Abstract
We consider the linear stability of 4-dimensional hairy black holes with mixed boundary conditions in Anti-de Sitter spacetime. We focus on the mass of scalar fields around the maximally supersymmetric vacuum of the gauged \( \mathcal{N}=8 \) supergravity in four dimensions, m 2 = −2l −2. It is shown that the Schrödinger operator on the half-line, governing the S 2, H 2 or \( {\mathbb{R}}^2 \) invariant mode around the hairy black hole, allows for non-trivial self-adjoint extensions and each of them corresponds to a class of mixed boundary conditions in the gravitational theory. Discarding the self-adjoint extensions with a negative mode impose a restriction on these boundary conditions. The restriction is given in terms of an integral of the potential in the Schrödinger operator resembling the estimate of Simon for Schrödinger operators on the real line. In the context of AdS/CFT duality, our result has a natural interpretation in terms of the field theory dual effective potential.
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References
J.D. Bekenstein, Exact solutions of Einstein conformal scalar equations, Annals Phys. 82 (1974) 535 [INSPIRE].
M. Heusler, A no hair theorem for selfgravitating nonlinear σ-models, J. Math. Phys. 33 (1992) 3497 [INSPIRE].
D. Sudarsky, A simple proof of a no hair theorem in Einstein Higgs theory, Class. Quant. Grav. 12 (1995) 579 [INSPIRE].
U. Nucamendi and M. Salgado, Scalar hairy black holes and solitons in asymptotically flat space-times, Phys. Rev. D 68 (2003) 044026 [gr-qc/0301062] [INSPIRE].
A. Anabalon and J. Oliva, Exact hairy black holes and their modification to the universal law of gravitation, Phys. Rev. D 86 (2012) 107501 [arXiv:1205.6012] [INSPIRE].
A. Anabalon, D. Astefanesei and R. Mann, Exact asymptotically flat charged hairy black holes with a dilaton potential, JHEP 10 (2013) 184 [arXiv:1308.1693] [INSPIRE].
M. Cadoni and E. Franzin, Asymptotically flat black holes sourced by a massless scalar field, Phys. Rev. D 91 (2015) 104011 [arXiv:1503.04734] [INSPIRE].
K.A. Bronnikov and Yu. N. Kireev, Instability of black holes with scalar charge, Phys. Lett. A 67 (1978) 95 [INSPIRE].
T.J.T. Harper, P.A. Thomas, E. Winstanley and P.M. Young, Instability of a four-dimensional de Sitter black hole with a conformally coupled scalar field, Phys. Rev. D 70 (2004) 064023 [gr-qc/0312104] [INSPIRE].
G. Dotti, R.J. Gleiser and C. Martinez, Static black hole solutions with a self interacting conformally coupled scalar field, Phys. Rev. D 77 (2008) 104035 [arXiv:0710.1735] [INSPIRE].
K.A. Bronnikov, J.C. Fabris and A. Zhidenko, On the stability of scalar-vacuum space-times, Eur. Phys. J. C 71 (2011) 1791 [arXiv:1109.6576] [INSPIRE].
A. Anabalon and N. Deruelle, On the mechanical stability of asymptotically flat black holes with minimally coupled scalar hair, Phys. Rev. D 88 (2013) 064011 [arXiv:1307.2194] [INSPIRE].
A. Anabalon, J. Bičák and J. Saavedra, Hairy black holes: stability under odd-parity perturbations and existence of slowly rotating solutions, Phys. Rev. D 90 (2014) 124055 [arXiv:1405.7893] [INSPIRE].
C.A.R. Herdeiro and E. Radu, Asymptotically flat black holes with scalar hair: a review, Int. J. Mod. Phys. D 24 (2015) 1542014 [arXiv:1504.08209] [INSPIRE].
C.A.R. Herdeiro and E. Radu, Kerr black holes with scalar hair, Phys. Rev. Lett. 112 (2014) 221101 [arXiv:1403.2757] [INSPIRE].
C. Herdeiro and E. Radu, Construction and physical properties of Kerr black holes with scalar hair, Class. Quant. Grav. 32 (2015) 144001 [arXiv:1501.04319] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Positive energy in anti-de Sitter backgrounds and gauged extended supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
A. Ishibashi and R.M. Wald, Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time, Class. Quant. Grav. 21 (2004) 2981 [hep-th/0402184] [INSPIRE].
G. Bonneau, J. Faraut and G. Valent, Selfadjoint extensions of operators and the teaching of quantum mechanics, Am. J. Phys. 69 (2001) 322 [quant-ph/0103153] [INSPIRE].
T. Hertog and G.T. Horowitz, Designer gravity and field theory effective potentials, Phys. Rev. Lett. 94 (2005) 221301 [hep-th/0412169] [INSPIRE].
T. Torii, K. Maeda and M. Narita, Scalar hair on the black hole in asymptotically anti-de Sitter space-time, Phys. Rev. D 64 (2001) 044007 [INSPIRE].
M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Black holes and asymptotics of 2 + 1 gravity coupled to a scalar field, Phys. Rev. D 65 (2002) 104007 [hep-th/0201170] [INSPIRE].
D. Sudarsky and J.A. Gonzalez, On black hole scalar hair in asymptotically anti-de Sitter space-times, Phys. Rev. D 67 (2003) 024038 [gr-qc/0207069] [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].
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].
T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, A new class of exact hairy black hole solutions, Gen. Rel. Grav. 43 (2011) 163 [arXiv:0911.1711] [INSPIRE].
F. Correa, C. Martinez and R. Troncoso, Scalar solitons and the microscopic entropy of hairy black holes in three dimensions, JHEP 01 (2011) 034 [arXiv:1010.1259] [INSPIRE].
A. Anabalon, F. Canfora, A. Giacomini and J. Oliva, Black holes with primary hair in gauged N = 8 supergravity, JHEP 06 (2012) 010 [arXiv:1203.6627] [INSPIRE].
A. Anabalon, Exact black holes and universality in the backreaction of non-linear σ-models with a potential in (A)dS 4, JHEP 06 (2012) 127 [arXiv:1204.2720] [INSPIRE].
A. Acena, A. Anabalon and D. Astefanesei, Exact hairy black brane solutions in AdS 5 and holographic RG flows, Phys. Rev. D 87 (2013) 124033 [arXiv:1211.6126] [INSPIRE].
A. Anabalon, Exact hairy black holes, Springer Proc. Phys. 157 (2014) 3 [arXiv:1211.2765] [INSPIRE].
J. Aparicio, D. Grumiller, E. Lopez, I. Papadimitriou and S. Stricker, Bootstrapping gravity solutions, JHEP 05 (2013) 128 [arXiv:1212.3609] [INSPIRE].
L. Zhao, W. Xu and B. Zhu, Novel rotating hairy black hole in (2 + 1)-dimensions, Commun. Theor. Phys. 61 (2014) 475 [arXiv:1305.6001] [INSPIRE].
H. Lü, Y. Pang and C.N. Pope, AdS dyonic black hole and its thermodynamics, JHEP 11 (2013) 033 [arXiv:1307.6243] [INSPIRE].
P.A. González, E. Papantonopoulos, J. Saavedra and Y. Vásquez, Four-dimensional asymptotically AdS black holes with scalar hair, JHEP 12 (2013) 021 [arXiv:1309.2161] [INSPIRE].
P. Bueno and C.S. Shahbazi, The violation of the no-hair conjecture in four-dimensional ungauged supergravity, Class. Quant. Grav. 31 (2014) 145005 [arXiv:1310.6379] [INSPIRE].
A. Aceña, A. Anabalón, D. Astefanesei and R. Mann, Hairy planar black holes in higher dimensions, JHEP 01 (2014) 153 [arXiv:1311.6065] [INSPIRE].
A. Anabalon and D. Astefanesei, Black holes in ω-defomed gauged N = 8 supergravity, Phys. Lett. B 732 (2014) 137 [arXiv:1311.7459] [INSPIRE].
X.-H. Feng, H. Lü and Q. Wen, Scalar hairy black holes in general dimensions, Phys. Rev. D 89 (2014) 044014 [arXiv:1312.5374] [INSPIRE].
X. Zhang and H. Lü, Exact black hole formation in asymptotically (A)dS and flat spacetimes, Phys. Lett. B 736 (2014) 455 [arXiv:1403.6874] [INSPIRE].
P.A. González, E. Papantonopoulos, J. Saavedra and Y. Vásquez, Extremal hairy black holes, JHEP 11 (2014) 011 [arXiv:1408.7009] [INSPIRE].
X. Zhang and H. Lü, Critical behavior in a massless scalar field collapse with self-interaction potential, Phys. Rev. D 91 (2015) 044046 [arXiv:1410.8337] [INSPIRE].
F. Faedo, D. Klemm and M. Nozawa, Hairy black holes in N = 2 gauged supergravity, arXiv:1505.02986 [INSPIRE].
Z.-Y. Fan and H. Lü, Static and dynamic hairy planar black holes, Phys. Rev. D 92 (2015) 064008 [arXiv:1505.03557] [INSPIRE].
Z.-Y. Fan and H. Lü, Charged black holes with scalar hair, JHEP 09 (2015) 060 [arXiv:1507.04369] [INSPIRE].
T. Hertog, Towards a novel no-hair theorem for black holes, Phys. Rev. D 74 (2006) 084008 [gr-qc/0608075] [INSPIRE].
T. Faulkner, G.T. Horowitz and M.M. Roberts, New stability results for Einstein scalar gravity, Class. Quant. Grav. 27 (2010) 205007 [arXiv:1006.2387] [INSPIRE].
W. Boucher, Positive energy without supersymmetry, Nucl. Phys. B 242 (1984) 282 [INSPIRE].
P.K. Townsend, Positive energy and the scalar potential in higher dimensional (super)gravity theories, Phys. Lett. B 148 (1984) 55 [INSPIRE].
T. Regge and J.A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. 108 (1957) 1063 [INSPIRE].
F.J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations, Phys. Rev. Lett. 24 (1970) 737 [INSPIRE].
H. Kodama and A. Ishibashi, A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions, Prog. Theor. Phys. 110 (2003) 701 [hep-th/0305147] [INSPIRE].
A. Ishibashi and H. Kodama, Stability of higher dimensional Schwarzschild black holes, Prog. Theor. Phys. 110 (2003) 901 [hep-th/0305185] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
I. Papadimitriou, Multi-trace deformations in AdS/CFT: exploring the vacuum structure of the deformed CFT, JHEP 05 (2007) 075 [hep-th/0703152] [INSPIRE].
T. Hertog and G.T. Horowitz, Holographic description of AdS cosmologies, JHEP 04 (2005) 005 [hep-th/0503071] [INSPIRE].
T. Hertog and S. Hollands, Stability in designer gravity, Class. Quant. Grav. 22 (2005) 5323 [hep-th/0508181] [INSPIRE].
C. Martinez, Instability of three-dimensional conformally dressed black hole, Phys. Rev. D 58 (1998) 027501 [gr-qc/9801091] [INSPIRE].
C. Martinez and J. Zanelli, Conformally dressed black hole in (2 + 1)-dimensions, Phys. Rev. D 54 (1996) 3830 [gr-qc/9604021] [INSPIRE].
T. Hertog and K. Maeda, Stability and thermodynamics of AdS black holes with scalar hair, Phys. Rev. D 71 (2005) 024001 [hep-th/0409314] [INSPIRE].
M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Asymptotic behavior and Hamiltonian analysis of anti-de Sitter gravity coupled to scalar fields, Annals Phys. 322 (2007) 824 [hep-th/0603185] [INSPIRE].
G. Dotti, Nonmodal linear stability of the Schwarzschild black hole, Phys. Rev. Lett. 112 (2014) 191101 [arXiv:1307.3340] [INSPIRE].
A. Anabalon, D. Astefanesei and C. Martinez, Mass of asymptotically antide Sitter hairy spacetimes, Phys. Rev. D 91 (2015) 041501 [arXiv:1407.3296] [INSPIRE].
B. Simon, The bound state of weakly coupled Schrödinger operators in one and two-dimensions, Annals Phys. 97 (1976) 279 [INSPIRE].
V. Balasubramanian and P. Kraus, A stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
B. de Wit and H. Nicolai, N = 8 supergravity, Nucl. Phys. B 208 (1982) 323 [INSPIRE].
G. Dall’Agata and G. Inverso, On the vacua of N = 8 gauged supergravity in 4 dimensions, Nucl. Phys. B 859 (2012) 70 [arXiv:1112.3345] [INSPIRE].
M. Cvetič, S.S. Gubser, H. Lü and C.N. Pope, Symmetric potentials of gauged supergravities in diverse dimensions and Coulomb branch of gauge theories, Phys. Rev. D 62 (2000) 086003 [hep-th/9909121] [INSPIRE].
I. Papadimitriou, Non-supersymmetric membrane flows from fake supergravity and multi-trace deformations, JHEP 02 (2007) 008 [hep-th/0606038] [INSPIRE].
G. Dall’Agata, G. Inverso and M. Trigiante, Evidence for a family of SO(8) gauged supergravity theories, Phys. Rev. Lett. 109 (2012) 201301 [arXiv:1209.0760] [INSPIRE].
J. Tarrío and O. Varela, Electric/magnetic duality and RG flows in AdS4/CFT3, JHEP 01 (2014) 071 [arXiv:1311.2933] [INSPIRE].
Y. Pang, C.N. Pope and J. Rong, Holographic RG flow in a new SO(3) × SO(3) sector of ω-deformed SO(8) gauged N = 8 supergravity, JHEP 08 (2015) 122 [arXiv:1506.04270] [INSPIRE].
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Anabalón, A., Astefanesei, D. & Oliva, J. Hairy black hole stability in AdS, quantum mechanics on the half-line and holography. J. High Energ. Phys. 2015, 68 (2015). https://doi.org/10.1007/JHEP10(2015)068
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DOI: https://doi.org/10.1007/JHEP10(2015)068