Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

This question already has an answer here:

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 $r=R$.

share|improve this question
    
The region inside the shell should have a Minkowski (flat) metric, certainly not a zero metric. –  Trimok Jun 30 '13 at 16:54
    
By $G_{\mu\nu}$ I meant to Einstein tensor, the equation above is the field equation under energy-momentum tensor $T_{\mu\nu}=0$ . –  Franz Unberlaude Jun 30 '13 at 17:02
    
This is worked quite beautifully in Poisson's book. The interesting case is when the shell is spinning. –  Jerry Schirmer Jun 30 '13 at 17:09
    
@FranzUnberlaude : Aaaah! Sorry. I think that the metrics inside has to be Minkowski. –  Trimok Jun 30 '13 at 17:10
1  
Possible duplicate: physics.stackexchange.com/q/43626/2451 –  Qmechanic Jun 30 '13 at 17:54
show 2 more comments

marked as duplicate by Chris White, Willie Wong, Waffle's Crazy Peanut, Emilio Pisanty, Qmechanic Jul 5 '13 at 23:51

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

1 Answer

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: $$ds^2~=~-dt^2 + dr^2 +r^2 d\Omega^2$$

share|improve this answer
add comment

Not the answer you're looking for? Browse other questions tagged or ask your own question.