(B) Symmetric tensors
An arbitrary symmetric tensor field Fab (C {Ihs, hs4, hs}) may be repre-
sented by a scalar field FTrace and a vector field Fa as
1
Fb = 2abrace,, + 2V F) abVF, (5-55)
where
FTrace a- ( bbT (5-56)
and the vector Fb is a solution of
1 1
VaVaFb + RFb = Va (b ab. (5-57)
and R is the scalar curvature of the geometry ag Fb is unique up to the
addition of a vector field ka which satisfies the conformal Killing equation,
Vak, + Vbka a Vka = 0. (5-58)
Further, Fib may be represented in terms of the scalar fields Fev and Fod by
substituting Eq. (5-52) into Eq. (5-55),
Fb b Face (2+V(2,) agV2) Fev + 2c (Vb)VRFod, (5-59)
where the three scalar fields Frace, Fev and Fod are referred to as the trace,
the even parity potential and the odd parity potential, respectively, and they
are determined by Eqs. (5-56), (5-53) and (5-54) via Eqs. (5-57) and (5-52).
Once their proper functional forms are found, the regularization parameters for
hs hs s Ih,}
vectors {h t, h hr h} and tensors {hj, hS,, h } are calculated similarly to
the case of scalars {Ih, h,, h ,}. Below we present the key steps of calculating the
leading order regularization parameters for hb for a general orbit.
Scalars: {hs,, h,, h s}.
hb (scalar) can be decomposed into spherical harmonic components,