Some properties of the Schwarzschild–de Sitter and Schwarzschild–anti-de Sitter spacetimes

Abstract

Properties of the Schwarzschild--de Sitter and Schwarzschild--anti-de Sitter spacetimes are characterized by three phenomena, namely, by the ``effective potential'' of the motion of test particles and photons, the photon escape cones, and the embedding diagrams of $t=\mathrm{const}$ sections of central planes of both the ordinary and optical reference geometry of these spacetimes. The phenomena are related to the corresponding phenomena of the Schwarzschild spacetime, and differences caused by the asymptotic structure of the spacetimes with a nonzero cosmological constant are discussed. The properties of the embedding diagrams of the optical geometry are related to the dynamical behavior of test particles. The limits of the embeddability of the optical geometry are given and compared with the limits on the outer radius of the interior solutions of Einstein's equations with a nonzero cosmological constant for static, spherically symmetric configurations of uniform density. It is shown that, contrary to the pure Schwarzschild case, these limits do not fully coincide for repulsive cosmological constants.