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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.

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Your idea seems right, the solution includes function $R(t)$ as a parameter. It should be solved for in such a way that the $T_{\mu\nu}$ on the shell would have the correct properties. Also, the time in the inner Schwarzschild metric is slowed (w.r.t. infinity) by a variable factor (dependent on $R(t)$). –  user23660 Feb 13 at 9:19
    
I think you're right, and that the EOM for $R$ becomes the geodesic equation for the particles on the shell in the limit where the rest mass of the shell goes to zero. –  Matthew Feb 13 at 18:44

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