Does anyone know where I can find the solution for a spherically symmetric thin shell of timelike matter falling into a Schwarzschild black hole? The matter should be pressureless, so that each particle on the shell follows a radial geodesic. I am interested in the case where the shell is released from rest at a given Schwarzschild radius and time.
I think this is a special case of something called the Bondi-Tolman metric, but I'd rather not try to unpack that solution if there is a simpler way to find what I'm looking for. For null matter the answer is known (and simple), it's the Vaidya metric.
Edit: After thinking about this a little more I think it might be trivial; is it just Schwarzschild inside the shell with the black hole mass $M$, and Schwarzschild outside the shell with mass $M+E$, where $E$ is the energy of the wave?
Edit to the edit: The first edit looks wrong, I don't think it's a solution to Einstein's equations.