Abstract
Horndeski gravities are theories of gravity coupled to a scalar field, in which the action contains an additional non-minimal quadratic coupling of the scalar, through its first derivative, to the Einstein tensor or the analogous higher-derivative tensors coming from the variation of Gauss-Bonnet or Lovelock terms. In this paper we study the thermodynamics of the static black hole solutions in n dimensions, in the simplest case of a Horndeski coupling to the Einstein tensor. We apply the Wald formalism to calculate the entropy of the black holes, and show that there is an additional contribution over and above those that come from the standard Wald entropy formula. The extra contribution can be attributed to unusual features in the behaviour of the scalar field. We also show that a conventional regularisation to calculate the Euclidean action leads to an expression for the entropy that disagrees with the Wald results. This seems likely to be due to ambiguities in the subtraction procedure. We also calculate the viscosity in the dual CFT, and show that the viscosity/entropy ratio can violate the η/S ≥ 1/(4π) bound for appropriate choices of the parameters.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, The Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [INSPIRE].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
P. Kovtun, D.T. Son and A.O. Starinets, Holography and hydrodynamics: diffusion on stretched horizons, JHEP 10 (2003) 064 [hep-th/0309213] [INSPIRE].
P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].
N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].
R.-G. Cai, Z.-Y. Nie and Y.-W. Sun, Shear viscosity from effective couplings of gravitons, Phys. Rev. D 78 (2008) 126007 [arXiv:0811.1665] [INSPIRE].
R.-G. Cai, Z.-Y. Nie, N. Ohta and Y.-W. Sun, Shear viscosity from Gauss-Bonnet gravity with a dilaton coupling, Phys. Rev. D 79 (2009) 066004 [arXiv:0901.1421] [INSPIRE].
R. Brustein, D. Gorbonos and M. Hadad, Wald’s entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling, Phys. Rev. D 79 (2009) 044025 [arXiv:0712.3206] [INSPIRE].
H.-S. Liu, H. Lü and C.N. Pope, Generalized Smarr formula and the viscosity bound for Einstein-Maxwell-dilaton black holes, Phys. Rev. D 92 (2015) 064014 [arXiv:1507.02294] [INSPIRE].
A. Buchel and J.T. Liu, Universality of the shear viscosity in supergravity, Phys. Rev. Lett. 93 (2004) 090602 [hep-th/0311175] [INSPIRE].
A. Buchel, On universality of stress-energy tensor correlation functions in supergravity, Phys. Lett. B 609 (2005) 392 [hep-th/0408095] [INSPIRE].
P. Benincasa, A. Buchel and R. Naryshkin, The shear viscosity of gauge theory plasma with chemical potentials, Phys. Lett. B 645 (2007) 309 [hep-th/0610145] [INSPIRE].
K. Landsteiner and J. Mas, The shear viscosity of the non-commutative plasma, JHEP 07 (2007) 088 [arXiv:0706.0411] [INSPIRE].
S. Cremonini, The shear viscosity to entropy ratio: a status report, Mod. Phys. Lett. B 25 (2011) 1867 [arXiv:1108.0677] [INSPIRE].
Y. Kats and P. Petrov, Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory, JHEP 01 (2009) 044 [arXiv:0712.0743] [INSPIRE].
M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity bound violation in higher derivative gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [INSPIRE].
M. Natsuume and M. Ohta, The shear viscosity of holographic superfluids, Prog. Theor. Phys. 124 (2010) 931 [arXiv:1008.4142] [INSPIRE].
J. Erdmenger, P. Kerner and H. Zeller, Non-universal shear viscosity from Einstein gravity, Phys. Lett. B 699 (2011) 301 [arXiv:1011.5912] [INSPIRE].
O. Ovdat and A. Yarom, A modulated shear to entropy ratio, JHEP 11 (2014) 019 [arXiv:1407.6372] [INSPIRE].
X.-H. Ge, Y. Ling, C. Niu and S.-J. Sin, Holographic transports and stability in anisotropic linear axion model, arXiv:1412.8346 [INSPIRE].
F.-W. Shu, The quantum viscosity bound in Lovelock gravity, Phys. Lett. B 685 (2010) 325 [arXiv:0910.0607] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, AdS 7 /CFT 6 , Gauss-Bonnet gravity and viscosity bound, JHEP 03 (2010) 087 [arXiv:0910.5347] [INSPIRE].
X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [INSPIRE].
C. Brans and R.H. Dicke, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. 124 (1961) 925 [INSPIRE].
G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, The galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
S.W. Hawking, Black hole explosions, Nature 248 (1974) 30 [INSPIRE].
S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
R.M. Wald, Black hole entropy is the 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].
A. Anabalon, A. Cisterna and J. Oliva, Asymptotically locally AdS and flat black holes in Horndeski theory, Phys. Rev. D 89 (2014) 084050 [arXiv:1312.3597] [INSPIRE].
M. Rinaldi, Black holes with non-minimal derivative coupling, Phys. Rev. D 86 (2012) 084048 [arXiv:1208.0103] [INSPIRE].
E. Babichev and C. Charmousis, Dressing a black hole with a time-dependent Galileon, JHEP 08 (2014) 106 [arXiv:1312.3204] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
H.-S. Liu and H. Lü, Scalar charges in asymptotic AdS geometries, Phys. Lett. B 730 (2014) 267 [arXiv:1401.0010] [INSPIRE].
H. Lü, C.N. Pope and Q. Wen, Thermodynamics of AdS black holes in Einstein-Scalar gravity, JHEP 03 (2015) 165 [arXiv:1408.1514] [INSPIRE].
H.-S. Liu, H. Lü and C.N. Pope, Thermodynamics of Einstein-Proca AdS black holes, JHEP 06 (2014) 109 [arXiv:1402.5153] [INSPIRE].
Z.-Y. Fan and H. Lü, SU(2)-Colored (A)dS black holes in conformal gravity, JHEP 02 (2015) 013 [arXiv:1411.5372] [INSPIRE].
Z.-Y. Fan and H. Lü, Thermodynamical first laws of black holes in quadratically-extended gravities, Phys. Rev. D 91 (2015) 064009 [arXiv:1501.00006] [INSPIRE].
H.-S. Liu and H. Lü, Thermodynamics of Lifshitz black holes, JHEP 12 (2014) 071 [arXiv:1410.6181] [INSPIRE].
W. Chen, H. Lü and C.N. Pope, Mass of rotating black holes in gauged supergravities, Phys. Rev. D 73 (2006) 104036 [hep-th/0510081] [INSPIRE].
L.F. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].
H. Lü, Y. Pang and C.N. Pope, AdS dyonic black hole and its thermodynamics, JHEP 11 (2013) 033 [arXiv:1307.6243] [INSPIRE].
D.D.K. Chow and G. Compère, Dyonic AdS black holes in maximal gauged supergravity, Phys. Rev. D 89 (2014) 065003 [arXiv:1311.1204] [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: 1509.07142
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
Feng, XH., Liu, HS., Lü, H. et al. Black hole entropy and viscosity bound in Horndeski gravity. J. High Energ. Phys. 2015, 176 (2015). https://doi.org/10.1007/JHEP11(2015)176
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2015)176