), & and v(p,p') on the right hand side of the equation, and combine

them to give an approximate expression for bs.

 First, for a vacuum spacetime (RAB 0) which is nearly flat, according to

Thorne and Kovhcs [22] we have


u(p,p') = 1 + (p4/ 4).


(4-62)