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)