Static black hole uniqueness theorems
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Perhaps the oldest uniqueness result in general relativity is the famous Birkhoff theorem, discovered merely a few years after general relativity itself. Birkhoff’s theorem states that the only spherically symmetric solution of the vacuum Einstein equation is the Schwarzschild solution. All subsequent uniqueness theorems essentially follow the same format - assume some symmetry and prove the symmetry to be so restrictive that Einstein’s equation has only one solution.
Despite not being assumed a priori, the Schwarszschild solution possesses the property that it is static. Naturally, one may wonder if a kind of converse to Birkhoff’s theorem is true. Does every static, vacuum spacetime have to be spherically symmetric and thus Schwarzschild? If not, what further assumptions are required? These issues are more interesting when studied in the context of spacetimes containing black holes and this is the main topic explored in this report.