Thus we must impose Bo~e~m = ii4Km = 0, which translates into e = m = 0 and
we have established that the angular dependence of the metric perturbation is purely
Yoo(B, 4) = constant. Also, in order to keep the perturbation static, the time dependence
of P(t) must be, at most, linear. Without loss of generality, we set P(t) = a~t. Finally,
our falloff conlditionls inl Equations 6-19 required: Q(r) = 0)) Thus we hlave arrived at
a description of source-free mass perturbations in the Schwarzschild spacetime in a family
of .l-i-!!!!1'1;1 I11y flat gauges that preserve the form of the metric. The physical spacetime
(gab 9 g~bh +ab) has components
2M~Q(r)Yoo 26M~
get = f (1 2a~Yoo) (6-33)
i~( l 2mQYoo + r2 00 nirQ 2rbMlrf) 6
goe= -72 2Q)Yoo (6-35)
g4 = -T2S 2 1- (6-36)
We can give an interpretation to a~ by considering Equation 6-33 with 6M~ = Q = 0, in
which case it is clear that a~ is just a rescaling of the time coordinate.
In order to perform the matching, we need to adapt our generic perturbation to the
interior and exterior spacetimes and choose a particular gauge to perform the matching.
We will begin with the description of the metric on the interior, g~b. Here 6M~ = 0, so
the perturbation is pure gauge. Furthermore, on the interior there is no need to impose
.I-i-inidll'lc flatness. Instead, we will choose Q-(r) so that the interior metric is regular on
the horizon and leave the form of P-(t) untouched. A suitable choice is
P-(t) = a~-t,
&- (> liro 2M~; 67
where r = ro is the location of E, and p is a constant inserted for dimensional reasons and
i > 0. The values of a~- and P will be determined from the jump conditions. Proceeding to