Thus, we reshape the quadratic part of Eq. (4-144) into
9Fr2
(E2 f)(t_ f 2_ 2 ot (t
f (ro + J2)
f2 (r + J2)
0-W^j -
- t)A 2EJ(t to)(
'/)2 +r2( )2
with
JA
7 #o -(4-149)
f (r + J2) (4-49)
where A = r ro, and an identity ,2 = E2 (1 + J2/r ) is used for simplifying
the coefficient of A2. Here, taking the coincidence limit A -+ 0, we have t' -+ Oo.
This same idea is found in Mino, Nakano, and Sasaki [26]. Also, for the multiple
decomposition the quadratic part must be analytic and smooth over the entire
two-sphere, and we write
2Err2
(E2 -f)(t )2 2E (t o)A-
f (ro + 2)
E2 2r2
2 A2 + (r2 + J2) sin2 0sin2(o
+ [(x- xo)4].
+O[(x X0)4].
2E.J(t t) sin 0 sin( 0')
0') + ro cos2
(4-150)
Here we have used the elementary approximations = sin(O 0') + O[( 0')3]
and 1 sinO + 0[(0- 7/2)2].
To aid in the multiple decomposition we rotate the usual Schwarzschild
coordinates by following the approach of Barack and Ori [27] such that the
coordinate location of the particle is moved from the equatorial plane 0 = r/2
to the new polar axis. The new angles O and 4 defined in terms of the usual
Schwarzschild angles are
sin 0 cos(0 0')
sin 0 sin(O 0')
cos 0
cos 0
sin O cos 4
sin O sin 4.
(4-148)
(4-148)
(4-151)