Abstract
We study a black hole of mass M, enclosed within a spherical box, in equilibrium with its Hawking radiation. We show that the spacetime geometry inside the box is described by the Oppenheimer-Volkoff equations for radiation, except for a thin shell around the horizon. We use the maximum entropy principle to show that the invariant width of the shell is of order \( \sqrt{M} \), its entropy is of order M and its temperature of order \( 1/\sqrt{M} \) (in Planck units). Thus, the width of the shell is much larger than the Planck length. Our approach is to insist on thermodynamic consistency when classical general relativity coexists with the Hawking temperature in the description of a gravitating system. No assumptions about an underlying theory are made and no restrictions are placed on the origins of the new physics near the horizon. We only employ classical general relativity and the principles of thermodynamics. Our result is strengthened by an analysis of the trace anomaly associated to the geometry inside the box, i.e., the regime where quantum field effects become significant correspond to the shells of maximum entropy around the horizon.
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References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
C. Anastopoulos and N. Savvidou, The thermodynamics of self-gravitating systems in equilibrium is holographic, Class. Quant. Grav. 31 (2014) 055003 [arXiv:1302.4407] [INSPIRE].
C. Anastopoulos and N. Savvidou, Entropy of singularities in self-gravitating radiation, Class. Quant. Grav. 29 (2012) 025004 [arXiv:1103.3898] [INSPIRE].
G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [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].
A. Casher, F. Englert, N. Itzhaki, S. Massar and R. Parentani, Black hole horizon fluctuations, Nucl. Phys. B 484 (1997) 419 [hep-th/9606106] [INSPIRE].
P.C.W. Davies, Thermodynamics of black holes, Rept. Prog. Phys. 41 (1978) 1313 [INSPIRE].
J.W. York Jr., Black hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33 (1986) 2092 [INSPIRE].
R.D. Sorkin, R.M. Wald and Z.J. Zhang, Entropy of selfgravitating radiation, Gen. Rel. Grav. 13 (1981) 1127 [INSPIRE].
W.H. Zurek and D.N. Page, Black-hole thermodynamics and singular solutions of the Tolman-Oppenheimer-Volkoff equation, Phys. Rev. D 29 (1984) 628 [arXiv:1511.07051] [INSPIRE].
P.-H. Chavanis, Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes, Astron. Astrophys. 483 (2008) 673 [arXiv:0707.2292] [INSPIRE].
J. Smoller and B. Temple, On the Oppenheimer-Volkoff equations in General Relativity, Arch. Rational Mech. Anal. 142 (1998) 177.
H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, John Wiley, New York U.S.A. (1985).
J. Katz and Y. Manor, Entropy Extremum of Relativistic Selfbound Systems-a Geometric Approach, Phys. Rev. D 12 (1975) 956 [INSPIRE].
D.M. Capper and M.J. Duff, Trace anomalies in dimensional regularization, Nuovo Cim. A 23 (1974) 173.
D.M. Capper and M.J. Duff, Conformal Anomalies and the Renormalizability Problem in Quantum Gravity, Phys. Lett. A 53 (1975) 361 [INSPIRE].
S. Deser, M.J. Duff and C.J. Isham, Nonlocal Conformal Anomalies, Nucl. Phys. B 111 (1976) 45 [INSPIRE].
D.N. Page and C.D. Geilker, Indirect Evidence for Quantum Gravity, Phys. Rev. Lett. 47 (1981) 979 [INSPIRE].
N.G. Phillips and B.L. Hu, Fluctuations of the vacuum energy density of quantum fields in curved space-time via generalized zeta functions, Phys. Rev. D 55 (1997) 6123 [gr-qc/9611012] [INSPIRE].
N.G. Phillips and B.L. Hu, Noise kernel in stochastic gravity and stress energy bitensor of quantum fields in curved space-times, Phys. Rev. D 63 (2001) 104001 [gr-qc/0010019] [INSPIRE].
N.G. Phillips and B.L. Hu, Noise kernel and stress energy bitensor of quantum fields in hot flat space and Gaussian approximation in the optical Schwarzschild metric, Phys. Rev. D 67 (2003) 104002 [gr-qc/0209056] [INSPIRE].
C. Anastopoulos and B.-L. Hu, Probing a Gravitational Cat State, Class. Quant. Grav. 32 (2015) 165022 [arXiv:1504.03103] [INSPIRE].
B.L. Hu and E. Verdaguer, Stochastic Gravity: Theory and Applications, Living Rev. Rel. 11 (2008) 3.
M. Gell-Mann and J.B. Hartle, in Complexity, Entropy and the Physics of Information, W. Zurek eds., Addison Wesley, Reading U.S.A. (1990).
M. Gell-Mann and J.B. Hartle, Classical equations for quantum systems, Phys. Rev. D 47 (1993) 3345 [gr-qc/9210010] [INSPIRE].
J.B. Hartle, Spacetime quantum mechanics and the quantum mechanics of spacetime, in Proceedings on the 1992 Les Houches School, Gravitation and Quantization, Les Houches France (1992), Elsevier, Amsterdam The Netherlands (1993).
T.A. Brun and J.J. Halliwell, Decoherence of hydrodynamic histories: A Simple spin model, Phys. Rev. D 54 (1996) 2899 [quant-ph/9601004] [INSPIRE].
J.J. Halliwell, Decoherent histories and hydrodynamic equations, Phys. Rev. D 58 (1998) 105015 [quant-ph/9805062] [INSPIRE].
J.J. Halliwell, Decoherence of histories and hydrodynamic equations for a linear oscillator chain, Phys. Rev. D 68 (2003) 025018 [INSPIRE].
C. Anastopoulos, Quantum correlation functions and the classical limit, Phys. Rev. D 63 (2001) 125024 [gr-qc/0011111] [INSPIRE].
D.N. Page, Thermal Stress Tensors in Static Einstein Spaces, Phys. Rev. D 25 (1982) 1499 [INSPIRE].
M.R. Brown and A.C. Ottewill, Photon Propagators and the Definition and Approximation of Renormalized Stress Tensors in Curved Space-time, Phys. Rev. D 34 (1986) 1776 [INSPIRE].
V.P. Frolov and A.I. Zel’nikov, Killing Approximation for Vacuum and Thermal Stress-Energy Tensor in Static Space-times, Phys. Rev. D 35 (1987) 3031 [INSPIRE].
P.R. Anderson, W.A. Hiscock and D.A. Samuel, Stress-energy tensor of quantized scalar fields in static spherically symmetric space-times, Phys. Rev. D 51 (1995) 4337 [INSPIRE].
J.D. Bekenstein and L. Parker, Path Integral Evaluation of Feynman Propagator in Curved Space-time, Phys. Rev. D 23 (1981) 2850 [INSPIRE].
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Anastopoulos, C., Savvidou, N. The thermodynamics of a black hole in equilibrium implies the breakdown of Einstein equations on a macroscopic near-horizon shell. J. High Energ. Phys. 2016, 144 (2016). https://doi.org/10.1007/JHEP01(2016)144
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DOI: https://doi.org/10.1007/JHEP01(2016)144