These results are to be expected because of the axisymmetric nature of both the
perturbations and the background spacetime. We now turn our attention to the mass
and angular momentum perturbations in the Schwarzschild background.
6.1 Schwarzschild
The Schwarzschild spacetime provides the perfect tested for our technique. 1\oreover,
because of the spherical symmetry of the background, matching the spacetime is quite
straightforward. In this case we can ah-li-s choose the matching hypersurface, Ez>, to be
a (round) 2-sphere and exploit the orthogonality and completeness of the spin-weighted
spherical harmonics to smear out the delta source on Ez,. The only caveat is that we must
choose Ez, outside of the innermost stable circular orbit. If the location of Ez, is ro, then
this amounts to requiring ro > 6Af.
6.1.1 Mass perturbations
Our first task is to construct a suitable description of source-free mass perturbations of
the Schwarzschild spacetime. We will then glue two such spacetimes together, as described
above. We will write the Schwarzschild metric as
d~s2 = d2 f-1 2 ,2 d2 Sin2 8d 2) (628)
where f = 1 2Af/r. According to Birkhoff's theorem, the only static, spherically
symmetric solution to the Einstein equations is the Schwarzschild solution. Thus, we
are assured from the outset that perturbing the mass will simply lead us to another
Schwarzschild spacetime with a mass At + 61f. The nonzero components of the
corresponding metric perturbation are given by
htt -26M
r (6-29y)
which is easily obtained hv linearizing a mass perturbation of Equation 6-28. In order
to characterize mass perturbations more generally, we will introduce more freedom by