where A, E and P, S = 0, 1,2, 3, and I = 1,2, 3. For the Schwarzschild
spacetime as the background, we may take the equatorial plane 0o = r/2 and
describe the source point in Eq. (4-131) as
x (to, ro, o) (4-136)
We also have
MA diag [fl/2 f-1/2, r ] (4-137)
together with all non-zero F% o for the background taken from Eq. (4-85). Describ-
ing the four-velocities,
i f-1/2E, f-1/2 -, (4-138)
( ro
(1)A -12 1+ f 0 (4-139)
0 t + fl2E+f' ro(E+f/2)' l
(2)A 1+ 0)} (4-140)
) ro o (E + f1/2)' + 2 (f-/2E + 1) 4
(3)A = (0, 0, 0, 1), (4-141)
where f 2M), and E -ut (1 2M/r) (dt/dr)o (T: proper time)
and J uu = ro (d/dr)o are the conserved energy and angular momentum
in the background, respectively, and i u" = (dr/dr)o. These are also used
to compute 7tAB and hAB via Eqs. (4-117) and (4-118). Also, one can evaluate
BC,D BCD C,DE ad ^a BCDE using Eqs. (4-93)-(4-96) along with
(4-91) and (4-92).
4.3.4 Determination of the Singular Field
We determined the THZ coordinates in terms of the background coordinates
in the previous Subsection. Using these results, we are now able to specify the
singular source field bs in terms of the background coordinates. In the approxima-
tion of bs as represented by Eq. (4-45), p is defined as the proper distance from
p to F measured along the spatial geodesic which is orthogonal to F. In the THZ