Abstract
We construct an entropy current and establish a local version of the classical second law of thermodynamics for dynamical black holes in Chern-Simons (CS) theories of gravity. We work in a chosen set of Gaussian null coordinates and assume the dynamics to be small perturbations around the Killing horizon. In explicit examples of both purely gravitational and mixed gauge gravity CS theories in (2 + 1) and (4 + 1)-dimensions, the entropy current is obtained by studying the off-shell structure of the equations of motion evaluated on the horizon. For the CS theory in (2 + 1) dimensions, we argue that the second law holds to quadratic order in perturbations by considering it as a low energy effective field theory with the leading piece given by Einstein gravity. In all such examples, we show that the construction of entropy current is invariant under the reparameterization of the null horizon coordinates. Finally, extending an existing formalism for diffeomorphism invariant theories, we construct an abstract proof for the linearised second law in arbitrary Chern-Simons theories in any given odd dimensions by studying the off-shell equations of motion. As a check of consistency, we verify that the outcome of this algorithmic proof matches precisely with the results obtained in explicit examples.
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References
J.M. Bardeen, B. Carter and S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31 (1973) 161 [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
J.F. Donoghue, General relativity as an effective field theory: The leading quantum corrections, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057] [INSPIRE].
D.G. Boulware and S. Deser, String Generated Gravity Models, Phys. Rev. Lett. 55 (1985) 2656 [INSPIRE].
B. Zwiebach, Curvature Squared Terms and String Theories, Phys. Lett. B 156 (1985) 315 [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) R3427 [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].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
A.C. Wall, A Second Law for Higher Curvature Gravity, Int. J. Mod. Phys. D 24 (2015) 1544014 [arXiv:1504.08040] [INSPIRE].
T. Jacobson and R.C. Myers, Black hole entropy and higher curvature interactions, Phys. Rev. Lett. 70 (1993) 3684 [hep-th/9305016] [INSPIRE].
S. Chatterjee and M. Parikh, The second law in four-dimensional Einstein-Gauss-Bonnet gravity, Class. Quant. Grav. 31 (2014) 155007 [arXiv:1312.1323] [INSPIRE].
S. Sarkar and A.C. Wall, Generalized second law at linear order for actions that are functions of Lovelock densities, Phys. Rev. D 88 (2013) 044017 [arXiv:1306.1623] [INSPIRE].
S. Bhattacharjee, S. Sarkar and A.C. Wall, Holographic entropy increases in quadratic curvature gravity, Phys. Rev. D 92 (2015) 064006 [arXiv:1504.04706] [INSPIRE].
S. Bhattacharjee, A. Bhattacharyya, S. Sarkar and A. Sinha, Entropy functionals and c-theorems from the second law, Phys. Rev. D 93 (2016) 104045 [arXiv:1508.01658] [INSPIRE].
S. Sarkar, Black Hole Thermodynamics: General Relativity and Beyond, Gen. Rel. Grav. 51 (2019) 63 [arXiv:1905.04466] [INSPIRE].
X.-Y. Wang and J. Jiang, Investigating the Linearized Second Law in Horndeski Gravity, Phys. Rev. D 102 (2020) 084020 [arXiv:2008.09774] [INSPIRE].
J. Bhattacharya, S. Bhattacharyya, A. Dinda and N. Kundu, An entropy current for dynamical black holes in four-derivative theories of gravity, JHEP 06 (2020) 017 [arXiv:1912.11030] [INSPIRE].
S. Bhattacharyya et al., An entropy current and the second law in higher derivative theories of gravity, JHEP 09 (2021) 169 [arXiv:2105.06455] [INSPIRE].
P. Biswas, P. Dhivakar and N. Kundu, Non-minimal coupling of scalar and gauge fields with gravity: an entropy current and linearized second law, JHEP 12 (2022) 036 [arXiv:2206.04538] [INSPIRE].
S. Hollands, Á.D. Kovács and H.S. Reall, The second law of black hole mechanics in effective field theory, JHEP 08 (2022) 258 [arXiv:2205.15341] [INSPIRE].
I. Davies and H.S. Reall, Dynamical Black Hole Entropy in Effective Field Theory, JHEP 05 (2023) 006 [arXiv:2212.09777] [INSPIRE].
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
S.-S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals Math. 99 (1974) 48 [INSPIRE].
L. Alvarez-Gaume and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
M.B. Green and J.H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically Massive Gauge Theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [INSPIRE].
S.N. Solodukhin, Holography with gravitational Chern-Simons, Phys. Rev. D 74 (2006) 024015 [hep-th/0509148] [INSPIRE].
P. Kraus, Lectures on black holes and the AdS3/CFT2 correspondence, in Supersymmetric Mechanics — Vol. 3, Sperling Berling, Heidelberg (2008), p. 1–55 [https://doi.org/10.1007/978-3-540-79523-0_4] [hep-th/0609074] [INSPIRE].
B. Sahoo and A. Sen, BTZ black hole with Chern-Simons and higher derivative terms, JHEP 07 (2006) 008 [hep-th/0601228] [INSPIRE].
Y. Tachikawa, Black hole entropy in the presence of Chern-Simons terms, Class. Quant. Grav. 24 (2007) 737 [hep-th/0611141] [INSPIRE].
L. Bonora et al., Gravitational Chern-Simons Lagrangians and black hole entropy, JHEP 07 (2011) 085 [arXiv:1104.2523] [INSPIRE].
T. Azeyanagi, R. Loganayagam, G.S. Ng and M.J. Rodriguez, Covariant Noether Charge for Higher Dimensional Chern-Simons Terms, JHEP 05 (2015) 041 [arXiv:1407.6364] [INSPIRE].
N. Yunes and F. Pretorius, Dynamical Chern-Simons Modified Gravity. I. Spinning Black Holes in the Slow-Rotation Approximation, Phys. Rev. D 79 (2009) 084043 [arXiv:0902.4669] [INSPIRE].
S. Alexander and N. Yunes, Chern-Simons Modified General Relativity, Phys. Rept. 480 (2009) 1 [arXiv:0907.2562] [INSPIRE].
J.P. Gauntlett, R.C. Myers and P.K. Townsend, Black holes of D = 5 supergravity, Class. Quant. Grav. 16 (1999) 1 [hep-th/9810204] [INSPIRE].
J.P. Gauntlett and J.B. Gutowski, All supersymmetric solutions of minimal gauged supergravity in five-dimensions, Phys. Rev. D 68 (2003) 105009 [Erratum ibid. 70 (2004) 089901] [hep-th/0304064] [INSPIRE].
M. Banados, G. Barnich, G. Compere and A. Gomberoff, Three dimensional origin of Godel spacetimes and black holes, Phys. Rev. D 73 (2006) 044006 [hep-th/0512105] [INSPIRE].
D. Grumiller and N. Yunes, How do Black Holes Spin in Chern-Simons Modified Gravity?, Phys. Rev. D 77 (2008) 044015 [arXiv:0711.1868] [INSPIRE].
S. Chapman, Y. Neiman and Y. Oz, Fluid/Gravity Correspondence, Local Wald Entropy Current and Gravitational Anomaly, JHEP 07 (2012) 128 [arXiv:1202.2469] [INSPIRE].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
S. Bhattacharyya et al., Local Fluid Dynamical Entropy from Gravity, JHEP 06 (2008) 055 [arXiv:0803.2526] [INSPIRE].
C. Eling, A. Meyer and Y. Oz, Local Entropy Current in Higher Curvature Gravity and Rindler Hydrodynamics, JHEP 08 (2012) 088 [arXiv:1205.4249] [INSPIRE].
A. Chandranathan, S. Bhattacharyya, M. Patra and S. Roy, Entropy current and fluid-gravity duality in Gauss-Bonnet theory, JHEP 09 (2023) 070 [arXiv:2208.07856] [INSPIRE].
S. Bhattacharyya et al., Towards a second law for Lovelock theories, JHEP 03 (2017) 065 [arXiv:1612.04024] [INSPIRE].
S. Bhattacharyya, P. Jethwani, M. Patra and S. Roy, Reparametrization Symmetry of Local Entropy Production on a Dynamical Horizon, arXiv:2204.08447 [INSPIRE].
S. Bhattacharyya, P. Biswas, A. Dinda and N. Kundu, The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity, JHEP 10 (2022) 013 [arXiv:2205.01648] [INSPIRE].
M. Gadioux and H.S. Reall, Creases, corners, and caustics: Properties of nonsmooth structures on black hole horizons, Phys. Rev. D 108 (2023) 084021 [arXiv:2303.15512] [INSPIRE].
C. Copetti and J. Fernández-Pendás, Membrane paradigm and RG flows for anomalous holographic theories, JHEP 04 (2018) 134 [arXiv:1712.06628] [INSPIRE].
S. Gao and R.M. Wald, The ‘Physical process’ version of the first law and the generalized second law for charged and rotating black holes, Phys. Rev. D 64 (2001) 084020 [gr-qc/0106071] [INSPIRE].
A.J. Amsel, D. Marolf and A. Virmani, The Physical Process First Law for Bifurcate Killing Horizons, Phys. Rev. D 77 (2008) 024011 [arXiv:0708.2738] [INSPIRE].
A. Chatterjee and S. Sarkar, Physical process first law and increase of horizon entropy for black holes in Einstein-Gauss-Bonnet gravity, Phys. Rev. Lett. 108 (2012) 091301 [arXiv:1111.3021] [INSPIRE].
S. Kolekar, T. Padmanabhan and S. Sarkar, Entropy Increase during Physical Processes for Black Holes in Lanczos-Lovelock Gravity, Phys. Rev. D 86 (2012) 021501 [arXiv:1201.2947] [INSPIRE].
S. Bhattacharjee and S. Sarkar, Physical process first law and caustic avoidance for Rindler horizons, Phys. Rev. D 91 (2015) 024024 [arXiv:1412.1287] [INSPIRE].
A. Mishra, S. Chakraborty, A. Ghosh and S. Sarkar, On the physical process first law for dynamical black holes, JHEP 09 (2018) 034 [arXiv:1709.08925] [INSPIRE].
Acknowledgments
We thank Parthajit Biswas for the initial collaboration. We are especially grateful to Sayantani Bhattacharyya for various enlightening discussions and collaboration on related projects. We would like to thank Jyotirmoy Bhattacharya and Harvey Reall for their useful comments on our draft. We would also like to thank Diptarka Das, Anirban Dinda, S. Shankaranarayanan, and Yogesh Kumar Srivastava for useful discussions. PD would like to thank NISER Bhubaneshwar for their warm hospitality during a visit where partial progress of this work was presented in a talk. PD also thanks the organizers of FTAG 2023 for giving the opportunity to present the results of this work in a poster. PD duly acknowledges the Council of Scientific and Industrial Research (CSIR), New Delhi, for financial assistance through the Senior Research Fellowship (SRF) scheme. The work of NK is supported by a MATRICS grant (MTR/2022/000794) from the Science and Engineering Research Board (SERB), India. We acknowledge our debt to the people of India for their steady support of research in basic sciences.
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Deo, I., Dhivakar, P. & Kundu, N. Entropy-current for dynamical black holes in Chern-Simons theories of gravity. J. High Energ. Phys. 2023, 114 (2023). https://doi.org/10.1007/JHEP11(2023)114
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DOI: https://doi.org/10.1007/JHEP11(2023)114