Abstract

We give a simple proof of the uniqueness of de Sitter and Schwarzschildde-Sitter spacetime without assuming extra conditions on the conformal boundary at infinity. Such spacetimes are the only solutions in the static class satisfying Einstein equations 4 Rαβ = Λ 4 gαβ, where the cosmological constant Λ is positive under appropriate boundary conditions. In the absence of black holes, that is, when the event horizon has only one component the unique solution is de Sitter solution. In the presence of a black hole we get Schwarzschild-de-Sitter spacetime. The problem has important relevance in differential geometry.

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