complicated. This being the case, we will focus our attention on the general features of the
angular momentum perturbation that can he obtained independently of a good matching.
We begin by noting that the nonzero components on the metric perturbation are given by
4M~ar cos2 86a
htt= (6-85)
(r2 2Afr + a2 2'
2a(r2 Sin2 0 + 2rM~ cos2 8)6
her (6-86)
(r2 2Af~r + a2 2
2M~ar sin2 8(2 a2 cos2 8
hte (6-87)
(T2 + 2 COS2 H 2
hoo = -2a Cos2 86a (688)
2a sin2 8 [2 p2 2 a"COS2 :)+3(r. + 2M~ sin2 8 16a
he (6-89)
(r2 + 2 COS2 H 2
The corresponding tetrad components (in the symmetric tetrad) are given by
aba[(r2 a2) Sin2 0 2Afr(cos2 8 p
hit = h,z (6-90)
p"2
aba sin2 8
hi,z (6-91)
aba(cos2 8 p
h,waz= (6-92)
-iba(a2 Iff) Sin2
bi,~ = h,z,>= (6-93)
(r + i cos H) ii
aba sin2 8
h,,n = (6-94)
(r + ia cos 8)2
where we have omitted the complex conjugates. Though it is not immediately obvious,
this perturbation makes no contribution to I~,, or 2/4, enSuring that this is a valid angular
momentum perturbation.
In light of relatively straightforward results for mass perturbations, the nontrivial
form of Equations 6-90-694 comes as a surprise. Unlike mass perturbations, angular
momentum perturbations are not confined to a single s sector, whereas one might expect
them to be exclusively s = 1, as intuition from working in the Schwarzschild background
would lead us to believe. Note that although the perturbation appears in the s = +2
sector of the metric, the vanishing of the s = +2 components of the Weyl curvature keep