Flat foliation of the Schwarzschild-anti-de Sitter metric

Abstract

Abstract Hypersurfaces used to specify a foliation are required to satisfy some geometric property. This restriction provides a way to derive a differential equation satisfied by those hypersurfaces. In this paper, a complete foliation of the Schwarzschild-anti-de Sitter spacetime by flat spacelike hypersurfaces is provided. A simple procedure based on the fact that geodesics are orthogonal to such hypersurfaces is adopted. There is a barrier found for the hypersurfaces to reach r = ∞ . The Schwarzschild-anti-de Sitter geometry is completely foliated by the analytic continuation of the hypersurfaces beyond the barrier.