performing a gauge transformation. To that end we introduce the gauge vector
I= (PGt, r)e m(8, 4), Q00, r)e m(0, 4), 0, 0), (6-30)
where we've taken a cue from R;--- & Wheeler and decomposed the gauge vector into
spherical harmonics. Note the absence of Co and (4 components in our gauge vector.
We have deliberately omitted these components on the grounds that they interfere
with the form invariance of the metric. In order to determine the functions P(t, r) and
Q(t, r) as well as the appropriate e and m, we'll look at their contribution to the metric
perturbation. Our gauge transformation, (ab = @agb, has the form
-2(fatP +Mlr-2Q _-d, r p d,) -f -1~d
sym 2 f-l(8,Q Mr-2 -1& f-1d f-1 E, 6-1
sym sym 2rQ 0
sym sym sym 2r sin2 8
where "sym" means symmetric and we've dropped the functional dependencies for
simplicity. First, we'll further specialize the gauge transformation by insisting on
preserving the form of Equation 6-28. A consequence of this is that
her = (-fdrP(t, r) + / 8tQ&(t, r)) = 0.
Because Q(t, r) appears in other parts of the metric perturbation, allowing it to carry a
time dependence would destroy the static nature of the perturbation and put us at odds
with Birkhoff's theorem. Therefore we require Q = Q(r), which immediately leads us to
P = P(t). Further consequences of our form invariance requirement are
hts = -f Pdeoh = 0,
hs = -f PdY, = 0,
(6-32)
bro = / 08oh~m = 0,
hts = / 0845me = 0.