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Area Inequalities for Stable Marginally Trapped Surfaces

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Recent Trends in Lorentzian Geometry

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 26))

Abstract

We discuss a family of inequalities involving the area, angular momentum and charges of stably outermost marginally trapped surfaces in generic non-vacuum dynamical spacetimes, with non-negative cosmological constant and matter sources satisfying the dominant energy condition. These inequalities provide lower bounds for the area of spatial sections of dynamical trapping horizons, namely hypersurfaces offering quasi-local models of black hole horizons. In particular, these inequalities represent particular examples of the extension to a Lorentzian setting of tools employed in the discussion of minimal surfaces in Riemannian contexts.

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Notes

  1. 1.

    Alternatively, one could start characterizing MOTS stability in terms of the principal eigenvalue λ X . Then, the expression of λ X in a Rayleigh–Ritz type characterization [4] leads essentially to the integral inequality. See M. Mars’ contribution, where the role of α is played by a function u.

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Acknowledgments

This work is fully indebted to the close scientific collaboration with S. Dain, M.E. Gabach Clément, M. Reiris and W. Simon. I would like to express here my gratitude to them. I would also like to thank A. Aceña, M. Ansorg, C. Barceló, M. Mars and J.M.M. Senovilla for useful discussions. I thank M.E. Gabach Clément for her careful reading of this chapter. I acknowledge the support of the Spanish MICINN (FIS2008-06078-C03-01) and the Junta de Andalucía (FQM2288/219).

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Authors and Affiliations

  1. Max-Planck-Institut für Gravitationsphysik, Albert Einstein Institut, Am Mühlenberg 1, D-14476, Potsdam, Germany

    José Luis Jaramillo

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  1. José Luis Jaramillo
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Correspondence to José Luis Jaramillo .

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Editors and Affiliations

  1. , Dept of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E18071, Granada, Spain

    Miguel Sánchez

  2. Departamento de Mátematica Aplicada, Universidad de Granada, Campus de Fuentenueva, Granada, 18071, Spain

    MIguel Ortega

  3. , Dept. of Geometry and Topology, University of Granada, Campus Fuentenueva s/n, Granada, E-18071, Granada, Spain

    Alfonso Romero

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Jaramillo, J.L. (2012). Area Inequalities for Stable Marginally Trapped Surfaces. In: Sánchez, M., Ortega, M., Romero, A. (eds) Recent Trends in Lorentzian Geometry. Springer Proceedings in Mathematics & Statistics, vol 26. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4897-6_5

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