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So there are so many solutions for general relativity - then how does one "choose" the solution that is right one? By checking with observation? (though I also know that it is currently unknown which one is the correct solution.)

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What do you mean by 'solutions' for general relativity? Do you mean metrics for how the spacetime is shaped? – DilithiumMatrix Feb 28 '13 at 14:48
The answer to the title question (v1) is By imposing appropriate initial (or boundary) conditions and fitting adjustable parameters of the theory with experimental data. These conditions and parameters depend on the details of the specific type of theory and problem that OP is considering. – Qmechanic Feb 28 '13 at 18:58

Generally speaking we start with a known stress-energy tensor and boundary conditions and look for solutions for the curvature. When doing this we're not usually overloaded with possible solutions, and it's normally pretty obvious which solutions are physically relevant.

Where multiple physically relevant solutions exist we select the appropriate one by comparing with experiment. For example we select the FLRW rather than the Gödel metric because experiment suggests the universe has no net rotation. Likewise the value of $\Lambda$ has to be fixed by experiment, as was done in 1998 by Perlmutter and Riese's groups.

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