Horizon area-angular momentum inequality in higher dimensional spacetimes

Abstract

We consider $n$-dimensional spacetimes which are axisymmetric--but not necessarily stationary (!)--in the sense of having isometry group $U(1{)}^{n-3}$, and which satisfy the Einstein equations with a non-negative cosmological constant. We show that any black hole horizon must have area $A \ge 8π|J_+ J_-|^\half$, where $J_{\pm }$ are distinguished components of the angular momentum corresponding to linear combinations of the rotational Killing fields that vanish somewhere on the horizon. In the case of $n=4$, where there is only one angular momentum component $J_{+}=J_{-}$, we recover an inequality of 1012.2413 [gr-qc]. Our work can hence be viewed as a generalization of this result to higher dimensions. In the case of $n=5$ with horizon of topology $S^1\times S^2$, the quantities $J_{+}=J_{-}$ are the same angular momentum component (in the $S^2$ direction). In the case of $n=5$ with horizon topology $S^3$, the quantities $J_{+},J_{-}$ are the distinct components of the angular momentum. We also show that, in all dimensions, the inequality is saturated if the metric is a so-called ``near horizon geometry''. Our argument is entirely quasi-local, and hence also applies e.g. to any stably outer marginally trapped surface.

16 pages, Latex, no figures

References in corpus (4)

• Near-horizon symmetries of extremal black holes
• A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric
• Uniqueness theorem for 5-dimensional black holes with two axial Killing fields
• A classification of near-horizon geometries of extremal vacuum black holes

Cited by in corpus (11)

• Classification of near-horizon geometries of extremal black holes
• Spacetime near isolated and dynamical trapping horizons
• Geometrical inequalities bounding angular momentum and charges in General Relativity
• Black hole Area-Angular momentum-Charge inequality in dynamical non-vacuum spacetimes
• Geometric inequalities for black holes
• Area-angular momentum-charge inequality for stable marginally outer trapped surfaces in 4D Einstein-Maxwell-dilaton theory
• Rotating black rings on Taub-NUT
• Mass angular momentum and charge inequalities for black holes in Einstein-Maxwell-axion-dilaton gravity
• On a mass functional for initial data in 4+1 dimensional spacetime
• Bounding Horizon Area by Angular Momentum, Charge, and Cosmological Constant in 5-Dimensional Minimal Supergravity
• Small deformations of extreme five dimensional Myers-Perry black hole initial data