2 j A3/2[1/2 2 + (1 f)E2 l /2
7/2- 2 4fA1/2
[(f + 9)2 (13f + 9)E2 + 3f(f + 8)] F3/2
8fA3/2
+ [(9 f)r4 + ((10f 18)E2 f(5f + 8)) r2
+9(1 f)E4 3f(13f 8)E2 + 21f3] F5/
8f2A5/2
5 [-(2f + 6)>4 + ((12 4f)E2 + 12f2) P2 + 6(f 1)E4 3f2E2] F7/2
35- t /2F/2 (4-43)
8A 7/2 I
Ao = Boe Co = Do = 0, (4-44)
where A r r- and E -ut = (1 2M/or) (dt/d'r) ('r: proper time)
and J u =- ro (df/dr)o are the conserved energy and angular momentum,
respectively, and = u" = (dr/dr)o, f = (1- 2M/ro), A (1 + J2/r2). The
subscript o denotes evaluation at the location of the particle. Also, shorthand for
the hypergeometric function is Fp 2F1 (p, 2; 1; J2/(r + J2)) (see Appendix A
for more details about the hypergeometric functions and the representations of the
regularization parameters in terms of them).
4.3 Description of Singular Field and THZ Coordinates
4.3.1 Introduction of THZ Coordinates
Intuitively, one may expect that the singular source field bs will behave like
a Coulomb scalar field due to its analogy to Dirac's mean of the advanced and
retarded fields in flat spacetime [2]. In fact, this intuition can be supported via
some local analysis of is. If bs resembles the Coulomb q/r piece of the scalar
potential near the particle with scalar charge q, where r is the distance between a
source point p' and a nearby field point p, we can think of bs as the field measured
by a local observer sitting on the particle, to whom the background geometry in the
vicinity of his location looks flat. The description of bs will then be advantageously
simple in this observer's frame of reference, and we are motivated to use some