XA + MA(Xa ) + MAa _
'A 2 a c -lo ()xb x-)(xC- 4), (4-83)
where we may choose XA 0 and
MAa diag [Mt,, Mxr, M Mz ]
= diag [(1 2M/ro)1/2, (1 2M/ro)-1/2 rosino, -ro] (4-84)
for convenience as this choice re-centers and re-scales the Schwarzschild coordinates
to = (1 2M/ro)1/2 (t to), X = (1 2M/ro)-1/2 (r- ro), To sin 0o Oo),
Z = -ro( 0o). Taking advantage of the spherical symmetry of the background,
we may take only the equatorial plane 0o = r/2 for this transformation. Then, for
Eq. (4-83) the non-zero C'!hi -1' !!. symbols evaluated at xo are
M f M M
tr o r2 to 2 Io 2 o 0o,
Fr, -fro, FOo (4-85)
o ro
where f 1 2M ) These coordinates are static and we have no information
about the particle's motion.
Following Ref. [24], the metric expansions in these coordinates can be deter-
mined via
axa aXb
gAB = gab (4-86)
8XA OX B
where the background metric gab can be expanded about xa in the Taylor series
1
gab = gab + gab,c o (xc X +2 g ab,cd j (xc- X )(xd d)
+6 9gab,cde 0 ( )(d )(e X) + 0[(x )4]. (4-87)