where the second expression follows from taking the prime of the first. We will refer to
these relations as the first form of the Teukolsky-Starobinsky identities. Note that the use
of V as a commuting operator restricts the validity of these relations to non-accelerating
type D metrics. In the an~ ll-k- of Torres del Castillo [65] and Ortigoza [66], where explicit
coordinate expressions were used, Equations 5-5 and 5-6 both appear to be true. This
fact appears to be coincidental since it is unclear how it follows in general from the
fundamental equations of perturbation theory. The remainder of the identities we will
present have not appeared in the literature in this form and we can only claim they hold
for non-accelerating type D spacetimes.
Before we continue, we'll take a look at the content of Equations 5-5 and 5-6 in
the context of th~e Ker~r spacetimre. If wve write i',, ~ R+2(r)S+2(0, Q) and 2 4"/3 4 ,
R-2(r)S-2(0, 4) and understand the time dependence of each to be given by e-ist, then
Equation 5-5 tells us: (1) the result of four radial derivatives on R+2 is proportional to
R-2 and (2) the result of four angular derivatives on S-2 is proportional to S+2. The same
is true of Equation 5-6 with the +'s and -'s swapped. Note that Equations 5-1 and 5-2
(and their primes in the ORG) ;?i essentially the same thing with the subtle difference
that the angular and radial functions are not obviously solutions to the same perturbation.
No such ambiguity arises in Equations 5-5 and 5-6.
Remarkably, we can actually take things a step further and arrive at expressions for
I'n, and ~4 independently. We begin by acting /4 7-4/3 o qain55
p14/4-4/34 -434 pl4 -4/3 14 1,-4/3 i 3Pl4 -4/3 ,,; (5-7)
By recalling that Ir has. th same~,,,. type,. as 2-4/3 i4 '; Car~rieS HO Weight), weit can simply
take the prime and conjugate of Equation 5-4, and use it to commute the derivatives on