and the scalar potentials Fev and Fod must satisfy the differential equations
according to Eq. (5-57) along with Eq. (5-52),
Vb (V2Fev) + CV (V2Fod) + R (V6lev + Cb od) v g F(rabRT.) ,
(5-131)
where R is the scalar curvature of the geometry ab. However, one can also
decompose Fev and Fod into spherical harmonic components,
Fev = Fev m(o,), (5-132)
&m
Fod F Ymr(0, ). (5-133)
&m
Then, we may solve the differential equations (5-131) in terms of spherical harmon-
ics via Eqs. (5-132) and (5-133), and obtain
L(f + 1)FvO VgY2m(0, ) e( + 1)RF db OV nm, b)
im im
+RE ^ [FiVYm(O, 0) + FRdm Y O, )]
&m
= Va (b 2-abFTrce (5-134)
To single out each expression of Fm and Fd, we contract Eq. (5-134) with Vb and
.v we contract Eq. (5-t34) with Vg and
bdVd, respectively, and obtain
S[ +)2 + F]vtv (0, ) V Fa abFT e (5-135)
f [( + 1)2 R ( + 1)] R yny(O, ) T bddV"a Fa- rabTF (5-136)
Integrating Eqs. (5-135) and (5-136) over dQ-,, (0, ), we further obtain
Fem 1 i dQ Ve,' (T 1 y*em/ (0(
Fe 2( + 1)2 2( + 1) Fab,- 2a a'b'Tcer Y e ')
(5-137)