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Gravity

Relativity: The Special and General Theory by Albert Einstein is part of HackerNoon’s Book Blog Post series. You can jump to any chapter in this book here. CHAPTER XIX. THE GRAVITATIONAL FIELD “If we pick up a stone and then let it go, why does it fall to the ground?” The usual answer to this question is: “Because it is attracted by the earth.” Modern physics formulates the answer rather differently for the following reason. As a result of the more careful study of electromagnetic phenomena, we have come to regard action at a distance as a process impossible without the intervention of some intermediary medium. If, for instance, a magnet attracts a piece of iron, we cannot be content to regard this as meaning that the magnet acts directly on the iron through the intermediate empty space, but we are constrained to imagine—after the manner of Faraday—that the magnet always calls into being something physically real in the space around it, that something being what we call a “magnetic field.” In its turn this magnetic field operates on the piece of iron, so that the latter strives to move towards the magnet. We shall not discuss here the justification for this incidental conception, which is indeed a somewhat arbitrary one. We shall only mention that with its aid electromagnetic phenomena can be theoretically represented much more satisfactorily than without it, and this applies particularly to the transmission of electromagnetic waves. The effects of gravitation also are regarded in an analogous manner. The action of the earth on the stone takes place indirectly. The earth produces in its surrounding a gravitational field, which acts on the stone and produces its motion of fall. As we know from experience, the intensity of the action on a body diminishes according to a quite definite law, as we proceed farther and farther away from the earth. From our point of view this means: The law governing the properties of the gravitational field in space must be a perfectly definite one, in order correctly to represent the diminution of gravitational action with the distance from operative bodies. It is something like this: The body (e.g. the earth) produces a field in its immediate neighbourhood directly; the intensity and direction of the field at points farther removed from the body are thence determined by the law which governs the properties in space of the gravitational fields themselves. In contrast to electric and magnetic fields, the gravitational field exhibits a most remarkable property, which is of fundamental importance for what follows. Bodies which are moving under the sole influence of a gravitational field receive an acceleration, which does not in the least depend either on the material or on the physical state of the body. For instance, a piece of lead and a piece of wood fall in exactly the same manner in a gravitational field (in vacuo), when they start off from rest or with the same initial velocity. This law, which holds most accurately, can be expressed in a different form in the light of the following consideration. According to Newton’s law of motion, we have where the “inertial mass” is a characteristic constant of the accelerated body. If now gravitation is the cause of the acceleration, we then have where the “gravitational mass” is likewise a characteristic constant for the body. From these two relations follows: If now, as we find from experience, the acceleration is to be independent of the nature and the condition of the body and always the same for a given gravitational field, then the ratio of the gravitational to the inertial mass must likewise be the same for all bodies. By a suitable choice of units we can thus make this ratio equal to unity. We then have the following law: The gravitational mass of a body is equal to its inertial mass. It is true that this important law had hitherto been recorded in mechanics, but it had not been interpreted. A satisfactory interpretation can be obtained only if we recognise the following fact: The same quality of a body manifests itself according to circumstances as “inertia” or as “weight” (lit. “heaviness”). In the following section we shall show to what extent this is actually the case, and how this question is connected with the general postulate of relativity. About HackerNoon Book Series: We bring you the most important technical, scientific, and insightful public domain books. This book is part of the public domain. Einstein, Albert, 2004. Relativity: The Special and General Theory. Urbana, Illinois: Project Gutenberg. Retrieved May 2022 from https://www.gutenberg.org/files/5001/5001-h/5001-h.htm#ch19 This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org, located at https://www.gutenberg.org/policy/license.html. Photo by Andrew George on Unsplash

The Gravitational Field

Table of Links Abstract and 1. Introduction Clock in Orbit
2.1 Coordinate Time
2.2 Local Frame for the Moon


Clock Rate Differences Between Earth and Moon


Clocks at Earth-Moon Lagrance Points
4.1 Clock at Lagrange point L1
4.2. Clock at Lagrange point L2
4.3. Clock at Lagrange point L4 or L5


Conclusions Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 3: Equations of Motion of Earth and Moon Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Acknowledgments and References 3. CLOCK RATE DIFFERENCES BETWEEN EARTH AND MOON The Earth and the Moon orbit around their mutual center of mass in different Keplerian orbits. Meanwhile, the center of mass of the Earth-Moon system orbits around the Sun in an approximately Keplerian orbit. To calculate the rate differences between clocks on Earth and on the Moon, a fictitious locally freely-falling inertial frame is introduced at the Earth-Moon center of mass. This makes it convenient to calculate the proper times elapsed on moving clocks in terms of Keplerian motions of the Earth and the Moon. The Sun’s contribution is only tidal effects. If we omit the tidal potential of the Sun, the scalar invariant takes a simple form (Appendix 2) Consider a clock fixed on the surface of the rotating geoid of Earth. Since the geoid is a surface of approximate hydrostatic equilibrium, if such clocks are viewed from the local inertial frame they beat at the same rate, which can be evaluated at the equator. The proper time on the Earth-based clock becomes The first two contributions can be identified with the quantity Φ0. The contribution from the Moon’s potential can be well approximated by setting where D is the Earth-Moon distance. The dot product term between velocities will depend on the specific position of the clock and will vary with a daily period; this variation is similar to the corrections to the gravitational potential contribution from the Moon arising from the fact that the clock is not at the center of the Earth. Omitting such contributions give A similar argument applied to a clock fixed on the rotating Moon’s surface of hydrostatic equilibrium gives the proper time The following combination of quantities occurs frequently and can be reduced to a simpler expression The difference of squares of velocities is then Using Eq.(27) in Eq.(18), we obtain This term will have a period of approximately 27 days. Cross terms between velocities of the center of mass and rotational velocities have been omitted. One such term is This contribution will also vary with the rotation period of the Earth. Another such term is Since the Moon is tidally locked to the Earth, its center of mass velocity and the velocity of a clock on the selenoid will be highly correlated. Therefore, this term might give rise to a constant long-term average. Omitting the position-dependent terms, we have used the constants listed in Table I to evaluate the constant contribution and the amplitude of the periodic term. We find: None of the above estimates include tidal effects. This omission is because as a tidal force pushes back and forth on a satellite, two other side effects have to be accounted for. These are a change in the position of the satellite that entails a change in the gravitational potential of the body about which the satellite is orbiting and a change in the velocity of the satellite clock that changes its second-order Doppler shift. The residuals of the gravitational potential and second-order Doppler shift for the Earth-Moon system obtained by subtracting the Keplerian model from that obtained from DE440 are graphed in Fig. 1. Previous work on such problems has shown that these changes are of similar orders of magnitude. Summarizing, there are three contributions to the frequency shift of a clock in a satellite that are of similar orders of magnitude: (1) the perturbing tidal potential itself; (2) the perturbed position that changes the contribution from the main potential; (3) the perturbed velocity that changes the time dilation contribution. Although the perturbing tidal potential can easily be estimated, calculating the other two contributions is more complicated. When the Keplerian model is compared with DE440 ephemerides, the effects of solar tides are plotted in Fig. 2. It might appear that the second-order Doppler contribution to the rate difference depends on the coordinate system used. In the center-of-mass coordinates, a difference of squares of velocities appears, but in a system in which the Earth is at rest, the square of the relative velocity appears. Appendix 4 shows that contributions from the centrifugal potential, which occurs in a rotating coordinate system resolve the apparent discrepancy. Authors:
(1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 (Neil.Ashby@Colorado.Edu);
(2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 (bijunath.patla@nist.gov). This paper is available on arxiv under CC0 1.0 license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Clock in Orbit
2.1 Coordinate Time
2.2 Local Frame for the Moon Clock Rate Differences Between Earth and Moon Clocks at Earth-Moon Lagrance Points
4.1 Clock at Lagrange point L1
4.2. Clock at Lagrange point L2
4.3. Clock at Lagrange point L4 or L5 Conclusions Clock in Orbit 2.1 Coordinate Time 2.2 Local Frame for the Moon Clock in Orbit Clock in Orbit 2.1 Coordinate Time 2.1 Coordinate Time 2.2 Local Frame for the Moon 2.2 Local Frame for the Moon Clock Rate Differences Between Earth and Moon Clock Rate Differences Between Earth and Moon Clock Rate Differences Between Earth and Moon Clocks at Earth-Moon Lagrance Points 4.1 Clock at Lagrange point L1 4.2. Clock at Lagrange point L2 4.3. Clock at Lagrange point L4 or L5 Clocks at Earth-Moon Lagrance Points 4.1 Clock at Lagrange point L1 4.1 Clock at Lagrange point L1 4.2. Clock at Lagrange point L2 4.2. Clock at Lagrange point L2 4.3. Clock at Lagrange point L4 or L5 4.3. Clock at Lagrange point L4 or L5 Conclusions Conclusions Conclusions Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 1: Fermi Coordinates with Origin at the Center of the Moon Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 2: Construction of Freely Falling Center of Mass Frame Appendix 3: Equations of Motion of Earth and Moon Appendix 3: Equations of Motion of Earth and Moon Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Appendix 4: Comparing Results in Rotating and Non-Rotating Coordinate Systems Acknowledgments and References Acknowledgments and References 3. CLOCK RATE DIFFERENCES BETWEEN EARTH AND MOON The Earth and the Moon orbit around their mutual center of mass in different Keplerian orbits. Meanwhile, the center of mass of the Earth-Moon system orbits around the Sun in an approximately Keplerian orbit. To calculate the rate differences between clocks on Earth and on the Moon, a fictitious locally freely-falling inertial frame is introduced at the Earth-Moon center of mass. This makes it convenient to calculate the proper times elapsed on moving clocks in terms of Keplerian motions of the Earth and the Moon. The Sun’s contribution is only tidal effects. If we omit the tidal potential of the Sun, the scalar invariant takes a simple form (Appendix 2) Consider a clock fixed on the surface of the rotating geoid of Earth. Since the geoid is a surface of approximate hydrostatic equilibrium, if such clocks are viewed from the local inertial frame they beat at the same rate, which can be evaluated at the equator. The proper time on the Earth-based clock becomes The first two contributions can be identified with the quantity Φ0. The contribution from the Moon’s potential can be well approximated by setting where D is the Earth-Moon distance. The dot product term between velocities will depend on the specific position of the clock and will vary with a daily period; this variation is similar to the corrections to the gravitational potential contribution from the Moon arising from the fact that the clock is not at the center of the Earth. Omitting such contributions give A similar argument applied to a clock fixed on the rotating Moon’s surface of hydrostatic equilibrium gives the proper time The following combination of quantities occurs frequently and can be reduced to a simpler expression The difference of squares of velocities is then Using Eq.(27) in Eq.(18), we obtain This term will have a period of approximately 27 days. Cross terms between velocities of the center of mass and rotational velocities have been omitted. One such term is This contribution will also vary with the rotation period of the Earth. Another such term is Since the Moon is tidally locked to the Earth, its center of mass velocity and the velocity of a clock on the selenoid will be highly correlated. Therefore, this term might give rise to a constant long-term average. Omitting the position-dependent terms, we have used the constants listed in Table I to evaluate the constant contribution and the amplitude of the periodic term. We find: None of the above estimates include tidal effects. This omission is because as a tidal force pushes back and forth on a satellite, two other side effects have to be accounted for. These are a change in the position of the satellite that entails a change in the gravitational potential of the body about which the satellite is orbiting and a change in the velocity of the satellite clock that changes its second-order Doppler shift. The residuals of the gravitational potential and second-order Doppler shift for the Earth-Moon system obtained by subtracting the Keplerian model from that obtained from DE440 are graphed in Fig. 1. Previous work on such problems has shown that these changes are of similar orders of magnitude. Summarizing, there are three contributions to the frequency shift of a clock in a satellite that are of similar orders of magnitude: (1) the perturbing tidal potential itself; (2) the perturbed position that changes the contribution from the main potential; (3) the perturbed velocity that changes the time dilation contribution. Although the perturbing tidal potential can easily be estimated, calculating the other two contributions is more complicated. When the Keplerian model is compared with DE440 ephemerides, the effects of solar tides are plotted in Fig. 2. It might appear that the second-order Doppler contribution to the rate difference depends on the coordinate system used. In the center-of-mass coordinates, a difference of squares of velocities appears, but in a system in which the Earth is at rest, the square of the relative velocity appears. Appendix 4 shows that contributions from the centrifugal potential, which occurs in a rotating coordinate system resolve the apparent discrepancy. Authors: (1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 (Neil.Ashby@Colorado.Edu); (2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 (bijunath.patla@nist.gov). Authors: Authors: (1) Neil Ashby, National Institute of Standards and Technology, Boulder, CO 80305 ( Neil.Ashby@Colorado.Edu ); Neil.Ashby@Colorado.Edu (2) Bijunath R. Patla, National Institute of Standards and Technology, Boulder, CO 80305 ( bijunath.patla@nist.gov ). bijunath.patla@nist.gov This paper is available on arxiv under CC0 1.0 license. This paper is available on arxiv under CC0 1.0 license. available on arxiv available on arxiv

The Role of Gravity and Motion in Space-Time Coordination

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 3 Quantum gravitational corrections to the Schwarzschild metric In order to solve (2.9), we start with the Ansatz where The Ricci scalar has the structure Case I We compare our solutions with the ones of Xiao and Tian [32], which read With all these terms the final expression for the metric is which generalizes the result of [32]. Notice that the coefficients (3.14) are formally divergent, but the solutions (3.15)-(3.17) are finite. Case II We compare now our solutions with the ones of Calmet and Kuipers, [22], which read With all these terms the final expression for the metric is hich generalizes the result of [22]. Again, notice that the coefficients (3.23) are formally divergent, but the solutions (3.24)-(3.26) are finite. 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 3 Quantum gravitational corrections to the Schwarzschild metric In order to solve (2.9), we start with the Ansatz where The Ricci scalar has the structure Case I We compare our solutions with the ones of Xiao and Tian [32], which read With all these terms the final expression for the metric is which generalizes the result of [32]. Notice that the coefficients (3.14) are formally divergent, but the solutions (3.15)-(3.17) are finite. Case II We compare now our solutions with the ones of Calmet and Kuipers, [22], which read With all these terms the final expression for the metric is hich generalizes the result of [22]. Again, notice that the coefficients (3.23) are formally divergent, but the solutions (3.24)-(3.26) are finite. 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: Quantum 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 2 Modified Einstein equations with the Jacobson’s approach The Barrow hypothesis suggests that the entropy has the form The general solution of (2.5) is where Λ′ is the integration constant. This expression matches the result of [30]. Since our goal is to see what happens when ∆ is a radial function, thus we expand where c is some constant. By doing a series expansion of (2.6), integrating by parts and collecting powers of ǫ, we find The limit ǫ → 0 reduces to general relativity. 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 2 Modified Einstein equations with the Jacobson’s approach The Barrow hypothesis suggests that the entropy has the form The general solution of (2.5) is where Λ′ is the integration constant. This expression matches the result of [30]. Since our goal is to see what happens when ∆ is a radial function, thus we expand where c is some constant. By doing a series expansion of (2.6), integrating by parts and collecting powers of ǫ, we find The limit ǫ → 0 reduces to general relativity. 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: Modified Einstein Equations

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

This paper is available on arxiv under CC 4.0 license.
Authors:
(1) A. Oliveros, Programa de F´ısica, Universidad del Atl´antico;
(2) Mario A. Acero, Programa de F´ısica, Universidad del Atl´antico. Table of Links Abstract and Intro
The f(Q) gravity: a brief review
Cosmological dynamics in late-time
Parameter constraints
Conclusions
Acknowledgments and References 2. The f(Q) gravity: a brief review In general, the action for an f(Q) gravity model in the presence of matter components is given by with the Levi-Civita connection of the metric given by the contortion tensor is Considering the flat Friedman-Robertson-Walker (FRW) metric, with a(t) representing the scale factor, the time and spatial components of Eq. (10) are given, respectively, by [9, 10] and It should be noted that in this scenario, the standard Friedmann equations of General Relativity (GR) plus the cosmological constant are recovered when assuming that f(Q) = Q + 2Λ. If there is no interaction between non-relativistic matter and radiation, then these components obey separately the conservation laws As usual in the literature, it is possible to rewrite the field equations (13) and (14) in the Einstein-Hilbert form: In this case, we assume that the equation of state parameter for this effective dark energy satisfies the following relation: This paper is available on arxiv under CC 4.0 license. Authors: (1) A. Oliveros, Programa de F´ısica, Universidad del Atl´antico; (2) Mario A. Acero, Programa de F´ısica, Universidad del Atl´antico. This paper is available on arxiv under CC 4.0 license. Authors: Authors: (1) A. Oliveros, Programa de F´ısica, Universidad del Atl´antico; (2) Mario A. Acero, Programa de F´ısica, Universidad del Atl´antico. Table of Links Abstract and Intro The f(Q) gravity: a brief review Cosmological dynamics in late-time Parameter constraints Conclusions Acknowledgments and References Abstract and Intro Abstract and Intro The f(Q) gravity: a brief review The f(Q) gravity: a brief review Cosmological dynamics in late-time Cosmological dynamics in late-time Parameter constraints Parameter constraints Conclusions Conclusions Acknowledgments and References Acknowledgments and References 2. The f(Q) gravity: a brief review In general, the action for an f(Q) gravity model in the presence of matter components is given by with the Levi-Civita connection of the metric given by the contortion tensor is Considering the flat Friedman-Robertson-Walker (FRW) metric, with a(t) representing the scale factor, the time and spatial components of Eq. (10) are given, respectively, by [9, 10] and It should be noted that in this scenario, the standard Friedmann equations of General Relativity (GR) plus the cosmological constant are recovered when assuming that f(Q) = Q + 2Λ. If there is no interaction between non-relativistic matter and radiation, then these components obey separately the conservation laws As usual in the literature, it is possible to rewrite the field equations (13) and (14) in the Einstein-Hilbert form: In this case, we assume that the equation of state parameter for this effective dark energy satisfies the following relation:

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