the first term on the right as follows
pl4 /-4/3 14 -,-4/3 ; __14 ,-4/p4-4/3pl --/
adL:a124/3:~ 4/ 4/3 -4
= : 314i3" :i 4 2 4 1 2 ( -8
where we made use of Equation 5-6 in the second line and commuted everything through
V in the third line. The second term in Equation 5-7 becomes
4/3','"' 3p474/3
3/4 --' TI l4 II
=3Vij5/4 2 i" 4 +t 9VV 4, (5-9)
where we made use of the complex conjugate of Equation 5-6. Combining these results
gives us
1, 4 p~ 4 / 3 ,4 / 4 4 i 4 3 4 9V V 4 ( 5 -1 0 )
4 --4/3 1 -/ 4 -43 1 -/ <, 9 ,, (5-11)
where we took the prime of the first equation to get the second one "for free." These are
the second form of the Teukolsky-Starobinsky identities. We note in passing that in the
context of the separated solutions of e',, and 2-4/3 4, Ithe relations above allows for the
determination of the magnitude of the proportionality constant relating R+2 and R-2 [29].
Surprisingly, this isn't the end of the story. Recall that in a type D spacetime we also
have at our disposal the outgoing radiation gauge where
21, ,, 84/ 4/399, (5-12)
~4 pl 1 (5-13)