Proof of the Mass-Angular Momentum Inequality for Bi-Axisymmetric Black Holes With Spherical Topology

TL;DR

This paper proves a mass-angular momentum inequality for 5-dimensional bi-axisymmetric black holes, showing extreme Myers-Perry data minimize mass and characterize equality cases.

Contribution

It establishes the mass-angular momentum inequality for 5D black holes with spherical topology, extending previous results and characterizing the equality case.

Findings

Extreme Myers-Perry data realize the minimum mass.

All data satisfy the mass-angular momentum inequality.

Equality holds only for extreme Myers-Perry black holes.

Abstract

We show that extreme Myers-Perry initial data realize the unique absolute minimum of the total mass in a physically relevant (Brill) class of maximal, asymptotically flat, bi-axisymmetric initial data for the Einstein equations with fixed angular momenta. As a consequence, we prove the relevant mass-angular momentum inequality in this setting for 5-dimensional spacetimes. That is, all data in this class satisfy the inequality $m^3\geq \frac{27\pi}{32}\left(|\mathcal{J}_1|+|\mathcal{J}_2|\right)^2$, where $m$ and $\mathcal{J}_i$, $i=1,2$ are the total mass and angular momenta of the spacetime. Moreover, equality holds if and only if the initial data set is isometric to the canonical slice of an extreme Myers-Perry black hole.