The general Kerr–de Sitter metrics in all dimensions

Abstract

We give the general Kerr-de Sitter metric in arbitrary spacetime 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.

Black rings in (Anti)-de Sitter space

“…The solutions were found in [12,13] and their physical parameters correctly computed in [21]. We consider the situation with only one spin turned on 7 , for which the solutions are parametrized by the mass and rotation parameters m and a, and the physical magnitudes are…”

“…The geometry in fact approaches that of a black membrane as the upper bound on J is approached. To see this, take the solution in Boyer-Lindquist coordinates (t, r, θ, φ, Ω d−4 ) of [13], and define a new mass parameter and new coordinateŝ…”

“…Here we discuss the phase space of the rotating AdS black hole solution of [13] with more than one angular momentum (the case of a single angular momentum is discussed in the main text). Their mass, angular momenta, area and surface gravity, as computed in [21], are…”

“…To avoid misnomers, we shall refer to the solutions that generalize the Kerr and MyersPerry black holes to include a cosmological constant [11,12,13] as rotating (A)dS black holes. Even if (A)dS black rings and pinched black holes are also (A)dS black holes, we hope that the distinction is clear from the context.…”

Killing-Yano tensors, rank-2 Killing tensors, and conserved quantities in higher dimensions

“…Upon the substitution of eq. (3.14) into (3.10), the dependence on w cancels out, and we recover the quadratic dependence on u which enters through u and v. Comparing with (3.2), we can write the tensorial relation between K (j) and h, which in components reads 15) JHEP02 (2007)004 where we have employed the definition (3.11), the identities (A.2) and (A.3), and the normalization (A.4). Recently [21] there have been found different conserved quantities, 16) which are, however, not quadratic in velocities.…”