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I'm able to derive the Schwarzschild solution under the assumptions that the metric is (1) static (2) spherically symmetric and that the space is the vacuum. However, I have read that the Schwarzschild solution can be found assuming only that the metric is a spherically symmetric vacuum. How would the Schwarzschild solution be derived under these weaker conditions?

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There is a theorem which states that any spherically symmetric solution to the vacuum equations is also necessarily static and asymptotically flat. It is known as Birkhoff's theorem. Chapter 4 of Straumann (2013) contains a full proof.

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    $\begingroup$ Also Carroll's GR book. $\endgroup$ – Danu Jan 13 '15 at 8:59
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    $\begingroup$ Indeed. OP has claimed to read this book, but I think it's hard to read a book and miss the content of 7 pages. $\endgroup$ – Ryan Unger Jan 13 '15 at 11:22
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    $\begingroup$ Yeah, never mind the bragging etc. Never take anything on the internet at face value ;) $\endgroup$ – Danu Jan 13 '15 at 12:08

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