In Relativity and Singularities, Natário states that

Any vacuum solution admitting $O(n)$ as an isometry group is locally isometric to $M=\mathbb{R}^2\times S^{n-1}$ (with the Schwarzchild metric).

Why is that so? It doesn't seem obvious to me, and, so far, all "proofs" on the web approach this problem from the physical point of view, so the connection to the above statement is blurred to me.

  • $\begingroup$ Search for "Birkhoff's theorem". $\endgroup$ – Heterotic Jun 29 '18 at 18:23
  • $\begingroup$ I did. I could not picture why it's the same statement as the one above. $\endgroup$ – big-lion Jun 29 '18 at 18:24
  • $\begingroup$ Birkhoff's theorem states that any spherically symmetric solution (ie a solution admitting O(n) as an isometry group) of the vacuum field equations is unique and is indeed the Schwarzchild solution. This is exactly the statement in the op. Any "other" solution will be isometric to the Schwarzchild solution. The Schwarzchild solution is unique. The answer to the op is the proof of Birkhoff's theorem. $\endgroup$ – Heterotic Jun 29 '18 at 18:29

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