be written
her pE 2r, sin 8 sin(4 Rt)G(r ro),
her = 2 pE r sin 8 cos(4 Rt)G(r ro),
p-E
bro = ~r, cos v cos(4/ -- Ot6(rV ro0),
Note that the singular nature of this metric perturbation inherently excludes it from
our analysis, as it destroys the continuity of the metric perturbation across E,. It is well
known [7] that the gauge transformation leading to this description can be interpreted as a
transformation from a non-inertial frame tethered to the central black hole to the center of
mass reference frame.
6.2 Kerr
In contrast to the situation in the Schwarzschild background, mass and angular
momentum perturbations in the K~err background are much more complicated. There
is, however, one simplifying feature of the mass and angular momentum perturbations.
Namely, the fact that both perturbations are stationary. Therefore the angular dependence
is not given by the spin-weighted spheroidal harmonics, sS(aw, 8, 4), but rather their
aw = 0 limit-the spin-weighted spherical harmonics.
The primary issue with treating the non-radiated multipoles in the context of
matched spacetimes is the choice of the matching surface, E,. Most of our discussion will
be focused on this issue.
6.2.1 Mass Perturbations
In place of Birkhoff's theorem there is Wald's theorem [16], described earlier, assuring
us that infinitesimal mass perturbations of the K~err solution lead to other K~err solutions
(with infinitesimally different masses, of course) because such perturbations do not
contribute the perturbations of I',, or tb4 (Which we will verify shortly). Thus we have the