The Petrov type D equation on genus $>0$ sections of isolated horizons

Abstract

The Petrov type D equation imposed on the 2-metric tensor and the rotation scalar of a cross-section of an isolated horizon can be used to uniquely distinguish the Kerr - (anti) de Sitter spacetime in the case the topology of the cross-section is that of a sphere. In the current paper we study that equation on closed 2-dimensional surfaces that have genus $>0$. We derive all the solutions assuming the embeddability in 4-dimensional spacetime that satisfies the vacuum Einstein equations with (possibly 0) cosmological constant. We prove all of them have constant Gauss curvature and zero rotation. Consequently, we provide a quazi-local argument for a black hole in 4-dimensional spacetime to have a topologically spherical cross-section.

References in corpus (31)

• Isolated and dynamical horizons and their applications
• General Laws of Black-Hole Dynamics
• The Extreme Kerr Throat Geometry: A Vacuum Analog of AdS_2 x S^2
• Mechanics of Rotating Isolated Horizons
• Geometry of Generic Isolated Horizons
• Classification of near-horizon geometries of extremal black holes
• Introduction to Superstring Theory
• A classification of near-horizon geometries of extremal vacuum black holes
• Uniqueness properties of the Kerr metric
• On non-existence of static vacuum black holes with degenerate components of the event horizon
• Extremal Isolated Horizons: A Local Uniqueness Theorem
• Space-Times Admitting Isolated Horizons
• General Kundt spacetimes in higher dimensions
• A spacetime characterization of the Kerr metric
• Quasi-local rotating black holes in higher dimension: geometry
• Mechanics of multidimensional isolated horizons
• Spacetimes foliated by Killing horizons
• Geometric Horizons
• Identification of black hole horizons using scalar curvature invariants
• Geometric Characterizations of the Kerr Isolated Horizon
• Stationary Black Holes as Holographs
• Physical interpretation of Kundt spacetimes using geodesic deviation
• Local version of the no-hair theorem
• Stationary Black Holes as Holographs II
• The Near Horizon Geometry Equation on Compact 2-Manifolds Including the General Solution for g > 0
• Spacetimes foliated by non-expanding and Killing horizons: higher dimension
• Towards a classification of vacuum near-horizons geometries
• The axial symmetry of Kerr without the rigidity theorem
• Uniqueness of extreme horizons in Einstein-Yang-Mills theory
• Near Horizon Geometries and Black Hole Holograph
• On the existence of Kundt's metrics with compact sections of null hypersurfaces

Cited by in corpus (7)

• The Near Horizon Geometry Equation on Compact 2-Manifolds Including the General Solution for g > 0
• The Petrov type D isolated null surfaces
• Non-singular Kerr-NUT-de Sitter spacetimes
• Projectively non-singular horizons in Kerr-NUT-de Sitter spacetimes
• Petrov type D equation on horizons of nontrivial bundle topology
• Isolated horizons of the Hopf bundle structure transversal to the null direction, the horizon equations and embeddability in NUT-like spacetimes
• Vacuum Petrov type D horizons of non-trivial $U(1)$ bundle structure over Riemann surfaces with genus $>0$