Why de sitter and Schwarzschild de sitter, and anti de sitter and Scharzschild anti de sitter are the only possible spherically symmetric solutions with a cosmological constant? I have read this fact but unable to prove it. (An intuitive proof may be even more helpful). (Please do note that I am from Physics background)
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Essentially a duplicate of http://physics.stackexchange.com/q/21705/2451 where the proof is sketched. – Qmechanic Feb 23 '17 at 03:35
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The Birkoff's theorem imposes strong constraints on spherically symmetric solutions in vacuum spacetime (with possible singularities at r= 0) - basically they have to be isometric to a maxim ally extended Schwarzschild (Sch) solution.(if I remember right if it's spherically symmetric and static it has to be Schwarzschild up to coordinate transformations). See https://en.wikipedia.org/wiki/Spherically_symmetric_spacetime
Since deSitter and AntideSitter (dS and AdS)come in for a lambda (cosmological constant) solution it's not unreasonable to think that adding lambda may bring in something similar. The paper below shows that the dS - Sch solution is the only one, with lambda any constant (so I'm guessing he includes the AdS but I did not read the full paper, you can read it in the link): Colin Rourke: Uniqueness of spherically-symmetric vacuum solutions to Einstein’s equations,
Still, I am not sure that the uniqueness has really been proven. The following pdf paper finds other non-equivalent solutions. See http://redshift.vif.com/JournalFiles/V09NO3PDF/V09N3ABA.pdf. I can't vouch for its correctness.
General Relativity, being very nonlinear, is not something where if you find one solution and combine it somehow with another it automatically works as another solution. Hard to get general theorems on uniqueness and manipulation so of solutions to get another.
This is not meant to be a complete answer, but maybe it helps some.
