which are easily obtained by taking the prime of Equations 5-1 and 5-2. Note that
whereas 1T satisfc,,~ie the, Teukolskyi. equation; fr (2-,i4/8,:3 4I, SliiSfies the adjoint equation-the
Teukolsky equation for I',, Fr-om the complex conjugate of the preceding equations and
their IR G counterparts, we get the following:
P49 __ 149'31 3 4:399' (514)
P/491 __ 4 2 ~r3'vW (5-15)
the first form of the Teukolsky-Starohinsky relationships for potentials. Note the difference
between the above and Equations 5-5 and 5-6, particularly II theDDII missing facor of 2-/
and the fact that V appears. As with Equations 5-5 and 5-6, we can obtain relations for
each potential individually by acting p/4~~l --4/ onquation 5-14and fur~ther exploitinlg
(the primed conjugate of) Equation 5-4. The result is that
p14 --4/3p49 __14 --4/3 49 4/:39, (5-16)16
p4 --4/3 14' / __4 --4/3 14 / 9V724/39/. (5-17)
We can summarize this last identity by writing
[p/l4 --4/p 4 1,4 --4/ 3 4 4/l~" 31 --4/":3 4, r9} = 01 (5-18)
[4 -4/3p l4 4 --4/3 1, 4 + 9 0 43 -43, .(5-19)
Bardeen has recently pointed out an issue in the standard treatment of the Teukolsky-
Starohinsky identities [67]. In particular, he finds that, in the Schwarzschild
background, there is a hitherto unnoticed relative sign difference between the odd-
and even-parity in the term proportional to 8, (alternatively w when time separation is
performed), which by continuity presumably persists in the K~err background. Bardeen
argues using standard techniques that don't make clear the difference between the ~'s
and their complex conjugates on the right-hand-sides of Equations 5-5 and 5-6. However,
recalling our discussion of parity in Chapter 3, a glance at these equations reveals that