Abstract
In this work we derive general quantum phenomenological equations of gravitational dynamics and analyse its features. The derivation uses the formalism developed in thermodynamics of spacetime and introduces low energy quantum gravity modifications to it. Quantum gravity effects are considered via modification of Bekenstein entropy by an extra logarithmic term in the area. This modification is predicted by several approaches to quantum gravity, including loop quantum gravity, string theory, AdS/CFT correspondence and generalised uncertainty principle phenomenology, giving our result a general character. The derived equations generalise classical equations of motion of unimodular gravity, instead of the ones of general relativity, and they contain at most second derivatives of the metric. We provide two independent derivations of the equations based on thermodynamics of local causal diamonds. First one uses Jacobson's maximal vacuum entanglement hypothesis, the second one Clausius entropy flux. Furthermore, we consider questions of diffeomorphism and local Lorentz invariance of the resulting dynamics and discuss its application to a simple cosmological model, finding a resolution of the classical singularity.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Kempf, Uncertainty relation in quantum mechanics with quantum group symmetry, J. Math. Phys. 35 (1994) 4483 [hep-th/9311147] [INSPIRE].
L.J. Garay, Quantum gravity and minimum length, Int. J. Mod. Phys. A 10 (1995) 145 [gr-qc/9403008] [INSPIRE].
L. Smolin, Four principles for quantum gravity, Fundam. Theor. Phys. 187 (2017) 427 [arXiv:1610.01968] [INSPIRE].
S. Chakraborty, D. Kothawala and A. Pesci, Raychaudhuri equation with zero point length, Phys. Lett. B 797 (2019) 134877 [arXiv:1904.09053] [INSPIRE].
R.J. Adler, P. Chen and D.I. Santiago, The generalized uncertainty principle and black hole remnants, Gen. Rel. Grav. 33 (2001) 2101 [gr-qc/0106080] [INSPIRE].
A. Awad and A.F. Ali, Minimal length, Friedmann equations and maximum density, JHEP 06 (2014) 093 [arXiv:1404.7825] [INSPIRE].
A. Alonso-Serrano, M.P. Dąbrowski and H. Gohar, Generalized uncertainty principle impact onto the black holes information flux and the sparsity of Hawking radiation, Phys. Rev. D 97 (2018) 044029 [arXiv:1801.09660] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
T. Jacobson and R. Parentani, Horizon entropy, Found. Phys. 33 (2003) 323 [gr-qc/0302099] [INSPIRE].
T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].
C. Eling, R. Guedens and T. Jacobson, Non-equilibrium thermodynamics of spacetime, Phys. Rev. Lett. 96 (2006) 121301 [gr-qc/0602001] [INSPIRE].
T. Padmanabhan, Thermodynamical aspects of gravity: new insights, Rept. Prog. Phys. 73 (2010) 046901 [arXiv:0911.5004] [INSPIRE].
G. Chirco and S. Liberati, Non-equilibrium thermodynamics of spacetime: the role of gravitational dissipation, Phys. Rev. D 81 (2010) 024016 [arXiv:0909.4194] [INSPIRE].
R. Guedens, T. Jacobson and S. Sarkar, Horizon entropy and higher curvature equations of state, Phys. Rev. D 85 (2012) 064017 [arXiv:1112.6215] [INSPIRE].
V. Baccetti and M. Visser, Clausius entropy for arbitrary bifurcate null surfaces, Class. Quant. Grav. 31 (2014) 035009 [arXiv:1303.3185] [INSPIRE].
T. Jacobson, Entanglement equilibrium and the Einstein equation, Phys. Rev. Lett. 116 (2016) 201101 [arXiv:1505.04753] [INSPIRE].
M. Parikh and A. Svesko, Einstein’s equations from the stretched future light cone, Phys. Rev. D 98 (2018) 026018 [arXiv:1712.08475] [INSPIRE].
P. Bueno, V.S. Min, A.J. Speranza and M.R. Visser, Entanglement equilibrium for higher order gravity, Phys. Rev. D 95 (2017) 046003 [arXiv:1612.04374] [INSPIRE].
A. Svesko, Equilibrium to Einstein: entanglement , thermodynamics, and gravity, Phys. Rev. D 99 (2019) 086006 [arXiv:1810.12236] [INSPIRE].
A. Alonso-Serrano and M. Liška, New perspective on thermodynamics of spacetime: the emergence of unimodular gravity and the equivalence of entropies, Phys. Rev. D 102 (2020) 104056 [arXiv:2008.04805] [INSPIRE].
R.K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84 (2000) 5255 [gr-qc/0002040] [INSPIRE].
K.A. Meissner, Black hole entropy in loop quantum gravity, Class. Quant. Grav. 21 (2004) 5245 [gr-qc/0407052] [INSPIRE].
A. Sen, Logarithmic corrections to Schwarzschild and other non-extremal black hole entropy in different dimensions, JHEP 04 (2013) 156 [arXiv:1205.0971] [INSPIRE].
S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Logarithmic corrections to N = 4 and N = 8 black hole entropy: a one loop test of quantum gravity, JHEP 11 (2011) 143 [arXiv:1106.0080] [INSPIRE].
S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
R. Fareghbal and P. Karimi, Logarithmic correction to BMSFT entanglement entropy, Eur. Phys. J. C 78 (2018) 267 [arXiv:1709.01804] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].
C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Princeton University Press, Princeton, NJ, U.S.A. (2017).
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
R.M. Wald, The thermodynamics of black holes, Living Rev. Rel. 4 (2001) 6 [gr-qc/9912119] [INSPIRE].
S. Das, S. Shankaranarayanan and S. Sur, Black hole entropy from entanglement: a review, arXiv:0806.0402 [INSPIRE].
T. Jacobson, Black hole entropy and induced gravity, gr-qc/9404039 [INSPIRE].
S.N. Solodukhin, The conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51 (1995) 609 [hep-th/9407001] [INSPIRE].
S. Das, P. Majumdar and R.K. Bhaduri, General logarithmic corrections to black hole entropy, Class. Quant. Grav. 19 (2002) 2355 [hep-th/0111001] [INSPIRE].
R.B. Mann and S.N. Solodukhin, Universality of quantum entropy for extreme black holes, Nucl. Phys. B 523 (1998) 293 [hep-th/9709064] [INSPIRE].
S.N. Solodukhin, Logarithmic terms in entropy of Schwarzschild black holes in higher loops, Phys. Lett. B 802 (2020) 135235 [arXiv:1907.07916] [INSPIRE].
S. Hod, High-order corrections to the entropy and area of quantum black holes, Class. Quant. Grav. 21 (2004) 197 [hep-th/0405235] [INSPIRE].
A.J.M. Medved, A follow-up to ‘does nature abhor a logarithm?’ (and apparently she doesn’t), Class. Quant. Grav. 22 (2005) 5195 [gr-qc/0411065] [INSPIRE].
A.J.M. Medved and E.C. Vagenas, When conceptual worlds collide: the GUP and the BH entropy, Phys. Rev. D 70 (2004) 124021 [hep-th/0411022] [INSPIRE].
T. Jacobson, J.M.M. Senovilla and A.J. Speranza, Area deficits and the Bel-Robinson tensor, Class. Quant. Grav. 35 (2018) 085005 [arXiv:1710.07379] [INSPIRE].
J. Wang, Geometry of small causal diamonds, Phys. Rev. D 100 (2019) 064020 [arXiv:1904.01034] [INSPIRE].
G.W. Gibbons and S.N. Solodukhin, The geometry of small causal diamonds, Phys. Lett. B 649 (2007) 317 [hep-th/0703098] [INSPIRE].
L. Brewin, Riemann normal coordinate expansions using Cadabra, Class. Quant. Grav. 26 (2009) 175017 [arXiv:0903.2087] [INSPIRE].
T. Jacobson and M. Visser, Gravitational thermodynamics of causal diamonds in (A)dS, SciPost Phys. 7 (2019) 079 [arXiv:1812.01596] [INSPIRE].
F. Scardigli, M. Blasone, G. Luciano and R. Casadio, Modified Unruh effect from generalized uncertainty principle, Eur. Phys. J. C 78 (2018) 728 [arXiv:1804.05282] [INSPIRE].
G.G. Luciano and L. Petruzziello, GUP parameter from maximal acceleration, Eur. Phys. J. C 79 (2019) 283 [arXiv:1902.07059] [INSPIRE].
M. Visser, Essential and inessential features of Hawking radiation, Int. J. Mod. Phys. D 12 (2003) 649 [hep-th/0106111] [INSPIRE].
R.M. Wald, Quantum field theory in curved spacetime and black hole thermodynamics, The University of Chicago Press, Chicago, IL, U.S.A. and London, U.K. (1994).
E. Di Casola, S. Liberati and S. Sonego, Nonequivalence of equivalence principles, Am. J. Phys. 83 (2015) 39 [arXiv:1310.7426] [INSPIRE].
S. Ghosh, Quantum gravity effects in geodesic motion and predictions of equivalence principle violation, Class. Quant. Grav. 31 (2014) 025025 [arXiv:1303.1256] [INSPIRE].
V.M. Tkachuk, Galilean and Lorentz transformations in a space with generalized uncertainty principle, Found. Phys. 46 (2016) 1666 [arXiv:1310.6243] [INSPIRE].
S. Pramanik, Implication of the geodesic equation in the generalized uncertainty principle framework, Phys. Rev. D 90 (2014) 024023 [arXiv:1404.2567] [INSPIRE].
R. Casadio and F. Scardigli, Generalized uncertainty principle, classical mechanics, and general relativity, Phys. Lett. B 807 (2020) 135558 [arXiv:2004.04076] [INSPIRE].
E.E. Caianiello, Maximal acceleration as a consequence of Heisenberg’s uncertainty relations, Lett. Nuovo Cim. 41 (1984) 370.
C. Barceló, R. Carballo-Rubio and L.J. Garay, Unimodular gravity and general relativity from graviton self-interactions, Phys. Rev. D 89 (2014) 124019 [arXiv:1401.2941] [INSPIRE].
E. Álvarez and M. Herrero-Valea, Unimodular gravity with external sources, JCAP 01 (2013) 014 [arXiv:1209.6223] [INSPIRE].
R. Carballo-Rubio, Longitudinal diffeomorphisms obstruct the protection of vacuum energy, Phys. Rev. D 91 (2015) 124071 [arXiv:1502.05278] [INSPIRE].
C. Barceló, R. Carballo-Rubio and L.J. Garay, Absence of cosmological constant problem in special relativistic field theory of gravity, Annals Phys. 398 (2018) 9 [arXiv:1406.7713] [INSPIRE].
P. Jiroušek and A. Vikman, New Weyl-invariant vector-tensor theory for the cosmological constant, JCAP 04 (2019) 004 [arXiv:1811.09547] [INSPIRE].
K. Hammer, P. Jirousek and A. Vikman, Axionic cosmological constant, arXiv:2001.03169 [INSPIRE].
M. Henneaux and C. Teitelboim, The cosmological constant and general covariance, Phys. Lett. B 222 (1989) 195 [INSPIRE].
R.M. Wald, General relativity , The University of Chicago Press, Chicago, IL, U.S.A. and London, U.K. (1984).
T. Josset, A. Perez and D. Sudarsky, Dark energy from violation of energy conservation, Phys. Rev. Lett. 118 (2017) 021102 [arXiv:1604.04183] [INSPIRE].
S. Liberati, Tests of Lorentz invariance: a 2013 update, Class. Quant. Grav. 30 (2013) 133001 [arXiv:1304.5795] [INSPIRE].
E. Di Casola, S. Liberati and S. Sonego, Weak equivalence principle for self-gravitating bodies: a sieve for purely metric theories of gravity, Phys. Rev. D 89 (2014) 084053 [arXiv:1401.0030] [INSPIRE].
A. Kegeles and D. Oriti, Generalized conservation laws in non-local field theories, J. Phys. A 49 (2016) 135401 [arXiv:1506.03320] [INSPIRE].
A. Ashtekar and B. Gupt, Generalized effective description of loop quantum cosmology, Phys. Rev. D 92 (2015) 084060 [arXiv:1509.08899] [INSPIRE].
E.P. Verlinde, On the origin of gravity and the laws of Newton, JHEP 04 (2011) 029 [arXiv:1001.0785] [INSPIRE].
J.F. da Rocha-Neto and B.R. Morais, Gravitational pressure, apparent horizon and thermodynamics of FLRW universe in the teleparallel gravity, Gen. Rel. Grav. 50 (2018) 35 [arXiv:1802.06062] [INSPIRE].
B. Majumder, The effects of minimal length in entropic force approach, Adv. High Energy Phys. 2013 (2013) 296836 [arXiv:1310.1165] [INSPIRE].
Z.-W. Feng, S.-Z. Yang, H.-1. Li and X.-T. Zu, The effects of minimal length, maximal momentum and minimal momentum in entropic force, Adv. High Energy Phys. 2016 (2016) 2341879 [arXiv:1607.04114] [INSPIRE].
S. Kibaroğlu, Generalized entropic gravity from modified Unruh temperature, Int. J. Mod. Phys. A 34 (2019) 1950119 [arXiv:1901.01946] [INSPIRE].
M. Salah, F. Hammad, M. Faizal and A.F. Ali, Non-singular and cyclic universe from the modified CUP, JCAP 02 (2017) 035 [arXiv:1608.00560] [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: 2009.03826
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
Alonso-Serrano, A., Liška, M. Quantum phenomenological gravitational dynamics: a general view from thermodynamics of spacetime. J. High Energ. Phys. 2020, 196 (2020). https://doi.org/10.1007/JHEP12(2020)196
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)196