Quantum BTZ black hole
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Springer
Received: September 7, 2020 Accepted: October 21, 2020 Published: November 25, 2020
Quantum BTZ black hole
a
Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís Companys 23, E-08010 Barcelona, Spain b Departament de Física Quàntica i Astrofísica, Institut de Ciències del Cosmos, Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain
E-mail: [email protected], [email protected], [email protected] Abstract: We study a holographic construction of quantum rotating BTZ black holes that incorporates the exact backreaction from strongly coupled quantum conformal fields. It is based on an exact four-dimensional solution for a black hole localized on a brane in AdS 4 , first discussed some years ago but never fully investigated in this manner. Besides quantum CFT effects and their backreaction, we also investigate the role of higher-curvature corrections in the effective three-dimensional theory. We obtain the quantum-corrected geometry and the renormalized stress tensor. We show that the quantum black hole entropy, which includes the entanglement of the fields outside the horizon, satisfies the first law of thermodynamics exactly, even in the presence of backreaction and with higher-curvature corrections, while the Bekenstein-Hawking-Wald entropy does not. This result, which involves a rather non-trivial bulk calculation, shows the consistency of the holographic interpretation of braneworlds. We compare our renormalized stress tensor to results derived for free conformal fields, and for a previous holographic construction without backreaction effects, which is shown to be a limit of the solutions in this article. Keywords: AdS-CFT Correspondence, Black Holes ArXiv ePrint: 2007.15999
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)137
JHEP11(2020)137
Roberto Emparan,a,b Antonia Micol Frassinob and Benson Wayb
Contents 1 Introduction
1 5 5 7 10 12 14 17 17 18 20
3 Rotating quBTZ 3.1 Geometry, M , J, and hT a b i of quBTZ 3.2 Branches of solutions and bounds on M and J 3.3 Comparison to other calculations 3.4 Quantum black hole thermodynamics
23 24 28 29 30
4 Discussion and outlook
33
A Glossary of main symbols
36
B Limit of no backreaction
37
C Stress tensor in BTZ parameters
38
1
Introduction
Despite the lack of a precise definition of a quantum black hole within a complete quantum theory of gravity, one can still gain insight through semi-classical approximations. One sensible approach is to treat gravity classically while fully accounting for the backreaction of all other quantum fields. This is the study of the ‘semi-classical Einstein equations’ Gµν (gαβ ) = 8πGhTµν (gαβ )i ,
(1.1)
where Gµν is the gravitational Einstein tensor (possibly with a cosmological constant) for a spacetime metric gαβ , and hTµν i is the renormalized stress tensor of the (non-gravitational) quantum matter fields in that spacetime. Quantum fluctuations of the metric can be comparatively suppressed by including a large numbe
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