Null hypersurfaces in Kerr–(A)dS spacetimes

Abstract

For many purposes, a three-dimensional foliation of spacetime is more advantageous to understanding its light cone structure. We derive the equations describing such foliations for the Kerr geometry with non-zero cosmological constant, and show that they reduce to null hypersurfaces in vacuum (anti-)de Sitter spacetime in the limit of zero mass. Furthermore, we find that these null hypersurfaces are free of caustics everywhere for r > 0. Our construction has applications in numerical studies of rotating black holes, and in defining Kruskal coordinates for rotating black holes with non-zero cosmological constant.The discovery of the Kerr solution in 1963 [1] and its extension to the Kerr-Newman [2] family of solutions of Einstein equations have allowed accurate analytical studies of astrophysical black holes. These solutions furthermore have opened the door for theoretical studies of rotating black hole thermodynamics [3,4], and quantum gravity within the context of the AdS/CFT correspondence [5,6] and the Kerr/CFT correspondence [7].Null hypersurfaces of the Kerr geometry were first systematically studied in [8], where a three-dimensional null slicing of the spacetime was obtained and its properties studied. These hypersurfaces were found to possess no caustics, which make them ideal for studying initial-value problems and wave propagation in Kerr geometry.This type of analysis has never been extended to other rotating black holes, and it is the purpose of this paper to address this deficit by investigating null hypersurfaces for the Kerr-Anti de Sitter (Kerr-AdS) and Kerr-de Sitter (Kerr-dS) black holes. We find that a similar threedimensional null foliation of Kerr-(A)dS spacetimes can be obtained and prove that it also develops no caustics. * a2albalu@uwaterloo.ca † rbmann@uwaterloo.ca arXiv:1909.06419v1 [gr-qc]

Kerr-AdS black hole behaviors from dark energy

“…Concretely, the second order phase transition of four-dimensional Kerr-AdS black holes has been investigated in [18,19]. Critical phenomena of extended Kerr-AdS black holes in the phase space have been also studied in [20].…”

Holographic complexity of rotating black holes

“…In the case of rotating spacetimes, carrying out a computation of the action for a WDW patch (or of the volume of a spacelike slice) is a technically formidable task. The description of null hypersurfaces is somewhat complicated even for 4 spacetime dimensions [49], and no generalization to higher-dimensional cases presently exists. Fortunately there is a special case that renders the computations tractable: Myers-Perry-AdS spacetimes in odd dimensions with equal angular momenta in each orthogonal rotation plane.…”

“…However, for the special case of odd-dimensional rotating black holes with equal angular momenta some of these difficulties can be circumvented, as first emphasized in [30]. Let us illustrate this, following the methods of [49,77,78]. For convenience we will focus on the non-extreme case r + = r − .…”

“…For the most general rotating black holes the light cones can be defined using PDEs as discussed in[49,77,78], though they must be solved numerically.…”

Holographic complexity in charged supersymmetric black holes

“…where we have denoted are the affinely null generator of the past right and past left null segments separately. The null hypersurfaces in Kerr-Newman-(A)dS spacetimes have also been studied in [75,76]. The transformation parameter of this joint is given by η ¼ lnðjk 1 • k 2 j=2Þ [6].…”