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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 I found no precise proof (from the mathematical point of view) around the web.

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    $\begingroup$ This is addressed by Birkhoff's theorem: en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity) $\endgroup$ – Danu Jun 26 '18 at 18:36
  • $\begingroup$ Indeed, I had looked into Birkhoff's theorem, but found no clear proof of it. Wikipedia, for example, only loosely states it and proceeds to discussion of its implications. $\endgroup$ – big-lion Jun 26 '18 at 18:38
  • $\begingroup$ The Wikipedia page features precise references (e.g. to Birkhoff's original work), so that should cover it. You might also find this in mathematically-oriented general relativity books (Ellis & Hawking, or Wald?) $\endgroup$ – Danu Jun 26 '18 at 18:39
  • $\begingroup$ Possible duplicate: physics.stackexchange.com/q/21705/2451 $\endgroup$ – Qmechanic Jun 26 '18 at 20:55
  • $\begingroup$ Looking at it I'm not sure why having $O(n)$ as isometry group implies local isometry to $\mathbb{R}^2\times S^{n-2}$, nor why this is equivalent to Birkhoff's theorem. $\endgroup$ – big-lion Jun 27 '18 at 13:13