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