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