Proof of the local mass-angular momenta inequality for $U(1)^2$ invariant black holes

TL;DR

This paper proves that certain extreme black hole geometries with specific symmetries are local minima of mass among nearby initial data with fixed angular momenta, supporting the mass-angular momentum inequality in this context.

Contribution

It establishes the local minimality of extreme $U(1)^2$ invariant black hole geometries for the mass functional, under specific symmetry and initial data conditions.

Findings

Extreme geometries are local minima of mass for fixed angular momenta.

The ADM mass of nearby data is bounded below by a function of angular momenta.

Conditions satisfied by extreme Myers-Perry and black ring data are crucial for the proof.

Abstract

We consider initial data for extreme vacuum asymptotically flat black holes with $\mathbb{R} \times U(1)^2$ symmetry. Such geometries are critical points of a mass functional defined for a wide class of asymptotically flat, `$(t-\phi^i)$' symmetric maximal initial data for the vacuum Einstein equations. We prove that the above extreme geometries are local minima of mass amongst nearby initial data (with the same interval structure) with fixed angular momenta. Thus the ADM mass of nearby data $m\geq f(J_1,J_2)$ for some function $f$ depending on the interval structure. The proof requires that the initial data of the critical points satisfy certain conditions that are satisfied by the extreme Myers-Perry and extreme black ring data.