# Solutions locally isometric to Schwarzchild [duplicate]

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.

• This is addressed by Birkhoff's theorem: en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity) – Danu Jun 26 '18 at 18:36
• 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. – big-lion Jun 26 '18 at 18:38
• 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?) – Danu Jun 26 '18 at 18:39
• Possible duplicate: physics.stackexchange.com/q/21705/2451 – Qmechanic Jun 26 '18 at 20:55
• 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. – big-lion Jun 27 '18 at 13:13