Abstract
In this paper, we analyze the static solutions for the U(1)4 consistent truncation of the maximally supersymmetric gauged supergravity in four dimensions. Using a new parametrization of the known solutions it is shown that for fixed charges there exist three possible black hole configurations according to the pattern of symmetry breaking of the (scalars sector of the) Lagrangian. Namely a black hole without scalar fields, a black hole with a primary hair and a black hole with a secondary hair respectively. This is the first, exact, example of a black hole with a primary scalar hair, where both the black hole and the scalar fields are regular on and outside the horizon. The configurations with secondary and primary hair can be interpreted as a spontaneous symmetry breaking of discrete permutation and reflection symmetries of the action. It is shown that there exist a triple point in the thermodynamic phase space where the three solution coexist. The corresponding phase transitions are discussed and the free energies are written explicitly as function of the thermodynamic coordinates in the uncharged case. In the charged case the free energies of the primary hair and the hairless black hole are also given as functions of the thermodynamic coordinates.
Similar content being viewed by others
References
W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967) 1776 [INSPIRE].
W. Israel, Event horizons in static electrovac space-times, Commun. Math. Phys. 8 (1968) 245 [INSPIRE].
B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26 (1971) 331 [INSPIRE].
R.M. Wald, Final states of gravitational collapse, Phys. Rev. Lett. 26 (1971) 1653 [INSPIRE].
R. Ruffini and J.A. Wheeler, Introducing the black hole, Physics Today 24 (1971) 30.
D. Robinson, Four decades of black holes uniqueness theorems, slides presented at Kerr Fest: black holes in astrophysics, general relativity and quantum gravity, August 26-28, Christchurch, New Zealand (2004).
M. Heusler, Black hole uniqueness theorems, Cambridge Lecture Notes in Physics, Cambridge University Press, Cambridge U.K. (1996).
T. Hertog, Towards a novel no-hair theorem for black holes, Phys. Rev. D 74 (2006) 084008 [gr-qc/0608075] [INSPIRE].
J.D. Bekenstein, ‘No hair’: twenty-five years after, chapter in the proceedings of the Second International Andrei D. Sakharov Conference in Physics, I.M. Dremin and A.M. Semikhatov eds., World Scientific, Singapore (1997).
N. Bocharova, K. Bronnikov and V. Melnikov, An exact solution of the system of Einstein equations and mass free scalar field, Vestn. Mosk. Univ. Fiz. Astron. 6 (1970) 706.
J. Bekenstein, Exact solutions of Einstein conformal scalar equations, Annals Phys. 82 (1974) 535 [INSPIRE].
J. Bekenstein, Black holes with scalar charge, Annals Phys. 91 (1975) 75 [INSPIRE].
K. Virbhadra and J. Parikh, A conformal scalar dyon black hole solution, Phys. Lett. B 331 (1994) 302 [Erratum ibid. B 340 (1994) 265] [hep-th/9407121] [INSPIRE].
J. Bekenstein, Novel ’no scalar hair’ theorem for black holes, Phys. Rev. D 51 (1995) 6608 [INSPIRE].
E. Ayon-Beato, ’No scalar hair’ theorems for nonminimally coupled fields with quartic selfinteraction, Class. Quant. Grav. 19 (2002) 5465 [gr-qc/0212050] [INSPIRE].
T. Johannsen and D. Psaltis, Testing the no-hair theorem with observations in the electromagnetic spectrum: I. Properties of a quasi-Kerr spacetime, Astrophys. J. 716 (2010) 187 [arXiv:1003.3415] [INSPIRE].
L. Sadeghian and C.M. Will, Testing the black hole no-hair theorem at the galactic center: Perturbing effects of stars in the surrounding cluster, Class. Quant. Grav. 28 (2011) 225029 [arXiv:1106.5056] [INSPIRE].
R. Emparan and H.S. Reall, A rotating black ring solution in five-dimensions, Phys. Rev. Lett. 88 (2002) 101101 [hep-th/0110260] [INSPIRE].
H. Elvang and P. Figueras, Black Saturn, JHEP 05 (2007) 050 [hep-th/0701035] [INSPIRE].
R. Emparan, Rotating circular strings and infinite nonuniqueness of black rings, JHEP 03 (2004) 064 [hep-th/0402149] [INSPIRE].
C. Martínez, R. Troncoso and J. Zanelli, De Sitter black hole with a conformally coupled scalar field in four-dimensions, Phys. Rev. D 67 (2003) 024008 [hep-th/0205319] [INSPIRE].
C. Martínez, J.P. Staforelli and R. Troncoso, Topological black holes dressed with a conformally coupled scalar field and electric charge, Phys. Rev. D 74 (2006) 044028 [hep-th/0512022] [INSPIRE].
E. Radu and E. Winstanley, Conformally coupled scalar solitons and black holes with negative cosmological constant, Phys. Rev. D 72 (2005) 024017 [gr-qc/0503095] [INSPIRE].
A. Anabalon and H. Maeda, New charged black holes with conformal scalar hair, Phys. Rev. D 81 (2010) 041501 [arXiv:0907.0219] [INSPIRE].
C. Charmousis, T. Kolyvaris and E. Papantonopoulos, Charged C-metric with conformally coupled scalar field, Class. Quant. Grav. 26 (2009) 175012 [arXiv:0906.5568] [INSPIRE].
M. Duff and J.T. Liu, Anti-de Sitter black holes in gauged N = 8 supergravity, Nucl. Phys. B 554 (1999) 237 [hep-th/9901149] [INSPIRE].
T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, Einstein hair, arXiv:1111.0263 [INSPIRE].
A. Anabalon and A. Cisterna, Asymptotically (anti) de Sitter black holes and wormholes with a self interacting scalar field in four dimensions, Phys. Rev. D 85 (2012) 084035 [arXiv:1201.2008] [INSPIRE].
S.G. Saenz and C. Martinez, Anti-de Sitter massless scalar field spacetimes in arbitrary dimensions, arXiv:1203.4776 [INSPIRE].
C. Martinez, R. Troncoso and J. Zanelli, Exact black hole solution with a minimally coupled scalar field, Phys. Rev. D 70 (2004) 084035 [hep-th/0406111] [INSPIRE].
M. Duff, TASI lectures on branes, black holes and Anti-de Sitter space, hep-th/9912164 [INSPIRE].
T. Hertog and K. Maeda, Black holes with scalar hair and asymptotics in N = 8 supergravity, JHEP 07 (2004) 051 [hep-th/0404261] [INSPIRE].
I.Z. Stefanov, S.S. Yazadjiev and M.D. Todorov, Phases of 4D scalar-tensor black holes coupled to Born-Infeld nonlinear electrodynamics, Mod. Phys. Lett. A 23 (2008) 2915 [arXiv:0708.4141] [INSPIRE].
D.D. Doneva, S.S. Yazadjiev, K.D. Kokkotas and I.Z. Stefanov, Quasi-normal modes, bifurcations and non-uniqueness of charged scalar-tensor black holes, Phys. Rev. D 82 (2010) 064030 [arXiv:1007.1767] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a holographic superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, Exact gravity dual of a gapless superconductor, JHEP 07 (2009) 026 [arXiv:0902.0733] [INSPIRE].
A. Ashtekar and A. Magnon, Asymptotically Anti-de Sitter space-times, Class. Quant. Grav. 1 (1984) L39 [INSPIRE].
A. Ashtekar and S. Das, Asymptotically Anti-de Sitter space-times: conserved quantities, Class. Quant. Grav. 17 (2000) L17 [hep-th/9911230] [INSPIRE].
R.M. Wald, General relativity, University of Chicago Press, Chicago U.S.A. (1984).
W. Chen, H. Lü and C. Pope, Mass of rotating black holes in gauged supergravities, Phys. Rev. D 73 (2006) 104036 [hep-th/0510081] [INSPIRE].
A. Anabalon, F. Canfora, A. Giacomini and J. Oliva, Black holes with gravitational hair in higher dimensions, Phys. Rev. D 84 (2011) 084015 [arXiv:1108.1476] [INSPIRE].
D. Gross and E. Witten, Possible third order phase transition in the large-N lattice gauge theory, Phys. Rev. D 21 (1980) 446 [INSPIRE].
S.R. Wadia, N = ∞ phase transition in a class of exactly soluble model lattice gauge theories, Phys. Lett. B 93 (1980) 403 [INSPIRE].
W. Janke, D.A. Johnston and R. Kenna, Properties of higher-order phase transitions, Nucl. Phys. B 736 (2006) 319 [INSPIRE].
T. Ortín, Nonsupersymmetric (but) extreme black holes, scalar hair and other open problems, hep-th/9705095 [INSPIRE].
E. Alvarez, P. Meessen and T. Ortín, Transformation of black hole hair under duality and supersymmetry, Nucl. Phys. B 508 (1997) 181 [hep-th/9705094] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1203.6627
Rights and permissions
About this article
Cite this article
Anabalón, A., Canfora, F., Giacomini, A. et al. Black holes with primary hair in gauged N = 8 supergravity. J. High Energ. Phys. 2012, 10 (2012). https://doi.org/10.1007/JHEP06(2012)010
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2012)010