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Given a spherically symmetric mass distribution

  • the shell theorem states that a test particle at radius $r$ experiences no net force from shells at a radius $R > r$;

  • the Birkhoff theorem states that (with the additional condition of asymptotic flatness) the metric is static, and any vacuum shell corresponds to a radial branch of the Schwarzschild solution.

But the Birkhoff theorem does not explicitly say that geodesics of test particles at radius $r$ follow entirely from an „inner solution“ and are not affected by the mass distribution at $R > r$.

Question: Under which conditions is it possible to derive a similar statement like the shell theorem? Or do I overlook anything, and this „shell theorem“ is implicitly included in the Birkhoff theorem?

Qmechanic
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TomS
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    To be clear, are you asking whether the mass parameter in Sch. metric at given $r$ (where there is vacuum) would change if a shell of matter were introduced at some larger $r$? – Andrew Steane Sep 22 '21 at 06:51
  • In some sense yes. Given that any interior solution with density $\rho(r)$ and mass $m(r) = \int_0^r \rho$, plus vacuum $\rho(R) = 0$ for $R > r$ determines the metric and the geodesics at $r$ uniquely, the question is if the metric and the geodesics at $r$ can affected by adding densities $\rho(R) \neq 0$ for for $R > r$. – TomS Sep 22 '21 at 07:07
  • Related: https://physics.stackexchange.com/q/43626/2451 – Qmechanic Sep 22 '21 at 07:13
  • thanks; I am aware of that result, but it doesn’t answer my question – TomS Sep 22 '21 at 17:47

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