Communications in Analysis and Geometry

Volume 23 (2015)

Number 2

Uniqueness of de Sitter and Schwarzschild-de Sitter spacetimes

Pages: 377 – 387

DOI: http://dx.doi.org/10.4310/CAG.2015.v23.n2.a7

Authors

A.K.M. Masood-ul-Alam (Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Wenhua Yu (Mathematical Sciences Center and Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Abstract

We give a simple proof of the uniqueness of de Sitter and Schwarzschild-de 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 $\overset{4}{R}_{\alpha \beta} = \Lambda \overset{4}{g}_{\alpha \beta}$, where the cosmological constant $\Lambda$ 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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