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 , where J± 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 Acena et al. 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 S1 × S2, the quantities J+ = J− are the same angular momentum component (in the S2-direction). In the case of n = 5 with horizon topology S3, 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.