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The area-angular momentum inequality for black holes in cosmological spacetimes

Clement, M. E. G., Reiris, M., & Simon, W. (2015). The area-angular momentum inequality for black holes in cosmological spacetimes. Classical and quantum gravity, 32(14): 145006. doi:10.1088/0264-9381/32/14/145006.

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Genre: Journal Article

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1501.07243.pdf (Preprint), 267KB
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CQG_32_14_145006.pdf (Publisher version), 494KB
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Creators:
Clement, Maria Eugenia Gabach, Author
Reiris, Martin1, Author
Simon, Walter, Author
Affiliations:
1Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, escidoc:24012

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Free keywords: General Relativity and Quantum Cosmology, gr-qc,Mathematics, Differential Geometry, math.DG,Mathematics, Functional Analysis, math.FA
Abstract: For a stable marginally outer trapped surface (MOTS) in an axially symmetric spacetime with cosmological constant $\Lambda > 0$ and with matter satisfying the dominant energy condition, we prove that the area $A$ and the angular momentum $J$ satisfy the inequality $8\pi |J| \le A\sqrt{(1-\Lambda A/4\pi)(1-\Lambda A/12\pi)}$ which is saturated precisely for the extreme Kerr-deSitter family of metrics. This result entails a universal upper bound $|J| \le J_{\max} \approx 0.17/\Lambda$ for such MOTS, which is saturated for one particular extreme configuration. Our result sharpens the inequality $8\pi |J| \le A$, [7,14] and we follow the overall strategy of its proof in the sense that we estimate the area from below in terms of the energy corresponding to a "mass functional", which is basically a suitably regularised harmonic map $\mathbb{S}^2 \rightarrow \mathbb{H}^2$. However, in the cosmological case this mass functional acquires an additional potential term which itself depends on the area. To estimate the corresponding energy in terms of the angular momentum and the cosmological constant we use a subtle scaling argument, a generalised "Carter-identity", and various techniques from variational calculus, including the mountain pass theorem.

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Dates: 2015-01-282015-02-022015
Publication Status: Published in print
Pages: 24p; minor corrections to v1
Publishing info: -
Rev. Method: -
Identifiers: arXiv: 1501.07243
DOI: 10.1088/0264-9381/32/14/145006
Degree: -

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### Source 1

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Title: Classical and quantum gravity
Source Genre: Journal
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Publ. Info: Bristol, U.K. : Institute of Physics
Pages: - Volume / Issue: 32 (14) Sequence Number: 145006 Start / End Page: - Identifier: ISSN: 0264-9381
CoNE: http://pubman.mpdl.mpg.de/cone/journals/resource/954925513480_1