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.
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.
References
Acena, A., Dain, S., Clement, M.E.G.: Horizon area–angular momentum inequality for a class of axially symmetric black holes. Class. Quant. Grav. 28, 105,014 (2011). DOI 10.1088/0264-9381/28/10/105014
Andersson, L., Eichmair, M., Metzger, J.: Jang’s equation and its applications to marginally trapped surfaces (2010)
Andersson, L., Mars, M., Simon, W.: Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 95, 111,102 (2005)
Andersson, L., Mars, M., Simon, W.: Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes. Adv. Theor. Math. Phys. 12, 853–888 (2008)
Ansorg, M., Hennig, J., Cederbaum, C.: Universal properties of distorted Kerr-Newman black holes. Gen. Rel. Grav. 43, 1205–1210 (2011). DOI 10.1007/s10714-010-1136-8
Ansorg, M., Pfister, H.: A universal constraint between charge and rotation rate for degenerate black holes surrounded by matter. Class. Quant. Grav. 25, 035,009 (2008). DOI 10.1088/0264-9381/25/3/035009
Arms, J., Marsden, J., Moncrief, V.: The structure of the space of solutions of Einstein’s equations. II. Several Killing fields and the Einstein Yang-Mills equations. Annals Phys. 144, 81–106 (1982). DOI 10.1016/0003-4916(82)90105-1
Ashtekar, A., Beetle, C., Lewandowski, J.: Mechanics of rotating isolated horizons. Phys.Rev. D64, 044,016 (2001). DOI 10.1103/PhysRevD.64.044016
Ashtekar, A., Engle, J., Pawlowski, T., Van Den Broeck, C.: Multipole moments of isolated horizons. Class. Quant. Grav. 21, 2549 (2004)
Ashtekar, A., Fairhurst, S., Krishnan, B.: Isolated horizons: Hamiltonian evolution and the first law. Phys. Rev. D62, 104,025 (2000)
Ashtekar, A., Fairhurst, S., Krishnan, B.: Isolated horizons: Hamiltonian evolution and the first law. Phys.Rev. D62, 104,025 (2000). DOI 10.1103/PhysRevD.62.104025
Ashtekar, A., Galloway, G.J.: Some uniqueness results for dynamical horizons. Adv. Theor. Math. Phys. 9, 1 (2005)
Ashtekar, A., Krishnan, B.: Dynamical horizons: Energy, angular momentum, fluxes and balance laws. Phys. Rev. Lett. 89, 261,101 (2002)
Ashtekar, A., Krishnan, B.: Dynamical horizons and their properties. Phys. Rev. D 68, 104,030 (2003)
Ashtekar, A., Krishnan, B.: Isolated and dynamical horizons and their applications. Liv. Rev. Relat. 7, 10 (2004). http://www.livingreviews.org/lrr-2004-10. URL (cited on 9 March 2012) http://www.livingreviews.org/lrr-2004-10
Booth, I., Fairhurst, S.: Isolated, slowly evolving, and dynamical trapping horizons: geometry and mechanics from surface deformations. Phys. Rev. D75, 084,019 (2007)
Booth, I., Fairhurst, S.: Extremality conditions for isolated and dynamical horizons. Phys. Rev. D77, 084,005 (2008). DOI 10.1103/PhysRevD.77.084005
Cao, L.M.: Deformation of Codimension-2 Surface and Horizon Thermodynamics. JHEP 1103, 112 (2011). DOI 10.1007/JHEP03(2011)112
Carter, B.: Republication of: Black hole equilibrium states part ii. general theory of stationary black hole states. Gen. Relat. Gravit. 42, 653–744 (2010). http://dx.doi.org/10.1007/s10714-009-0920-9. 10.1007/s10714-009-0920-9
Chrusciel, P., Kondracki, W.: Some global charges in classical Yang-Mills theory. Phys.Rev. D36, 1874–1881 (1987). DOI 10.1103/PhysRevD.36.1874
Chruściel, P.T.: Mass and angular-momentum inequalities for axi-symmetric initial data sets I. Positivity of mass. Annals Phys. 323, 2566–2590 (2008). DOI 10.1016/j.aop.2007.12.010
Chruściel, P.T., Li, Y., Weinstein, G.: Mass and angular-momentum inequalities for axi-symmetric initial data sets. II. Angular-momentum. Annals Phys. 323, 2591–2613 (2008). DOI 10.1016/j.aop.2007.12.011
Chrusciel, P.T., Lopes Costa, J.: Mass, angular-momentum, and charge inequalities for axisymmetric initial data. Class. Quant. Grav. 26, 235,013 (2009). DOI 10.1088/0264-9381/26/23/235013
Costa, J.L.: A Dain Inequality with charge. arXiv:0912.0838 (2009)
Dain, S.: Angular-momentum-mass inequality for axisymmetric black holes. Phys. Rev. Lett. 96, 101,101 (2006)
Dain, S.: Proof of the (local) angular momemtum-mass inequality for axisymmetric black holes. Class. Quant. Grav. 23, 6845–6856 (2006). DOI 10.1088/0264-9381/23/23/015
Dain, S.: Proof of the angular momentum-mass inequality for axisymmetric black holes. J. Diff. Geom. 79, 33–67 (2008)
Dain, S.: Extreme throat initial data set and horizon area-angular momentum inequality for axisymmetric black holes. Phys. Rev. D82, 104,010 (2010). DOI 10.1103/PhysRevD.82.104010
Dain, S.: Geometric inequalities for axially symmetric black holes. Class. Quant. Grav. 29, 073,001 (2012). DOI 10.1088/0264-9381/29/7/073001
Dain, S., Jaramillo, J.L., Reiris, M.: Area-charge inequality for black holes. Class.Quant.Grav. 29, 035,013 (2012). DOI 10.1088/0264-9381/29/3/035013
Dain, S., Reiris, M.: Area - Angular momentum inequality for axisymmetric black holes. Phys.Rev.Lett. 107, 051,101 (2011). DOI 10.1103/PhysRevLett.107.051101
Fischer, A., Marsden, J., Moncrief, V.: The structure of the space of solutions of Einsteins equations I. One Killing field. Ann. Inst. H. Poincaré 33, 147–194 (1980)
Gabach Clement, M.E.: Comment on Horizon area-Angular momentum inequality for a class of axially symmetric black holes. arXiv:1102.3834 (2011)
Gabach Clement, M.E., Jaramillo, J.L.: Black hole Area-Angular momentum-Charge inequality in dynamical non-vacuum spacetimes. Accepted in Phys. Rev. D. arXiv:1111.6248 (2012)
Gabach Clement, M.E., Jaramillo, J.L., Reiris, M.: Proof of the area-angular momentum-charge inequality for axisymmetric black holes. arXiv:1207.6761 (2012)
Galloway, G.J., Schoen, R.: A Generalization of Hawking’s black hole topology theorem to higher dimensions. Commun.Math.Phys. 266, 571–576 (2006). DOI 10.1007/s00220-006-0019-z
Hayward, S.: General laws of black-hole dynamics. Phys. Rev. D 49, 6467 (1994)
Hayward, S.: Energy conservation for dynamical black holes. Phys. Rev. Lett. 93, 251,101 (2004)
Hayward, S.A.: Energy and entropy conservation for dynamical black holes. Phys. Rev. D 70, 104,027 (2004)
Hennig, J., Ansorg, M., Cederbaum, C.: A universal inequality between the angular momentum and horizon area for axisymmetric and stationary black holes with surrounding matter. Class. Quant. Grav. 25, 162,002 (2008)
Hennig, J., Cederbaum, C., Ansorg, M.: A universal inequality for axisymmetric and stationary black holes with surrounding matter in the Einstein-Maxwell theory. Commun. Math. Phys. 293, 449–467 (2010). DOI 10.1007/s00220-009-0889-y
Hollands, S.: Horizon area-angular momentum inequality in higher dimensional spacetimes. Class.Quant.Grav. 29, 065,006 (2012). DOI 10.1088/0264-9381/29/6/065006
Jaramillo, J.L., Reiris, M., Dain, S.: Black hole Area-Angular momentum inequality in non-vacuum spacetimes. Phys.Rev. D84, 121,503 (2011). DOI 10.1103/PhysRevD.84.121503
Jaramillo, J.L.: A note on degeneracy, marginal stability and extremality of black hole horizons. Class.Quant.Grav. 29 177,001 (2012). DOI 10.1088/0264-9381/29/17/177001
Mars, M.: Stability of MOTS in totally geodesic null horizons. Class.Quant.Grav. 29 145,019 (2012). DOI 10.1088/0264-9381/29/14/145019
Penrose, R.: Gravitational collapse: The role of general relativity. Riv. Nuovo Cim. 1, 252 (1969)
Penrose, R.: Naked singularities. Annals N. Y. Acad. Sci. 224, 125 (1973)
Racz, I.: A simple proof of the recent generalisations of Hawking’s black hole topology theorem. Class. Quant. Grav. 25, 162,001 (2008). DOI 10.1088/0264-9381/25/16/162001
Ryder, L.: Dirac monopoles and the Hopf map S(3) to S(2). J. Phys. A A13, 437–447 (1980). DOI 10.1088/0305-4470/13/2/012
Simon, W.: Gravitational field strength and generalized Komar integral. Gen. Rel. Grav. 17, 439 (1985). DOI 10.1007/BF00761903
Simon, W.: Bounds on area and charge for marginally trapped surfaces with a cosmological constant. Class. Quant. Grav. 29, 062,001 (2012). DOI 10.1088/0264-9381/29/6/062001
Sudarsky, D., Wald, R.M.: Extrema of mass, stationarity, and staticity, and solutions to the Einstein Yang-Mills equations. Phys. Rev. D46, 1453–1474 (1992). DOI 10.1103/PhysRevD.46.1453
Wald, R.M.: General Relativity. Chicago University Press, Chicago (1984)
Weinberg, S.: The quantum theory of fields. Modern applications, vol. 2. Cambridge University Press, New York (1996)
Wu, T.T., Yang, C.N.: Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D 12, 3845–3857 (1975). DOI 10.1103/PhysRevD.12.3845. http://link.aps.org/doi/10.1103/PhysRevD.12.3845
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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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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