Gauge-invariant and coordinate-independent perturbations of stellar   collapse II: matching to the exterior
  In Paper I in this series we constructed evolution equations for the complete gauge-invariant linear perturbations of a time-dependent spherically symmetric perfect fluid spacetime. A key application of this formalism is the interior of a collapsing star. Here we derive boundary conditions at the surface of the star, matching the interior perturbations to the well-known perturbations of the vacuum Schwarzschild spacetime outside the star. 
Jose M. Martin-Garcia and Carsten Gundlach