As a consequence of Equation :3-35, the actual perturbed Weyl scalars follow directly from
Equations :327 and :328.1 The expressions are at first sight quite complicated, but by
commuting derivatives so that they appear in a standard order and using the fact that the
potential satisfies the Teukolsky equation, they become:
< 94, (5-1)
('414 r_ 4/ /37'a 87a p'P fp 24'2)]W .) (5 2)
The term in square brackets [] in Equation 5-2 is actually just the operator form of the
(generally complex) K~illing vector (acting on 9, which has type {-4,0}) discussed in
C'!s Ilter 2. We can further combine the relations in Equations 5-1 and 5-2 to eliminate
a~ny reference to\ the po~tent~ials. Tphe firt s~tep is to\ act o~n Equaition 5-2 with > #'2-4/3
which gives us
p4 -,-4/:% 4 4 -/ 4 _qpV 53
Commuting the eight derivatives on the first term (using GHPtools, of course) yields the
useful identity
p394 -- 149 __ i14 -- 4!p4W, (5-4)
which we will have occasion to exploit again. Commuting the derivatives in the second
term of Equation 5-3 poses no problem because V commutes with everything. Now it is a
simple matter to identify the resulting expression with the terms in Equations 5-1 and 5-2
to arrive at the following
p4 -4: % 4 1 -4/ 7,, (5-5)
p/4 -,-4/3 __i ~4 --4/:% 4 + V4, (5-6)
i We thanlk Joh~n Friedmanl and Toby Keidl for noting missing factors of in? several
earlier papers. Stewart [21] and C'!,l~!!. 1,.---1:! [18] have these factors correct, the latter
with different sign conventions.