Annales Henri Poincaré

, Volume 20, Issue 2, pp 481–525 | Cite as

Bounding Horizon Area by Angular Momentum, Charge, and Cosmological Constant in 5-Dimensional Minimal Supergravity

  • Aghil Alaee
  • Marcus Khuri
  • Hari KunduriEmail author


We establish a class of area–angular momentum–charge inequalities satisfied by stable marginally outer trapped surfaces in 5-dimensional minimal supergravity which admit a \(U(1)^2\) symmetry. A novel feature is the fact that such surfaces can have the non-trivial topologies \(S^1 \times S^2\) and L(pq). In addition to two angular momenta, they may be characterized by ‘dipole charge’ as well as electric charge. We show that the unique geometries which saturate the inequalities are the horizon geometries corresponding to extreme black hole solutions. Analogous inequalities which also include contributions from a positive cosmological constant are also presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alaee, A., Khuri, M., Kunduri, H.: Relating mass to angular momentum and charge in 5-dimensional minimal supergravity. Ann. Henri Poincaré 18(5), 1703–1753 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alaee, A., Khuri, M., Kunduri, H.: Existence and uniqueness of near-horizon geometries for 5-dimensional black holes. In preparation (2018)Google Scholar
  3. 3.
    Breckenridge, J.C., Myers, R.C., Peet, A.W., Vafa, C.: D-branes and spinning black holes. Phys. Lett. B. 391, 93–98 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bryden, E.T., Khuri, M.A.: The area-angular momentum–charge inequality for black holes with positive cosmological constant. Class. Quantum Gravity 34, 125017 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chong, Z., Cvetič, M., Lü, H., Pope, C.N.: Non-extremal rotating black holes in five-dimensional gauged supergravity. Phys. Lett. B 644(2), 192–197 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chruściel, P.T., Costa, J.L., Heusler, M.: Stationary black holes: uniqueness and beyond. Living Rev. Relativ. 15, 7 (2012)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Chruściel, P.T., Nguyen, L.: A uniqueness theorem for degenerate Kerr–Newman black holes. Ann. Henri Poincaré 11, 585–609 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dain, S.: Geometric inequalities for axially symmetric black holes. Class. Quantum Gravity 29(7), 073001 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dain, S., Gabach-Clement, M.E.: Geometrical inequalities bounding angular momentum and charges in general relativity. Living Rev. Relativ. to appear arXiv:1710.04457 (2017)
  10. 10.
    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. D 88(2), 024048 (2013)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Elvang, H., Emparan, R., Mateos, D., Reall, H.S.: A supersymmetric black ring. Phys. Rev. Lett. 93, 211302 (2004)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Emparan, R.: Rotating circular strings, and infinite nonuniqueness of black rings. JHEP 03, 064 (2004)ADSCrossRefGoogle Scholar
  13. 13.
    Emparan, R., Reall, H.S.: A rotating black ring solution in five dimensions. Phys. Rev. Lett. 88(10), 101101 (2002)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Emparan, R., Reall, H.S.: Black holes in higher dimensions. Living Rev. Relativ. 11(6), 0801–3471 (2008)zbMATHGoogle Scholar
  15. 15.
    Fajman, D., Simon, W.: Area inequalities for stable marginally outer trapped surfaces in Einstein–Maxwell-dilaton theory. Adv. Theor. Math. Phys. 18(3), 687–707 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gabach-Clement, M.E., Jaramillo, J.L., Reiris, M.: Proof of the area-angular momentum–charge inequality for axisymmetric black holes. Class. Quantum Gravity 30(6), 065017 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gabach-Clement, M.E., Reiris, M., Simon, W.: The area-angular momentum inequality for black holes in cosmological spacetimes. Class. Quantum Gravity 32(14), 145006 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Galloway, G.J.: Rigidity of marginally trapped surfaces and the topology of black holes. Commun. Anal. Geom. 16(1), 217–229 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Galloway, G.J., Schoen, R.: A generalization of Hawkings black hole topology theorem to higher dimensions. Commun. Math. Phys. 266(2), 571–576 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gibbons, G.W., Kastor, D., London, L.A.J., Townsend, P.K., Traschen, J.H.: Supersymmetric selfgravitating solitons. Nucl. Phys. B 416, 850–880 (1994)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Hollands, S.: Horizon area-angular momentum inequality in higher-dimensional spacetimes. Class. Quantum Gravity 29(6), 065006 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Hollands, S., Ishibashi, A.: All vacuum near horizon geometries in D-dimensions with (D-3) commuting rotational symmetries. Ann. Henri Poincaré 10(8), 1537–1557 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hollands, S., Ishibashi, A.: Black hole uniqueness theorems in higher dimensional spacetimes. Class. Quantum Gravity 29(16), 163001 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hollands, S., Ishibashi, A., Wald, R.M.: A higher dimensional stationary rotating black hole must be axisymmetric. Commun. Math. Phys. 271(3), 699–722 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hollands, S., Yazadjiev, S.: Uniqueness theorem for 5-dimensional black holes with two axial killing fields. Commun. Math. Phys. 283(3), 749–768 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Khuri, M., Woolgar, E.: Nonexistence of extremal de sitter black rings. Class. Quantum Gravity 34, 22LT01 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kunduri, H.K., Lucietti, J.: A classification of near-horizon geometries of extremal vacuum black holes. J. Math. Phys. 50(8), 082502 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kunduri, H.K., Lucietti, J.: Uniqueness of near-horizon geometries of rotating extremal AdS(4) black holes. Class. Quantum Gravity 26, 055019 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kunduri, H.K., Lucietti, J.: Constructing near-horizon geometries in supergravities with hidden symmetry. JHEP 2011(7), 1–31 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kunduri, H.K., Lucietti, J.: Classification of near-horizon geometries of extremal black holes. Living Rev. Relativ. 16, 8 (2013)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Kunduri, H.K., Lucietti, J.: Black hole non-uniqueness via spacetime topology in five dimensions. JHEP 1410, 82 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kunduri, H.K., Lucietti, J.: Supersymmetric black holes with lens-space topology. Phys. Rev. Lett. 113(21), 211101 (2014)ADSCrossRefGoogle Scholar
  33. 33.
    Kunduri, H.K., Lucietti, J., Reall, H.S.: Near-horizon symmetries of extremal black holes. Class. Quantum Gravity 24, 4169–4190 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rácz, I.: A simple proof of the recent generalizations of Hawking’s black hole topology theorem. Class. Quantum Gravity 25(16), 162001 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Reall, H.S.: Higher dimensional black holes and supersymmetry. Phys. Rev. D 68, 024024 (2003). [Erratum: Phys. Rev. D 70, 089902 (2004)]ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Rogatko, M.: Mass angular momentum and charge inequalities for black holes in Einstein–Maxwell-axion-dilaton gravity. Phys. Rev. D 89, 044020 (2014)ADSCrossRefGoogle Scholar
  37. 37.
    Schoen, R., Zhou, X.: Convexity of reduced energy and mass angular momentum inequalities. Ann. Henri Poincaré 14(7), 1747–1773 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Strominger, A., Vafa, C.: Microscopic origin of the Bekenstein–Hawking entropy. Phys. Lett. B 379(1), 99–104 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Tomizawa, S., Nozawa, M.: Supersymmetric black lenses in five dimensions. Phys. Rev. D 94(4), 044037 (2016)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Yazadjiev, S.: Area-angular momentum–charge inequality for stable marginally outer trapped surfaces in 4D Einstein–Maxwell-dilaton theory. Phys. Rev. D 87(2), 024016 (2013)ADSMathSciNetCrossRefGoogle Scholar
  41. 41.
    Yazadjiev, S.: Horizon area-angular momentum–charge–magnetic flux inequalities in the 5D Einstein–Maxwell-dilaton gravity. Class. Quantum Gravity 30(11), 115010 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada
  2. 2.Department of MathematicsStony Brook UniversityStony BrookUSA
  3. 3.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations