Abstract
We study entanglement entropy in a particular tensor-scalar theory: Horndeski gravity. Our goal is two-fold: investigate the Lewkowycz-Maldacena proposal for entanglement entropy in the presence of a tensor-scalar coupling and address a puzzle existing in the literature regarding the thermal entropy of asymptotically AdS Horndeski black holes. Using the squashed cone method, i.e. turning on a conical singularity in the bulk, we derive the functional for entanglement entropy in Horndeski gravity. We analyze the divergence structure of the bulk equation of motion. Demanding that the leading divergence of the transverse component of the equation of motion vanishes we identify the surface where to evaluate the entanglement functional. We show that the surface obtained is precisely the one that minimizes said functional. By evaluating the entanglement entropy functional on the horizon we obtain the thermal entropy for Horndeski black holes; this result clarifies discrepancies in the literature. As an application of the functional derived we find the minimal surfaces numerically and study the entanglement plateaux.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].
D.N. Kabat, Black hole entropy and entropy of entanglement, Nucl. Phys. B 453 (1995) 281 [hep-th/9503016] [INSPIRE].
M. Hotta, T. Kato and K. Nagata, A comment on geometric entropy and conical space, Class. Quant. Grav. 14 (1997) 1917 [gr-qc/9611058] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
X. Dong, Holographic Entanglement Entropy for General Higher Derivative Gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].
J. Camps, Generalized entropy and higher derivative Gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Entanglement Entropy in Lovelock Gravities, JHEP 07 (2011) 109 [arXiv:1101.5781] [INSPIRE].
L.-Y. Hung, R.C. Myers and M. Smolkin, On Holographic Entanglement Entropy and Higher Curvature Gravity, JHEP 04 (2011) 025 [arXiv:1101.5813] [INSPIRE].
A. Bhattacharyya and M. Sharma, On entanglement entropy functionals in higher derivative gravity theories, JHEP 10 (2014) 130 [arXiv:1405.3511] [INSPIRE].
E. Caceres, M. Sanchez and J. Virrueta, Holographic Entanglement Entropy in Time Dependent Gauss-Bonnet Gravity, JHEP 09 (2017) 127 [arXiv:1512.05666] [INSPIRE].
A. Bhattacharyya, A. Kaviraj and A. Sinha, Entanglement entropy in higher derivative holography, JHEP 08 (2013) 012 [arXiv:1305.6694] [INSPIRE].
S.N. Solodukhin, Nonminimal coupling and quantum entropy of black hole, Phys. Rev. D 56 (1997) 4968 [hep-th/9612061] [INSPIRE].
F. Larsen and F. Wilczek, Renormalization of black hole entropy and of the gravitational coupling constant, Nucl. Phys. B 458 (1996) 249 [hep-th/9506066] [INSPIRE].
G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].
X.-M. Kuang and E. Papantonopoulos, Building a Holographic Superconductor with a Scalar Field Coupled Kinematically to Einstein Tensor, JHEP 08 (2016) 161 [arXiv:1607.04928] [INSPIRE].
G. Papallo and H.S. Reall, On the local well-posedness of Lovelock and Horndeski theories, Phys. Rev. D 96 (2017) 044019 [arXiv:1705.04370] [INSPIRE].
W.-J. Jiang, H.-S. Liu, H. Lü and C.N. Pope, DC Conductivities with Momentum Dissipation in Horndeski Theories, JHEP 07 (2017) 084 [arXiv:1703.00922] [INSPIRE].
X.-H. Feng, H.-S. Liu, W.-T. Lu and H. Lu, Horndeski Gravity and the Violation of Reverse Isoperimetric Inequality, arXiv:1705.08970 [INSPIRE].
M. Rinaldi, Black holes with non-minimal derivative coupling, Phys. Rev. D 86 (2012) 084048 [arXiv:1208.0103] [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].
X.-H. Feng, H.-S. Liu, H. Lü and C.N. Pope, Black Hole Entropy and Viscosity Bound in Horndeski Gravity, JHEP 11 (2015) 176 [arXiv:1509.07142] [INSPIRE].
R. Emparan, AdS/CFT duals of topological black holes and the entropy of zero energy states, JHEP 06 (1999) 036 [hep-th/9906040] [INSPIRE].
M. Minamitsuji, Causal structure in the scalar-tensor theory with field derivative coupling to the Einstein tensor, Phys. Lett. B 743 (2015) 272 [INSPIRE].
T. Kobayashi, H. Motohashi and T. Suyama, Black hole perturbation in the most general scalar-tensor theory with second-order field equations. II. The even-parity sector, Phys. Rev. D 89 (2014) 084042 [arXiv:1402.6740] [INSPIRE].
M.R. Mohammadi Mozaffar, A. Mollabashi, M.M. Sheikh-Jabbari and M.H. Vahidinia, Holographic Entanglement Entropy, Field Redefinition Invariance and Higher Derivative Gravity Theories, Phys. Rev. D 94 (2016) 046002 [arXiv:1603.05713] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, Extremality, Euclidean Variations and the Equations of Motion, arXiv:1705.08453 [INSPIRE].
R.-X. Miao and W.-z. Guo, Holographic Entanglement Entropy for the Most General Higher Derivative Gravity, JHEP 08 (2015) 031 [arXiv:1411.5579] [INSPIRE].
J. Camps and W.R. Kelly, Generalized gravitational entropy without replica symmetry, JHEP 03 (2015) 061 [arXiv:1412.4093] [INSPIRE].
M. Headrick and T. Takayanagi, A holographic proof of the strong subadditivity of entanglement entropy, Phys. Rev. D 76 (2007) 106013 [arXiv:0704.3719] [INSPIRE].
W. Nelson, A comment on black hole entropy in string theory, Phys. Rev. D 50 (1994) 7400 [hep-th/9406011] [INSPIRE].
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] [INSPIRE].
V.E. Hubeny, H. Maxfield, M. Rangamani and E. Tonni, Holographic entanglement plateaux, JHEP 08 (2013) 092 [arXiv:1306.4004] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
C. Martinez and J. Zanelli, Conformally dressed black hole in (2+1)-dimensions, Phys. Rev. D 54 (1996) 3830 [gr-qc/9604021] [INSPIRE].
M. Henneaux, C. Martinez, R. Troncoso and J. Zanelli, Black holes and asymptotics of 2+1 gravity coupled to a scalar field, Phys. Rev. D 65 (2002) 104007 [hep-th/0201170] [INSPIRE].
G. Giribet, M. Leoni, J. Oliva and S. Ray, Hairy black holes sourced by a conformally coupled scalar field in D dimensions, Phys. Rev. D 89 (2014) 085040 [arXiv:1401.4987] [INSPIRE].
M. Chernicoff et al., Black hole thermodynamics, conformal couplings and R 2 terms, JHEP 06 (2016) 159 [arXiv:1604.08203] [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: 1707.06322
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
Caceres, E., Mohan, R. & Nguyen, P.H. On holographic entanglement entropy of Horndeski black holes. J. High Energ. Phys. 2017, 145 (2017). https://doi.org/10.1007/JHEP10(2017)145
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2017)145