type D spaces in C'!s Ilter 1. Perhaps the best example of this is our proof of the existence
of radiation gauges in sourcefree regions of spacetime. Our form of the Einstein equations
and Held's integration technique is a powerful combination that allowed us to prove the
result in arbitrary type II backgrounds, where the background integration isn't even
complete.
Finally, our treatment of the non-radiated multipoles demonstrates the power
of our framework when combined with existing techniques. Our results in the K~err
spacetime represent the first attempt at treating this part of the perturbation. Though we
were unable to obtain the description in terms of a matched spacetime, we nevertheless
provided a perturbation suitable for use in metric reconstruction.
7.2 Future Work
For all the generality inherent in the framework we developed, the applications we
presented were narrowly focused around the problem of metric reconstruction in the K~err
spacetime. This leaves many problems to be explored, both within the realm of metric
perturbations of K~err and otherwise. We detail some of these below.
Perhaps most pressing is the generalization of our result for the non-radiated
multipoles in the K~err spacetime to encompass more general orbits. In particular, orbits
not lying in the equatorial plane are of particular interest. Such orbits necessarily contain
off-axis angular momentum, which in turn are widely thought to be related to Carter's
constant (associated with the K~illing tensor). For such orbits the K~omar formulae fail to
completely characterize these off-axis angular momentum components, so it is clear that
we must look elsewhere for a solution. One potential avenue for progress is the Einstein
equations themselves. As we noted in the previous chapter, mass and angular momentum
perturbations are both stationary perturbations with angular dependence characterized
by the spin-weighted spherical harmonics. The simplifications this brings for working with
the Einstein equations is immense and may prove to make the problem tractable, without
recourse to purely numerical methods. In any case, it seems clear that our framework,