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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1https://en.wikipedia.org/wiki/Birkhoff%27s_theorem_(relativity) If you are actually 13, good job. – Ryan Unger Jan 13 '15 at 04:29
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Thanks for the link. I was 13, though it was a week before by 14th birthday. – Bob Bobby Jan 13 '15 at 04:30
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How are you learning GR? Texts or online resources? – Ryan Unger Jan 13 '15 at 04:33
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GR texts and pdfs that I have found – Bob Bobby Jan 13 '15 at 04:34
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Related: http://physics.stackexchange.com/q/21705/2451 and links therein. – Qmechanic Jan 13 '15 at 10:12
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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.
Ryan Unger
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1Indeed. OP has claimed to read this book, but I think it's hard to read a book and miss the content of 7 pages. – Ryan Unger Jan 13 '15 at 11:22
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1Yeah, never mind the bragging etc. Never take anything on the internet at face value ;) – Danu Jan 13 '15 at 12:08