nonzero components of the mass perturbation given by
2r6M~
htt= (6-66)
r~2 + 2 cos2 H)
2r(r2 + 2 COS2 )1
her= (6-67)
(r2 2Af~r + a2 2
2ar sin2 81
he (6-68)
r~2 + 2 cos2 H
2a2T Sin4 81
hee = .(6-69)
r2 + 2 COS2 H
Because the calculations in the K~err spacetime are significantly more complicated, we will
take a shortcut to determining the angular dependence of the perturbation by looking at
the tetrad components of the metric perturbation, a result which we will in any case use
shortly. In the symmetric tetrad (Equations 6-13) we have
2r6M~
ha =(6-70)
2r6M~
h,z~ (6-71)
with all other components vanishing. Because both hit and h,z, are spin-weight 0, they
have a natural decomposition into -e = 0, m = 0 scalar (ordinary) spherical harmonics.
Furthermore, utilizing Equation :327 we see that
I~,, = (B r')(B r')hit = 82 11 2r'Bhy, = 0, (6-72)
and similarly for 1/4. Therefore, according to Wald's theorem, we are ensured that
Equations 6-70 and 6-71 are a perturbation towards another K~err solution.
With the angular dependence determined, we are led to consider a gauge vector of the
form
(a = (P(t), Q(r), 0, S(t, 4)), (6-73)