Semiclassical S $$ \mathcal{S} $$ -matrix and black hole entropy in dilaton gravity
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Springer
Received: June 11, Revised: July 29, Accepted: August 1, Published: August 27,
2020 2020 2020 2020
Maxim Fitkevich,a,b Dmitry Levkova,c and Sergey Sibiryakovd,e,a a
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia b Moscow Institute of Physics and Technology, Dolgoprudny 141700, Moscow Region, Russia c Institute for Theoretical and Mathematical Physics, MSU, Moscow 119991, Russia d Institute of Physics, LPTP, Ecole Polytechnique Federale de Lausanne, CH-1015, Lausanne, Switzerland e Theoretical Physics Department, CERN, CH-1211 Geneva 23, Switzerland
E-mail: [email protected], [email protected], [email protected] 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. Keywords: Black Holes, 2D Gravity, Models of Quantum Gravity ArXiv ePrint: 2006.03606
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP08(2020)142
JHEP08(2020)142
Semiclassical S-matrix and black hole entropy in dilaton gravity
Contents 1 Introduction
1
2 The setup 2.1 Dilaton gravity 2.2 Classical scattering
5 5 6 7 7 8 10
4 Relation to black entropy 4.1 Euclidean calculation of entropy: a puzzle 4.2 Experiments with the thermal gas 4.3 Correcting the Euclidean calculation
12 12 14 15
5 Conclusions
16
A Classical solutions A.1 Birkhoff theorem A.2 Junction conditions and equation of motion for the particle A.3 Boundary condition and reflection law
18 18 19 20
B Regularization method
20
C Computing the action
21
D Constrained instantons for the entropy
25
1
Introduction
Black hole (BH) information paradox [1, 2] has long history ever since the discovery of BH evaporation [3]. Recently there has been a remarkable progress towards its resolution. Within the framework of the AdS/CFT correspondence, refs. [4, 5] performed semiclassical calculations of the entanglement entropy of an evaporating BH and demonstrated that it follows the Page curve [6, 7], consistent with unitarity. To derive the expression for the entanglement entropy these calculations use complex saddle points of the gravitational path integral — replica wormholes [8–12]. It has been suggested that this approach applies also beyond the holographic setting
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