Skip to main content
Download PDF

Semiclassical \( \mathcal{S} \)-matrix and black hole entropy in dilaton gravity

Journal of High Energy Physics volume 2020, Article number: 142 (2020) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We use complex semiclassical method to compute scattering amplitudes of a point particle in dilaton gravity with a boundary. This model has nonzero minimal black hole mass Mcr. We find that at energies below Mcr the particle trivially scatters off the boundary with unit probability. At higher energies the scattering amplitude is exponentially suppressed. The corresponding semiclassical solution is interpreted as formation of an intermediate black hole decaying into the final-state particle. Relating the suppression of the scattering probability to the number of the intermediate black hole states, we find an expression for the black hole entropy consistent with thermodynamics. In addition, we fix the constant part of the entropy which is left free by the thermodynamic arguments. We rederive this result by modifying the standard Euclidean entropy calculation.

References

  1. S.W. Hawking, Breakdown of Predictability in Gravitational Collapse, Phys. Rev. D 14 (1976) 2460 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  2. D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, Rev. Mod. Phys. 88 (2016) 015002 [arXiv:1409.1231] [INSPIRE].

    ADS  Google Scholar 

  3. S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].

  4. G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, arXiv:1905.08255 [INSPIRE].

  5. 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].

    ADS  MathSciNet  MATH  Google Scholar 

  6. D.N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 (1993) 3743 [hep-th/9306083] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  7. P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 09 (2007) 120 [arXiv:0708.4025] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  8. V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  9. T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].

    ADS  MATH  Google Scholar 

  10. N. Engelhardt and A.C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].

    ADS  Google Scholar 

  11. G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, arXiv:1911.11977 [INSPIRE].

  12. 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].

    ADS  MathSciNet  MATH  Google Scholar 

  13. 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].

    ADS  MathSciNet  MATH  Google Scholar 

  14. F.F. Gautason, L. Schneiderbauer, W. Sybesma and L. Thorlacius, Page Curve for an Evaporating Black Hole, JHEP 05 (2020) 091 [arXiv:2004.00598] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  15. T. Hartman, E. Shaghoulian and A. Strominger, Islands in Asymptotically Flat 2D Gravity, JHEP 07 (2020) 022 [arXiv:2004.13857] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  16. J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  17. A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  18. 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] [INSPIRE].

  19. S.B. Giddings and R.A. Porto, The Gravitational S-matrix, Phys. Rev. D 81 (2010) 025002 [arXiv:0908.0004] [INSPIRE].

    ADS  Google Scholar 

  20. F. Bezrukov, D. Levkov and S. Sibiryakov, Semiclassical S-matrix for black holes, JHEP 12 (2015) 002 [arXiv:1503.07181] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  21. V.A. Berezin, A. Boyarsky and A. Neronov, On the Mechanism of Hawking radiation, Grav. Cosmol. 5 (1999) 16 [gr-qc/0605099] [INSPIRE].

  22. M.K. Parikh and F. Wilczek, Hawking radiation as tunneling, Phys. Rev. Lett. 85 (2000) 5042 [hep-th/9907001] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  23. F.L. Bezrukov and D. Levkov, Dynamical tunneling of bound systems through a potential barrier: complex way to the top, J. Exp. Theor. Phys. 98 (2004) 820 [quant-ph/0312144] [INSPIRE].

  24. D.G. Levkov, A.G. Panin and S.M. Sibiryakov, Unstable Semiclassical Trajectories in Tunneling, Phys. Rev. Lett. 99 (2007) 170407 [arXiv:0707.0433] [INSPIRE].

    ADS  MATH  Google Scholar 

  25. C.G. Callan Jr., S.B. Giddings, J.A. Harvey and A. Strominger, Evanescent black holes, Phys. Rev. D 45 (1992) 1005 [hep-th/9111056] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  26. A. Strominger, Les Houches lectures on black holes, hep-th/9501071 [INSPIRE].

  27. J.G. Russo, L. Susskind and L. Thorlacius, The endpoint of Hawking radiation, Phys. Rev. D 46 (1992) 3444 [hep-th/9206070] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  28. T.D. Chung and H.L. Verlinde, Dynamical moving mirrors and black holes, Nucl. Phys. B 418 (1994) 305 [hep-th/9311007] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  29. A. Strominger and L. Thorlacius, Conformally invariant boundary conditions for dilaton gravity, Phys. Rev. D 50 (1994) 5177 [hep-th/9405084] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  30. S.R. Das and S. Mukherji, Boundary dynamics in dilaton gravity, Mod. Phys. Lett. A 9 (1994) 3105 [hep-th/9407015] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  31. M. Fitkevich, D. Levkov and Y. Zenkevich, Exact solutions and critical chaos in dilaton gravity with a boundary, JHEP 04 (2017) 108 [arXiv:1702.02576] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  32. C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  33. R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].

    ADS  Google Scholar 

  34. D. Cangemi and R. Jackiw, Gauge invariant formulations of lineal gravity, Phys. Rev. Lett. 69 (1992) 233 [hep-th/9203056] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  35. M. Fitkevich, D. Levkov and Y. Zenkevich, Dilaton gravity with a boundary: from unitarity to black hole evaporation, JHEP 06 (2020) 184 [arXiv:2004.13745] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  36. T.M. Fiola, J. Preskill, A. Strominger and S.P. Trivedi, Black hole thermodynamics and information loss in two-dimensions, Phys. Rev. D 50 (1994) 3987 [hep-th/9403137] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  37. R.C. Myers, Black hole entropy in two-dimensions, Phys. Rev. D 50 (1994) 6412 [hep-th/9405162] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  38. J.D. Hayward, Entropy in the RST model, Phys. Rev. D 52 (1995) 2239 [gr-qc/9412065] [INSPIRE].

  39. S.N. Solodukhin, Two-dimensional quantum corrected eternal black hole, Phys. Rev. D 53 (1996) 824 [hep-th/9506206] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  40. G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].

    ADS  Google Scholar 

  41. A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  42. S.P. de Alwis, Quantization of a theory of 2-D dilaton gravity, Phys. Lett. B 289 (1992) 278 [hep-th/9205069] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  43. D. Louis-Martinez and G. Kunstatter, On Birckhoff ’s theorem in 2-D dilaton gravity, Phys. Rev. D 49 (1994) 5227 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  44. V.A. Berezin, V.A. Kuzmin and I.I. Tkachev, Dynamics of Bubbles in General Relativity, Phys. Rev. D 36 (1987) 2919 [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  45. D. Levkov, A. Panin and S. Sibiryakov, Complex trajectories in chaotic dynamical tunneling, Phys. Rev.E 76 (2007) 046209 [nlin/0701063].

  46. D.G. Levkov, A.G. Panin and S.M. Sibiryakov, Signatures of unstable semiclassical trajectories in tunneling, J. Phys. A 42 (2009) 205102 [arXiv:0811.3391] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  47. M.K. Parikh, A secret tunnel through the horizon, Gen. Rel. Grav. 36 (2004) 2419 [hep-th/0405160] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  48. T. Banks, A. Dabholkar, M.R. Douglas and M. O’Loughlin, Are horned particles the climax of Hawking evaporation?, Phys. Rev. D 45 (1992) 3607 [hep-th/9201061] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  49. J.G. Russo, L. Susskind and L. Thorlacius, Black hole evaporation in (1+1)-dimensions, Phys. Lett. B 292 (1992) 13 [hep-th/9201074] [INSPIRE].

    ADS  Google Scholar 

  50. L. Thorlacius, Black hole evolution, Nucl. Phys. B Proc. Suppl. 41 (1995) 245 [hep-th/9411020] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  51. R. Landauer and T. Martin, Barrier interaction time in tunneling, Rev. Mod. Phys. 66 (1994) 217 [INSPIRE].

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, 117312, Russia

    Maxim Fitkevich, Dmitry Levkov & Sergey Sibiryakov

  2. Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Moscow Region, Russia

    Maxim Fitkevich

  3. Institute for Theoretical and Mathematical Physics, MSU, Moscow, 119991, Russia

    Dmitry Levkov

  4. Institute of Physics, LPTP, Ecole Polytechnique Federale de Lausanne, CH-1015, Lausanne, Switzerland

    Sergey Sibiryakov

  5. Theoretical Physics Department, CERN, CH-1211, Geneva 23, Switzerland

    Sergey Sibiryakov

Authors
  1. Maxim Fitkevich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry Levkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sergey Sibiryakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxim Fitkevich.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2006.03606

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fitkevich, M., Levkov, D. & Sibiryakov, S. Semiclassical \( \mathcal{S} \)-matrix and black hole entropy in dilaton gravity. J. High Energ. Phys. 2020, 142 (2020). https://doi.org/10.1007/JHEP08(2020)142

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP08(2020)142

Keywords

  • Black Holes
  • 2D Gravity
  • Models of Quantum Gravity