Authors:
(1) Ankit Anand, Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India;
(2) Ruben Campos Delgado, Bethe Center for Theoretical Physics, Physikalisches Institut der Universit¨at Bonn, Nussallee 12, 53115 Bonn, Germany. Table of Links Abstract and 1 Introduction 2 Modified Einstein equations with the Jacobson’s approach 3 Quantum gravitational corrections to the Schwarzschild metric 4 Conclusions and References 4 Conclusions The Barrow hypothesis posits a fractal structure at the black hole horizon which modifies the form of the Bekenstein-Hawking entropy area law according to a parameter ∆. If ∆ is constant, then the Barrow hypothesis does not give any substantial modifications to general relativity. In this paper we showed that the assumption of ∆ as function of the radial distance leads to modified gravity theories beyond general relativity. All these theories have a common feature: they modify the classical Schwarzschild black hole metric with quantum gravitational corrections. Interestingly, we found corrections not only to the gtt and grr components, but also to the gθθ and gφφ components. This should not be a surprise, since the central point of the Barrow hypothesis is to replace spherical symmetry with fractal symmetry. The different theories that one can generate are encoded in the choice for I(r) (see (3.8)). However, the specific expression of I(r) remains indeterminate a priori, i.e. there is no best choice for I(r) and one can in principle obtain any form of quantum gravitational corrections. Furthermore, we assumed so far that the Barrow hypothesis is connected to already existing theories, but this does not necessarily have to be true. It may be that the Barrow hypothesis leads to completely new, still unknown quantum corrections. In this regard, an interesting future direction of research would be to compute some black hole thermodynamic quantities like temperature and pressure, and see whether the answer matches with the already available results. A similar calculation with constant ∆ has been recently carried out in [33]. Despite the aforementioned difficulties, we can rightly affirm that the Barrow hypothesis has become a rich and valid framework within the realm of quantum gravity. References [1] J. M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M theory, JHEP 12, 002 (1997) [arXiv:hep-th/9711053 [hep-th]]. [2] S. N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011), 8 [arXiv:1104.3712 [hep-th]]. [3] S. N. Solodukhin, The Conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51, 609-617 (1995) [arXiv:hep-th/9407001 [hep-th]]. [4] D. V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51, 5352-5355 (1995) [arXiv:hep-th/9412161 [hep-th]]. [5] A. Sen, Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions, JHEP 04, 156 (2013) [arXiv:1205.0971 [hep-th]]. [6] B. K. El-Menoufi, Quantum gravity of Kerr-Schild spacetimes and the logarithmic correction to Schwarzschild black hole entropy, JHEP 05 (2016), 035 [arXiv:1511.08816 [hep-th]]. [7] B. K. El-Menoufi, Quantum gravity effects on the thermodynamic stability of 4D Schwarzschild black hole, JHEP 08 (2017), 068 [arXiv:1703.10178 [gr-qc]]. [8] R. G. Cai, L. M. Cao and N. Ohta, Black Holes in Gravity with Conformal Anomaly and Logarithmic Term in Black Hole Entropy, JHEP 04, 082 (2010) [arXiv:0911.4379 [hep-th]]. [9] S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17, 4175-4186 (2000) [arXiv:gr-qc/0005017 [gr-qc]]. [10] R. Banerjee and B. R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 06, 095 (2008) [arXiv:0805.2220 [hep-th]]. [11] Quantum Tunneling, Trace Anomaly and Effective Metric, Phys. Lett. B 674, 218-222 (2009) [arXiv:0808.3688 [hep-th]]. [12] R. K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84, 5255-5257 (2000) [arXiv:gr-qc/0002040 [gr-qc]]. [13] C. Tsallis and L. J. L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73 (2013), 2487 doi:10.1140/epjc/s10052-013-2487-6 [arXiv:1202.2154 [cond-mat.stat-mech]]. [14] A. R´enyi, On measures of information and entropy, [15] C. Tsallis, Possible Generalization of Boltzmann-Gibbs Statistics, J. Statist. Phys. 52 (1988), 479-487 doi:10.1007/BF01016429 [16] J. D. Barrow, The Area of a Rough Black Hole, Phys. Lett. B 808 (2020), 135643 doi:10.1016/j.physletb.2020.135643 [arXiv:2004.09444 [gr-qc]]. [17] A. Sayahian Jahromi, S. A. Moosavi, H. Moradpour, J. P. Morais Gra¸ca, I. P. Lobo, I. G. Salako and A. Jawad, Generalized entropy formalism and a new holographic dark energy model, Phys. Lett. B 780 (2018), 21-24 doi:10.1016/j.physletb.2018.02.052 [arXiv:1802.07722 [gr-qc]]. [18] G. Kaniadakis, Statistical mechanics in the context of special relativity. II., Phys. Rev. E 72 (2005), 036108 doi:10.1103/PhysRevE.72.036108 [arXiv:cond-mat/0507311 [cond-mat]]. [19] N. Drepanou, A. Lymperis, E. N. Saridakis and K. Yesmakhanova, Kaniadakis holographic dark energy and cosmology, Eur. Phys. J. C 82 (2022) no.5, 449 doi:10.1140/epjc/s10052- 022-10415-9 [arXiv:2109.09181 [gr-qc]]. [20] S. Nojiri, S. D. Odintsov and V. Faraoni, Area-law versus R´enyi and Tsallis black hole entropies, Phys. Rev. D 104 (2021) no.8, 084030 doi:10.1103/PhysRevD.104.084030 [arXiv:2109.05315 [gr-qc]]. [21] J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994), 3874-3888 doi:10.1103/PhysRevD.50.3874 [arXiv:gr-qc/9405057 [gr-qc]]. [22] X. Calmet and F. Kuipers, Quantum gravitational corrections to the entropy of a Schwarzschild black hole, Phys. Rev. D 104 (2021) no.6, 066012 doi:10.1103/PhysRevD.104.066012 [arXiv:2108.06824 [hep-th]]. [23] R. C. Delgado, Quantum gravitational corrections to the entropy of a Reissner–Nordstr¨om black hole, Eur. Phys. J. C 82 (2022) no.3, 272 [erratum: Eur. Phys. J. C 83 (2023) no.6, 468] doi:10.1140/epjc/s10052-022-10232-0 [arXiv:2201.08293 [hep-th]]. [24] R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) no.8, R3427-R3431 doi:10.1103/PhysRevD.48.R3427 [arXiv:gr-qc/9307038 [gr-qc]]. [25] P. A. Cano, S. Chimento, R. Linares, T. Ort´ın and P. F. Ramırez, α  corrections of Reissner-Nordstrom black holes, JHEP 02 (2020), 031 doi:10.1007/JHEP02(2020)031 [arXiv:1910.14324 [hep-th]]. [26] M. Yoon, J. Ha and W. Kim, Entropy of Reissner-Nordstrom Black Holes with Minimal Length Revisited, Phys. Rev. D 76 (2007), 047501 doi:10.1103/PhysRevD.76.047501 [arXiv:0706.0364 [gr-qc]]. [27] M. M. Akbar and S. Das, Entropy corrections for Schwarzschild and Reissner-Nordstr¨om black holes, Class. Quant. Grav. 21 (2004), 1383-1392 doi:10.1088/0264-9381/21/6/007 [arXiv:hep-th/0304076 [hep-th]]. [28] J. Sadeghi, B. Pourhassan and F. Rahimi, Logarithmic corrections of charged hairy black holes in (2 + 1) dimensions, Can. J. Phys. 92 (2014) no.12, 1638-1642 doi:10.1139/cjp2014-0229 [arXiv:1708.07383 [gr-qc]]. [29] T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett. 75 (1995), 1260-1263 doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [grqc]]. [30] S. Di Gennaro, H. Xu and Y. C. Ong, How barrow entropy modifies gravity: with comments on Tsallis entropy, Eur. Phys. J. C 82 (2022) no.11, 1066 doi:10.1140/epjc/s10052-022- 11040-2 [arXiv:2207.09271 [gr-qc]]. [31] T. Nutma, xTras : A field-theory inspired xAct package for mathematica, Comput. Phys. Commun. 185 (2014), 1719–1738 doi:10.1016/j.cpc.2014.02.006 [arXiv:1308.3493 [cs.SC]] [32] Y. Xiao and Y. Tian, Logarithmic correction to black hole entropy from the nonlocality of quantum gravity, Phys. Rev. D 105 (2022) no.4, 044013 doi:10.1103/PhysRevD.105.044013 [arXiv:2104.14902 [gr-qc]]. [33] E.M.C. Abreu, Surface gravity analysis in Gauss-Bonnet and Barrow black holes, [arXiv:2403.02540 [gr-qc]]. This paper is available on arxiv under CC BY 4.0 Deed license. Authors: (1) Ankit Anand, Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India; (2) Ruben Campos Delgado, Bethe Center for Theoretical Physics, Physikalisches Institut der Universit¨at Bonn, Nussallee 12, 53115 Bonn, Germany. Authors: Authors: (1) Ankit Anand, Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India; (2) Ruben Campos Delgado, Bethe Center for Theoretical Physics, Physikalisches Institut der Universit¨at Bonn, Nussallee 12, 53115 Bonn, Germany. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Modified Einstein equations with the Jacobson’s approach 2 Modified Einstein equations with the Jacobson’s approach 3 Quantum gravitational corrections to the Schwarzschild metric 3 Quantum gravitational corrections to the Schwarzschild metric 4 Conclusions and References 4 Conclusions and References 4 Conclusions The Barrow hypothesis posits a fractal structure at the black hole horizon which modifies the form of the Bekenstein-Hawking entropy area law according to a parameter ∆. If ∆ is constant, then the Barrow hypothesis does not give any substantial modifications to general relativity. In this paper we showed that the assumption of ∆ as function of the radial distance leads to modified gravity theories beyond general relativity. All these theories have a common feature: they modify the classical Schwarzschild black hole metric with quantum gravitational corrections. Interestingly, we found corrections not only to the gtt and grr components, but also to the gθθ and gφφ components. This should not be a surprise, since the central point of the Barrow hypothesis is to replace spherical symmetry with fractal symmetry. The different theories that one can generate are encoded in the choice for I(r) (see (3.8)). However, the specific expression of I(r) remains indeterminate a priori, i.e. there is no best choice for I(r) and one can in principle obtain any form of quantum gravitational corrections. Furthermore, we assumed so far that the Barrow hypothesis is connected to already existing theories, but this does not necessarily have to be true. It may be that the Barrow hypothesis leads to completely new, still unknown quantum corrections. In this regard, an interesting future direction of research would be to compute some black hole thermodynamic quantities like temperature and pressure, and see whether the answer matches with the already available results. A similar calculation with constant ∆ has been recently carried out in [33]. Despite the aforementioned difficulties, we can rightly affirm that the Barrow hypothesis has become a rich and valid framework within the realm of quantum gravity. References [1] J. M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M theory, JHEP 12, 002 (1997) [arXiv:hep-th/9711053 [hep-th]]. [2] S. N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011), 8 [arXiv:1104.3712 [hep-th]]. [3] S. N. Solodukhin, The Conical singularity and quantum corrections to entropy of black hole, Phys. Rev. D 51, 609-617 (1995) [arXiv:hep-th/9407001 [hep-th]]. [4] D. V. Fursaev, Temperature and entropy of a quantum black hole and conformal anomaly, Phys. Rev. D 51, 5352-5355 (1995) [arXiv:hep-th/9412161 [hep-th]]. [5] A. Sen, Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions, JHEP 04, 156 (2013) [arXiv:1205.0971 [hep-th]]. [6] B. K. El-Menoufi, Quantum gravity of Kerr-Schild spacetimes and the logarithmic correction to Schwarzschild black hole entropy, JHEP 05 (2016), 035 [arXiv:1511.08816 [hep-th]]. [7] B. K. El-Menoufi, Quantum gravity effects on the thermodynamic stability of 4D Schwarzschild black hole, JHEP 08 (2017), 068 [arXiv:1703.10178 [gr-qc]]. [8] R. G. Cai, L. M. Cao and N. Ohta, Black Holes in Gravity with Conformal Anomaly and Logarithmic Term in Black Hole Entropy, JHEP 04, 082 (2010) [arXiv:0911.4379 [hep-th]]. [9] S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17, 4175-4186 (2000) [arXiv:gr-qc/0005017 [gr-qc]]. [10] R. Banerjee and B. R. Majhi, Quantum Tunneling Beyond Semiclassical Approximation, JHEP 06, 095 (2008) [arXiv:0805.2220 [hep-th]]. [11] Quantum Tunneling, Trace Anomaly and Effective Metric, Phys. Lett. B 674, 218-222 (2009) [arXiv:0808.3688 [hep-th]]. [12] R. K. Kaul and P. Majumdar, Logarithmic correction to the Bekenstein-Hawking entropy, Phys. Rev. Lett. 84, 5255-5257 (2000) [arXiv:gr-qc/0002040 [gr-qc]]. [13] C. Tsallis and L. J. L. Cirto, Black hole thermodynamical entropy, Eur. Phys. J. C 73 (2013), 2487 doi:10.1140/epjc/s10052-013-2487-6 [arXiv:1202.2154 [cond-mat.stat-mech]]. [14] A. R´enyi, On measures of information and entropy, [15] C. Tsallis, Possible Generalization of Boltzmann-Gibbs Statistics, J. Statist. Phys. 52 (1988), 479-487 doi:10.1007/BF01016429 [16] J. D. Barrow, The Area of a Rough Black Hole, Phys. Lett. B 808 (2020), 135643 doi:10.1016/j.physletb.2020.135643 [arXiv:2004.09444 [gr-qc]]. [17] A. Sayahian Jahromi, S. A. Moosavi, H. Moradpour, J. P. Morais Gra¸ca, I. P. Lobo, I. G. Salako and A. Jawad, Generalized entropy formalism and a new holographic dark energy model, Phys. Lett. B 780 (2018), 21-24 doi:10.1016/j.physletb.2018.02.052 [arXiv:1802.07722 [gr-qc]]. [18] G. Kaniadakis, Statistical mechanics in the context of special relativity. II., Phys. Rev. E 72 (2005), 036108 doi:10.1103/PhysRevE.72.036108 [arXiv:cond-mat/0507311 [cond-mat]]. [19] N. Drepanou, A. Lymperis, E. N. Saridakis and K. Yesmakhanova, Kaniadakis holographic dark energy and cosmology, Eur. Phys. J. C 82 (2022) no.5, 449 doi:10.1140/epjc/s10052- 022-10415-9 [arXiv:2109.09181 [gr-qc]]. [20] S. Nojiri, S. D. Odintsov and V. Faraoni, Area-law versus R´enyi and Tsallis black hole entropies, Phys. Rev. D 104 (2021) no.8, 084030 doi:10.1103/PhysRevD.104.084030 [arXiv:2109.05315 [gr-qc]]. [21] J. F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994), 3874-3888 doi:10.1103/PhysRevD.50.3874 [arXiv:gr-qc/9405057 [gr-qc]]. [22] X. Calmet and F. Kuipers, Quantum gravitational corrections to the entropy of a Schwarzschild black hole, Phys. Rev. D 104 (2021) no.6, 066012 doi:10.1103/PhysRevD.104.066012 [arXiv:2108.06824 [hep-th]]. [23] R. C. Delgado, Quantum gravitational corrections to the entropy of a Reissner–Nordstr¨om black hole, Eur. Phys. J. C 82 (2022) no.3, 272 [erratum: Eur. Phys. J. C 83 (2023) no.6, 468] doi:10.1140/epjc/s10052-022-10232-0 [arXiv:2201.08293 [hep-th]]. [24] R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) no.8, R3427-R3431 doi:10.1103/PhysRevD.48.R3427 [arXiv:gr-qc/9307038 [gr-qc]]. [25] P. A. Cano, S. Chimento, R. Linares, T. Ort´ın and P. F. Ramırez, α  corrections of Reissner-Nordstrom black holes, JHEP 02 (2020), 031 doi:10.1007/JHEP02(2020)031 [arXiv:1910.14324 [hep-th]]. [26] M. Yoon, J. Ha and W. Kim, Entropy of Reissner-Nordstrom Black Holes with Minimal Length Revisited, Phys. Rev. D 76 (2007), 047501 doi:10.1103/PhysRevD.76.047501 [arXiv:0706.0364 [gr-qc]]. [27] M. M. Akbar and S. Das, Entropy corrections for Schwarzschild and Reissner-Nordstr¨om black holes, Class. Quant. Grav. 21 (2004), 1383-1392 doi:10.1088/0264-9381/21/6/007 [arXiv:hep-th/0304076 [hep-th]]. [28] J. Sadeghi, B. Pourhassan and F. Rahimi, Logarithmic corrections of charged hairy black holes in (2 + 1) dimensions, Can. J. Phys. 92 (2014) no.12, 1638-1642 doi:10.1139/cjp2014-0229 [arXiv:1708.07383 [gr-qc]]. [29] T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett. 75 (1995), 1260-1263 doi:10.1103/PhysRevLett.75.1260 [arXiv:gr-qc/9504004 [grqc]]. [30] S. Di Gennaro, H. Xu and Y. C. Ong, How barrow entropy modifies gravity: with comments on Tsallis entropy, Eur. Phys. J. C 82 (2022) no.11, 1066 doi:10.1140/epjc/s10052-022- 11040-2 [arXiv:2207.09271 [gr-qc]]. [31] T. Nutma, xTras : A field-theory inspired xAct package for mathematica, Comput. Phys. Commun. 185 (2014), 1719–1738 doi:10.1016/j.cpc.2014.02.006 [arXiv:1308.3493 [cs.SC]] [32] Y. Xiao and Y. Tian, Logarithmic correction to black hole entropy from the nonlocality of quantum gravity, Phys. Rev. D 105 (2022) no.4, 044013 doi:10.1103/PhysRevD.105.044013 [arXiv:2104.14902 [gr-qc]]. [33] E.M.C. Abreu, Surface gravity analysis in Gauss-Bonnet and Barrow black holes, [arXiv:2403.02540 [gr-qc]]. This paper is available on arxiv under CC BY 4.0 Deed license. This paper is available on arxiv under CC BY 4.0 Deed license. available on arxiv

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