Abstract
We use a microcanonical path integral closely related to that introduced by Brown and York in 1992 to add new entries to the AdS/CFT dictionary concerning the interiors of small black holes. Stationary points of such path integrals are also stationary points of more standard canonical-type path integrals with fixed boundary metric, but the condition for dominance is now maximizing Hubeny-Rangamani-Takayanagi entropy at fixed energy. As a result, such path integrals can bring to the fore saddles that fail to dominate in more familiar contexts. We use this feature to argue that the standard Kruskal-like two-sided extension of small AdS black holes with energy E0 is dual to a microcanonical version of the thermofield double state for AdS black holes that maximize the microcanonical bulk entropy at this energy. We also comment on entanglement in such states and on quantum effects that become large when the energy-width of the microcanonical ensemble is sufficiently small.
References
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
M. Botta-Cantcheff, P. Martínez and G.A. Silva, On excited states in real-time AdS/CFT, JHEP 02 (2016) 171 [arXiv:1512.07850] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
A. Christodoulou and K. Skenderis, Holographic Construction of Excited CFT States, JHEP 04 (2016) 096 [arXiv:1602.02039] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
D. Marolf, O. Parrikar, C. Rabideau, A. Izadi Rad and M. Van Raamsdonk, From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories, JHEP 06 (2018) 077 [arXiv:1709.10101] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
K. Skenderis and B.C. van Rees, Holography and wormholes in 2+1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].
V. Balasubramanian, P. Hayden, A. Maloney, D. Marolf and S.F. Ross, Multiboundary Wormholes and Holographic Entanglement, Class. Quant. Grav. 31 (2014) 185015 [arXiv:1406.2663] [INSPIRE].
D. Marolf, H. Maxfield, A. Peach and S.F. Ross, Hot multiboundary wormholes from bipartite entanglement, Class. Quant. Grav. 32 (2015) 215006 [arXiv:1506.04128] [INSPIRE].
H. Maxfield, S. Ross and B. Way, Holographic partition functions and phases for higher genus Riemann surfaces, Class. Quant. Grav. 33 (2016) 125018 [arXiv:1601.00980] [INSPIRE].
D. Marolf, M. Rota and J. Wien, Handlebody phases and the polyhedrality of the holographic entropy cone, JHEP 10 (2017) 069 [arXiv:1705.10736] [INSPIRE].
D. Marolf and J. Wien, The Torus Operator in Holography, JHEP 01 (2018) 105 [arXiv:1708.03048] [INSPIRE].
Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal, JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
C.T. Asplund and D. Berenstein, Small AdS black holes from SYM, Phys. Lett. B 673 (2009) 264 [arXiv:0809.0712] [INSPIRE].
M. Hanada and J. Maltz, A proposal of the gauge theory description of the small Schwarzschild black hole in AdS 5×S 5, JHEP 02 (2017) 012 [arXiv:1608.03276] [INSPIRE].
L.G. Yaffe, Large N phase transitions and the fate of small Schwarzschild-AdS black holes, Phys. Rev. D 97 (2018) 026010 [arXiv:1710.06455] [INSPIRE].
D. Berenstein, Submatrix deconfinement and small black holes in AdS, JHEP 09 (2018) 054 [arXiv:1806.05729] [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].
T. Klosch and T. Strobl, Classical and quantum gravity in (1+1)-dimensions. Part 2: The universal coverings, Class. Quant. Grav. 13 (1996) 2395 [gr-qc/9511081] [INSPIRE].
P. Gao, D.L. Jafferis and A. Wall, Traversable Wormholes via a Double Trace Deformation, JHEP 12 (2017) 151 [arXiv:1608.05687] [INSPIRE].
B. Freivogel, V.E. Hubeny, A. Maloney, R.C. Myers, M. Rangamani and S. Shenker, Inflation in AdS/CFT, JHEP 03 (2006) 007 [hep-th/0510046] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
S. Fischetti, D. Marolf and A.C. Wall, A paucity of bulk entangling surfaces: AdS wormholes with de Sitter interiors, Class. Quant. Grav. 32 (2015) 065011 [arXiv:1409.6754] [INSPIRE].
D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, Rev. Mod. Phys. 88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].
D. Marolf, The Black Hole information problem: past, present, and future, Rept. Prog. Phys. 80 (2017) 092001 [arXiv:1703.02143] [INSPIRE].
W.G. Unruh and R.M. Wald, Information Loss, Rept. Prog. Phys. 80 (2017) 092002 [arXiv:1703.02140] [INSPIRE].
J.D. Brown and J.W. York Jr., The microcanonical functional integral. 1. The gravitational field, Phys. Rev. D 47 (1993) 1420 [gr-qc/9209014] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP 11 (2016) 028 [arXiv:1607.07506] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].
D. Marolf and A.C. Wall, Eternal Black Holes and Superselection in AdS/CFT, Class. Quant. Grav. 30 (2013) 025001 [arXiv:1210.3590] [INSPIRE].
K. Papadodimas and S. Raju, Local Operators in the Eternal Black Hole, Phys. Rev. Lett. 115 (2015) 211601 [arXiv:1502.06692] [INSPIRE].
I. Bena, M. Shigemori and N.P. Warner, Black-Hole Entropy from Supergravity Superstrata States, JHEP 10 (2014) 140 [arXiv:1406.4506] [INSPIRE].
O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].
G.T. Horowitz, Comments on black holes in string theory, Class. Quant. Grav. 17 (2000) 1107 [hep-th/9910082] [INSPIRE].
J.D. Brown and J.W. York Jr., Microcanonical action and the entropy of a rotating black hole, Math. Phys. Stud. 15 (1994) 23 [gr-qc/9303012] [INSPIRE].
G.T. Horowitz and D. Marolf, Counting states of black strings with traveling waves, Phys. Rev. D 55 (1997) 835 [hep-th/9605224] [INSPIRE].
G.T. Horowitz and D. Marolf, Counting states of black strings with traveling waves. 2., Phys. Rev. D 55 (1997) 846 [hep-th/9606113] [INSPIRE].
G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].
T. Andrade and D. Marolf, AdS/CFT beyond the unitarity bound, JHEP 01 (2012) 049 [arXiv:1105.6337] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Diving into traversable wormholes, Fortsch. Phys. 65 (2017) 1700034 [arXiv:1704.05333] [INSPIRE].
N. Bao, S. Nezami, H. Ooguri, B. Stoica, J. Sully and M. Walter, The Holographic Entropy Cone, JHEP 09 (2015) 130 [arXiv:1505.07839] [INSPIRE].
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Marolf, D. Microcanonical path integrals and the holography of small black hole interiors. J. High Energ. Phys. 2018, 114 (2018). https://doi.org/10.1007/JHEP09(2018)114
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DOI: https://doi.org/10.1007/JHEP09(2018)114
Keywords
- AdS-CFT Correspondence
- Black Holes in String Theory