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 $A \ge 8\pi |J_+ J_-|^\half$, 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.