Schwarzschild background, where angular separation is not dependent on separation in
time. Nevertheless, with the formulation of the Hertz potentials being set in the context
of radiation gauges, Lousto and Whiting were effectively unable to introduce sources into
their treatment. Regardless, several results of their analysis are noteworthy. Similar to
the analysis of R. -~ and Wheeler, Lousto and Whiting made use of angular and parity
decompositions, two features that have eluded application in the K~err background.
One unexpected feature of Lousto and Whiting's work is how algebraically special
frequencies emerge in a fundamental way. Algebraically special solutions arise when one
of I<'n or ~4 is ZeoO While the other is not, and then only for specific (complex) frequencies.
While this is inherently a frequency domain phenomena, it ptil- a crucial role in this time
domain approach. The algebraically special equation here has a source term depending on
the initial data for the Hertz potential-this term effectively corresponds to that which
arises for a Laplace transform. For the Schwarzschild background, all the algebraically
special frequencies are known and the algebraically special solutions have been found
explicitly [26], so the equations for this analysis could be solved by quadrature [25].
Attempts to generalize this technique to the K~err background have to date remained
unsuccessful.
1.3.2.3 Working in the Regge-Wheeler gauge
The RW formalism has been extensively used for Schwarzschild perturbations,
and its implications have been thoroughly investigated. In particular, full sets of gauge
invariant quantities are known, and in chosen cases these have been directly related to the
perturbed Weyl scalars Iel, and #4, Which are naturally gauge invariant [27]. Lousto [28]
has recently chosen to work with such a formulation, rather than with a Hertz potential
formulation. This immediately gives him freedom over gauge choice and it circumvents
the problems previously encountered with the introduction of sources. Having calculated
explicitly the dependence on sources, and knowing also how to represent all relevant