At the retarded time, p' is on the past null cone emanating from p, where
a(p,p') = 0, and the first factor in the square brackets in Eq. (4-71) is
T' T + XI"Io p(l + 'oo) ~ p
(4-72)
due to the fact that T' T < 0 and IT' TI ~ p, and the second must be
T' T + X'N0o + p(l + Hoo) p-1 x O(p6/j4) O(p5/R4)
(4-73)
to cancel the term O(p6/TR4) in Eq. (4-71) such that a(p,p') = 0 precisely. Then,
the differentiation of Eq. (4-71) with respect to T', evaluated at the retarded time
is dominated by the differentiation of the second factor,
da(p,p')] -( oo) [7"- 7 + XIio p(l + oo)] + O(p6/75)
dT- I ret ret
1
2(1 Noo) [-2p( + Noo) + O(p5/R4)]
p [1 + O(p4/4)] ,
(4-74)
where the second equality follows from the fact that T -p for the retarded
time, and the third equality from Eq. (4-65) [18].
Similarly, at the advanced time, p' lies on the future null cone of p, where
a(p,p') = 0, and the first and second factors in the square brackets in Eq. (4-71)
now reverse their roles,
T' T + X',Ho p(l + ,oo) ~ O(p5/R4)
(4-75)
and
T' T + XI7-po + p(1 + -oo) ~ p,
(4-76)
due to the facts that T' T > 0 and 7T' TI ~ p and that r(p,p') = 0. Then, the
differentiation of Eq. (4-71) with respect to 7', evaluated at the advanced time is