Geometric inequalities for black holes

  • Sergio DainEmail author
Review Article
Part of the following topical collections:
  1. The First Century of General Relativity: GR20/Amaldi10


It is well known that the three parameters that characterize the Kerr black hole (mass, angular momentum and horizon area) satisfy several important inequalities. Remarkably, some of these inequalities remain valid also for dynamical black holes. This kind of inequalities play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this article recent results in this subject are reviewed.


Black holes Geometric inequalities Mass Angular momentum Horizon area 



This work was supported by Grant PICT-2010-1387 of CONICET (Argentina) and grant Secyt-UNC (Argentina).


  1. 1.
    Aceña, A., Dain, S., Gabach Clément, M.E.: Horizon area: angular momentum inequality for a class of axially symmetric black holes. Class. Quantum Grav. 28(10), 105014 (2011).
  2. 2.
    Andersson, L., Mars, M., Simon, W.: Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes. Adv. Theor. Math. Phys. 12(4), 853–888 (2008).
  3. 3.
    Anglada, P., Dain, S., Ortiz, O.: In preparationGoogle Scholar
  4. 4.
    Ansorg, M., Hennig, J.: The Inner Cauchy horizon of axisymmetric and stationary black holes with surrounding matter. Class. Quantum Grav. 25, 222001 (2008). doi: 10.1088/0264-9381/25/22/222001 ADSCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ansorg, M., Hennig, J.: The Inner Cauchy horizon of axisymmetric and stationary black holes with surrounding matter in Einstein-Maxwell theory. Phys. Rev. Lett. 102, 221102 (2009). doi: 10.1103/PhysRevLett.102.221102 ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ansorg, M., Hennig, J., Cederbaum, C.: Universal properties of distorted Kerr–Newman black holes. Gen. Relativ. Gravit 43, 1205–1210 (2011). doi: 10.1007/s10714-010-1136-8 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Ansorg, M., Petroff, D.: Black holes surrounded by uniformly rotating rings. Phys. Rev. D 72, 024019 (2005)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ansorg, M., Pfister, H.: A universal constraint between charge and rotation rate for degenerate black holes surrounded by matter. Class. Quantum Grav. 25, 035009 (2008). doi: 10.1088/0264-9381/25/3/035009 ADSCrossRefMathSciNetGoogle Scholar
  9. 9.
    Arnowitt, R., Deser, S., Misner, C.W.: The dynamics of general relativity. In: Witten, L. (ed.) Gravitation: An Introduction to Current Research, pp. 227–265. Wiley, New York (1962)Google Scholar
  10. 10.
    Ashtekar, A., Krishnan, B.: Dynamical horizons: energy, angular momentum, fluxes and balance laws. Phys. Rev. Lett. 89, 261101 (2002)ADSCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ashtekar, A., Krishnan, B.: Dynamical horizons and their properties. Phys. Rev. D 68, 104030 (2003)ADSCrossRefMathSciNetGoogle Scholar
  12. 12.
    Baiotti, L., et al.: Three-dimensional relativistic simulations of rotating neutron-star collapse to a Kerr black hole. Phys. Rev. D71, 024035 (2005). doi: 10.1103/PhysRevD.71.024035 ADSGoogle Scholar
  13. 13.
    Bartnik, R.: The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39(5), 661–693 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Bizon, P., Malec, E., O’Murchadha, N.: Trapped surfaces in spherical stars. Phys. Rev. Lett. 61, 1147–1450 (1988). doi: 10.1103/PhysRevLett.61.1147 ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    Bizon, P., Malec, E., O’Murchadha, N.: Trapped surfaces due to concentration of matter in spherically symmetric geometries. Class. Quantum Grav. 6, 961–976 (1989). doi: 10.1088/0264-9381/6/7/004 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Booth, I., Fairhurst, S.: Isolated, slowly evolving, and dynamical trapping horizons: geometry and mechanics from surface deformations. Phys. Rev. D75, 084019 (2007). doi: 10.1103/PhysRevD.75.084019 ADSMathSciNetGoogle Scholar
  17. 17.
    Booth, I., Fairhurst, S.: Extremality conditions for isolated and dynamical horizons. Phys. Rev. D77, 084005 (2008). doi: 10.1103/PhysRevD.77.084005 ADSMathSciNetGoogle Scholar
  18. 18.
    Bray, H.L.: Proof of the riemannian penrose conjecture using the positive mass theorem. J. Differ. Geom. 59, 177–267 (2001)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Christodoulou, D.: Reversible and irreversible transforations in black-hole physics. Phys. Rev. Lett. 25, 1596–1597 (1970)ADSCrossRefGoogle Scholar
  20. 20.
    Chruściel, P.: Boundary conditions at spatial infinity from a Hamiltonian point of view. In: Topological properties and global structure of space-time (Erice, 1985), NATO Adv. Sci. Inst. Ser. B Phys., vol. 138, pp. 49–59. Plenum, New York (1986).
  21. 21.
    Chrusciel, P.T.: Mass and angular-momentum inequalities for axi-symmetric initial data sets I. Posit. Mass. Ann. Phys. 323, 2566–2590 (2008). doi: 10.1016/j.aop.2007.12.010 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Chrusciel, P.T., Eckstein, M., Nguyen, L., Szybka, S.J.: Existence of singularities in two-Kerr black holes. Class. Quantum Grav. 28, 245017 (2011). doi: 10.1088/0264-9381/28/24/245017 ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Chruściel, P.T., Li, Y., Weinstein, G.: Mass and angular-momentum inequalities for axi-symmetric initial data sets II. Angular-momentum. Ann. Phys. 323(10), 2591–2613 (2008)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Chrusciel, P.T., Lopes Costa, J.: Mass, angular-momentum, and charge inequalities for axisymmetric initial data. Class. Quant. Grav. 26, 235013 (2009). doi: 10.1088/0264-9381/26/23/235013 ADSCrossRefGoogle Scholar
  25. 25.
    Chruciel, P.T., Costa, J.L., Heusler, M.: Stationary black holes: uniqueness and beyond. Living Rev. Relativ. 15(7) (2012). doi: 10.12942/lrr-2012-7.
  26. 26.
    Costa, J.L.: Proof of a Dain inequality with charge. J. Phys. A Math. Theor. 43(28), 285202 (2010). Google Scholar
  27. 27.
    Cvetic, M., Gibbons, G., Pope, C.: Universal area product formulae for rotating and charged black holes in four and higher dimensions. Phys. Rev. Lett. 106, 121301 (2011). doi: 10.1103/PhysRevLett.106.121301 ADSCrossRefMathSciNetGoogle Scholar
  28. 28.
    Dain, S.: Angular momemtum–mass inequality for axisymmetric black holes. Phys. Rev. Lett. 96, 101101 (2006)ADSCrossRefMathSciNetGoogle Scholar
  29. 29.
    Dain, S.: Proof of the (local) angular momemtum–mass inequality for axisymmetric black holes. Class. Quantum Grav. 23, 6845–6855 (2006)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Dain, S.: A variational principle for stationary, axisymmetric solutions of Einstein’s equations. Class. Quantum Grav. 23, 6857–6871 (2006)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  31. 31.
    Dain, S.: The inequality between mass and angular momentum for axially symmetric black holes. Int. J. Mod. Phys. D 17(3–4), 519–523 (2008)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Dain, S.: Proof of the angular momentum–mass inequality for axisymmetric black holes. J. Differ. Geom. 79(1), 33–67 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Dain, S.: Extreme throat initial data set and horizon area-angular momentum inequality for axisymmetric black holes. Phys. Rev. D 82(10), 104010 (2010). doi: 10.1103/PhysRevD.82.104010 ADSCrossRefMathSciNetGoogle Scholar
  34. 34.
    Dain, S.: Geometric inequalities for axially symmetric black holes. Classical and Quantum Gravity 29(7), 073001 (2012).
  35. 35.
    Dain, S.: Inequality between size and angular momentum for bodies. Phys. Rev. Lett. 112, 041101 (2014). doi: 10.1103/PhysRevLett.112.041101 ADSCrossRefGoogle Scholar
  36. 36.
    Dain, S., Gentile de Austria, I.: On the linear stability of the extreme Kerr black hole under axially symmetric perturbations (2014)Google Scholar
  37. 37.
    Dain, S., Jaramillo, J.L., Reiris, M.: Area-charge inequality for black holes. Class. Quantum Grav. 29(3), 035013 (2012). Google Scholar
  38. 38.
    Dain, S., Khuri, M., Weinstein, G., Yamada, S.: Lower bounds for the area of black holes in terms of mass, charge, and angular momentum. Phys. Rev. D88, 024048 (2013). doi: 10.1103/PhysRevD.88.024048 ADSGoogle Scholar
  39. 39.
    Dain, S., Ortiz, O.E.: Numerical evidences for the angular momentum-mass inequality for multiple axially symmetric black holes. Phys. Rev. D80, 024045 (2009). doi: 10.1103/PhysRevD.80.024045 ADSGoogle Scholar
  40. 40.
    Dain, S., Reiris, M.: Area–angular-momentum inequality for axisymmetric black holes. Phys. Rev. Lett. 107(5), 051101 (2011). doi: 10.1103/PhysRevLett.107.051101 ADSCrossRefGoogle Scholar
  41. 41.
    Fajman, D., Simon, W.: Area inequalities for stable marginally outer trapped surfaces in Einstein–Maxwell–dilaton theory. Preprint (2013).
  42. 42.
    Flanagan, E.: Hoop conjecture for black-hole horizon formation. Phys. Rev. D 44, 2409–2420 (1991). doi: 10.1103/PhysRevD.44.2409 ADSCrossRefMathSciNetGoogle Scholar
  43. 43.
    Gabach Clément, M.E.: Comment on Horizon area-angular momentum inequality for a class of axially symmetric black holes. Preprint (2011).
  44. 44.
    Gabach Clément, M.E.: Bounds on the force between black holes. Class. Quantum Grav. 29, 165008 (2012). doi: 10.1088/0264-9381/29/16/165008 ADSCrossRefGoogle Scholar
  45. 45.
    Gabach Clément, M.E., Jaramillo, J.L.: Black hole area-angular momentum-charge inequality in dynamical non-vacuum spacetimes. Phys. Rev. D86, 064021 (2012). doi: 10.1103/PhysRevD.86.064021 ADSGoogle Scholar
  46. 46.
    Gabach Clément, M.E., Jaramillo, J.L., Reiris, M.: Proof of the area-angular momentum-charge inequality for axisymmetric black holes. Class. Quantum Grav. 30, 065017 (2013). doi: 10.1088/0264-9381/30/6/065017 ADSCrossRefGoogle Scholar
  47. 47.
    Gabach Clément, M.E., Reiris, M.: On the shape of rotating black-holes. Phys. Rev. D88, 044031 (2013). doi: 10.1103/PhysRevD.88.044031 ADSGoogle Scholar
  48. 48.
    Giacomazzo, B., Rezzolla, L., Stergioulas, N.: Collapse of differentially rotating neutron stars and cosmic censorship. Phys. Rev. D84, 024022 (2011). doi: 10.1103/PhysRevD.84.024022 ADSGoogle Scholar
  49. 49.
    Gibbons, G.: What is the shape of a black hole? AIP Conf. Proc. 1460, 90–100 (2012). doi: 10.1063/1.4733363 ADSCrossRefGoogle Scholar
  50. 50.
    Hennig, J., Ansorg, M.: The Inner Cauchy horizon of axisymmetric and stationary black holes with surrounding matter in Einstein–Maxwell theory: study in terms of soliton methods. Annales Henri Poincare 10, 1075–1095 (2009). doi: 10.1007/s00023-009-0012-0 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    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. Quantum. Grav. 25(16), 162002 (2008).
  52. 52.
    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 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  53. 53.
    Hollands, S.: Horizon area-angular momentum inequality in higher dimensional spacetimes. Class. Quant. Grav. 29, 065006 (2012). doi: 10.1088/0264-9381/29/6/065006
  54. 54.
    Huang, L.H., Schoen, R., Wang, M.T.: Specifying angular momentum and center of mass for vacuum initial data sets. Commun. Math. Phys. 306, 785–803 (2011). doi: 10.1007/s00220-011-1295-9 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  55. 55.
    Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 352–437 (2001)MathSciNetGoogle Scholar
  56. 56.
    Hek, P.: Three remarks on axisymmetric stationary horizons. Commun. Math. Phys. 36(4), 305–320 (1974). doi: 10.1007/BF01646202 ADSCrossRefGoogle Scholar
  57. 57.
    Jaramillo, J.L., Reiris, M., Dain, S.: Black hole area–angular momentum inequality in non-vacuum spacetimes. Phys. Rev. D84, 121503 (2011). doi: 10.1103/PhysRevD.84.121503 ADSGoogle Scholar
  58. 58.
    Khuri, M.A.: The hoop conjecture in spherically symmetric spacetimes. Phys. Rev. D80, 124025 (2009). doi: 10.1103/PhysRevD.80.124025 ADSMathSciNetGoogle Scholar
  59. 59.
    Khuri, M.A., Yamada, S., Weinstein, G.: On the Riemannian Penrose inequality with charge and the cosmic censorship conjecture (2013)Google Scholar
  60. 60.
    Kunduri, H.K., Lucietti, J.: A Classification of near-horizon geometries of extremal vacuum black holes. J. Math. Phys. 50, 082502 (2009). doi: 10.1063/1.3190480 ADSCrossRefMathSciNetGoogle Scholar
  61. 61.
    Kunduri, H.K., Lucietti, J.: Classification of near-horizon geometries of extremal black holes. Liv. Rev. Relativ. 16(8) (2013). doi: 10.12942/lrr-2013-8.
  62. 62.
    Lewandowski, J., Pawlowski, T.: Extremal isolated horizons: a local uniqueness theorem. Class. Quantum Grav. 20, 587–606 (2003). doi: 10.1088/0264-9381/20/4/303 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  63. 63.
    Malec, E.: Hoop conjecture and trapped surfaces in nonspherical massive systems. Phys. Rev. Lett. 67, 949–952 (1991). doi: 10.1103/PhysRevLett.67.949 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  64. 64.
    Malec, E.: Isoperimetric inequalities in the physics of black holes. Acta Phys. Polon. B22, 829 (1992)ADSMathSciNetGoogle Scholar
  65. 65.
    Mars, M.: Present status of the Penrose inequality. Class. Quantum Grav. 26, 193001 (2009). doi: 10.1088/0264-9381/26/19/193001 ADSCrossRefMathSciNetGoogle Scholar
  66. 66.
    Neugebauer, G., Hennig, J.: Stationary two-black-hole configurations: a non-existence proof. J. Geom. Phys. 62, 613–630 (2012). doi: 10.1016/j.geomphys.2011.05.008 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  67. 67.
    Neugebauer, G., Hennig, J.: Stationary black-hole binaries: a non-existence proof. Preprint (2013).
  68. 68.
    Osserman, R.: The isoperimetric inequality. Bull. Am. Math. Soc. 84(6), 1182–1238 (1978)CrossRefMathSciNetzbMATHGoogle Scholar
  69. 69.
    Paetz, T.T., Simon, W.: Marginally outer trapped surfaces in higher dimensions (2013). doi: 10.1088/0264-9381/30/23/235005
  70. 70.
    Penrose, R.: Naked singularities. Ann. NY. Acad. Sci. 224, 125–134 (1973)ADSCrossRefGoogle Scholar
  71. 71.
    Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65(1), 45–76 (1979)ADSCrossRefMathSciNetzbMATHGoogle Scholar
  72. 72.
    Schoen, R., Yau, S.T.: The energy and the linear momentum of space-times in general relativity. Comm. Math. Phys. 79(1), 47–51 (1981). Google Scholar
  73. 73.
    Schoen, R., Yau, S.T.: The existence of a black hole due to condensation of matter. Comm. Math. Phys. 90(4), 575–579 (1983).
  74. 74.
    Schoen, R., Zhou, X.: Convexity of reduced energy and mass angular momentum inequalities. Annales Henri Poincar (7), 1747–1773 (2013). doi: 10.1007/s00023-013-0240-1
  75. 75.
    Seifert, H.: Naked singularities and cosmic censorship: comment on the current situation. Gen. Relativ. Gravit. 10(12), 1065–1067 (1979). doi: 10.1007/BF00776539 ADSCrossRefMathSciNetGoogle Scholar
  76. 76.
    Senovilla, J.M.: A reformulation of the hoop conjecture. Eur. Lett. 81, 20004 (2008). doi: 10.1209/0295-5075/81/20004 ADSCrossRefMathSciNetGoogle Scholar
  77. 77.
    Szabados, L.B.: Quasi-local energy-momentum and angular momentum in GR: a review article. Living Rev. Rel. 7(4) (2004). Cited on 8 August 2005
  78. 78.
    Thorne, K.: Nonspherical gravitational collapse: a short review. In: J. Klauder (ed.) Magic without magic: John Archibald wheeler. A collection of essays in honor of his sixtieth birthday, pp. 231–258. W.H. Freeman, San Francisco (1972)Google Scholar
  79. 79.
    Visser, M.: Area products for black hole horizons. Phys. Rev. D88, 044014 (2013). doi: 10.1103/PhysRevD.88.044014 ADSGoogle Scholar
  80. 80.
    Wald, R.: Final states of gravitational collapse. Phys. Rev. Lett. 26(26), 1653–1655 (1971)ADSCrossRefGoogle Scholar
  81. 81.
    Wald, R.: Gravitational collapse and cosmic censorship. In: Iyer, B.R., Bhawal, B. (eds.) Black Holes, Gravitational Radiation and the Universe, Fundamental Theories of Physics, vol. 100, pp. 69–85. Kluwer Academic, Dorddrecht (1999)CrossRefGoogle Scholar
  82. 82.
    Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981). doi: 10.1007/BF01208277 ADSCrossRefMathSciNetzbMATHGoogle Scholar
  83. 83.
    Yazadjiev, S.: Horizon area–angular momentum–charge-magnetic fluxes inequalities in 5D Einstein–Maxwell–dilaton gravity. Class. Quantum Grav. 30, 115010 (2013). doi: 10.1088/0264-9381/30/11/115010 ADSCrossRefMathSciNetGoogle Scholar
  84. 84.
    Yazadjiev, S.S.: Area–angular momentum-charge inequality for stable marginally outer trapped surfaces in 4D Einstein–Maxwell-dilaton theory. Phys. Rev. D87, 024016 (2013). doi: 10.1103/PhysRevD.87.024016 ADSGoogle Scholar
  85. 85.
    Zhou, X.: Mass angular momentum inequality for axisymmetric vacuum data with small trace. ArXiv e-prints (2012)Google Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Facultad de Matemática, Astronomía y Física, FaMAF, Universidad Nacional de CórdobaInstituto de Física Enrique Gaviola, IFEG, CONICETCórdobaArgentina

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