comparatively simple derivation of the same result. This is the approach we will follow
here.
Wald's method is centered around the notion of adjoints. Consider some linear
differential operator, that takes n-index tensor fields into m-index tensor fields. Its
adjoint, Lt, which takes ni-index tensor fields into n-index tensor fields is defined by
no ...a, (/3)a,...a,, (gtCa)bl...b, /3by...b,, = aa, (3_30)
for some tensor fields c1 "l" and /3bl...b" and some vector field s". If Lt = then L
is self-adjoint. An important property of adjoints is that for two linear operators, L
and M./1 (MZ/)t = M2/t t. Now let 8 = S(hub) denoted the linear Einstein operator,
S the operator that gives either of the Teukolsky equations from 8 (Equation :32:3 or
:324), O = O( ~,, or