Abstract
For a general maximally symmetric (spherically, plane or hyperbola symmetric) holographic screen, we rewrite the equations of motion of general Lovelock gravity into the form of some generalized first law of thermodynamics, under certain ansatz. With this observation together with other two independent ways, exactly the same temperature and entropy on the screen are obtained. So it is argued that the thermodynamic interpretation of gravity is physically meaningful not only on the horizon, but also on a general maximally symmetric screen. Moreover, the formula of entropy is further checked in the (maximally symmetric) general static case and dynamical case. The entropy formula also holds for those cases. Finally, the method of conical singularity is used to calculate the entropy on such screen, and the result again confirms the entropy formula.
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J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [SPIRES].
J.D. Bekenstein, Extraction of energy and charge from a black hole, Phys. Rev. D 7 (1973) 949 [SPIRES].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [SPIRES].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [SPIRES].
T. Jacobson, Thermodynamics of space-time: the Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [SPIRES].
T. Padmanabhan, Thermodynamical aspects of gravity: new insights, Rept. Prog. Phys. 73 (2010) 046901 [arXiv:0911.5004] [SPIRES].
E.P. Verlinde, On the origin of gravity and the laws of Newton, arXiv:1001.0785 [SPIRES].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [SPIRES].
G. ’t Hooft, Dimensional reduction in quantum gravity, gr-qc/9310026 [SPIRES].
E. Witten, Anti de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [SPIRES].
T. Padmanabhan, Gravitational entropy of static spacetimes and microscopic density of states, Class. Quant. Grav. 21 (2004) 4485 [gr-qc/0308070] [SPIRES].
T. Padmanabhan, Equipartition of energy in the horizon degrees of freedom and the emergence of gravity, Mod. Phys. Lett. A 25 (2010) 1129 [arXiv:0912.3165] [SPIRES].
F.-W. Shu and Y. Gong, Equipartition of energy and the first law of thermodynamics at the apparent horizon, arXiv:1001.3237 [SPIRES].
R.-G. Cai, L.-M. Cao and N. Ohta, Friedmann equations from entropic force, Phys. Rev. D 81 (2010) 061501 [arXiv:1001.3470] [SPIRES].
Y. Zhang, Y.-g. Gong and Z.-H. Zhu, Modified gravity emerging from thermodynamics and holographic principle, arXiv:1001.4677 [SPIRES].
S.-W. Wei, Y.-X. Liu and Y.-Q. Wang, Friedmann equation of FRW universe in deformed Hoˇrava-Lifshitz gravity from entropic force, arXiv:1001.5238 [SPIRES].
Y. Ling and J.-P. Wu, A note on entropic force and brane cosmology, JCAP 08 (2010) 017 [arXiv:1001.5324] [SPIRES].
L. Smolin, Newtonian gravity in loop quantum gravity, arXiv:1001.3668 [SPIRES].
T. Wang, Coulomb force as an entropic force, Phys. Rev. D 81 (2010) 104045 [arXiv:1001.4965] [SPIRES].
M. Li and Y. Wang, Quantum UV/IR relations and holographic dark energy from entropic force, Phys. Lett. B 687 (2010) 243 [arXiv:1001.4466] [SPIRES].
J. Makea, Notes concerning ’On the origin of gravity and the laws of Newton’ by E. Verlinde, arXiv:1001.3808 [SPIRES].
F. Caravelli and L. Modesto, Holographic actions from black hole entropy, arXiv:1001.4364 [SPIRES].
J.-W. Lee, H.-C. Kim and J. Lee, Gravity from quantum information, arXiv:1001.5445 [SPIRES].
C. Gao, Modified entropic force, Phys. Rev. D 81 (2010) 087306 [arXiv:1001.4585] [SPIRES].
J. Munkhammar, Is holographic entropy and gravity the result of quantum mechanics?, arXiv:1003.1262 [SPIRES].
L. Zhao, Hidden symmetries for thermodynamics and emergence of relativity, Commun. Theor. Phys. 54 (2010) 641 [arXiv:1002.0488] [SPIRES].
Y. Zhao, Entropic force and its fluctuation in euclidian quantum gravity, arXiv:1002.4039 [SPIRES].
X. Kuang, Y. Ling and H. Zhang, On thermal force from holographic action, arXiv:1003.0195 [SPIRES].
X.-G. He and B.-Q. Ma, Black holes and photons with entropic force, Chin. Phys. Lett. 27 (2010) 070402 [arXiv:1003.1625] [SPIRES].
X. Li and Z. Chang, Debye entropic force and modified Newtonian dynamics, arXiv:1005.1169 [SPIRES].
H. Wei, Cosmological constraints on the modified entropic force model, Phys. Lett. B 692 (2010) 167 [arXiv:1005.1445] [SPIRES].
J.-W. Lee, On the origin of entropic gravity and inertia, arXiv:1003.4464 [SPIRES].
E. Chang-Young, M. Eune, K. Kimm and D. Lee, Surface gravity and Hawking temperature from entropic force viewpoint, Mod. Phys. Lett. A 25 (2010) 2825 [arXiv:1003.2049] [SPIRES].
Y.S. Myung, Entropic force in the presence of black hole, arXiv:1002.0871 [SPIRES].
Y.S. Myung and Y.-W. Kim, Entropic force and entanglement system, Phys. Rev. D 81 (2010) 105012 [arXiv:1002.2292] [SPIRES].
Y.S. Myung, Does entropic force always imply the newtonian force law?, arXiv:1003.5037 [SPIRES].
I.V. Vancea and M.A. Santos, Entropic force law, emergent gravity and the uncertainty principle, arXiv:1002.2454 [SPIRES].
J. Kowalski-Glikman, A note on gravity, entropy and BF topological field theory, Phys. Rev. D 81 (2010) 084038 [arXiv:1002.1035] [SPIRES].
R.A. Konoplya, Entropic force, holography and thermodynamics for static space-times, Eur. Phys. J. C 69 (2010) 555 [arXiv:1002.2818] [SPIRES].
A. Sheykhi, Entropic corrections to friedmann equations, Phys. Rev. D 81 (2010) 104011 [arXiv:1004.0627] [SPIRES].
S. Samanta, A note on the thermodynamical origin of gravity, arXiv:1003.5965 [SPIRES].
C.M. Ho, D. Minic and Y.J. Ng, Cold dark matter with MOND scaling, Phys. Lett. B 693 (2010) 567 [arXiv:1005.3537] [SPIRES].
M. Li and Y. Pang, A no-go theorem prohibiting inflation in the entropic force scenario, Phys. Rev. D 82 (2010) 027501 [arXiv:1004.0877] [SPIRES].
Y.-X. Liu, Y.-Q. Wang and S.-W. Wei, A note on temperature and energy of 4-dimensional black holes from entropic force, Class. Quant. Grav. 27 (2010) 185002 [arXiv:1002.1062] [SPIRES].
S. Ghosh, Planck scale effect in the entropic force law, arXiv:1003.0285 [SPIRES].
P. Nicolini, Entropic force, noncommutative gravity and ungravity, Phys. Rev. D 82 (2010) 044030 [arXiv:1005.2996] [SPIRES].
R. Banerjee and B.R. Majhi, Statistical origin of gravity, Phys. Rev. D 81 (2010) 124006 [arXiv:1003.2312] [SPIRES].
W. Gu, M. Li and R.-X. Miao, A new entropic force scenario and holographic thermodynamics, arXiv:1011.3419 [SPIRES].
T. Padmanabhan, Surface density of spacetime degrees of freedom from equipartition law in theories of gravity, Phys. Rev. D 81 (2010) 124040 [arXiv:1003.5665] [SPIRES].
Q. Pan and B. Wang, Influence on the entropic force by the virtual degree of freedom on the holographic screen, Phys. Lett. B 694 (2011) 456 [arXiv:1004.2954] [SPIRES].
V.V. Kiselev and S.A. Timofeev, The surface density of holographic entropy, Mod. Phys. Lett. A 25 (2010) 2223 [arXiv:1004.3418] [SPIRES].
J.-P. Lee, Comments on Verlinde’s entropic gravity, arXiv:1005.1347 [SPIRES].
Y. Tian and X.-N. Wu, Thermodynamics of black holes from equipartition of energy and holography, Phys. Rev. D 81 (2010) 104013 [arXiv:1002.1275] [SPIRES].
Y.-X. Chen and J.-L. Li, First law of thermodynamics on holographic screens in entropic force frame, arXiv:1006.1442 [SPIRES].
R.-G. Cai, L.-M. Cao and N. Ohta, Notes on entropy force in general spherically symmetric spacetimes, Phys. Rev. D 81 (2010) 084012 [arXiv:1002.1136] [SPIRES].
Y.-X. Chen, J.-L. Li and Y.-Q. Wang, Thermodynamics for Kodama observer in general spherically symmetric spacetimes, arXiv:1008.3215 [SPIRES].
Y.-X. Chen, J.-L. Li and Y.-Q. Wang, Thermodynamics for Kodama observer in general spherically symmetric spacetimes, arXiv:1008.3215 [SPIRES].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [SPIRES].
V. Iyer and R.M. Wald, A Comparison of Noether charge and euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52 (1995) 4430 [gr-qc/9503052] [SPIRES].
R.-G. Cai and L.-M. Cao, Unified first law and thermodynamics of apparent horizon in FRW universe, Phys. Rev. D 75 (2007) 064008 [gr-qc/0611071] [SPIRES].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [SPIRES].
R.-G. Cai and N. Ohta, Horizon thermodynamics and gravitational field equations in Hoˇrava-Lifshitz gravity, Phys. Rev. D 81 (2010) 084061 [arXiv:0910.2307] [SPIRES].
R.-G. Cai, L.-M. Cao, Y.-P. Hu and S.P. Kim, Generalized Vaidya spacetime in lovelock gravity and thermodynamics on apparent horizon, Phys. Rev. D 78 (2008) 124012 [arXiv:0810.2610] [SPIRES].
R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D 65 (2002) 084014 [hep-th/0109133] [SPIRES].
R.-G. Cai and K.-S. Soh, Topological black holes in the dimensionally continued gravity, Phys. Rev. D 59 (1999) 044013 [gr-qc/9808067] [SPIRES].
R.-G. Cai and Q. Guo, Gauss-Bonnet black holes in dS spaces, Phys. Rev. D 69 (2004) 104025 [hep-th/0311020] [SPIRES].
R.-G. Cai, A note on thermodynamics of black holes in Lovelock gravity, Phys. Lett. B 582 (2004) 237 [hep-th/0311240] [SPIRES].
V.V. Kiselev, Quintessence and black holes, Class. Quant. Grav. 20 (2003) 1187 [gr-qc/0210040] [SPIRES].
A. Paranjape, S. Sarkar and T. Padmanabhan, Thermodynamic route to field equations in Lancos-Lovelock gravity, Phys. Rev. D 74 (2006) 104015 [hep-th/0607240] [SPIRES].
C.W. Misner and D.H. Sharp, Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Phys. Rev. 136 (1964) B571.
B. Zwiebach, Curvature squared terms and string theories, Phys. Lett. B 156 (1985) 315 [SPIRES].
B. Zumino, Gravity theories in more than four-dimensions, Phys. Rept. 137 (1986) 109 [UCB-PTH-85-13] [SPIRES].
J.T. Wheeler, Symmetric solutions to the maximally Gauss-Bonnet extended Einstein equations, Nucl. Phys. B 273 (1986) 732 [SPIRES].
H. Maeda and M. Nozawa, Generalized Misner-Sharp quasi-local mass in Einstein-Gauss-Bonnet gravity, Phys. Rev. D 77 (2008) 064031 [arXiv:0709.1199] [SPIRES].
D.L. Wiltshire, Spherically symmetric solutions of Einstein-Maxwell theory with a Gauss-Bonnet term, Phys. Lett. B 169 (1986) 36 [SPIRES].
D.L. Wiltshire, Black holes in string generated gravity models, Phys. Rev. D 38 (1988) 2445 [SPIRES].
T. Jacobson and R.C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett. 70 (1993) 3684 [hep-th/9305016] [SPIRES].
S.A. Hayward, Unified first law of black-hole dynamics and relativistic thermodynamics, Class. Quant. Grav. 15 (1998) 3147 [gr-qc/9710089] [SPIRES].
S.A. Hayward, S. Mukohyama and M.C. Ashworth, Dynamic black-hole entropy, Phys. Lett. A 256 (1999) 347 [gr-qc/9810006] [SPIRES].
S.A. Hayward, R. Di Criscienzo, L. Vanzo, M. Nadalini and S. Zerbini, Local Hawking temperature for dynamical black holes, Class. Quant. Grav. 26 (2009) 062001 [arXiv:0806.0014] [SPIRES].
R.G. Cai, Gravity from thermodynamics, talk given at Peking University, Beijing, P.R. China (2010).
G. Abreu and M. Visser, Kodama time: geometrically preferred foliations of spherically symmetric spacetimes, Phys. Rev. D 82 (2010) 044027 [arXiv:1004.1456] [SPIRES].
M. Bañados, C. Teitelboim and J. Zanelli, Black hole entropy and the dimensional continuation of the Gauss-Bonnet theorem, Phys. Rev. Lett. 72 (1994) 957 [gr-qc/9309026] [SPIRES].
D.V. Fursaev and S.N. Solodukhin, On the description of the riemannian geometry in the presence of conical defects, Phys. Rev. D 52 (1995) 2133 [hep-th/9501127] [SPIRES].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [SPIRES].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [SPIRES].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, arXiv:1011.5819 [SPIRES].
A. Schwimmer and S. Theisen, Entanglement entropy, trace anomalies and holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [SPIRES].
G. Michalogiorgakis, Entanglement entropy of two dimensional systems and holography, JHEP 12 (2008) 068 [arXiv:0806.2661] [SPIRES].
J.-R. Sun, Note on Chern-Simons term correction to holographic entanglement entropy, JHEP 05 (2009) 061 [arXiv:0810.0967] [SPIRES].
D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [SPIRES].
D.V. Fursaev, Entanglement entropy in quantum gravity and the plateau problem, Phys. Rev. D 77 (2008) 124002 [arXiv:0711.1221] [SPIRES].
W. Nelson, A comment on black hole entropy in string theory, Phys. Rev. D 50 (1994) 7400 [hep-th/9406011] [SPIRES].
D.V. Fursaev, ‘Thermodynamics’ of minimal surfaces and entropic origin of gravity, Phys. Rev. D 82 (2010) 064013 [arXiv:1006.2623] [SPIRES].
J.D. Bekenstein, A universal upper bound on the entropy to energy ratio for bounded systems, Phys. Rev. D 23 (1981) 287 [SPIRES].
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Tian, Y., Wu, XN. Thermodynamics on the maximally symmetric holographic screen and entropy from conical singularities. J. High Energ. Phys. 2011, 150 (2011). https://doi.org/10.1007/JHEP01(2011)150
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DOI: https://doi.org/10.1007/JHEP01(2011)150