Abstract
It is sometimes believed that small quantum gravity effects can encode information as ‘delicate correlations’ in Hawking radiation, thus saving unitarity while maintaining a semi classical horizon. A recently derived inequality showed that this belief is incorrect: one must have order unity corrections to low energy evolution at the horizon (i.e. fuzzballs) to remove entanglement between radiation and the hole. In this paper we take several models of ‘small corrections’ and compute the entanglement entropy numerically; in each case this entanglement is seen to monotonically grow, in agreement with the general inequality. We also construct a model of ‘burning paper’, where the entanglement is found to rise and then return to zero, in agreement with the general arguments of Page. We then note that the fuzzball structure of string microstates offers a version of ‘complementarity’. Low energy evolution is modified by order unity, resolving the information problem, while for high energy infalling modes (E ≫ kT) we may be able to replace correlators by their ensemble averaged values. Israel (and others) have suggested that this ensemble sum can be represented in the thermo-field-dynamics language as an entangled sum over two copies of the system, giving the two sides of the extended black hole diagram. Thus high energy correlators in a microstate may be approximated by correlators in a spacetime with horizons, with the ensemble sum over microstates acting like the ‘sewing’ prescription of conformal field theory.
Similar content being viewed by others
References
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [SPIRES].
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [SPIRES].
W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967) 1776 [SPIRES].
B. Carter, Axisymmetric black hole has only two degrees of freedom, Phys. Rev. Lett. 26 (1971) 331 [SPIRES].
R.H. Price, Nonspherical perturbations of relativistic gravitational collapse. II. Integer-spin, zero-rest-mass fields, Phys. Rev. D 5 (1972) 2439 [SPIRES].
D.C. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett. 34 (1975) 905 [SPIRES].
A. Ashtekar and M. Bojowald, Black hole evaporation: a paradigm, Class. Quant. Grav. 22 (2005) 3349 [gr-qc/0504029] [SPIRES].
J.B. Hartle, Generalized quantum theory in evaporating black hole spacetimes, gr-qc/9705022 [SPIRES].
H. Nikolic, Resolving the black-hole information paradox by treating time on an equal footing with space, Phys. Lett. B 678 (2009) 218 [arXiv:0905.0538] [SPIRES].
C.G. Callan and J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472 (1996) 591 [hep-th/9602043] [SPIRES].
A. Dhar, G. Mandal and S.R. Wadia, Absorption vs decay of black holes in string theory and T-symmetry, Phys. Lett. B 388 (1996) 51 [hep-th/9605234] [SPIRES].
S.R. Das and S.D. Mathur, Comparing decay rates for black holes and D-branes, Nucl. Phys. B 478 (1996) 561 [hep-th/9606185] [SPIRES].
S.R. Das and S.D. Mathur, Interactions involving D-branes, Nucl. Phys. B 482 (1996) 153 [hep-th/9607149] [SPIRES].
J.M. Maldacena and A. Strominger, Black hole greybody factors and D-brane spectroscopy, Phys. Rev. D 55 (1997) 861 [hep-th/9609026] [SPIRES].
S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [SPIRES].
S.D. Mathur, The information paradox and the infall problem, Class. Quant. Grav. 28 (2011) 125010 [arXiv:1012.2101] [SPIRES].
D.N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291 [gr-qc/9305007] [SPIRES].
W. Israel, Thermo field dynamics of black holes, Phys. Lett. A 57 (1976) 107 [SPIRES].
H. Umezawa, H. Matsumoto and M. Tachiki, Thermo field dynamics and condensed states, North-Holland, Amsterdam The Netherlands (1982) [SPIRES].
J.M. Maldacena, Eternal black holes in Anti-de-Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [SPIRES].
M. Van Raamsdonk, Comments on quantum gravity and entanglement, arXiv:0907.2939 [SPIRES].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [Int. J. Mod. Phys. D 19 (2010) 2429] [ arXiv:1005.3035] [SPIRES].
G. ’t Hooft, The black hole interpretation of string theory, Nucl. Phys. B 335 (1990) 138 [SPIRES].
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [SPIRES].
G. ’t Hooft, The scattering matrix approach for the quantum black hole: an overview, Int. J. Mod. Phys. A 11 (1996) 4623 [gr-qc/9607022] [SPIRES].
L. Susskind, The world as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [SPIRES].
L. Susskind, L. Thorlacius and J. Uglum, The stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [SPIRES].
L. Susskind, String theory and the principles of black hole complementarity, Phys. Rev. Lett. 71 (1993) 2367 [hep-th/9307168] [SPIRES].
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] [SPIRES].
S.B. Giddings and W.M. Nelson, Quantum emission from two-dimensional black holes, Phys. Rev. D 46 (1992) 2486 [hep-th/9204072] [SPIRES].
O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [SPIRES].
O. Lunin and S.D. Mathur, Statistical interpretation of Bekenstein entropy for systems with a stretched horizon, Phys. Rev. Lett. 88 (2002) 211303 [hep-th/0202072] [SPIRES].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [SPIRES].
S.D. Mathur, The quantum structure of black holes, Class. Quant. Grav. 23 (2006) R115 [hep-th/0510180] [SPIRES].
S.D. Mathur, Fuzzballs and the information paradox: a summary and conjectures, arXiv:0810.4525 [SPIRES].
O. Lunin, J.M. Maldacena and L. Maoz, Gravity solutions for the D1-D5 system with angular momentum, hep-th/0212210 [SPIRES].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [SPIRES].
S.D. Mathur, A. Saxena and Y.K. Srivastava, Constructing ‘hair’ for the three charge hole, Nucl. Phys. B 680 (2004) 415 [hep-th/0311092] [SPIRES].
O. Lunin, Adding momentum to D1-D5 system, JHEP 04 (2004) 054 [hep-th/0404006] [SPIRES].
S. Giusto, S.D. Mathur and A. Saxena, Dual geometries for a set of 3-charge microstates, Nucl. Phys. B 701 (2004) 357 [hep-th/0405017] [SPIRES].
S. Giusto, S.D. Mathur and A. Saxena, 3-charge geometries and their CFT duals, Nucl. Phys. B 710 (2005) 425 [hep-th/0406103] [SPIRES].
I. Bena and N.P. Warner, One ring to rule them all… …and in the darkness bind them?, Adv. Theor. Math. Phys. 9 (2005) 667 [hep-th/0408106] [SPIRES].
V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries and D1-D5-P bound states, Phys. Rev. D 71 (2005) 124030 [hep-th/0504181] [SPIRES].
I. Bena and N.P. Warner, Bubbling supertubes and foaming black holes, Phys. Rev. D 74 (2006) 066001 [hep-th/0505166] [SPIRES].
P. Berglund, E.G. Gimon and T.S. Levi, Supergravity microstates for BPS black holes and black rings, JHEP 06 (2006) 007 [hep-th/0505167] [SPIRES].
A. Saxena, G. Potvin, S. Giusto and A.W. Peet, Smooth geometries with four charges in four dimensions, JHEP 04 (2006) 010 [hep-th/0509214] [SPIRES].
I. Bena, C.-W. Wang and N.P. Warner, The foaming three-charge black hole, Phys. Rev. D 75 (2007) 124026 [hep-th/0604110] [SPIRES].
V. Balasubramanian, E.G. Gimon and T.S. Levi, Four dimensional black hole microstates: from D-branes to spacetime foam, JHEP 01 (2008) 056 [hep-th/0606118] [SPIRES].
I. Bena, C.-W. Wang and N.P. Warner, Mergers and typical black hole microstates, JHEP 11 (2006) 042 [hep-th/0608217] [SPIRES].
J. Ford, S. Giusto and A. Saxena, A class of BPS time-dependent 3-charge microstates from spectral flow, Nucl. Phys. B 790 (2008) 258 [hep-th/0612227] [SPIRES].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [SPIRES].
I. Bena, N. Bobev and N.P. Warner, Bubbles on manifolds with a U(1) isometry, JHEP 08 (2007) 004 [arXiv:0705.3641] [SPIRES].
E.G. Gimon and T.S. Levi, Black ring deconstruction, JHEP 04 (2008) 098 [arXiv:0706.3394] [SPIRES].
I. Bena, C.-W. Wang and N.P. Warner, Plumbing the abyss: black ring microstates, JHEP 07 (2008) 019 [arXiv:0706.3786] [SPIRES].
S. Giusto, S.F. Ross and A. Saxena, Non-supersymmetric microstates of the D1-D5-KK system, JHEP 12 (2007) 065 [arXiv:0708.3845] [SPIRES].
V. Jejjala, O. Madden, S.F. Ross and G. Titchener, Non-supersymmetric smooth geometries and D1-D5-P bound states, Phys. Rev. D 71 (2005) 124030 [hep-th/0504181] [SPIRES].
V. Cardoso, O.J.C. Dias, J.L. Hovdebo and R.C. Myers, Instability of non-supersymmetric smooth geometries, Phys. Rev. D 73 (2006) 064031 [hep-th/0512277] [SPIRES].
B.D. Chowdhury and S.D. Mathur, Radiation from the non-extremal fuzzball, Class. Quant. Grav. 25 (2008) 135005 [arXiv:0711.4817] [SPIRES].
B.D. Chowdhury and S.D. Mathur, Pair creation in non-extremal fuzzball geometries, Class. Quant. Grav. 25 (2008) 225021 [arXiv:0806.2309] [SPIRES].
B.D. Chowdhury and S.D. Mathur, Non-extremal fuzzballs and ergoregion emission, Class. Quant. Grav. 26 (2009) 035006 [arXiv:0810.2951] [SPIRES].
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [SPIRES].
G.W. Gibbons and M.J. Perry, Black holes and thermal Green’s functions, Proc. Roy. Soc. Lond. A 358 (1978) 467 [SPIRES].
W.G. Unruh and N. Weiss, Acceleration radiation in interacting field theories, Phys. Rev. D 29 (1984) 1656 [SPIRES].
G.T. Horowitz and D. Marolf, A new approach to string cosmology, JHEP 07 (1998) 014 [hep-th/9805207] [SPIRES].
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] [SPIRES].
A. Sen, State operator correspondence and entanglement in AdS 2 /CFT 1, arXiv:1101.4254 [SPIRES].
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] [SPIRES].
V. Balasubramanian, B. Czech, Y.-H. He, K. Larjo and J. Simon, Typicality, black hole microstates and superconformal field theories, JHEP 03 (2008) 008 [arXiv:0712.2434] [SPIRES].
V. Balasubramanian et al., Typicality versus thermality: an analytic distinction, Gen. Rel. Grav. 40 (2008) 1863 [hep-th/0701122] [SPIRES].
V. Balasubramanian, J. de Boer, V. Jejjala and J. Simon, The library of Babel: on the origin of gravitational thermodynamics, JHEP 12 (2005) 006 [hep-th/0508023] [SPIRES].
E.P. Verlinde and H.L. Verlinde, A unitary S matrix and 2D black hole formation and evaporation, Nucl. Phys. B 406 (1993) 43 [hep-th/9302022] [SPIRES].
K. Schoutens, H.L. Verlinde and E.P. Verlinde, Black hole evaporation and quantum gravity, hep-th/9401081 [SPIRES].
E.P. Verlinde and H.L. Verlinde, High-energy scattering in quantum gravity, Class. Quant. Grav. 10 (1993) S175 [SPIRES].
Y. Kiem, H.L. Verlinde and E.P. Verlinde, Black hole horizons and complementarity, Phys. Rev. D 52 (1995) 7053 [hep-th/9502074] [SPIRES].
F. Englert and P. Spindel, The hidden horizon and black hole unitarity, JHEP 12 (2010) 065 [arXiv:1009.6190] [SPIRES].
S.D. Mathur, Membrane paradigm realized?, Gen. Rel. Grav. 42 (2010) 2331 [Int. J. Mod. Phys. D 19 (2010) 2423] [ arXiv:1005.3555] [SPIRES].
T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [SPIRES].
S.W. Hawking, Information loss in black holes, Phys. Rev. D 72 (2005) 084013 [hep-th/0507171] [SPIRES].
D. Marolf, Unitarity and holography in gravitational physics, Phys. Rev. D 79 (2009) 044010 [arXiv:0808.2842] [SPIRES].
D. Marolf, Holographic thought experiments, Phys. Rev. D 79 (2009) 024029 [arXiv:0808.2845] [SPIRES].
S.D. Mathur, Tunneling into fuzzball states, Gen. Rel. Grav. 42 (2010) 113 [arXiv:0805.3716] [SPIRES].
S.D. Mathur, How fast can a black hole release its information?, Int. J. Mod. Phys. D 18 (2009) 2215 [arXiv:0905.4483] [SPIRES].
S.D. Mathur, Black hole size and phase space volumes, arXiv:0706.3884 [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1101.4899
Rights and permissions
About this article
Cite this article
Mathur, S.D., Plumberg, C.J. Correlations in Hawking radiation and the infall problem. J. High Energ. Phys. 2011, 93 (2011). https://doi.org/10.1007/JHEP09(2011)093
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2011)093