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)