CHAPTER 5
THE TEUK(OLSK(Y-STAROBINSK(Y IDENTITIES
Having established the conditions for the existence of the radiation gauges, we will use
the corresponding metric perturbations to establish some useful relationships between the
perturbed Weyl scalars known generally (and quite loosely) as the Teukolsky-St arohinsky
identities. Because Hertz potentials are solutions to the Teukolsky equation, these
identities have immediate relevance for metric reconstruction in the IRG, both in the
time-domain approach of Lousto and Whiting [25] and the frequency domain approach of
Ori [2:3].
The original analysis of Teukolsky [11, 12] was based on the .I- i-i np u'tic form of the
solutions of the separated angular and radial functions in the K~err spacetime as well as a
theorem due to Starohinsky and Churilov [64]. Only later did C'I .!1.4 I-ekhar provide a
full analysis, which is nicely summarized in his book [29]. Our analysis, however, will be
entirely symbolic, involving only GHP quantities. This approach has the advantage not
only of applying to a larger class of spacetimes, but displaying the structure inherent in
the identities in a much more obvious way. A similar analysis of some of the identities we
will discuss was previously undertaken in the NP formalism by Torres del Castillo [65] and
later translated into GHP hy Ortigoza [66]. These prior analyses made use of the most
general type D spacetime and translated back and forth between coordinate-based and
coordinate-free expressions. In contrast, our approach will not make any reference to the
choice of coordinates or a tetrad (other than requiring it to be aligned with the principal
null directions). Because of this, our approach will showcase one of GHPtools' greatest
strengths-the ability to commute several derivatives with relative ease.
Our starting point is the (source-free) IR G metric perturbation given by