Geometric and Causal Structure of Schwarzschild–de Sitter and Kerr–de Sitter Spacetimes

Abstract

We investigate the geometric and causal structure of Schwarzschild-de Sitter and Kerr-de Sitter space times as solutions of Einstein’s field equations with a non-zero cosmological constant. Emphasis is placed on the role of the cosmological constant as an active geometric parameter influencing horizon formation, global curvature, and causal accessibility. For the Schwarzschild-de Sitter case, we analyze the existence of multiple horizons and the emergence of a finite static region bounded by the black hole and cosmological horizons, including the Nariai limit as a critical configuration. The Kerr-de Sitter space time is examined to assess the combined effects of rotation and cosmological expansion, highlighting the presence of additional horizons and ergo regions that modify causal structure. Using conformal compactification, Penrose and projection diagrams are constructed to elucidate the global properties of these space times. A particular contribution of this work is the unified comparative treatment of Schwarzschild-de Sitter and Kerr-de Sitter geometries within a common conformal framework, allowing direct comparison of their causal structures and horizon configurations. The results provide a unified geometric framework for understanding Λ-modified black hole space times.