Abstract
We propose a 1+1 dimensional CFT dual structure for quantum gravity and matter on the extended 2 + 1 dimensional BTZ black hole, realized as a quotient of the Poincaré patch of AdS3. The quotient spacetime includes regions beyond the singularity, “whiskers”, containing timelike and lightlike closed curves, which at first sight seem unphysical. The spacetime includes the usual AdS-asymptotic boundaries outside the horizons as well as boundary components inside the whiskers. We show that local boundary correlators with some endpoints in the whisker regions: (i) are a protected class of amplitudes, dominated by effective field theory even when the associated Witten diagrams appear to traverse the singularity, (ii) describe well-defined diffeomorphism-invariant quantum gravity amplitudes in BTZ, (iii) sharply probe some of the physics inside the horizon but outside the singularity, and (iv) are equivalent to correlators of specific non-local CFT operators in the standard thermofield entangled state of two CFTs. In this sense, the whisker regions can be considered as purely auxiliary spacetimes in which these useful non-local CFT correlators can be rendered as local boundary correlators, and their diagnostic value more readily understood. Our results follow by first performing a novel reanalysis of the Rindler view of standard AdS/CFT duality on the Poincaré patch of AdS, followed by exploiting the simple quotient structure of BTZ which turns the Rindler horizon into the BTZ black hole horizon. While most of our checks are within gravitational effective field theory, we arrive at a fully non-perturbative CFT proposal to probe the UV-sensitive approach to the singularity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, An Apologia for Firewalls, JHEP 09 (2013) 018 [arXiv:1304.6483] [INSPIRE].
D. Marolf and J. Polchinski, Gauge/Gravity Duality and the Black Hole Interior, Phys. Rev. Lett. 111 (2013) 171301 [arXiv:1307.4706] [INSPIRE].
S.L. Braunstein, S. Pirandola and K. Życzkowski, Better Late than Never: Information Retrieval from Black Holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [INSPIRE].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
J.M. Maldacena, TASI 2003 lectures on AdS/CFT, hep-th/0309246 [INSPIRE].
R. Sundrum, From Fixed Points to the Fifth Dimension, Phys. Rev. D 86 (2012) 085025 [arXiv:1106.4501] [INSPIRE].
D. Bigatti and L. Susskind, TASI lectures on the holographic principle, hep-th/0002044 [INSPIRE].
S.D. Mathur, The Information paradox: A Pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [gr-qc/9302012] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [INSPIRE].
G.T. Horowitz and D. Marolf, A New approach to string cosmology, JHEP 07 (1998) 014 [hep-th/9805207] [INSPIRE].
V. Balasubramanian, P. Kraus, A.E. Lawrence and S.P. Trivedi, Holographic probes of anti-de Sitter space-times, Phys. Rev. D 59 (1999) 104021 [hep-th/9808017] [INSPIRE].
P.C. Martin and J.S. Schwinger, Theory of many particle systems. 1., Phys. Rev. 115 (1959) 1342 [INSPIRE].
J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].
K.T. Mahanthappa, Multiple production of photons in quantum electrodynamics, Phys. Rev. 126 (1962) 329 [INSPIRE].
P.M. Bakshi and K.T. Mahanthappa, Expectation value formalism in quantum field theory. 1., J. Math. Phys. 4 (1963) 1 [INSPIRE].
P.M. Bakshi and K.T. Mahanthappa, Expectation value formalism in quantum field theory. 2., J. Math. Phys. 4 (1963) 12 [INSPIRE].
L.V. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [INSPIRE].
Y. Takahasi and H. Umezawa, Thermo field dynamics, Collect. Phenom. 2 (1975) 55 [INSPIRE].
J.B. Hartle and S.W. Hawking, Path Integral Derivation of Black Hole Radiance, Phys. Rev. D 13 (1976) 2188 [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, hep-th/0106109 [INSPIRE].
I. Ichinose and Y. Satoh, Entropies of scalar fields on three-dimensional black holes, Nucl. Phys. B 447 (1995) 340 [hep-th/9412144] [INSPIRE].
E. Keski-Vakkuri, Bulk and boundary dynamics in BTZ black holes, Phys. Rev. D 59 (1999) 104001 [hep-th/9808037] [INSPIRE].
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
M. Parikh and P. Samantray, Rindler-AdS/CFT, arXiv:1211.7370 [INSPIRE].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, Rindler Quantum Gravity, Class. Quant. Grav. 29 (2012) 235025 [arXiv:1206.1323] [INSPIRE].
S. Elitzur, A. Giveon, D. Kutasov and E. Rabinovici, From big bang to big crunch and beyond, JHEP 06 (2002) 017 [hep-th/0204189] [INSPIRE].
A.R. Steif, The Quantum stress tensor in the three-dimensional black hole, Phys. Rev. D 49 (1994) 585 [gr-qc/9308032] [INSPIRE].
G. Lifschytz and M. Ortiz, Scalar field quantization on the (2+1)-dimensional black hole background, Phys. Rev. D 49 (1994) 1929 [gr-qc/9310008] [INSPIRE].
B.S. Kay, M.J. Radzikowski and R.M. Wald, Quantum field theory on space-times with a compactly generated Cauchy horizon, Commun. Math. Phys. 183 (1997) 533 [gr-qc/9603012] [INSPIRE].
M. Visser, The Quantum physics of chronology protection, gr-qc/0204022 [INSPIRE].
M. Berkooz and D. Reichmann, A Short Review of Time Dependent Solutions and Space-like Singularities in String Theory, Nucl. Phys. Proc. Suppl. 171 (2007) 69 [arXiv:0705.2146] [INSPIRE].
J. McGreevy and E. Silverstein, The Tachyon at the end of the universe, JHEP 08 (2005) 090 [hep-th/0506130] [INSPIRE].
A. de la Fuente and R. Sundrum, in preparation.
P. Kraus, H. Ooguri and S. Shenker, Inside the horizon with AdS/CFT, Phys. Rev. D 67 (2003) 124022 [hep-th/0212277] [INSPIRE].
J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].
J. Schwinger, The special canonical group, Proc. Natl. Acad. Sci. Unit. States Am. 46 (1960) 1401. http://www.pnas.org/content/46/10/1401.short.
S. Weinberg, Quantum contributions to cosmological correlations, Phys. Rev. D 72 (2005) 043514 [hep-th/0506236] [INSPIRE].
V. Balasubramanian and S.F. Ross, Holographic particle detection, Phys. Rev. D 61 (2000) 044007 [hep-th/9906226] [INSPIRE].
J. Louko, D. Marolf and S.F. Ross, On geodesic propagators and black hole holography, Phys. Rev. D 62 (2000) 044041 [hep-th/0002111] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [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].
A. Hamilton, D.N. Kabat, G. Lifschytz and D.A. Lowe, Local bulk operators in AdS/CFT: A Holographic description of the black hole interior, Phys. Rev. D 75 (2007) 106001 [Erratum ibid. D 75 (2007) 129902] [hep-th/0612053] [INSPIRE].
I. Heemskerk, D. Marolf, J. Polchinski and J. Sully, Bulk and Transhorizon Measurements in AdS/CFT, JHEP 10 (2012) 165 [arXiv:1201.3664] [INSPIRE].
K. Papadodimas and S. Raju, An Infalling Observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
S. Hemming, E. Keski-Vakkuri and P. Kraus, Strings in the extended BTZ space-time, JHEP 10 (2002) 006 [hep-th/0208003] [INSPIRE].
J. Kaplan, Extracting data from behind horizons with the AdS/CFT correspondence, hep-th/0402066 [INSPIRE].
V. Balasubramanian and T.S. Levi, Beyond the veil: Inner horizon instability and holography, Phys. Rev. D 70 (2004) 106005 [hep-th/0405048] [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].
E.J. Martinec and W. McElgin, Exciting AdS orbifolds, JHEP 10 (2002) 050 [hep-th/0206175] [INSPIRE].
F. Loran and M.M. Sheikh-Jabbari, O-BTZ: Orientifolded BTZ Black Hole, Phys. Lett. B 693 (2010) 184 [arXiv:1003.4089] [INSPIRE].
F. Loran and M.M. Sheikh-Jabbari, Orientifolded Locally AdS 3 Geometries, Class. Quant. Grav. 28 (2011) 025013 [arXiv:1008.0462] [INSPIRE].
S. Carlip, The (2 + 1)-Dimensional black hole, Class. Quant. Grav. 12 (1995) 2853 [gr-qc/9506079] [INSPIRE].
U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Vacua, propagators and holographic probes in AdS/CFT, JHEP 01 (1999) 002 [hep-th/9812007] [INSPIRE].
C.W. Misner, The Flatter regions of Newman, Unti and Tamburino’s generalized Schwarzschild space, J. Math. Phys. 4 (1963) 924 [INSPIRE].
C.R. Cramer and B.S. Kay, The Thermal and two particle stress - energy must be ill defined on the 2 − D Misner space chronology horizon, Phys. Rev. D 57 (1998) 1052 [gr-qc/9708028] [INSPIRE].
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press, Cambridge U.K. (1998).
A. de la Fuente and R. Sundrum, in preparation.
J. Bisognano and E.H. Wichmann, On the Duality Condition for a Hermitian Scalar Field, J. Math. Phys. 16 (1975) 985 [INSPIRE].
J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
G. Sewell, Relativity of temperature and the Hawking effect, Phys. Lett. A 79 (1980) 23.
G.L. Sewell, Quantum fields on manifolds: PCT and gravitationally induced thermal states, Annals Phys. 141 (1982) 201 [INSPIRE].
W.G. Unruh and N. Weiss, Acceleration Radiation in Interacting Field Theories, Phys. Rev. D 29 (1984) 1656 [INSPIRE].
T. Inami and H. Ooguri, One Loop Effective Potential in Anti-de Sitter Space, Prog. Theor. Phys. 73 (1985) 1051 [INSPIRE].
C.P. Burgess and C.A. Lütken, Propagators and Effective Potentials in Anti-de Sitter Space, Phys. Lett. B 153 (1985) 137 [INSPIRE].
S. Aminneborg, I. Bengtsson, S. Holst and P. Peldan, Making anti-de Sitter black holes, Class. Quant. Grav. 13 (1996) 2707 [gr-qc/9604005] [INSPIRE].
D.R. Brill, J. Louko and P. Peldan, Thermodynamics of (3 + 1)-dimensional black holes with toroidal or higher genus horizons, Phys. Rev. D 56 (1997) 3600 [gr-qc/9705012] [INSPIRE].
S. Holst and P. Peldan, Black holes and causal structure in anti-de Sitter isometric space-times, Class. Quant. Grav. 14 (1997) 3433 [gr-qc/9705067] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1307.7738
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
de la Fuente, A., Sundrum, R. Holography of the BTZ black hole, inside and out. J. High Energ. Phys. 2014, 73 (2014). https://doi.org/10.1007/JHEP09(2014)073
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2014)073