When the source point p' is on F, Synge's world function a(p,p') is particularly
easy to evaluate in THZ coordinates for p close to p'. The world function is shown
by Thorne and Kovhcs [22] to be
a(p,p')= 1xAB (TAB + HABdA + O(p6/T4), (4-63)
where XA is the THZ coordinate representation of the field point p while the source
point p' is represented by (T', 0, 0, 0) [18]. The integration of the coordinate along a
straight path is given by
(A() A(XXA '6A), (4-64)
where A runs from 0 to 1.
Working through only the lower order expansions of the perturbed field,
namely HAB 2HAB + 3HAB + O(p4/R4), the integral of HAB along the straight
path C is evaluated with the help of Eq. (4-64) to be [18]
Hoo /HoodA
Jc
SKL + SKLM K L ] dA + O(P/Z4)
SSKLXKXL -- KLMXKXLX + O(p44/R), (4-65)
3 12
Hoi HoidA
I 2 eKP8 L KL 1_ KLoK L
21 3 J
+ 4 KI% K + IKP8 LM LM d A+ (p/4)
2 5
SIKP8 L KX L KLX XKLX
9 42
1 1
+ tSKIXK 2 + KPi LMX KXLX M + O(p4/R4) (4-66)
21 12