Abstract
For a long time, it is believed that black hole horizon is thermal and quantum mechanical in nature. The microscopic origin of this thermality is the main question behind our present investigation, which reveals possible importance of near horizon symmetry. It is this symmetry which is assumed to be spontaneously broken by the background spacetime, generates the associated Goldstone modes. In this paper, we construct a suitable classical action for those Goldstone modes and show that all the momentum modes experience nearly the same inverted harmonic potential, leading to an instability. Thanks to the recent conjectures on the chaos and thermal quantum system, particularly in the context of an inverted harmonic oscillator system. Going into the quantum regime, the system of large number of Goldstone modes with the aforementioned instability is shown to be inherently thermal. Interestingly, the temperature of the system also turns out to be proportional to that of the well-known horizon temperature. Therefore, we hope our present study can illuminate an intimate connection between the horizon symmetries and the associated Goldstone modes as a possible mechanism of the microscopic origin of the horizon thermality.
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H.A. Al-Kuwari, M.O. Taha, Noether’s theorem and local gauge invariance. Am. J. Phys. 59, 363 (1991)
D.L. Karatas, K.L. Kowalski, Noether’s theorem for local gauge transformations. Am. J. Phys. 58, 123 (1990)
S. Weinberg, “The quantum theory of fields. Vol. 2: Modern applications,” Cambridge University Press, (2005)
H. Bondi, M.G.J. van der Burg, A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems. Proc. R. Soc. A 269, 21 (1962)
R. Sachs, Asymptotic symmetries in gravitational theory. Phys. Rev. 128, 2851 (1962)
R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times. Proc. R. Soc. A 270, 103 (1962)
E.T. Newman, R. Penrose, Note on the Bondi–Metzner–Sachs group. J. Math. Phys. 7, 863 (1966)
D. Kapec, M. Pate, A. Strominger, New symmetries of QED. Adv. Theor. Math. Phy. 21, 1769 (2017)
T. He, P. Mitra, A.P. Porfyriadis, A. Strominger, New symmetries of massless QED. J. High Energy Phys. 10, 112 (2014)
A. Strominger, Asymptotic symmetries of Yang–Mills theory. J. High Energy Phys. 07, 151 (2014)
C. Troessaert, The BMS4 algebra at spatial infinity. Class. Quantum Gravity 35(7), 074003 (2018)
M. Campiglia, R. Eyheralde, Asymptotic \(U(1)\) charges at spatial infinity. J. High Energy Phys. 11, 168 (2017)
G. Barnich, P.H. Lambert, P. Mao, Three-dimensional asymptotically flat Einstein–Maxwell theory. Class. Quantum Gravity 32, 245001 (2015)
G. Barnich, P.H. Lambert, Einstein–Yang–Mills theory: asymptotic symmetries. Phys. Rev. D 88, 103006 (2013)
R.G. Cai, S.M. Ruan, Y.L. Zhang, Horizon supertranslation and degenerate black hole solutions. J. High Eergy Phys. 09, 163 (2016)
L. Donnay, G. Giribet, H.A. Gonzalez, M. Pino, Supertranslations and superrotations at the black hole horizon. Phys. Rev. Lett. 116, 091101 (2016)
E.T. Akhmedov, M. Godazgar, Symmetries at the black hole horizon. Phys. Rev. D 96, 104025 (2017)
V. Iyer, R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D 50, 846 (1994)
J.D. Brown, M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity. Commun. Math. Phys. 104, 207 (1986)
A. Strominger, Black hole entropy from near horizon microstates. J. High Energy Phys. 02, 009 (1998)
S. Carlip, Entropy from conformal field theory at Killing horizons. Class. Quantum Gravity 16, 3327 (1999)
S. Carlip, Black hole entropy from conformal field theory in any dimension. Phys. Rev. Lett. 82, 2828 (1999)
B.R. Majhi, T. Padmanabhan, Noether current, horizon virasoro algebra and entropy. Phys. Rev. D 85, 084040 (2012)
B.R. Majhi, T. Padmanabhan, Noether current from the surface term of gravitational action, Virasoro algebra and horizon entropy. Phys. Rev. D 86, 101501 (2012)
B.R. Majhi, Noether current of the surface term of Einstein-Hilbert action, Virasoro algebra and entropy. Adv. High Energy Phys. 2013, 386342 (2013)
B.R. Majhi, S. Chakraborty, Anomalous effective action, noether current, virasoro algebra and horizon entropy. Eur. Phys. J. C 74, 2867 (2014)
B.R. Majhi, Conformal transformation, near horizon symmetry, virasoro algebra and entropy. Phys. Rev. D 90, 044020 (2014)
B.R. Majhi, Near horizon hidden symmetry and entropy of Sultana–Dyer black hole: a time dependent case. Phys. Rev. D 92, 064026 (2015)
S. Chakraborty, K. Parattu, T. Padmanabhan, Gravitational field equations near an arbitrary null surface expressed as a thermodynamic identity. J. High Energy Phys. 10, 097 (2015)
B.R. Majhi, Noncommutativity in near horizon symmetries in gravity. Phys. Rev. D 95(4), 044020 (2017). arXiv:1701.07952 [gr-qc]
K. Bhattacharya, B.R. Majhi, Noncommutative Heisenberg algebra in the neighbourhood of a generic null surface. Nucl. Phys. B 934, 557 (2018). [arXiv:1802.02862 [gr-qc]]
H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez, D. Tempo, R. Troncoso, Soft Heisenberg hair on black holes in three dimensions. Phys. Rev. D 93, 101503 (2016)
D. Grumiller, A. Perez, S. Prohazka, D. Tempo, R. Troncoso, Higher spin black holes with soft hair. J. High Energy Phys. 10, 119 (2016)
M.M. Sheikh-Jabbari, H. Yavartanoo, Horizon fluffs: near horizon soft hairs as microstates of generic AdS3 black holes. Phys. Rev. D 95, 044007 (2017)
D. Grumiller, M.M. Sheikh-Jabbari, Membrane paradigm from near horizon soft hair. Int. J. Mod. Phys. D 27(14), 1847006 (2018)
K. Hajian, M.M. Sheikh-Jabbari, H. Yavartanoo, Extreme Kerr black hole microstates with horizon fluff. Phys. Rev. D 98(2), 026025 (2018)
M.R. Setare, H. Adami, Horizon fluffs: in the context of generalized minimal massive gravity. EPL 121(4), 41001 (2018)
S. Weinberg, Infrared photons and gravitons. Phys. Rev. 140, B516 (1965)
T. He, V. Lysov, P. Mitra, A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem. JHEP 1505, 151 (2015)
A. Strominger, On BMS invariance of gravitational scattering. JHEP 1407, 152 (2014)
A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” arXiv:1703.05448 [hep-th]
M. Campiglia, A. Laddha, New symmetries for the gravitational S-matrix. JHEP 1504, 076 (2015)
M. Campiglia, A. Laddha, Asymptotic symmetries of QED and Weinberg’s soft photon theorem. JHEP 1507, 115 (2015)
A. Ashtekar, M. Campiglia, A. Laddha, Null infinity, the BMS group and infrared issues. Gen. Relat. Gravit. 50(11), 140 (2018)
A. Strominger, A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems. JHEP 1601, 086 (2016)
L. Donnay, G. Giribet, H.A. González, A. Puhm, Black hole memory effect. Phys. Rev. D 98, 124016 (2018)
J. Koga, Asymptotic symmetries on Killing horizons. Phys. Rev. D 64, 124012 (2001)
M.Z. Iofa, Near-horizon symmetries of the Schwarzschild black holes with supertranslation field. Phys. Rev. D 99(6), 064052 (2019)
G. Dvali, C. Gomez, Black hole’s quantum N-portrait. Fortsch. Phys. 61, 742 (2013)
G. Dvali, A. Franca, C. Gomez, N. Wintergerst, Nambu–Goldstone effective theory of information at quantum criticality. Phys. Rev. D 92(12), 125002 (2015)
A. Averin, G. Dvali, C. Gomez, D. Lust, Gravitational black hole hair from event horizon supertranslations. JHEP 1606, 088 (2016)
A. Averin, G. Dvali, C. Gomez, D. Lust, Goldstone origin of black hole hair from supertranslations and criticality. Mod. Phys. Lett. A 31, 1630045 (2016)
G. Dvali, C. Gomez, D. Lüst, Classical limit of black hole quantum N-portrait and BMS symmetry. Phys. Lett. B 753, 173 (2016)
C. Eling, Y. Oz, On the membrane paradigm and spontaneous breaking of horizon BMS symmetries. J. High Energy Phys. 07, 065 (2016)
C. Eling, Spontaneously broken asymptotic symmetries and an effective action for horizon dynamics. JHEP 1702, 052 (2017). https://doi.org/10.1007/JHEP02(2017)052
S.W. Hawking, M.J. Perry, A. Strominger, Soft hair on black holes. Phys. Rev. Lett. 116, 231301 (2016)
S.W. Hawking, M.J. Perry, A. Strominger, Superrotation charge and supertranslation hair on black holes. J. High Energy Phys. 05, 161 (2017)
M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, M.A., 1995). Chap 11
M. Maitra, D. Maity, B.R. Majhi, Symmetries near a generic charged null surface and associated algebra: an off-shell analysis. Phys. Rev. D 97(12), 124065 (2018)
L. Donnay, G. Giribet, H.A. González, M. Pino, Extended symmetries at the black hole horizon. JHEP 1609, 100 (2016). https://doi.org/10.1007/JHEP09(2016)100
C. Bunster, M. Henneaux, A. Perez, D. Tempo, R. Troncoso, Generalized black holes in three-dimensional spacetime. JHEP 1405, 031 (2014). https://doi.org/10.1007/JHEP05(2014)031
A. Pérez, D. Tempo, R. Troncoso, Boundary conditions for general relativity on AdS\(_{3}\) and the KdV hierarchy. JHEP 1606, 103 (2016). https://doi.org/10.1007/JHEP06(2016)103
T. Morita, Semi-classical bound on Lyapunov exponent and acoustic Hawking radiation in \(c=1\) matrix model. arXiv:1801.00967 [hep-th]
T. Morita, Thermal emission from semi-classical dynamical systems. Phys. Rev. Lett. 122(10), 101603 (2019)
J. Maldacena, S.H. Shenker, D. Stanford, A bound on chaos. JHEP 1608, 106 (2016)
J. Kurchan, Quantum bound to chaos and the semiclassical limit. arXiv:1612.01278 [cond-mat.stat-mech]
G. Barton, Quantum mechanics of the inverted oscillator potential. Ann. Phys. 166, 322 (1986). https://doi.org/10.1016/0003-4916(86)90142-9
K. Hashimoto, N. Tanahashi, Universality in chaos of particle motion near black hole horizon. Phys. Rev. D 95(2), 024007 (2017). arXiv:1610.06070 [hep-th]
L. Bombelli, E. Calzetta, Chaos around a black hole. Class. Quantum Gravity 9, 2573 (1992)
S. Dalui, B.R. Majhi, P. Mishra, Presence of horizon makes particle motion chaotic. Phys. Lett. B 788, 486 (2019). [arXiv:1803.06527 [gr-qc]]
S. Dalui, B. R. Majhi and P. Mishra, Role of acceleration in inducing chaotic fluctuations in particle dynamics. arXiv:1904.11760 [gr-qc]
S. Dalui, B. R. Majhi and P. Mishra, Horizon induces instability and creates quantum thermality. arXiv:1910.07989 [gr-qc]
W.G. Unruh, Notes on black hole evaporation. Phys. Rev. D 14, 870 (1976)
S.W. Hawking, Black hole explosions. Nature 248, 30 (1974)
T. Padmanabhan, Equipartition energy, Noether energy and boundary term in gravitational action. Gen. Relat. Gravit. 44, 2681 (2012). [arXiv:1205.5683 [gr-qc]]
B.R. Majhi, T. Padmanabhan, Thermality and Heat Content of horizons from infinitesimal coordinate transformations. Eur. Phys. J. C 73(12), 2651 (2013). arXiv:1302.1206 [gr-qc]
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Appendices
Appendix A: Near horizon analysis of Schwarzschild black hole
As mentioned in the main text, in this section we will argue that same dynamical equations and solution for the Goldstone modes can be obtained starting from the near horizon metric of the Schwarzschild black hole.
Now, we can easily check that the action constructed from the near horizon metric will contain three types of terms. The ones which are independent and linear in F, can be traced back from their origin which can be transmitted to the fact that the near horizon geometry of the Schwarzschild black hole is Rindler times a sphere, and it does not satisfy the background Einstein’s equation. We, therefore, ignore those terms as they can also be made total derivative. Non-trivial dynamics of the Goldstone modes are attributed to second-order term in F in the action, and it can be easily checked that those terms are exactly the same as in (39) up to a total derivative. As a result with a proper prescription, full spacetime geometry as well as near horizon geometry of the Schwarzschild background are giving rise to the same Goldstone mode dynamics.
Appendix B: Surface Hamiltonian and heat content
In the main text, we have constructed the GHY boundary term in the action formulation which did not contribute in the dynamics of the Goldstone mode. However, an important analysis is left to be discussed there. It is well-known that the boundary term of the Einstein–Hilbert action in gravitational theory leads to surface Hamiltonian which is directly related to the heat content of the Horizon (detail discussion is given in ([75, 76])). The expression of the surface Hamiltonian comes out to be the product of temperature and entropy of the horizon. Keeping this in mind, we can write surface Hamiltonian corresponding to the GHY boundary term (18):
Now, substituting the solution (23) in the expression (18) and integrating the boundary term , the Hamiltonian (55) comes out:
\( \bar{A}\) denotes the transverse area of the Rindler horizon. In the second part function \(f_3(\alpha )\) comes from the first and higher order time derivative of F in the boundary term of the action. Whereas \(f_2 (\alpha )\) denotes all the square, cubic and higher order terms in the expression. Now, near horizon (where \(v \rightarrow -\infty \)) the second terms in the above expression vanish. Hence, the result comes out as:
where \(S = \bar{A} /4 G \) and \(T = \alpha /2\pi \) are the Horizon entropy and temperature, respectively. The result clearly indicates that irrespective of Goldstone modes \(F_{m,n}\), the GHY boundary term in the action is related to the heat content of the horizon. Similar conclusion can be drawn for Schwarzschild black hole horizon also.
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Maitra, M., Maity, D. & Majhi, B.R. Near horizon symmetries, emergence of Goldstone modes and thermality. Eur. Phys. J. Plus 135, 483 (2020). https://doi.org/10.1140/epjp/s13360-020-00451-3
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DOI: https://doi.org/10.1140/epjp/s13360-020-00451-3