Also, under this coordinate rotation, a spherical harmonic Yem(0, 0) becomes
Ym(,) rnmm, '( ), (4-152)
where the coefficients aim, depend on the rotation (0, 4) (6, E) as well as on
, m and m', and the index f is preserved under the rotation [28]. As recognized
in Ref. [27], there is a great advantage of using the rotated angles (6, 4): after
expanding a,(q/p) into a sum of spherical harmonic components, we take the
coincidence limit A -- 0, -- 0. Then, finally only the m = 0 components
contribute to the self-force at = 0 because Ym(0, ) = 0 for m / 0. Thus, the
regularization parameters of Eq. (4-33) are just (, m = 0) spherical harmonic
components of a,(q/p) evaluated at x0.
Now, with these rotated angles, PuI is re-expressed as
o2Epyr2
P =I (E2 f) (t_ to)2 2E (t o)A 2EJ(t to) sin 0 cos 4
f (r + J2)
+ -2 o 9 2 J2 2
r E2 2
+2(J)(1 n )
x r2 2 + 1 -cosO + O[(x )4], (4-153)
2f2(r2 + J2)2 ( 2 )
where the elementary approximation sin2 ( = 2(1 cos () + 0(94) is used. We may
now define
2Err2
p2 (E2 f)(t 2 )2 2E ( to)A- 2EJ(t to) sin cos
+2 2 of ( + J2)
o0 +
( J2)( 2 1 sin2 i)
r E2A2
x 0f-(r-- r2EA 2 + 1 cos (4-154)
In particular, when fixing t t we defin
In particular, when fixing t to, we define
p2 p2 t = 2(r + 2)X (2 + COS )
(4-155)