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Page Curves of Reissner-Nordström Black Hole in HD Gravity

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

Modified Gravity Theories From the Barrow Hypothesis: Conclusions and References

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 Abstract 1 Introduction Black holes are intriguing entities for several reasons. Hawking’s discovery that black holes have a radiation spectrum comparable to that of a black body gives them a perfect laboratory for studying the interactions between quantum mechanics, gravity, and thermodynamics. The concept of Bekenstein-Hawking entropy, often known as black hole entropy, has gained popularity in recent decades. The exploration of black hole entropy is essential for gaining insights into the intricate microscopic aspects of quantum gravity. The renowned Bekenstein–Hawking entropy, which is proportional to the area of the horizon, represents a foundational expression [1,2] in this realm. However, within the broader context of quantum gravity, this entropy is regarded merely as the tree-level outcome of black hole entropy, with subsequent refinements revealing a leading-order quantum gravitational correction in the form of a logarithmic term. This logarithmic correction is deemed fundamental and universal due to its derivation from diverse methods. These approaches encompass conical singularity and entanglement entropy [2,3], Euclidean action method [4–7], conformal anomaly [8], Cardy formula [9], quantum tunneling [10, 11], and quantum geometry [12]. This ubiquity underscores the significance of the logarithmic correction in our comprehension of black hole entropy. The exploration of black hole entropy is fundamental to grasp the intricate nature of the microscopic constituents within the framework of quantum gravity. One of the most renowned expressions in this realm is the Bekenstein-Hawking entropy, denoted as where A is the event horizon area of the black hole. This formula represents a pivotal link between the geometric properties of black holes, such as their event horizon area, and the underlying quantum structure, providing invaluable insights into the nature of spacetime at the smallest scales. The Bekenstein-Hawking entropy differs from conventional thermodynamics in that it is proportional to the black hole horizon area instead of the volume. In classical thermodynamics, a system entropy is proportional to its mass and volume, indicating an extensive and additive quantity. However, the cause for black hole entropy being nonextensive is unknown [13]. Based on the Bekenstein-Hawking entropy, there are several developments of alternative construction based on nonextensive statistics, such as Renyi [14] and Tsallis [15]. Recent studies on entropy include Barrow’s entropy [16], Sharma-Mittal’s [17], and Kaniadakis’ [18, 19]. The first law of thermodynamics links entropy, temperature, internal energy, and heat transport. Changing the definition of entropy can affect these other numbers, as in [20]. As we don’t yet have a complete theory of quantum gravity that works for all energy scales, using effective field theory (EFT) techniques on general relativity gives us a reliable theory of quantum gravity up to an energy scale near the Planck scale [21]. By applying EFT methods, we can recognize certain common aspects of quantum gravity. One of the most fascinating features is the dynamic non-locality of space-time caused by quantum effects when we look at very short distances. Quantum corrections are expected to adjust the black hole metric to match the corrected entropy since it is calculated by the black hole horizon area and event horizon radius. The starting point for the EFT approach is a curvature expansion from which equations of motion can be derived. Previous studies found that the metric of a Schwarzschild black hole acquires quantum corrections at the third level of curvature [22], while the metric of a Reissner-Nordstrom black hole receives modifications already at second order [23]. The authors employed the same strategy to derive a revised geometry and corrected entropy using the Wald formula [24], despite various methods available [25–28]. 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 Abstract 1 Introduction Black holes are intriguing entities for several reasons. Hawking’s discovery that black holes have a radiation spectrum comparable to that of a black body gives them a perfect laboratory for studying the interactions between quantum mechanics, gravity, and thermodynamics. The concept of Bekenstein-Hawking entropy, often known as black hole entropy, has gained popularity in recent decades. The exploration of black hole entropy is essential for gaining insights into the intricate microscopic aspects of quantum gravity. The renowned Bekenstein–Hawking entropy, which is proportional to the area of the horizon, represents a foundational expression [1,2] in this realm. However, within the broader context of quantum gravity, this entropy is regarded merely as the tree-level outcome of black hole entropy, with subsequent refinements revealing a leading-order quantum gravitational correction in the form of a logarithmic term. This logarithmic correction is deemed fundamental and universal due to its derivation from diverse methods. These approaches encompass conical singularity and entanglement entropy [2,3], Euclidean action method [4–7], conformal anomaly [8], Cardy formula [9], quantum tunneling [10, 11], and quantum geometry [12]. This ubiquity underscores the significance of the logarithmic correction in our comprehension of black hole entropy. The exploration of black hole entropy is fundamental to grasp the intricate nature of the microscopic constituents within the framework of quantum gravity. One of the most renowned expressions in this realm is the Bekenstein-Hawking entropy, denoted as where A is the event horizon area of the black hole. This formula represents a pivotal link between the geometric properties of black holes, such as their event horizon area, and the underlying quantum structure, providing invaluable insights into the nature of spacetime at the smallest scales. The Bekenstein-Hawking entropy differs from conventional thermodynamics in that it is proportional to the black hole horizon area instead of the volume. In classical thermodynamics, a system entropy is proportional to its mass and volume, indicating an extensive and additive quantity. However, the cause for black hole entropy being nonextensive is unknown [13]. Based on the Bekenstein-Hawking entropy, there are several developments of alternative construction based on nonextensive statistics, such as Renyi [14] and Tsallis [15]. Recent studies on entropy include Barrow’s entropy [16], Sharma-Mittal’s [17], and Kaniadakis’ [18, 19]. The first law of thermodynamics links entropy, temperature, internal energy, and heat transport. Changing the definition of entropy can affect these other numbers, as in [20]. As we don’t yet have a complete theory of quantum gravity that works for all energy scales, using effective field theory (EFT) techniques on general relativity gives us a reliable theory of quantum gravity up to an energy scale near the Planck scale [21]. By applying EFT methods, we can recognize certain common aspects of quantum gravity. One of the most fascinating features is the dynamic non-locality of space-time caused by quantum effects when we look at very short distances. Quantum corrections are expected to adjust the black hole metric to match the corrected entropy since it is calculated by the black hole horizon area and event horizon radius. The starting point for the EFT approach is a curvature expansion from which equations of motion can be derived. Previous studies found that the metric of a Schwarzschild black hole acquires quantum corrections at the third level of curvature [22], while the metric of a Reissner-Nordstrom black hole receives modifications already at second order [23]. The authors employed the same strategy to derive a revised geometry and corrected entropy using the Wald formula [24], despite various methods available [25–28]. 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

Modified Gravity Theories From the Barrow Hypothesis: Abstract and Introduction

Physical Properties of KIC 9851944: Conclusion, Data Availability, Acknowledgments, and References

Ever feel like your phone knows you better than your best friend? Does your newsfeed seem less like a window to the world and more like some weird fun-house mirror? Mine too. They may even be a little too good. And they’re only going to get better. But here's the secret: you can opt-out. A growing cadre of tech gurus, renegade artists and unstoppable activists are throwing a wrench in the system. Big Tech's algorithms are good. There was no escape. Breathe, eat, and sleep. That was all there was to life in the machine." - Aldous Huxley, Brave New World is not a word commonly used by the masses. The idea of independence feels like a betrayal. Talking about “freedom” sets off triggers. But autonomy is a dynamic force. Today it’s making a global comeback.  If you look closely you can spot it in  tech, art and activism. Autonomy It adapts and evolves with our ever-changing world. It’s a mindset that says “ ” I’m not going to be controlled by outside forces and external interests. Here’s some non-breaking news. The traditional work compact, in which employees traded hard work and loyalty in exchange for job security is shattered. Thought so. How many times have you gone job-hunting in your life? But consider this. boasts . He remixed the for the most recent Super Bowl in Las Vegas. DJ Steve Aoki more than 3 billion music streams NFL on CBS theme Here’s what he said at Gala Music: “In the 10 years I’ve been making music … six albums, and all those advances … in one drop last year in NFTs, I made more money.” - Steve Aoki Want to opt-out and join the revolution? Connect with techs and developers building a . Why give your personal information to some faceless corporation? web where you are in control Learn from artists minting their creations as . For the first time, it’s possible to cut out the middleman and take ownership of your work. Discover how activists bypass the financial gatekeepers. Now you can use crowdfunding to fuel your movements directly from the people. NFTs Keep reading to see how the coolest tools and platforms are putting you back in control. Let’s get this show on the road. Get ready for an empowered future. The Rise of Autonomy Early in my entrepreneurial journey, a millionaire mentor gave me some advice on autonomy. He said, “ How often are your decisions based on your own values and goals? Norm whoever controls your time controls your income. I suggest it be you who does it.” The concept drove my parents. “Get an education. Get a good job. Retire there.” In their day people often stayed with the same company for their entire career. They were even called “a company man” or “company gal.” Life goals consisted of building deep connections with colleagues and a strong sense of pride in your work. company loyalty https://www.youtube.com/watch?v=a4F9u8Z84YI&embedable=true Back then companies invested in their employees through training programs and full benefits. When I started in sales my big tech employer had a “no-layoff” policy (it ended while I was there). The global creator economy is projected to reach $104.2 billion by 2028, reflecting the increasing number of individuals leveraging online platforms to express themselves creatively and generate income independently. - Creator Economy Over the course of several decades, cracks began to appear in the model. Tech advancements led to increased job insecurity. One of the first job killers I remember was voicemail. Yep. It eliminated the need for receptionists to answer phones, take messages, and write them down on those little pink slips for salespeople returning to the office. Wage freezes and layoffs increased and the "good job" became less common. As the internet took hold in the 90’s globalization became sexy. The social contract between employer and employee collapsed, Companies chased cheap labor and maximum profits for shareholders -  wherever they could find them. But in public, they sought to maintain the illusion of the Golden Age. Their corporate PR departments were quite busy. Today despite the perception of freedom your autonomy is limited. own social media networks. They control the search engines and video production platforms. Their TOS and centralization secure their dominance. Maximize profits for the few. Sound familiar? This traditional model locks out millions of creators worldwide. Big tech giants “The goal of most people for the last half century was to get the good job that paid well with benefits..” - Anthony Spadafore, Director Pathfinders Autonomy Success Story My friend Sarah is a graphic designer with six years of experience. During the COVID-19 pandemic, she lost her job at a big ad agency. After chasing hundreds of job posts she decided to be her own boss. Now she works remotely as a freelance designer. Instead of being assigned clients, she sets her own schedule. Her boring projects got replaced by gigs that aligned with her interests. She earns more on her own than she did going into the office. And now she finds personal meaning in her work. The Back-to-Office Push and the Autonomy Challenge A remote work revolution is sweeping around the world. Set free from the confines of the physical office, employees are discovering a newfound sense of autonomy. Do you thrive in an environment tailored to your work style? Can you manage your schedule and responsibilities better with more flexibility? For most of us, the answer is YES! This newfound control increases productivity and boosts morale. Many companies are actively encouraging, some even demanding, a return to “in-office” work. Their  push-back against remote work poses a challenge to employee autonomy. But big business CEOs, government bureaucrats, and even Elon Musk don’t like it. Here's why: :  Remote work helps employees manage their work schedules around personal needs and responsibilities. If your productivity is solid and profits are up why poke the bear? A forced return to the office disrupts this balance and increases stress. Goodbye Flexibility, Hello Frustration : It’s the time and money stupid. We all have 24 hours in our day. Having to commute to the office, pay to park, and upgrade your wardrobe eats into personal time and monthly budgets. The impact on working parents or people with long commutes is huge. The Commute Crisis : Most employees I speak with enjoy the flexibility and convenience of working from home. Some never had this opportunity before COVID-19. Now they realize they can do their job more conveniently. Forcing them back into the office feels like a slap in the face. Morale Killer I think companies that are inflexible with remote work options will lose valuable talent. The job market is competitive. The creator economy allows people like my friend Sarah to level up their skills and create jobs for themselves. The pull of autonomy is strong. Collaboration: Tech, Art and Activism Disruption accelerates when tech, art and activism come together. From interactive installations to digital art platforms, technology has enabled artists to reach wider audiences. You can also spark important conversations about social issues and reform. Techpreneurs are launching tools that empower artists and activists. Modern artists are wielding technology to create art that raises awareness for social causes. Activist groups are reaching a global audience with every social media post. This is the future we're building. A world where creative expression meets social justice. And the power of technology is at the core. - Norm Bond Tech Creators Build Worlds Beyond Reality Tech developers and creators are building alternative platforms to traditional activities. Can you imagine stepping into a painting instead of just looking at it in a museum? You can with . Van Gogh Exhibit: The Immersive Experience The exhibition has been ! This blend of art and cutting-edge technology ranks among the 12 best immersive experiences in the world by CNN. touring since 2017 with +5,000,000 visitors Take a walk alongside Van Gogh during a visually rich journey. It’s a 20,000 square feet sound and light spectacular. The two-story projections leave visitors in awe. You discover the inspiration behind eight of Van Gogh’s iconic works. is one of my favorites. The Starry Night Artists Go Indie with Tech Blockchain technology provides innovative tools for artists to monetize their work through smart contracts, NFTs and crypto. This strategy allows creators to sell their work directly to collectors . Artists are creating digital art and minting it as NFTs. without the need for traditional galleries or intermediaries London's glitchy artist , averaging over $23,000 per sale and topping the charts. His work captivates collectors and redefines the limits of digital art. His signature style is a mesmerizing blend of flashing visuals and dark themes (his website includes a flashing media warning). XCOPY is minting millions with his dystopian NFTs Here’s the thing. XCOPY’s work isn't just visually arresting. Sure there are lots of flashy NFTs and dystopian styles. But his NFT’s spark conversations. He challenges the status quo in the art world. He’s using his creativity to provide a unique commentary on our digital age. . Is a sign of a future where artists control their destinies, or a blip in a system that still favors the powerful? Don’t just wait to see what happens. Take action on YOUR dream. Telling your own story and controlling distribution is an expression of autonomy XCOPY How Activists Use Tech to Win The murder of George Floyd was recorded and spread virally across social media. It ignited Black Lives Matter like a rocket. This wasn't just a tragic event. It was a spark that . The case study reveals how activists can use social media to amplify their message. Consider this. lit the fuse of a movement already simmering online In the month Floyd’s murder, was tweeted more than 400 thousand times. But in the month , the . before #BlackLivesMatter after hashtag spiked and was shared more than 100 million times Nordstrom posted #BlackLivesMatter to Facebook and received more than 5-million likes. The from the brand. engagement was nearly 90 times the level of typical posts Colleges and universities joined in. More than 107 shared . Healthcare systems, nonprofits, corporations, and sports teams also climbed on the #BlackLivesMatter juggernaut. #BlackLivesMatter The entire internet was ablaze with social posts, videos, CEO statements, and brand platitudes to support this trending social movement. It shook the system. The Future of Autonomy The dream of dumping the 9-to-5 and living life on your terms is more than just a hashtag. It’s a call to action. One fueled by a deep desire for autonomy – the freedom to create, work, and live by your own rules. Don’t let it just be a pipe dream. The total value locked (TVL) in DeFi protocols surpassed $300 billion in 2023, demonstrating the growing interest in alternative financial systems that promote individual control over assets and transactions. - DeFi Pulse Revolutionary art movements, tech-driven innovation, and modern activism are converging. This shift provides a path to an exciting future. Break free from traditional structures. Now is your chance to build something empowering. It's a world where you can set the pace, call the shots, and shape your days around your passions. There are lots of challenges ahead. Working together is often necessary for maximum impact. Finding the right balance between maintaining individual artistic vision and working within a collective - can be difficult. But let's keep it real. Breaking free from traditional structures means leaving the financial security that these systems can provide. Artists, activists and techs still have bills to pay. Do you have alternative means of funding your work in place? With the rise of new technologies, there’s a need to ensure that ethical practices are followed. You need to address concerns regarding data privacy. There are questions regarding intellectual property rights to be sorted. The environmental impact of tech-driven projects like crypto and NFTs are a dealbreaker for some people. These challenges will have to be met to increase adoption and scale. They require innovative solutions, resilience and funding. To build a more equitable and creative future demands continuous learning and adaptation. Conclusion We’ve covered a lot. You know the journey toward autonomy is an exciting yet challenging one. This trend in tech, art and activism is attracting more people looking for a way out. The digital age provides new tools to take control of your life. It has never been easier to express yourself and reach a target audience. Artists can leverage blockchain technology. Tech creators can build alternative platforms. Activists can fuel movements without traditional limits. Working collectively can exponentially grow results for both individuals and organizations. But the road ahead is not all roses and sunshine. Challenges include navigating concerns about the environment. Ethical considerations riddle the increased use of AI and ML. Building tools that engage marginalized communities is critical in a market-based economy. Long-established traditional systems are not going to just roll over and embrace change. So take the long view if you want to embrace this autonomy trend. If you’re an artist, technologist or activist it’s time to redefine the possible. I’m in. Let’s create and cause the change you want to see. right here on HN to connect. Visit my profile

Breaking Rules: Autonomy in Tech, Art and Activism

নিয়ম ভাঙা: প্রযুক্তি, শিল্প এবং সক্রিয়তায় স্বায়ত্তশাসন

Regeln brechen: Autonomie in Technologie, Kunst und Aktivismus

Rompiendo reglas: autonomía en tecnología, arte y activismo

Briser les règles : autonomie dans la technologie, l'art et l'activisme

नियम तोड़ना: तकनीक, कला और सक्रियता में स्वायत्तता

ルールの破り: テクノロジー、アート、アクティビズムにおける自律性

규칙 위반: 기술, 예술 및 행동주의의 자율성

Quebrando regras: autonomia em tecnologia, arte e ativismo

Нарушение правил: автономия в технологиях, искусстве и активизме

Kuralları Çiğnemek: Teknolojide, Sanatta ve Aktivizmde Özerklik

Phá vỡ các quy tắc: Quyền tự chủ trong công nghệ, nghệ thuật và hoạt động

打破规则：科技、艺术和行动主义的自主性

Author(s):
(1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Table of Links Abstract & Introduction Brief Review of Wedge Holography Emerging Multiverse from Wedge Holography Application to Information Paradox Application to Grandfather Paradox Conclusion Acknowledgements and References 6 Conclusion In this work, we propose the existence of a multiverse in the Karch-Randall braneworld using the idea of wedge holography. Multiverse is described in the sense that if we talk about 2n universes, then those will be represented by Karch-Randall branes embedded in the bulk. These branes will contain black holes or not that can be controlled by gravitational action. We studied three cases: • We constructed mutiverse from d-dimensional Karch-Randall branes embedded in AdSd+1 in section 3.1. The geometry of these branes is AdSd. In this case, the multiverse consists of 2n anti de-Sitter branes and all are connected to each other at the defect via transparent boundary conditions. Multiverse consists of AdS branes exists forever once created. • We constructed multiverse from d-dimensional de-Sitter spaces on Karch-Randall branes embedded in (d + 1)-dimensional bulk AdSd+1 in 3.2. Multiverse made up of 2n de-Sitter branes has a short lifetime. All the de-Sitter branes in this setup should be created and annihilated at the same time. Defect CFT is a non-unitary conformal field theory becauseof dS/CFT correspondence. • We also discussed why it is not possible to describe multiverse as a mixture of d-dimensional de-Sitter and anti de-Sitter spacetimes in the same bulk in section 3.3. We can have the multiverse with anti de-Sitter branes (M1) or de-Sitter branes (M2) but not the mixture of the two. Because AdS branes intersect at “time-like” boundary and de-Sitter branes intersect at “space-like” boundary of the bulk AdSd+1. Universes in M1 can communicate with each other, similarly, M2 consists of communicating de-Sitter branes but M1 can’t communicate with M2 As a consistency check of the proposal, we calculated the Page curves of two black holes for n = 2 multiverse. We assumed that black hole and bath systems between −2ρ ≤ r ≤ 2ρ and −ρ ≤ r ≤ ρ. In this case, we found that entanglement entropy contribution from the Hartman-Maldacena surfaces has a linear dependence on time for the AdS and Schwarzschild black holes and it is zero for the de-Sitter black hole, whereas island surfaces contributions are constant. Therefore this reproduces the Page curve. Using this idea, we obtain the Page curve of Schwarzschild de-Sitter black hole and one can also do the same for Reissner-Nordström deSitter black hole. This proposal is helpful in the computation of the Page curve of black holes with multiple horizons from wedge holography. We also discussed the possibility of getting a Page curve of these black holes using two Karch-Randall branes, one as a black hole and the other as a bath. In this case, there will be an issue in defining the island surface and identifying what kind of radiation we are getting. For example, when a Karch-Randall brane consists of black hole and cosmological event horizons, i.e., Schwarzschild de-Sitter black hole on the brane, the observer collecting the radiation will not be able to identify clearly whether it is Hawking radiation or Gibbons-Hawking radiation. We checked our proposal for very simple examples without DGP term on the Karch-Randall branes, but one can also talk about massless gravity by adding the DGP term on the KarchRandall branes [35]. In this case, tensions of the branes will recieve correction from the extra term in (11). Further, we argued that one could resolve the “grandfather paradox” using this setup where all universes communicate via transparent boundary conditions at the interface point. To avoid the paradox, one can travel to another universe where his grandfather is not living, so he can’t kill his grandfather. We have given a qualitative idea to resolve the “grandfather paradox” but detailed analysis requires more research in this direction using wedge holography. This paper is available on arxiv under CC 4.0 license. Author(s): (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Author(s): Author(s): (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Table of Links Abstract & Introduction Abstract & Introduction Brief Review of Wedge Holography Brief Review of Wedge Holography Emerging Multiverse from Wedge Holography Emerging Multiverse from Wedge Holography Application to Information Paradox Application to Information Paradox Application to Grandfather Paradox Application to Grandfather Paradox Conclusion Conclusion Acknowledgements and References Acknowledgements and References 6 Conclusion In this work, we propose the existence of a multiverse in the Karch-Randall braneworld using the idea of wedge holography. Multiverse is described in the sense that if we talk about 2n universes, then those will be represented by Karch-Randall branes embedded in the bulk. These branes will contain black holes or not that can be controlled by gravitational action. We studied three cases: • We constructed mutiverse from d-dimensional Karch-Randall branes embedded in AdSd+1 in section 3.1. The geometry of these branes is AdSd. In this case, the multiverse consists of 2n anti de-Sitter branes and all are connected to each other at the defect via transparent boundary conditions. Multiverse consists of AdS branes exists forever once created. • We constructed multiverse from d-dimensional de-Sitter spaces on Karch-Randall branes embedded in (d + 1)-dimensional bulk AdSd+1 in 3.2. Multiverse made up of 2n de-Sitter branes has a short lifetime. All the de-Sitter branes in this setup should be created and annihilated at the same time. Defect CFT is a non-unitary conformal field theory becauseof dS/CFT correspondence. • We also discussed why it is not possible to describe multiverse as a mixture of d-dimensional de-Sitter and anti de-Sitter spacetimes in the same bulk in section 3.3. We can have the multiverse with anti de-Sitter branes (M1) or de-Sitter branes (M2) but not the mixture of the two. Because AdS branes intersect at “time-like” boundary and de-Sitter branes intersect at “space-like” boundary of the bulk AdSd+1. Universes in M1 can communicate with each other, similarly, M2 consists of communicating de-Sitter branes but M1 can’t communicate with M2 As a consistency check of the proposal, we calculated the Page curves of two black holes for n = 2 multiverse. We assumed that black hole and bath systems between −2ρ ≤ r ≤ 2ρ and −ρ ≤ r ≤ ρ. In this case, we found that entanglement entropy contribution from the Hartman-Maldacena surfaces has a linear dependence on time for the AdS and Schwarzschild black holes and it is zero for the de-Sitter black hole, whereas island surfaces contributions are constant. Therefore this reproduces the Page curve. Using this idea, we obtain the Page curve of Schwarzschild de-Sitter black hole and one can also do the same for Reissner-Nordström deSitter black hole. This proposal is helpful in the computation of the Page curve of black holes with multiple horizons from wedge holography. We also discussed the possibility of getting a Page curve of these black holes using two Karch-Randall branes, one as a black hole and the other as a bath. In this case, there will be an issue in defining the island surface and identifying what kind of radiation we are getting. For example, when a Karch-Randall brane consists of black hole and cosmological event horizons, i.e., Schwarzschild de-Sitter black hole on the brane, the observer collecting the radiation will not be able to identify clearly whether it is Hawking radiation or Gibbons-Hawking radiation. We checked our proposal for very simple examples without DGP term on the Karch-Randall branes, but one can also talk about massless gravity by adding the DGP term on the KarchRandall branes [35]. In this case, tensions of the branes will recieve correction from the extra term in (11). Further, we argued that one could resolve the “grandfather paradox” using this setup where all universes communicate via transparent boundary conditions at the interface point. To avoid the paradox, one can travel to another universe where his grandfather is not living, so he can’t kill his grandfather. We have given a qualitative idea to resolve the “grandfather paradox” but detailed analysis requires more research in this direction using wedge holography. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Multiverse in Karch-Randall Braneworld: Conclusion

কার্চ-র্যান্ডাল ব্রেনওয়ার্ল্ডে মাল্টিভার্স: উপসংহার

Multiversum in Karch-Randall Braneworld: Fazit

Multiverso en Karch-Randall Braneworld: Conclusión

Multivers dans Karch-Randall Braneworld : conclusion

कर्च-रान्डेल ब्रैनवर्ल्ड में मल्टीवर्स: निष्कर्ष

カーチ・ランダル・ブレーンワールドの多元宇宙: 結論

Karch-Randall Braneworld의 다중우주: 결론

Multiverso em Karch-Randall Braneworld: Conclusão

Мультивселенная в мире бран Карча-Рэндалла: заключение

Karch-Randall Braneworld'deki Çoklu Evren: Sonuç

Đa vũ trụ trong Karch-Randall Braneworld: Kết luận

卡奇-兰德尔薄膜世界中的多重宇宙：结论

Authors:
(1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Table of Links Abstract Acknowledgment PART I Chapter 1: Introduction Chapter 2: SU(3) LECs from Type IIA String Theory Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation Chapter 4: Conclusion and Future Outlook PART II Chapter 5: Introduction Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times Chapter 9: Multiverse in Karch-Randall Braneworld Chapter 10: Conclusion and Future outlook APPENDIX A APPENDIX B APPENDIX C Bibliography APPENDIX C C.1 Complex Extrinsic Curvatures and Riemann Tensors We can go from complex Riemann tensors to real Riemann tensors by using the following coordinate transformations: Similarly to calculate the second term is holographic entanglement entropy formula we use the following coordinate transformations: Hence by using equation (C.4) we can obtain the second term appearing in holographic entanglement entropy formula in terms of the real coordinates (t, x) using the coordinate transformations given below: C.2 HM-Like/IS Analytics/Numerics In the present appendix, we describe (i) the angular integrations utilized throughout the computation and the equation describing the area of a Hartman-Maldacena-like surface, and (ii) the process by which we estimate the turning point in the context of Island Surface entanglement entropy. C.2.1 HM-Like Surface Area C.2.2 IS Turning Point C.3 Hartman-Maldacena-like Surface Miscellania We present numerous r dependent functions that appear in the entanglement entropy formula to describe the Hartman-Maldacena-like surface at O(β 0 ) and O(β) in this appendix. We also calculated the equation of motion pertaining to the embedding function associated with the Hartman-Maldacena-like surface. • The r dependent functions appearing in equation (7.57) are given as: • In the case of the existence of higher derivative terms, the following have been used to compute the first and second terms of holographic entanglement entropy computation: and where • To derive the equation of motion associated with embedding t(r), we must compute the derivatives that result from equation (7.56): and where C.4 Island Surface Miscellania We enumerate the many functions that emerge from the entanglement entropy of the island surface at O(β) in this appendix. We additionally calculated the derivatives of the Lagrangian associated with the island surface here, which were utilized to calculate the equation of motion concerning the island surface’s embedding. • The following terms (λ3,4(r)) exist in the Wald entanglement entropy term (first term of holographic entanglement entropy (7.78)) of the island surface: where κλ3 and κλ4 being the numerical factors. • The r dependent functions appeared in the anomaly term (the second term of holographic entanglement entropy (7.78)) are as follows: • For the action associated with the island surface’s embedding (7.79), we obtained: and, and, C.5 Possible Terms Appearing in Holographic Entanglement Entropies Within this appendix, we provided all of the terms that could possibly be obtained by differentiating the Lagrangian of the M-theory dual, including O(R4 ) corrections. We have included all of the words associated with the Hartman-Maldacena-like surface in C.5.1, and all of the available terms of the island surface in C.5.2. C.5.1 Hartman-Maldacena-like Surface • √ −g Associated with the Induced Metric (7.23): The equation (C.30) suggests that in the large-N limit, (iii) and (iv) terms correspond to the most dominant terms. where there is no N dependence in G1(r) and G2(r). The terms (v) and (xii) are the most prominent in the large-N limit among all the terms stated above. where G3(r) and G4(r) have no N dependence. The terms (v) and (xii) are the most dominant in the large-N limit. Since, only first term is non-zero, and hence this contributes to the Lagrangian. For the above equation, we found that all the three non-zero terms scale as N, and hence we kept all the non-zero terms in the Lagrangian. C.5.2 Island Surface • √ −g Associated with the Induced Metric (7.40): In the large N limit, terms (iii) and (iv) turn out to be the most dominant. where G5(r) and G6(r) do not depend on N. We found that, terms (v) and (xii) dominate in comparison to other terms in the large N limit. where G7(r) and G8(r) independent of N and the most dominant terms are (v) and (xii) in the larg-N limit. This paper is available on arxiv under CC 4.0 license. Authors: (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Authors: Authors: (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Table of Links Abstract Abstract Acknowledgment Acknowledgment PART I PART I Chapter 1: Introduction Chapter 1: Introduction Chapter 2: SU(3) LECs from Type IIA String Theory Chapter 2: SU(3) LECs from Type IIA String Theory Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation Chapter 4: Conclusion and Future Outlook Chapter 4: Conclusion and Future Outlook PART II PART II Chapter 5: Introduction Chapter 5: Introduction Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times Chapter 9: Multiverse in Karch-Randall Braneworld Chapter 9: Multiverse in Karch-Randall Braneworld Chapter 10: Conclusion and Future outlook Chapter 10: Conclusion and Future outlook APPENDIX A APPENDIX A APPENDIX B APPENDIX B APPENDIX C APPENDIX C Bibliography Bibliography APPENDIX C C.1 Complex Extrinsic Curvatures and Riemann Tensors We can go from complex Riemann tensors to real Riemann tensors by using the following coordinate transformations: Similarly to calculate the second term is holographic entanglement entropy formula we use the following coordinate transformations: Hence by using equation (C.4) we can obtain the second term appearing in holographic entanglement entropy formula in terms of the real coordinates (t, x) using the coordinate transformations given below: C.2 HM-Like/IS Analytics/Numerics In the present appendix, we describe (i) the angular integrations utilized throughout the computation and the equation describing the area of a Hartman-Maldacena-like surface, and (ii) the process by which we estimate the turning point in the context of Island Surface entanglement entropy. C.2.1 HM-Like Surface Area C.2.2 IS Turning Point C.3 Hartman-Maldacena-like Surface Miscellania We present numerous r dependent functions that appear in the entanglement entropy formula to describe the Hartman-Maldacena-like surface at O(β 0 ) and O(β) in this appendix. We also calculated the equation of motion pertaining to the embedding function associated with the Hartman-Maldacena-like surface. • The r dependent functions appearing in equation (7.57) are given as: • In the case of the existence of higher derivative terms, the following have been used to compute the first and second terms of holographic entanglement entropy computation: and where • To derive the equation of motion associated with embedding t(r), we must compute the derivatives that result from equation (7.56): and where C.4 Island Surface Miscellania We enumerate the many functions that emerge from the entanglement entropy of the island surface at O(β) in this appendix. We additionally calculated the derivatives of the Lagrangian associated with the island surface here, which were utilized to calculate the equation of motion concerning the island surface’s embedding. • The following terms (λ3,4(r)) exist in the Wald entanglement entropy term (first term of holographic entanglement entropy (7.78)) of the island surface: where κλ3 and κλ4 being the numerical factors. • The r dependent functions appeared in the anomaly term (the second term of holographic entanglement entropy (7.78)) are as follows: • For the action associated with the island surface’s embedding (7.79), we obtained: and, and, C.5 Possible Terms Appearing in Holographic Entanglement Entropies Within this appendix, we provided all of the terms that could possibly be obtained by differentiating the Lagrangian of the M-theory dual, including O(R4 ) corrections. We have included all of the words associated with the Hartman-Maldacena-like surface in C.5.1, and all of the available terms of the island surface in C.5.2. C.5.1 Hartman-Maldacena-like Surface • √ −g Associated with the Induced Metric (7.23): Associated with the Induced Metric The equation (C.30) suggests that in the large-N limit, (iii) and (iv) terms correspond to the most dominant terms. where there is no N dependence in G1(r) and G2(r). The terms (v) and (xii) are the most prominent in the large-N limit among all the terms stated above. where G3(r) and G4(r) have no N dependence. The terms (v) and (xii) are the most dominant in the large-N limit. Since, only first term is non-zero, and hence this contributes to the Lagrangian. For the above equation, we found that all the three non-zero terms scale as N, and hence we kept all the non-zero terms in the Lagrangian. C.5.2 Island Surface • √ −g Associated with the Induced Metric (7.40): Associated with the Induced Metric In the large N limit, terms (iii) and (iv) turn out to be the most dominant. where G5(r) and G6(r) do not depend on N. We found that, terms (v) and (xii) dominate in comparison to other terms in the large N limit. where G7(r) and G8(r) independent of N and the most dominant terms are (v) and (xii) in the larg-N limit. This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

Aspects of Thermal QCD Phenomenology at Intermediate Gauge/'t Hooft Coupling: Appendix C

Aspects of Thermal QCD Phenomenology at Intermediate Gauge/'t Hooft Coupling: Bibliography

Authors:
(1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Table of Links Abstract Acknowledgment PART I Chapter 1: Introduction Chapter 2: SU(3) LECs from Type IIA String Theory Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation Chapter 4: Conclusion and Future Outlook PART II Chapter 5: Introduction Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times Chapter 9: Multiverse in Karch-Randall Braneworld Chapter 10: Conclusion and Future outlook APPENDIX A APPENDIX B APPENDIX C Bibliography Similarly, the profile function of the π-meson is given as [2, 83]: We showed that UV valued profile functions are [2, 83]: This paper is available on arxiv under CC 4.0 license. Authors: (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Authors: Authors: (1) Gopal Yadav, Department of Physics, Indian Institute of Technology & Chennai Mathematical Institute. Table of Links Abstract Abstract Acknowledgment Acknowledgment PART I PART I Chapter 1: Introduction Chapter 1: Introduction Chapter 2: SU(3) LECs from Type IIA String Theory Chapter 2: SU(3) LECs from Type IIA String Theory Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation Chapter 3: Deconfinement Phase Transition in Thermal QCD-Like Theories at Intermediate Coupling in the Absence and Presence of Rotation Chapter 4: Conclusion and Future Outlook Chapter 4: Conclusion and Future Outlook PART II PART II Chapter 5: Introduction Chapter 5: Introduction Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity Chapter 6: Page Curves of Reissner-Nordström Black Hole in HD Gravity Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling Chapter 7: Entanglement Entropy and Page Curve from the M-Theory Dual of Thermal QCD Above Tc at Intermediate Coupling Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times Chapter 8: Black Hole Islands in Multi-Event Horizon Space-Times Chapter 9: Multiverse in Karch-Randall Braneworld Chapter 9: Multiverse in Karch-Randall Braneworld Chapter 10: Conclusion and Future outlook Chapter 10: Conclusion and Future outlook APPENDIX A APPENDIX A APPENDIX B APPENDIX B APPENDIX C APPENDIX C Bibliography Bibliography Similarly, the profile function of the π-meson is given as [2, 83]: We showed that UV valued profile functions are [2, 83]: This paper is available on arxiv under CC 4.0 license. This paper is available on arxiv under CC 4.0 license. available on arxiv

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