either alone or in conjunction with various other techniques, will help to clarify the
problem enormously.
Another avenue worth pursuing is the commuting operator associated with the K~illing
tensor due to Beyer [44] (cf. ('! .pter 1 ). Recall that Beyer's operator commutes with
the scalar wave equation in K~err. It is very tempting to think that such an operator
would exist for the Teukolsky equation as well. The GHP formalism, and GHPtools (of
course), provide the ideal environment in which to study such questions. Furthermore,
in the context of work performed by Jeffryes [79] concerning the implications of the
existence of the K~illing spinor (which includes a discussion of the Teukolsky-Starobinksy
identities), it is natural to think that such an operator may in fact shed some new light
on the Teukolsky-Starobinsky identities in the form presented in ('! .pter 5. Additionally,
the existence of a generalization of Beyer's operator carries with it the possibility of new
decomposition of functions in the K~err spacetime--just as the existence of the K~illing
vectlors and lead to separation in t and cf according to e-ime and e""m* (respectively),l
the eigenfunctions of a generalized Beyer operator may provide a new separation of
variables in the K~err spacetime. This is certainly a possibility worth pursuing.
Finally, both GHPtools and our form of the perturbed Einstein equations are
entirely general and ready for use by researchers interested in more general (or even
more specialized) backgrounds than Petrov type D. In particular, the class of type II
spacetimes seems a likely candidate for further analysis, especially with the aid of the
integration technique of Held. We have only begun to scratch the surface of the wide
v-1I r ii of problems these tools can help solve.