Imposing the radial jump conditions then gives us
i6y il6xpLZ
f1/2 1'-o 101/2 6CS@ f
OJ l= 0 m= -
After exploiting the orthogonality of the spin-weight 1 spherical harmonics and using
Equation 6-57 we obtain
7 = .(6-64)
It then follows from Equation 6-60 that
6J = 6aM~= pL, (6-65)
which is precisely what is to be expected-all of the angular momentum in the (otherwise
non-rotating) spacetime comes from the angular momentum of the particle. Once again
we can verify directly from Equation 6-23 that we have correctly identified the angular
momentum of the spacetime.
6.1.2.2 Even-parity dipole perturbations
First let's review what we already know: (1) We only need consider the -e = 1
pieces. This was established above when we found the angular momentum to have -e = 1
angular dependence. (2) The even-parity perturbation cannot contribute to the mass or
angular momentum of the spacetime. These perturbations have already been accounted
for. Furthermore, it is clear that in the absence of a source, there would be no e = 1
perturbation in the even parity sector.
In contrast to the situation with mass and angular momentum perturbations, where it
was easy to write down the general form of the perturbations, we have no general form for
the metric perturbation. Without prior knowledge of the perturbation, we must resort to
solving the Einstein equations to determine the perturbation. This has been carried out by
both Zerilli [7]and Detweiler & Poisson [17]. The result is a metric perturbation that can