As we noted previously, angular separation is dependent on separation in time
in the K~err spacetime. Ori's analysis therefore takes place in the frequency domain.
Ori's construction is effective in the vacuum situation, for which 9 satisfies Teukolsky's
Equation 1-17 with a = -2, so it does provide a complete solution in the frequency
domain.
For incorporating sources, Ori continues to take Equation 1-21 as correct, where now
I, n is a source-dependent, non-vacuum solution. Equation 1-21 allows the freedom to add
to WIRG any function that is killed by the four derivatives there. Ori utilizes this freedom
to choose functions that reproduce the discontinuity at the source and, by extension,
',, However, Equation 1-17 no longer applies for 9, nor does Equation 1-22 for tb4 in
the form it has here.2 Furthermore, any metric reconstructed in the radiation gauges
is ill-defined at the location of the source, a fact that is proven in ('!s Ilter 4. Moreover,
Ori also finds a problem in the -1, Hol~w regions", which occur wherever null rays (;?i,
incoming from infinity) have been blocked by the source. Apparently, the shadow region
has to be thought of as being identified for each mode independently. For point sources,
discontinuities are thought to develop across the shadow regions, although they have not
been observed in simple flat spacetime model calculations. Nevertheless, no complete
proposal has yet been developed to deal with these expected discontinuities. Earlier work
by Barack and Ori [24] -11--- -; that gauge freedom may phI i- a role in resolving these
1SSUeS.
1.3.2.2 Time domain treatment for Schwarzschild
In a different approach to the inversion problem, Lousto and Whiting [25] have
chosen to work in the time domain. Because of this choice their result is only valid in the
2 The term multiplying W in Equation 5-2 arose by repeated use of the Teukolsky
equation in quite a complicated expression, initially given correctly by Stewart [21], and
also obtainable from the results of C'!s Ilters 2 and 3 here. The full form of the expression
may still apply here.