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I just found a paper "On a common misunderstanding of the Birkhoff theorem". This means that inside a spherically symmetric thin shell there is no gravitational force, BUT there is time dilation, and so the interior solution is NOT the Minkowski metric, right?

Dale
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Andrea Di Pinto
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1 Answers1

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In GR you use coordinate charts which cover some specified open region of spacetime. You are allowed to choose the region that you want to cover, and it may be all of spacetime or it may be some smaller region.

For a spherically symmetric thin shell, if the chosen region is strictly inside the shell then it is the Minkowski metric. If the region is strictly outside the shell then it is the Schwarzschild metric. If the region contains the shell then it is more complicated, and this complicated region is the focus of the paper.

There is gravitational time dilation between an observer on the interior of the shell and one exterior to the shell, but this is only relevant if you are covering both the interior and the exterior. Strictly within the interior it is not relevant and the straight Minkowski metric is valid strictly in the interior.

Dale
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  • So in general we can still use the Birkhoff Theorem, "as the Gauss Theorem"? – Andrea Di Pinto Jul 24 '20 at 15:11
  • @DiPintoAndrea I don’t know what you mean by that. You can use Birkhoff’s theorem as Birkhoff’s theorem. It just says that the only spherically symmetric vacuum spacetime is Schwarzschild. Any other use will be wrong. Gauss’ theorem does not require spherical symmetry, so the connection you are asking about is unclear to me. – Dale Jul 24 '20 at 15:53
  • My professor states: "We can analyze the property of the Universe inside a sphere of radius r without worry of the externe of that sphere, this is for the Birkhoff theorem (like the Gauss theorem in Electrostatic)".

    I meant this

     – Andrea Di Pinto Jul 24 '20 at 20:13
  • @AndreaDiPinto I think you should ask your professor for clarification on that. I don’t consider Birkhoff’s theorem to be generally analogous to Gauss’ law. But your professor may have a specific limited analogy in mind – Dale Jul 25 '20 at 00:09