Abstract
In the 1980’s, work by Coleman and by Giddings and Strominger linked the physics of spacetime wormholes to ‘baby universes’ and an ensemble of theories. We revisit such ideas, using features associated with a negative cosmological constant and asymptotically AdS boundaries to strengthen the results, introduce a change in perspective, and connect with recent replica wormhole discussions of the Page curve. A key new feature is an emphasis on the role of null states. We explore this structure in detail in simple topological models of the bulk that allow us to compute the full spectrum of associated boundary theories. The dimension of the asymptotically AdS Hilbert space turns out to become a random variable Z , whose value can be less than the naive number k of independent states in the theory. For k > Z , consistency arises from an exact degeneracy in the inner product defined by the gravitational path integral, so that many a priori independent states differ only by a null state. We argue that a similar property must hold in any consistent gravitational path integral. We also comment on other aspects of extrapolations to more complicated models, and on possible implications for the black hole information problem in the individual members of the above ensemble.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [INSPIRE].
T. Jacobson, Introduction to quantum fields in curved space-time and the Hawking effect, in School on quantum gravity, Springer, Boston, MA, U.S.A. (2003), pg. 39 [gr-qc/0308048] [INSPIRE].
S.D. Mathur, The information paradox: a pedagogical introduction, arXiv:0909.1038 [INSPIRE].
D. Harlow, Jerusalem lectures on black holes and quantum information, Rev. Mod. Phys. 88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].
W.G. Unruh and R.M. Wald, Information loss, Rept. Prog. Phys. 80 (2017) 092002 [arXiv:1703.02140] [INSPIRE].
D. Marolf, The black hole information problem: past, present, and future, Rept. Prog. Phys. 80 (2017) 092001 [arXiv:1703.02143] [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].
L. Susskind, String theory and the principles of black hole complementarity, Phys. Rev. Lett. 71 (1993) 2367 [hep-th/9307168] [INSPIRE].
L. Susskind, Strings, black holes and Lorentz contraction, Phys. Rev. D 49 (1994) 6606 [hep-th/9308139] [INSPIRE].
G. Chapline, E. Hohlfeld, R.B. Laughlin and D.I. Santiago, Quantum phase transitions and the breakdown of classical general relativity, Int. J. Mod. Phys. A 18 (2003) 3587 [gr-qc/0012094] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
P.O. Mazur and E. Mottola, Gravitational condensate stars: an alternative to black holes, gr-qc/0109035 [INSPIRE].
F. Winterberg, Gamma-ray bursters and Lorentzian relativity, Z. Naturforsch. A 56 (2001) 889.
G.T. Horowitz and J.M. Maldacena, The black hole final state, JHEP 02 (2004) 008 [hep-th/0310281] [INSPIRE].
S.W. Hawking, Information loss in black holes, Phys. Rev. D 72 (2005) 084013 [hep-th/0507171] [INSPIRE].
G.T. Horowitz and E. Silverstein, The inside story: quasilocal tachyons and black holes, Phys. Rev. D 73 (2006) 064016 [hep-th/0601032] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
S.B. Giddings, Black hole information, unitarity, and nonlocality, Phys. Rev. D 74 (2006) 106005 [hep-th/0605196] [INSPIRE].
S.D. Mathur, Fuzzballs and the information paradox: a summary and conjectures, arXiv:0810.4525 [INSPIRE].
S.B. Giddings, Models for unitary black hole disintegration, Phys. Rev. D 85 (2012) 044038 [arXiv:1108.2015] [INSPIRE].
A. Davidson, Holographic shell model: stack data structure inside black holes?, Int. J. Mod. Phys. D 23 (2014) 1450041 [arXiv:1108.2650] [INSPIRE].
S.B. Giddings, Nonviolent nonlocality, Phys. Rev. D 88 (2013) 064023 [arXiv:1211.7070] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
S.D. Mathur and D. Turton, Comments on black holes I: the possibility of complementarity, JHEP 01 (2014) 034 [arXiv:1208.2005] [INSPIRE].
K. Papadodimas and S. Raju, An infalling observer in AdS/CFT, JHEP 10 (2013) 212 [arXiv:1211.6767] [INSPIRE].
E. Verlinde and H. Verlinde, Black hole entanglement and quantum error correction, JHEP 10 (2013) 107 [arXiv:1211.6913] [INSPIRE].
Y. Nomura, J. Varela and S.J. Weinberg, Complementarity endures: no firewall for an infalling observer, JHEP 03 (2013) 059 [arXiv:1207.6626] [INSPIRE].
E. Verlinde and H. Verlinde, Passing through the firewall, arXiv:1306.0515 [INSPIRE].
S.D. Mathur and D. Turton, The flaw in the firewall argument, Nucl. Phys. B 884 (2014) 566 [arXiv:1306.5488] [INSPIRE].
S.B. Giddings, Nonviolent information transfer from black holes: a field theory parametrization, Phys. Rev. D 88 (2013) 024018 [arXiv:1302.2613] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
E. Silverstein, Backdraft: string creation in an old Schwarzschild black hole, arXiv:1402.1486 [INSPIRE].
C. Rovelli and F. Vidotto, Planck stars, Int. J. Mod. Phys. D 23 (2014) 1442026 [arXiv:1401.6562] [INSPIRE].
H.M. Haggard and C. Rovelli, Quantum-gravity effects outside the horizon spark black to white hole tunneling, Phys. Rev. D 92 (2015) 104020 [arXiv:1407.0989] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, Stringy horizons, JHEP 06 (2015) 064 [arXiv:1502.03633] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft hair on black holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
M. Christodoulou, C. Rovelli, S. Speziale and I. Vilensky, Planck star tunneling time: an astrophysically relevant observable from background-free quantum gravity, Phys. Rev. D 94 (2016) 084035 [arXiv:1605.05268] [INSPIRE].
A. Giveon and N. Itzhaki, Stringy information and black holes, JHEP 06 (2020) 117 [arXiv:1912.06538] [INSPIRE].
L. Amadei and A. Perez, Hawking’s information puzzle: a solution realized in loop quantum cosmology, arXiv:1911.00306 [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].
G. Penington, Entanglement wedge reconstruction and the information paradox, arXiv:1905.08255 [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].
A. Almheiri, R. Mahajan and J. Maldacena, Islands outside the horizon, arXiv:1910.11077 [INSPIRE].
A. Almheiri, R. Mahajan and J.E. Santos, Entanglement islands in higher dimensions, SciPost Phys. 9 (2020) 001 [arXiv:1911.09666] [INSPIRE].
H.Z. Chen, Z. Fisher, J. Hernandez, R.C. Myers and S.-M. Ruan, Information flow in black hole evaporation, JHEP 03 (2020) 152 [arXiv:1911.03402] [INSPIRE].
A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Replica wormholes and the entropy of Hawking radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, extremality, Euclidean variations, and the equations of motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].
S.R. Coleman, Black holes as red herrings: topological fluctuations and the loss of quantum coherence, Nucl. Phys. B 307 (1988) 867 [INSPIRE].
S.B. Giddings and A. Strominger, Loss of incoherence and determination of coupling constants in quantum gravity, Nucl. Phys. B 307 (1988) 854 [INSPIRE].
S.B. Giddings and A. Strominger, Baby universes, third quantization and the cosmological constant, Nucl. Phys. B 321 (1989) 481 [INSPIRE].
J.M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02 (2004) 053 [hep-th/0401024] [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].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
N. Arkani-Hamed, J. Orgera and J. Polchinski, Euclidean wormholes in string theory, JHEP 12 (2007) 018 [arXiv:0705.2768] [INSPIRE].
S.W. Hawking, Quantum coherence down the wormhole, Phys. Lett. B 195 (1987) 337 [INSPIRE].
S.W. Hawking, Wormholes in space-time, Phys. Rev. D 37 (1988) 904 [INSPIRE].
S.B. Giddings and A. Strominger, Axion induced topology change in quantum gravity and string theory, Nucl. Phys. B 306 (1988) 890 [INSPIRE].
G.V. Lavrelashvili, V.A. Rubakov and P.G. Tinyakov, Disruption of quantum coherence upon a change in spatial topology in quantum gravity, JETP Lett. 46 (1987) 167 [Pisma Zh. Eksp. Teor. Fiz. 46 (1987) 134] [INSPIRE].
J.B. Hartle and S.W. Hawking, Wave function of the universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].
J.J. Halliwell and J.B. Hartle, Wave functions constructed from an invariant sum over histories satisfy constraints, Phys. Rev. D 43 (1991) 1170 [INSPIRE].
P.A.M. Dirac, Lectures on quantum mechanics, Belfor Graduate School of Science, Yeshiva University, New York, NY, U.S.A. (1964).
N.P. Landsman, Rieffel induction as generalized quantum Marsden-Weinstein reduction, hep-th/9305088 [INSPIRE].
D. Marolf, Quantum observables and recollapsing dynamics, Class. Quant. Grav. 12 (1995) 1199 [gr-qc/9404053] [INSPIRE].
A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, Quantization of diffeomorphism invariant theories of connections with local degrees of freedom, J. Math. Phys. 36 (1995) 6456 [gr-qc/9504018] [INSPIRE].
D. Marolf, Group averaging and refined algebraic quantization: where are we now?, in 9th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories (MG 9), Rome, Italy, 2–8 July 2000 [gr-qc/0011112] [INSPIRE].
O.Y. Shvedov, On correspondence of BRST-BFV, Dirac and refined algebraic quantizations of constrained systems, Annals Phys. 302 (2002) 2 [hep-th/0111270] [INSPIRE].
D. Marolf, Path integrals and instantons in quantum gravity: minisuperspace models, Phys. Rev. D 53 (1996) 6979 [gr-qc/9602019] [INSPIRE].
M.P. Reisenberger and C. Rovelli, ‘Sum over surfaces’ form of loop quantum gravity, Phys. Rev. D 56 (1997) 3490 [gr-qc/9612035] [INSPIRE].
J.B. Hartle and D. Marolf, Comparing formulations of generalized quantum mechanics for reparametrization-invariant systems, Phys. Rev. D 56 (1997) 6247 [gr-qc/9703021] [INSPIRE].
A.S. Wightman, Quantum field theory in terms of vacuum expectation values, Phys. Rev. 101 (1956) 860 [INSPIRE].
R.F. Streater and A.S. Wightman, PCT, spin and statistics and all that, Princeton University Press, Princeton, NJ, U.S.A. (2016).
K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Commun. Math. Phys. 31 (1973) 83 [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Eigenbranes in Jackiw-Teitelboim gravity, arXiv:1911.11603 [INSPIRE].
D. Harlow, Wormholes, emergent gauge fields, and the weak gravity conjecture, JHEP 01 (2016) 122 [arXiv:1510.07911] [INSPIRE].
M. Guica and D.L. Jafferis, On the construction of charged operators inside an eternal black hole, SciPost Phys. 3 (2017) 016 [arXiv:1511.05627] [INSPIRE].
D. Harlow and D. Jafferis, The factorization problem in Jackiw-Teitelboim gravity, JHEP 02 (2020) 177 [arXiv:1804.01081] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, A semiclassical ramp in SYK and in gravity, arXiv:1806.06840 [INSPIRE].
D. Stanford and E. Witten, JT gravity and the ensembles of random matrix theory, arXiv:1907.03363 [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, Notes on topological string theory and 2D quantum gravity, in Cargese study institute: random surfaces, quantum gravity and strings, Cargese, France, 27 May–2 June 1990, pg. 91.
A.F. Möbius, Theorie der elementaren Verwandtschaft (in German), Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften, Mathematisch-physikalische Klasse 15 (1863) 19.
C. Jordan, Sur la déformation des surfaces (in French), J. Math. Pures Appl. (1866) 105.
K. Okuyama, Replica symmetry breaking in random matrix model: a toy model of wormhole networks, Phys. Lett. B 803 (2020) 135280 [arXiv:1903.11776] [INSPIRE].
Complex Wishart distribution. Wikipedia page, https://en.wikipedia.org/wiki/Complex_Wishart_distribution.
H. Massam, G. Letac and P. Graczyk, The complex Wishart distribution and the symmetric group, Annals Statist. 31 (2003) 287.
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-Sitter space, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
D. Stanford and E. Witten, Fermionic localization of the Schwarzian theory, JHEP 10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
D.L. Jafferis, Bulk reconstruction and the Hartle-Hawking wavefunction, arXiv:1703.01519 [INSPIRE].
J. Polchinski, String theory and black hole complementarity, in Strings ′95: future perspectives in string theory, (1995), pg. 417 [hep-th/9507094] [INSPIRE].
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
J. Polchinski and A. Strominger, A possible resolution of the black hole information puzzle, Phys. Rev. D 50 (1994) 7403 [hep-th/9407008] [INSPIRE].
W. Fischler, I.R. Klebanov, J. Polchinski and L. Susskind, Quantum mechanics of the Googolplexus, Nucl. Phys. B 327 (1989) 157 [INSPIRE].
J. Preskill, Wormholes in space-time and the constants of nature, Nucl. Phys. B 323 (1989) 141 [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
I. Kourkoulou and J. Maldacena, Pure states in the SYK model and nearly-AdS2 gravity, arXiv:1707.02325 [INSPIRE].
D.A. Lowe, J. Polchinski, L. Susskind, L. Thorlacius and J. Uglum, Black hole complementarity versus locality, Phys. Rev. D 52 (1995) 6997 [hep-th/9506138] [INSPIRE].
N. Goheer, M. Kleban and L. Susskind, The trouble with de Sitter space, JHEP 07 (2003) 056 [hep-th/0212209] [INSPIRE].
A. Maloney, Geometric microstates for the three dimensional black hole?, arXiv:1508.04079 [INSPIRE].
A. Almheiri, Holographic quantum error correction and the projected black hole interior, arXiv:1810.02055 [INSPIRE].
Z. Fu and D. Marolf, Bag-of-gold spacetimes, Euclidean wormholes, and inflation from domain walls in AdS/CFT, JHEP 11 (2019) 040 [arXiv:1909.02505] [INSPIRE].
C. Akers, N. Engelhardt, G. Penington and M. Usatyuk, Quantum maximin surfaces, arXiv:1912.02799 [INSPIRE].
J. Cotler and K. Jensen, Emergent unitarity in de Sitter from matrix integrals, arXiv:1911.12358 [INSPIRE].
D. Harlow and D. Stanford, Operator dictionaries and wave functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
S.D. Mathur, A model with no firewall, arXiv:1506.04342 [INSPIRE].
S.B. Giddings, Nonviolent unitarization: basic postulates to soft quantum structure of black holes, JHEP 12 (2017) 047 [arXiv:1701.08765] [INSPIRE].
S.D. Mathur, Resolving the black hole causality paradox, Gen. Rel. Grav. 51 (2019) 24 [arXiv:1703.03042] [INSPIRE].
P. Hayden, S. Nezami, X.-L. Qi, N. Thomas, M. Walter and Z. Yang, Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
X.-L. Qi and Z. Yang, Space-time random tensor networks and holographic duality, arXiv:1801.05289 [INSPIRE].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [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].
Y. Chen, Pulling out the island with modular flow, JHEP 03 (2020) 033 [arXiv:1912.02210] [INSPIRE].
R. Bousso, Firewalls from double purity, Phys. Rev. D 88 (2013) 084035 [arXiv:1308.2665] [INSPIRE].
R. Bousso, Violations of the equivalence principle by a nonlocally reconstructed vacuum at the black hole horizon, Phys. Rev. Lett. 112 (2014) 041102 [arXiv:1308.3697] [INSPIRE].
D. Marolf and J. Polchinski, Violations of the Born rule in cool state-dependent horizons, JHEP 01 (2016) 008 [arXiv:1506.01337] [INSPIRE].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, hep-th/0701216 [INSPIRE].
V. Balasubramanian, J. de Boer, S. El-Showk and I. Messamah, Black holes as effective geometries, Class. Quant. Grav. 25 (2008) 214004 [arXiv:0811.0263] [INSPIRE].
K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].
B.D. Chowdhury and A. Virmani, Modave lectures on fuzzballs and emission from the D1-D5 system, in 5th Modave Summer School in Mathematical Physics, (2010) [arXiv:1001.1444] [INSPIRE].
I. Bena and N.P. Warner, Resolving the structure of black holes: philosophizing with a hammer, arXiv:1311.4538 [INSPIRE].
T. Jacobson, Boundary unitarity and the black hole information paradox, Int. J. Mod. Phys. D 22 (2013) 1342002 [arXiv:1212.6944] [INSPIRE].
T. Jacobson and P. Nguyen, Diffeomorphism invariance and the black hole information paradox, Phys. Rev. D 100 (2019) 046002 [arXiv:1904.04434] [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
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2002.08950
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Marolf, D., Maxfield, H. Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information. J. High Energ. Phys. 2020, 44 (2020). https://doi.org/10.1007/JHEP08(2020)044
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2020)044