the description of the exterior spacetime, g b, We choose
P+(t) = cft,
&'(>=P ro 2M~(68
where, in anticipation of the nr -, W11111 we've chosen the same dimensional constant, P,
that we used in the description of the interior spacetime and j > 2. With both metrics
specified we now turn our attention to matching the spacetimes.
Because both background metrics are the same, it will suffice to match the perturbations
only. By imposing [hab] = 0, we arrive at three unique conditions:
S+ fo[ca]Yoo + 2 00Yo = 0, (6-39)
ro To
rifo~ Yoo M~[Q]Yoo ro61M = 0, (6-40)
[Q] = 0, (6-41)
where we used fo = f(ro). Our choices for Q+ and Q- (6-38,6-37) ensure that the third
condition is satisfied. We can solve Equations 6-39 and 6-40 to get equations for [c0] and
dQP(i + j)
[0][] (6-42)
6M~ = (ro 2M Yoo = -P(i + j)Yoo, (6-43)
where we've made use of Equations 6-38 and 6-37. Next we will use the jump conditions
to solve for p.
Application of the jump conditions (Equations 6-14-618) is simplified by the
fact that our metric perturbation is pure spin-0. Thus we only need consider the jump
conditions for the spin-0 components of the metric perturbation (hiz, hi,, he, and hmm).
For simplicity we will work with Equation 6-15, though it can be directly verified that the