Let the form of the metric outside to be such that
gamma_tt(r,t) =1
so that it is asymptoticaly Euclidean (or Minkowski). This will fix the metric right inside the shell to be also the same, but there will be a non-trivial
\partial_r gamma_tt
inside the shell which will be controlled by
S_tt (which is a function of t, and R)
It is easy to calculate gamma_tt, by writing a power series expansion for gamma_tt inside the shell:
1 + \Sigma_{n=1}^{\infty} a_n(t) (r-R)^n
I have set the first coefficient to 1, because at r=R, the metric should match the outside metric. The IRC will fix the a_1(t) i…