the perturbation from contributing to gravitational radiation. More importantly, this is
a sign that our intuition needs adjustment for working in the K~err spacetime. In further
contrast to our prior results, the complicated 8 dependence in the tetrad components
of the metric perturbation leads to mixing of the (spin-weighted) spherical harmonic
e-modes, a complication not previously encountered.
Another surprising feature is the fact that the perturbation is complex and thus
exhibits both types of 1. .. .ty". Although the static nature of the perturbation guarantees
spin-weighted spherical harmonic angular dependence, we must be careful not to speak
of parity in the Schwarzschild sense, but rather the real and imaginary parts of the
perturbation. In any case the implications of this fact are presently unclear and remain to
be determined in future work.
6.2.3 Discussion
In this section we discuss in more detail the possible problems with our matching by
looking more closely at the assumptions that we made. This will lead naturally to ideas
about future work that is beyond our present scope.
First off, one may speculate that our requirement of form invariance is perhaps too
strict to allow for a proper matching. This does not appear to be the case. A result of
Carter [76] implies that, due to stationarity and axial symmetry, the K~err metric (in
Bci-; r-Lindquist coordinates) has precisely the minimum number of nonzero components.
Having established independently that the mass and angular momentum perturbations
preserve theses properties of the background, Carter's result -II_ _- -; that the problem lies
elsewhere .
This leads us to consider whether the introduction of an infinitesimally thin shell of
matter (which is effectively what E, is), necessarily introduces non-K~err perturbations.
A shell (of some currently unspecified shape) would presumably be a differentially
rotating object. It is unclear whether this disrupts the stationarity or axial symmetry
of the exterior spacetime by the introduction of perturbations that we have neglected