Given a fixed shell with the mass of $M$ and a radius $R$ , what would be the metric tensor for $r<R$? I do know that using Birkhoff Theorem the metric for $r>R$ should be schwarzschild. I'm not sure how to solve $G_{\mu\nu}=0$ for the inner part, and I'm not sure if I can demand continuity at $r=R$.

  • $\begingroup$ The region inside the shell should have a Minkowski (flat) metric, certainly not a zero metric. $\endgroup$ – Trimok Jun 30 '13 at 16:54
  • $\begingroup$ By $G_{\mu\nu}$ I meant to Einstein tensor, the equation above is the field equation under energy-momentum tensor $T_{\mu\nu}=0$ . $\endgroup$ – Franz Unberlaude Jun 30 '13 at 17:02
  • $\begingroup$ This is worked quite beautifully in Poisson's book. The interesting case is when the shell is spinning. $\endgroup$ – Jerry Schirmer Jun 30 '13 at 17:09
  • $\begingroup$ @FranzUnberlaude : Aaaah! Sorry. I think that the metrics inside has to be Minkowski. $\endgroup$ – Trimok Jun 30 '13 at 17:10
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    $\begingroup$ Possible duplicate: physics.stackexchange.com/q/43626/2451 $\endgroup$ – Qmechanic Jun 30 '13 at 17:54

As Qmechanic said, this question been answered before . to sum it up, as the birkhoff theorem hold also in void, we look at the schwarzschild metric: $$\tag{1} ds^2~=~-\left(1-\frac{R}{r}\right)dt^2 + \left(1-\frac{R}{r}\right)^{-1}dr^2 +r^2 d\Omega^2$$ and take $M=0$ as there no mass inside the shell, we end up with the expected flat metric: $$ds^2~=~-dt^2 + dr^2 +r^2 d\Omega^2$$

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