Abstract
We study the internal structure of anisotropic black holes with charged vector hairs. Taking advantage of the scaling symmetries of the system, some radially conserved charges are found via the extension of the Noether theorem. Then, a general proof of no inner horizon of these black holes is presented and the geometry ends at a spacelike singularity. Before reaching the singularity, we find several intermediate regimes both analytically and numerically. In addition to the Einstein-Rosen bridge contracting towards the singularity, the instability triggered by the vector hair results in the oscillations of vector condensate and the anisotropy of spatial geometry. Moreover, the latter oscillates at twice the frequency of the condensate. Then, the geometry enters into Kasner epochs with spatial anisotropy. Due to the effects from vector condensate and U(1) gauge potential, there is generically a never-ending alternation of Kasner epochs towards the singularity. The character of evolution on approaching the singularity is found to be described by the Kasner epoch alternation with flipping of powers of the Belinskii-Khalatnikov-Lifshitz type.
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References
LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett. 875 (2019) L1 [arXiv:1906.11238] [INSPIRE].
Event Horizon Telescope collaboration, First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole, Astrophys. J. Lett. 875 (2019) L4 [arXiv:1906.11241] [INSPIRE].
R. Penrose, Gravitational collapse and space-time singularities, Phys. Rev. Lett. 14 (1965) 57 [INSPIRE].
E.M. Lifshitz and I.M. Khalatnikov, Investigations in relativistic cosmology, Adv. Phys. 12 (1963) 185 [INSPIRE].
V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 (1970) 525 [INSPIRE].
V.A. Belinski and I.M. Khalatnikov, Effect of Scalar and Vector Fields on the Nature of the Cosmological Singularity, Sov. Phys. JETP 36 (1973) 591 [INSPIRE].
L. Andersson and A.D. Rendall, Quiescent cosmological singularities, Commun. Math. Phys. 218 (2001) 479 [gr-qc/0001047] [INSPIRE].
J.K. Erickson, D.H. Wesley, P.J. Steinhardt and N. Turok, Kasner and mixmaster behavior in universes with equation of state w ≥ 1, Phys. Rev. D 69 (2004) 063514 [hep-th/0312009] [INSPIRE].
V.D. Ivashchuk, V.N. Melnikov and D. Singleton, On avoiding cosmological oscillating behavior for S-brane solutions with diagonal metrics, Phys. Rev. D 72 (2005) 103511 [gr-qc/0509065] [INSPIRE].
S. Nojiri and S.D. Odintsov, Regular multihorizon black holes in modified gravity with nonlinear electrodynamics, Phys. Rev. D 96 (2017) 104008 [arXiv:1708.05226] [INSPIRE].
C. Gao, Black holes with many horizons in the theories of nonlinear electrodynamics, Phys. Rev. D 104 (2021) 064038 [arXiv:2106.13486] [INSPIRE].
H. Ringström, Origins and development of the Cauchy problem in general relativity, Class. Quant. Grav. 32 (2015) 124003 [INSPIRE].
J. Isenberg, On Strong Cosmic Censorship, arXiv:1505.06390 [INSPIRE].
L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS/CFT, JHEP 02 (2004) 014 [hep-th/0306170] [INSPIRE].
G. Festuccia and H. Liu, Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I, JHEP 04 (2006) 044 [hep-th/0506202] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
R.-G. Cai, L. Li and R.-Q. Yang, No Inner-Horizon Theorem for Black Holes with Charged Scalar Hairs, JHEP 03 (2021) 263 [arXiv:2009.05520] [INSPIRE].
Y.-S. An, L. Li and F.-G. Yang, No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs, Phys. Rev. D 104 (2021) 024040 [arXiv:2106.01069] [INSPIRE].
S.A. Hartnoll, G.T. Horowitz, J. Kruthoff and J.E. Santos, Diving into a holographic superconductor, SciPost Phys. 10 (2021) 009 [arXiv:2008.12786] [INSPIRE].
S.A. Hartnoll, G.T. Horowitz, J. Kruthoff and J.E. Santos, Gravitational duals to the grand canonical ensemble abhor Cauchy horizons, JHEP 10 (2020) 102 [arXiv:2006.10056] [INSPIRE].
D.O. Devecioglu and M.-I. Park, No Scalar-Haired Cauchy Horizon Theorem in Einstein-Maxwell-Horndeski Theories, arXiv:2101.10116 [INSPIRE].
M. Van de Moortel, Violent nonlinear collapse in the interior of charged hairy black holes, arXiv:2109.10932 [INSPIRE].
S.A.H. Mansoori, L. Li, M. Rafiee and M. Baggioli, What’s inside a hairy black hole in massive gravity?, JHEP 10 (2021) 098 [arXiv:2108.01471] [INSPIRE].
N. Grandi and I. Salazar Landea, Diving inside a hairy black hole, JHEP 05 (2021) 152 [arXiv:2102.02707] [INSPIRE].
R.-Q. Yang, R.-G. Cai and L. Li, Constraining the number of horizons with energy conditions, Class. Quant. Grav. 39 (2022) 035005 [arXiv:2104.03012] [INSPIRE].
O.J.C. Dias, G.T. Horowitz and J.E. Santos, Inside an asymptotically flat hairy black hole, JHEP 12 (2021) 179 [arXiv:2110.06225] [INSPIRE].
R.-G. Cai, S. He, L. Li and L.-F. Li, A Holographic Study on Vector Condensate Induced by a Magnetic Field, JHEP 12 (2013) 036 [arXiv:1309.2098] [INSPIRE].
R.-G. Cai, L. Li and L.-F. Li, A Holographic P-wave Superconductor Model, JHEP 01 (2014) 032 [arXiv:1309.4877] [INSPIRE].
S.S. Gubser and S.S. Pufu, The Gravity dual of a p-wave superconductor, JHEP 11 (2008) 033 [arXiv:0805.2960] [INSPIRE].
R.-G. Cai, L. Li, L.-F. Li and Y. Wu, Vector Condensate and AdS Soliton Instability Induced by a Magnetic Field, JHEP 01 (2014) 045 [arXiv:1311.7578] [INSPIRE].
E.E. Donets, D.V. Galtsov and M.Y. Zotov, Internal structure of Einstein Yang-Mills black holes, Phys. Rev. D 56 (1997) 3459 [gr-qc/9612067] [INSPIRE].
P. Breitenlohner, G.V. Lavrelashvili and D. Maison, Mass inflation and chaotic behavior inside hairy black holes, Nucl. Phys. B 524 (1998) 427 [gr-qc/9703047] [INSPIRE].
Z.-Y. Nie, R.-G. Cai, X. Gao, L. Li and H. Zeng, Phase transitions in a holographic s + p model with back-reaction, Eur. Phys. J. C 75 (2015) 559 [arXiv:1501.00004] [INSPIRE].
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Cai, RG., Ge, C., Li, L. et al. Inside anisotropic black hole with vector hair. J. High Energ. Phys. 2022, 139 (2022). https://doi.org/10.1007/JHEP02(2022)139
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DOI: https://doi.org/10.1007/JHEP02(2022)139