Advances in Theoretical and Mathematical Physics

Volume 20 (2016)

Number 6

Proof of the mass-angular momentum inequality for bi-axisymmetric black holes with spherical topology

Pages: 1397 – 1441

DOI: http://dx.doi.org/10.4310/ATMP.2016.v20.n6.a4

Authors

Aghil Alaee (Dept. of Mathematical and Statistical Science, University of Alberta, Edmonton, AB, Canada; and Dept. of Mathematics and Statistics, Memorial University of Newfoundland, St John’s, NL, Canada)

Marcus Khuri (Department of Mathematics, Stony Brook University, Stony Brook, New York, U.S.A.)

Hari Kunduri (Department of Mathematics and Statistics, Memorial University of Newfoundland, St John’s, NL, Canada)

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 $m^3 \geq \frac{27 \pi}{32} {(\lvert \mathcal{J}_1 \rvert + \lvert \mathcal{J}_2 \rvert)}^2$, 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.

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Published 2 February 2017