interior perturbation is then characterized by
~1Y10 = r sin 0. (6-57)
Finally, it is easy to set
kg ~8,S 2r-S =0,
by imposing S(t, r) = T2S(t). Note that because of the quadratic dependence on r, we
cannot perform this gauge transformation in the exterior spacetime if we wish to preserve
.I-i-m!hlli' c flatness. This is not a problem because the angular momentum perturbation
provides the necessary freedom for matching. Finally, the piece in h4 is proportional to
the time derivative of S(t, r), which -II__- -is we choose S(t) = yt, to keep the perturbation
static. In summary, we have for the interior and exterior metric perturbations
h.,o =72 Sin 8 11, (6- 58)
26aM~ sin2 8
"t4 (6-59)
with all other components vanishing.
Continuity of the metric perturbation ([hab] = 0) requires
yra =Yl (6-60)
2M~ sin O'
where we've used the equality of azY~o to expand Y,$. As before, the radial jump
conditions will determine y. In this case we'll use the odd-parity (imaginary) part of
Equation 6-17. The relevant tetrad components are given by:
iyr2 1 10
him- him = hm him- 12(-1
&t h = ht h' ir YIO II (6-62)
Im Im am am p~2 1/2 '
752 (Yi"ilG~pL
16xr Tim Ti =r0/ O)s COS 8)6(4 Ot). (6-63)