The General Kerr-de Sitter Metrics in All Dimensions

TL;DR

The paper delivers the general Kerr-de Sitter solution in all dimensions $D\ge 4$ with the maximal set of rotations, presenting both a Kerr-Schild form around de Sitter space and a generalized Boyer-Lindquist form. It verifies the solutions satisfy the Einstein equations with cosmological constant for $D\le 11$, analyzes horizon structure including angular velocities, areas, and surface gravity, and discusses Euclidean-signature continuations. A major contribution is the construction of complete non-singular compact Einstein spaces on $S^{D-2}$ bundles over $S^2$, with infinitely many examples in each odd dimension $D\ge 5$, arising from intricate regularity and quantization conditions on the rotation parameters. The results provide a unifying framework for higher-dimensional rotating black holes with a cosmological constant and yield rich geometric families in the Euclidean regime, with implications for topology and gravitational thermodynamics.

Abstract

We give the general Kerr-de Sitter metric in arbitrary spacetime dimension D\ge 4, with the maximal number [(D-1)/2] of independent rotation parameters. We obtain the metric in Kerr-Schild form, where it is written as the sum of a de Sitter metric plus the square of a null geodesic vector, and in generalised Boyer-Lindquist coordinates. The Kerr-Schild form is simpler for verifying that the Einstein equations are satisfied, and we have explicitly checked our results for all dimensions D\le 11. We discuss the global structure of the metrics, and obtain formulae for the surface gravities and areas of the event horizons. We also obtain the Euclidean-signature solutions, and we construct complete non-singular compact Einstein spaces on associated S^{D-2} bundles over S^2, infinitely many for each odd D \ge 5.