Abstract
The deep connection between gravitational dynamics and horizon thermodynamics leads to several intriguing features both in general relativity and in Lanczos-Lovelock theories of gravity. Recently in arXiv:1312.3253 several additional results strengthening the above connection have been established within the framework of general relativity. In this work we provide a generalization of the above setup to Lanczos-Lovelock gravity as well. To our expectation it turns out that most of the results obtained in the context of general relativity generalize to Lanczos-Lovelock gravity in a straightforward but non-trivial manner. First, we provide an alternative and more general derivation of the connection between Noether charge for a specific time evolution vector field and gravitational heat density of the boundary surface. This will lead to holographic equipartition for static spacetimes in Lanczos-Lovelock gravity as well. Taking a cue from this, we have introduced naturally defined four-momentum current associated with gravity and matter energy momentum tensor for both Lanczos-Lovelock Lagrangian and its quadratic part. Then, we consider the concepts of Noether charge for null boundaries in Lanczos-Lovelock gravity by providing a direct generalization of previous results derived in the context of general relativity.
Another very interesting feature for gravity is that gravitational field equations for arbitrary static and spherically symmetric spacetimes with horizon can be written as a thermodynamic identity in the near horizon limit. This result holds in both general relativity and in Lanczos-Lovelock gravity as well. In a previous work [arXiv:1505.05297] we have shown that, for an arbitrary spacetime, the gravitational field equations near any null surface generically leads to a thermodynamic identity. In this work, we have also generalized this result to Lanczos-Lovelock gravity by showing that gravitational field equations for Lanczos-Lovelock gravity near an arbitrary null surface can be written as a thermodynamic identity. Our general expressions under appropriate limits reproduce previously derived results for both the static and spherically symmetric spacetimes in Lanczos-Lovelock gravity. Also by taking appropriate limit to general relativity we can reproduce the results presented in arXiv:1312.3253 and arXiv:1505.05297.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Lanczos, Electromagnetism as a natural Property of Riemannian Geometry, Z. Phys. 73 (1932) 147.
C. Lanczos, A Remarkable property of the Riemann-Christoffel tensor in four dimensions, Annals Math. 39 (1938) 842 [INSPIRE].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].
T. Padmanabhan and D. Kothawala, Lanczos-Lovelock models of gravity, Phys. Rept. 531 (2013) 115 [arXiv:1302.2151] [INSPIRE].
T. Padmanabhan, Classical and Quantum Thermodynamics of horizons in spherically symmetric spacetimes, Class. Quant. Grav. 19 (2002) 5387 [gr-qc/0204019] [INSPIRE]
T. Padmanabhan, Gravity and the thermodynamics of horizons, Phys. Rept. 406 (2005) 49 [gr-qc/0311036] [INSPIRE].
T. Padmanabhan, Thermodynamical Aspects of Gravity: New insights, Rept. Prog. Phys. 73 (2010) 046901 [arXiv:0911.5004] [INSPIRE].
R.-G. Cai and S.P. Kim, First law of thermodynamics and Friedmann equations of Friedmann-Robertson-Walker universe, JHEP 02 (2005) 050 [hep-th/0501055] [INSPIRE].
M. Akbar and R.-G. Cai, Friedmann equations of FRW universe in scalar-tensor gravity, f(R) gravity and first law of thermodynamics, Phys. Lett. B 635 (2006) 7 [hep-th/0602156] [INSPIRE].
M. Akbar and M. Jamil, Wormhole Thermodynamics at Apparent Horizons, arXiv:0911.2556.
M. Akbar, Thermodynamic interpretation of field equations at horizon of BTZ black hole, Chin. Phys. Lett. 24 (2007) 1158 [hep-th/0702029] [INSPIRE].
M. Akbar and A.A. Siddiqui, Charged rotating BTZ black hole and thermodynamic behavior of field equations at its horizon, Phys. Lett. B 656 (2007) 217 [arXiv:1009.3749] [INSPIRE].
R.G. Cai, L.M. Cao and Y.P. Hu, Corrected entropy-area relation and modified Friedmann equations, JHEP 08 (2008) 090 [arXiv:0807.1232] [INSPIRE].
M. Akbar and R.-G. Cai, Thermodynamic Behavior of Friedmann Equations at Apparent Horizon of FRW Universe, Phys. Rev. D 75 (2007) 084003 [hep-th/0609128] [INSPIRE].
R.-G. Cai and L.-M. Cao, Thermodynamics of Apparent Horizon in Brane World Scenario, Nucl. Phys. B 785 (2007) 135 [hep-th/0612144] [INSPIRE].
A. Sheykhi, B. Wang and R.-G. Cai, Thermodynamical properties of apparent horizon in warped DGP braneworld, Nucl. Phys. B 779 (2007) 1 [hep-th/0701198] [INSPIRE].
A. Sheykhi, B. Wang and R.-G. Cai, Deep Connection Between Thermodynamics and Gravity in Gauss-Bonnet Braneworld, Phys. Rev. D 76 (2007) 023515 [hep-th/0701261] [INSPIRE].
R.-G. Cai, Thermodynamics of apparent horizon in brane world scenarios, Prog. Theor. Phys. Suppl. 172 (2008) 100 [arXiv:0712.2142] [INSPIRE].
X.-H. Ge, First law of thermodynamics and Friedmann-like equations in braneworld cosmology, Phys. Lett. B 651 (2007) 49 [hep-th/0703253] [INSPIRE].
Y. Gong and A. Wang, The Friedmann equations and thermodynamics of apparent horizons, Phys. Rev. Lett. 99 (2007) 211301 [arXiv:0704.0793] [INSPIRE].
S.-F. Wu, G.-H. Yang and P.-M. Zhang, Cosmological equations and Thermodynamics on Apparent Horizon in Thick Braneworld, Gen. Rel. Grav. 42 (2010) 1601 [arXiv:0710.5394] [INSPIRE].
S.-F. Wu, B. Wang and G.-H. Yang, Thermodynamics on the apparent horizon in generalized gravity theories, Nucl. Phys. B 799 (2008) 330 [arXiv:0711.1209] [INSPIRE].
S.F. Wu et al., The generalized second law of thermodynamics in generalized gravity theories, Class. Quant. Grav. 25 (2008) 235018 [arXiv:0801.2688] [INSPIRE].
T. Zhu, J.-R. Ren and S.-F. Mo, Thermodynamics of Friedmann Equation and Masslike Function in Generalized Braneworlds, Int. J. Mod. Phys. A 24 (2009) 5877 [arXiv:0805.1162] [INSPIRE].
M. Akbar, Viscous Cosmology and Thermodynamics of Apparent Horizon, Chin. Phys. Lett. 25 (2008) 4199 [arXiv:0808.0169] [INSPIRE].
D. Kothawala, S. Sarkar and T. Padmanabhan, Einstein’s equations as a thermodynamic identity: The Cases of stationary axisymmetric horizons and evolving spherically symmetric horizons, Phys. Lett. B 652 (2007) 338 [gr-qc/0701002] [INSPIRE].
S. Chakraborty, R. Biswas and N. Mazumder, Unified First Law and Some Comments, Nuovo Cim. B 125 (2011) 1209 [arXiv:1006.1169] [INSPIRE].
N. Mazumder and S. Chakraborty, Does the validity of the first law of thermodynamics imply that the generalized second law of thermodynamics of the universe is bounded by the event horizon?, Class. Quant. Grav. 26 (2009) 195016 [INSPIRE].
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] [INSPIRE].
D. Kothawala and T. Padmanabhan, Thermodynamic structure of Lanczos-Lovelock field equations from near-horizon symmetries, Phys. Rev. D 79 (2009) 104020 [arXiv:0904.0215] [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
P.C.W. Davies, S.A. Fulling and W.G. Unruh, Energy Momentum Tensor Near an Evaporating Black Hole, Phys. Rev. D 13 (1976) 2720 [INSPIRE].
W.G. Unruh, Notes on black hole evaporation, Phys. Rev. D 14 (1976) 870 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett. 75 (1995) 1260 [gr-qc/9504004] [INSPIRE].
T. Padmanabhan, Gravitation: Foundation and Frontiers, Cambridge University Press, Cambridge U.K. (2010).
R.M. Wald, The Thermodynamics of Black Holes, Liv. Rev. Relt. 4 (2001) 6 [gr-qc/9912119].
T. Padmanabhan, Dark energy and gravity, Gen. Rel. Grav. 40 (2008) 529 [arXiv:0705.2533] [INSPIRE].
T. Padmanabhan, Is gravity an intrinsically quantum phenomenon? Dynamics of gravity from the entropy of space-time and the principle of equivalence, Mod. Phys. Lett. A 17 (2002) 1147 [hep-th/0205278] [INSPIRE].
T. Padmanabhan, The Holography of gravity encoded in a relation between entropy, horizon area and action for gravity, Gen. Rel. Grav. 34 (2002) 2029 [gr-qc/0205090] [INSPIRE].
A. Mukhopadhyay and T. Padmanabhan, Holography of gravitational action functionals, Phys. Rev. D 74 (2006) 124023 [hep-th/0608120] [INSPIRE].
S. Kolekar and T. Padmanabhan, Holography in Action, Phys. Rev. D 82 (2010) 024036 [arXiv:1005.0619] [INSPIRE].
S. Kolekar, D. Kothawala and T. Padmanabhan, Two Aspects of Black Hole Entropy in Lanczos-Lovelock Models of Gravity, Phys. Rev. D 85 (2012) 064031 [arXiv:1111.0973] [INSPIRE].
T. Padmanabhan, Entropy density of spacetime and the Navier-Stokes fluid dynamics of null surfaces, Phys. Rev. D 83 (2011) 044048 [arXiv:1012.0119] [INSPIRE].
S. Kolekar and T. Padmanabhan, Action Principle for the Fluid-Gravity Correspondence and Emergent Gravity, Phys. Rev. D 85 (2012) 024004 [arXiv:1109.5353] [INSPIRE].
T. Damour, Surface Effects in Black Hole Physics, in Proceedings of the second Marcel Grossmann Meeting on General Relativity, Trieste Italy (1979).
T. Padmanabhan and A. Paranjape, Entropy of null surfaces and dynamics of spacetime, Phys. Rev. D 75 (2007) 064004 [gr-qc/0701003] [INSPIRE].
K. Parattu, B.R. Majhi and T. Padmanabhan, Structure of the gravitational action and its relation with horizon thermodynamics and emergent gravity paradigm, Phys. Rev. D 87 (2013) 124011 [arXiv:1303.1535] [INSPIRE].
S. Chakraborty and T. Padmanabhan, Geometrical variables with direct thermodynamic significance in Lanczos-Lovelock gravity, Phys. Rev. D 90 (2014) 084021 [arXiv:1408.4791] [INSPIRE].
T. Padmanabhan and H. Padmanabhan, CosMIn: The Solution to the Cosmological Constant Problem, Int. J. Mod. Phys. D 22 (2013) 1342001 [arXiv:1302.3226] [INSPIRE].
T. Padmanabhan and H. Padmanabhan, Cosmological constant from the emergent gravity perspective, Int. J. Mod. Phys. D23 (2014) 1430011 [arXiv:1404.2284] [INSPIRE].
T. Padmanabhan, General Relativity from a Thermodynamic Perspective, Gen. Rel. Grav. 46 (2014) 1673 [arXiv:1312.3253] [INSPIRE].
S. Chakraborty and T. Padmanabhan, Evolution of Spacetime arises due to the departure from Holographic Equipartition in all Lanczos-Lovelock Theories of Gravity, Phys. Rev. D 90 (2014) 124017 [arXiv:1408.4679] [INSPIRE].
B.R. Majhi and S. Chakraborty, Anomalous effective action, Noether current, Virasoro algebra and Horizon entropy, Eur. Phys. J. C 74 (2014) 2867 [arXiv:1311.1324] [INSPIRE].
B.R. Majhi and T. Padmanabhan, Noether Current, Horizon Virasoro Algebra and Entropy, Phys. Rev. D 85 (2012) 084040 [arXiv:1111.1809] [INSPIRE].
R.M. Wald, Black Hole Entropy is Noether Charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
R.M. Wald and A. Zoupas, A General definition of ’conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett. 80 (1998) 904 [gr-qc/9710007] [INSPIRE].
J.M. Garcia-Islas, BTZ Black Hole Entropy: A Spin foam model description, Class. Quant. Grav. 25 (2008) 245001 [arXiv:0804.2082] [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
V. Moncrief and J. Isenberg, Symmetries of cosmological Cauchy horizons, Commun. Math. Phys. 89 (1983) 387 [INSPIRE].
E.M. Morales, On a Second Law of Black Hole Mechanics in a Higher Derivative Theory of Gravity Ph.D Thesis, Göttingen University, Göttingen Germany (2008).
K. Parattu, S. Chakraborty, B.R. Majhi and T. Padmanabhan, Null Surfaces: Counter-term for the Action Principle and the Characterization of the Gravitational Degrees of Freedom, arXiv:1501.01053 [INSPIRE].
N. Dadhich and J.M. Pons, Static pure Lovelock black hole solutions with horizon topology S (n) × S (n), JHEP 05 (2015) 067 [arXiv:1503.00974] [INSPIRE].
N. Dadhich and J.M. Pons, Probing pure Lovelock gravity by Nariai and Bertotti-Robinson solutions, J. Math. Phys. 54 (2013) 102501 [arXiv:1210.1109] [INSPIRE].
N. Dadhich, Characterization of the Lovelock gravity by Bianchi derivative, Pramana 74 (2010) 875 [arXiv:0802.3034] [INSPIRE].
S. Chakraborty, K. Parattu and T. Padmanabhan, Gravitational Field equations near an Arbitrary Null Surface expressed as a Thermodynamic Identity, arXiv:1505.05297 [INSPIRE].
D. Kothawala, The thermodynamic structure of Einstein tensor, Phys. Rev. D 83 (2011) 024026 [arXiv:1010.2207] [INSPIRE].
S.A. Hayward, Unified first law of black hole dynamics and relativistic thermodynamics, Class. Quant. Grav. 15 (1998) 3147 [gr-qc/9710089] [INSPIRE].
T. Jacobson and R.C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett. 70 (1993) 3684 [hep-th/9305016] [INSPIRE].
T. Clunan, S.F. Ross and D.J. Smith, On Gauss-Bonnet black hole entropy, Class. Quant. Grav. 21 (2004) 3447 [gr-qc/0402044] [INSPIRE].
B. Julia and S. Silva, Currents and superpotentials in classical gauge invariant theories. 1. Local results with applications to perfect fluids and general relativity, Class. Quant. Grav. 15 (1998) 2173 [gr-qc/9804029] [INSPIRE].
T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].
S. Carlip, Entropy from conformal field theory at Killing horizons, Class. Quant. Grav. 16 (1999) 3327 [gr-qc/9906126] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1505.07272
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chakraborty, S. Lanczos-Lovelock gravity from a thermodynamic perspective. J. High Energ. Phys. 2015, 29 (2015). https://doi.org/10.1007/JHEP08(2015)029
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2015)029