now dominated by the differentiation of the first factor,
d(p, p -) (1- 'oo) ) [T' + X"Hno o+p(lt + oo)] + O(p6/R5)
dL adv 2adv
(1 -oo) [2p(1 + Roo) + O(p5/i4)]
p [1 + O(p4/R4)], (4-77)
where the second equality results from the fact that T' T p for the advanced
time.
Dewitt and Brehme show that in general
1
v(p,p') = R(p') + O(p/n3), p F. (4-78)
12
However, in vacuum spacetime, where R = 0,
v(p,p') = O(p2/n4). (4-79)
When integrated over the proper time, the dominant contribution from this term is
O(p3/R4) in the coincidence limit p F.
Then substituting all the results in Eqs. (4-62), (4-74), (4-77) and (4-78) into
Eq. (4-17), we eventually obtain [18]
s + O(p3/R4). Q. E.D. (4-80)
P
From Eqs. (4-49)-(4-51), one notes that our expressions of THZ coordinates
are well defined up to the addition of a term O(p5/R4), which corresponds to a
term O(p4/R4) in the metric perturbation. The change in p due to the addition
of such a term is O(p3/R4), which would be consistent with the order term in
Eq. (4-80). The differentiability of the order term is of interest, and a term of
O(p3/R4) is C2 in the limit p --- 0. From the fact that R = ret _- s, where
'R is a homogeneous solution of the scalar wave equation, we find that Eq. (4-80)
clarifies the relationship between the accuracy of an approximation for bs and the