The general Kerr–de Sitter metrics in all dimensions

• Published in: Journal of geometry and physics Vol. 53; no. 1; pp. 49 - 73
• Main Authors: Gibbons, G.W., Lü, H., Page, Don N., Pope, C.N.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 2005
• Subjects: Einstein equations; Exact solutions; Kerr–de Sitter metrics; Compact Einstein spaces; Kerr–de Sitter metrics; Compact Einstein spaces; 83C15; Einstein equations; 53C25; Exact solutions
• ISSN: 0393-0440; 1879-1662
• DOI: 10.1016/j.geomphys.2004.05.001

We give the general Kerr–de Sitter metric in arbitrary space–time dimension D ≥ 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 ≤ 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 ≥ 5 .