Abstract
We consider a black hole with a stretched horizon as a toy model for a fuzzball microstate. The stretched horizon provides a cut-off, and therefore one can determine the normal (as opposed to quasi-normal) modes of a probe scalar in this geometry. For the BTZ black hole, we compute these as a function of the level n and the angular quantum number J. Conventional level repulsion is absent in this system, and yet we find that the Spectral Form Factor (SFF) shows clear evidence for a dip-ramp-plateau structure with a linear ramp of slope ~ 1 on a log-log plot, with or without ensemble averaging. We show that this is a robust feature of stretched horizons by repeating our calculations on the Rindler wedge (times a compact space). We also observe that this is not a generic feature of integrable systems, as illustrated by standard examples like integrable billiards and random 2-site coupled SYK model, among others. The origins of the ramp can be traced to the hierarchically weaker dependence of the normal mode spectrum on the quantum numbers of the compact directions, and the resulting quasi-degeneracy. We conclude by noting an analogy between the 4-site coupled SYK model and the quartic coupling responsible for the non-linear instability of capped geometries. Based on this, we speculate that incorporating probe self-interactions will lead to stronger connections to random matrix behavior.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Schwarzschild, On the gravitational field of a mass point according to Einstein’s theory, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916 (1916) 189 [physics/9905030] [INSPIRE].
C.H. McGruder and B.W. VanDerMeer, The 1916 Ph.D. thesis of Johannes Droste and the discovery of gravitational repulsion, arXiv:1801.07592 [INSPIRE].
S.W. Hawking, Breakdown of predictability in gravitational collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].
D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].
S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].
A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [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].
I. Kanitscheider, K. Skenderis and M. Taylor, Fuzzballs with internal excitations, JHEP 06 (2007) 056 [arXiv:0704.0690] [INSPIRE].
I. Bena, S. Giusto, R. Russo, M. Shigemori and N.P. Warner, Habemus Superstratum! A constructive proof of the existence of superstrata, JHEP 05 (2015) 110 [arXiv:1503.01463] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
I. Bena and N.P. Warner, Resolving the structure of black holes: philosophizing with a hammer, arXiv:1311.4538 [INSPIRE].
M. Shigemori, Superstrata, Gen. Rel. Grav. 52 (2020) 51 [arXiv:2002.01592] [INSPIRE].
V.S. Rychkov, D1-D5 black hole microstate counting from supergravity, JHEP 01 (2006) 063 [hep-th/0512053] [INSPIRE].
C. Krishnan and A. Raju, A note on D1-D5 entropy and geometric quantization, JHEP 06 (2015) 054 [arXiv:1504.04330] [INSPIRE].
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] [INSPIRE].
A. Sen, Two charge system revisited: small black holes or horizonless solutions?, JHEP 05 (2010) 097 [arXiv:0908.3402] [INSPIRE].
S.D. Mathur and D. Turton, The fuzzball nature of two-charge black hole microstates, Nucl. Phys. B 945 (2019) 114684 [arXiv:1811.09647] [INSPIRE].
I. Bena, M. Berkooz, J. de Boer, S. El-Showk and D. Van den Bleeken, Scaling BPS solutions and pure-Higgs states, JHEP 11 (2012) 171 [arXiv:1205.5023] [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
J.S. Cotler et al., Black holes and random matrices, JHEP 05 (2017) 118 [Erratum ibid. 09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
Y. Liu, M.A. Nowak and I. Zahed, Disorder in the Sachdev-Yee-Kitaev model, Phys. Lett. B 773 (2017) 647 [arXiv:1612.05233] [INSPIRE].
C. Krishnan, S. Sanyal and P.N. Bala Subramanian, Quantum chaos and holographic tensor models, JHEP 03 (2017) 056 [arXiv:1612.06330] [INSPIRE].
A. del Campo, J. Molina-Vilaplana and J. Sonner, Scrambling the spectral form factor: unitarity constraints and exact results, Phys. Rev. D 95 (2017) 126008 [arXiv:1702.04350] [INSPIRE].
C. Krishnan, K.V. Pavan Kumar and S. Sanyal, Random matrices and holographic tensor models, JHEP 06 (2017) 036 [arXiv:1703.08155] [INSPIRE].
A. Gaikwad and R. Sinha, Spectral form factor in non-Gaussian random matrix theories, Phys. Rev. D 100 (2019) 026017 [arXiv:1706.07439] [INSPIRE].
C. Krishnan, K.V. Pavan Kumar and D. Rosa, Contrasting SYK-like models, JHEP 01 (2018) 064 [arXiv:1709.06498] [INSPIRE].
R. Bhattacharya, S. Chakrabarti, D.P. Jatkar and A. Kundu, SYK model, chaos and conserved charge, JHEP 11 (2017) 180 [arXiv:1709.07613] [INSPIRE].
C.V. Johnson, F. Rosso and A. Svesko, Jackiw-Teitelboim supergravity as a double-cut matrix model, Phys. Rev. D 104 (2021) 086019 [arXiv:2102.02227] [INSPIRE].
Y. Chen, Spectral form factor for free large N gauge theory and strings, JHEP 06 (2022) 137 [arXiv:2202.04741] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
J.L.F. Barbon and E. Rabinovici, Very long time scales and black hole thermal equilibrium, JHEP 11 (2003) 047 [hep-th/0308063] [INSPIRE].
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
Microstate conference 2021: Monday discussion, YouTube video, https://www.youtube.com/watch?v=0BO-p58Pypc&t=3397s.
P.H.C. Lau, C.-T. Ma, J. Murugan and M. Tezuka, Randomness and chaos in qubit models, Phys. Lett. B 795 (2019) 230 [arXiv:1812.04770] [INSPIRE].
F.C. Eperon, H.S. Reall and J.E. Santos, Instability of supersymmetric microstate geometries, JHEP 10 (2016) 031 [arXiv:1607.06828] [INSPIRE].
E. Keski-Vakkuri, Bulk and boundary dynamics in BTZ black holes, Phys. Rev. D 59 (1999) 104001 [hep-th/9808037] [INSPIRE].
G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [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].
C. Yang, Singularities, geodesics and Green functions in the BTZ black hole, hep-th/0611049 [INSPIRE].
A. Pandey, A. Kumar and S. Puri, Quantum chaotic systems and random matrix theory, arXiv:1905.10596.
P.H.C. Lau, C.-T. Ma, J. Murugan and M. Tezuka, Correlated disorder in the SYK2 model, J. Phys. A 54 (2021) 095401 [arXiv:2003.05401] [INSPIRE].
G. Penington, Entanglement wedge reconstruction and the information paradox, JHEP 09 (2020) 002 [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].
N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
J. Pollack, M. Rozali, J. Sully and D. Wakeham, Eigenstate thermalization and disorder averaging in gravity, Phys. Rev. Lett. 125 (2020) 021601 [arXiv:2002.02971] [INSPIRE].
H. Liu and S. Vardhan, Entanglement entropies of equilibrated pure states in quantum many-body systems and gravity, PRX Quantum 2 (2021) 010344 [arXiv:2008.01089] [INSPIRE].
C. Krishnan and V. Mohan, Hints of gravitational ergodicity: Berry’s ensemble and the universality of the semi-classical Page curve, JHEP 05 (2021) 126 [arXiv:2102.07703] [INSPIRE].
J.-M. Schlenker and E. Witten, No ensemble averaging below the black hole threshold, JHEP 07 (2022) 143 [arXiv:2202.01372] [INSPIRE].
P. Basu, C. Krishnan and A. Saurabh, A stochasticity threshold in holography and the instability of AdS, Int. J. Mod. Phys. A 30 (2015) 1550128 [arXiv:1408.0624] [INSPIRE].
P. Basu, C. Krishnan and P.N. Bala Subramanian, AdS (in)stability: lessons from the scalar field, Phys. Lett. B 746 (2015) 261 [arXiv:1501.07499] [INSPIRE].
I.-S. Yang, Missing top of the AdS resonance structure, Phys. Rev. D 91 (2015) 065011 [arXiv:1501.00998] [INSPIRE].
O. Evnin and C. Krishnan, A hidden symmetry of AdS resonances, Phys. Rev. D 91 (2015) 126010 [arXiv:1502.03749] [INSPIRE].
D. Marolf, B. Michel and A. Puhm, A rough end for smooth microstate geometries, JHEP 05 (2017) 021 [arXiv:1612.05235] [INSPIRE].
O. Evnin and W. Piensuk, Quantum resonant systems, integrable and chaotic, J. Phys. A 52 (2019) 025102 [arXiv:1808.09173] [INSPIRE].
O. Evnin, Spectroscopy instead of scattering: particle experimentation in AdS spacetime, in 10th high-energy physics international conference in Madagascar, (2018) [arXiv:1812.07132] [INSPIRE].
B. Craps, M. De Clerck and O. Evnin, Time-periodicities in holographic CFTs, JHEP 09 (2021) 030 [arXiv:2103.12798] [INSPIRE].
O. Evnin, Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes, Class. Quant. Grav. 38 (2021) 203001 [arXiv:2104.09797] [INSPIRE].
P.N. Bala Subramanian, Applications of holography, Ph.D. thesis, Indian Inst. Sci., Bangalore, India (2018) [arXiv:1809.05482] [INSPIRE].
J. Zinn-Justin and U.D. Jentschura, Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions, Annals Phys. 313 (2004) 197 [quant-ph/0501136] [INSPIRE].
S. Muller, S. Heusler, P. Braun, F. Haake and A. Altland, Semiclassical foundation of universality in quantum chaos, Phys. Rev. Lett. 93 (2004) 014103 [nlin/0401021] [INSPIRE].
F. Haake, Quantum signatures of chaos, Springer (2010).
G. ’t Hooft, On the quantum structure of a black hole, Nucl. Phys. B 256 (1985) 727 [INSPIRE].
V. Balasubramanian, B. Craps, B. Czech and G. Sárosi, Echoes of chaos from string theory black holes, JHEP 03 (2017) 154 [arXiv:1612.04334] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
E. Witten, Gravity and the crossed product, JHEP 10 (2022) 008 [arXiv:2112.12828] [INSPIRE].
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: 2208.14744
Rights and permissions
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.
About this article
Cite this article
Das, S., Krishnan, C., Kumar, A.P. et al. Synthetic fuzzballs: a linear ramp from black hole normal modes. J. High Energ. Phys. 2023, 153 (2023). https://doi.org/10.1007/JHEP01(2023)153
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2023)153