Abstract
We employ the Noether procedure to derive a general formula for the radially conserved heat current in AdS planar black holes with certain transverse and traceless perturbations, for a general class of gravity theories. For Einstein gravity, the general higher-order Lovelock gravities and also a class of Horndeski gravities, we derive the boundary stress tensor and show that the resulting boundary heat current matches precisely the bulk Noether current.
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S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [INSPIRE].
S. Sachdev, What can gauge-gravity duality teach us about condensed matter physics?, Ann. Rev. Condensed Matter Phys. 3 (2012) 9 [arXiv:1108.1197] [INSPIRE].
J. McGreevy, TASI 2015 lectures on quantum matter (with a view toward holographic duality), arXiv:1606.08953 [INSPIRE].
J. Zaanen, Y.W. Sun, Y. Liu and K. Schalm, Holographic duality in condensed matter physics, Cambridge University Press, Cambridge U.K. (2015).
G.T. Horowitz, J.E. Santos and D. Tong, Optical conductivity with holographic lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].
G.T. Horowitz, J.E. Santos and D. Tong, Further evidence for lattice-induced scaling, JHEP 11 (2012) 102 [arXiv:1209.1098] [INSPIRE].
G.T. Horowitz and J.E. Santos, General relativity and the cuprates, JHEP 06 (2013) 087 [arXiv:1302.6586] [INSPIRE].
R.A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev. D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].
P. Chesler, A. Lucas and S. Sachdev, Conformal field theories in a periodic potential: results from holography and field theory, Phys. Rev. D 89 (2014) 026005 [arXiv:1308.0329] [INSPIRE].
M. Blake and D. Tong, Universal resistivity from holographic massive gravity, Phys. Rev. D 88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].
Y. Ling, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic lattice in Einstein-Maxwell-Dilaton gravity, JHEP 11 (2013) 006 [arXiv:1309.4580] [INSPIRE].
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
A. Donos and J.P. Gauntlett, Novel metals and insulators from holography, JHEP 06 (2014) 007 [arXiv:1401.5077] [INSPIRE].
A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities from black hole horizons, JHEP 11 (2014) 081 [arXiv:1406.4742] [INSPIRE].
A. Donos and J.P. Gauntlett, The thermoelectric properties of inhomogeneous holographic lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].
Y. Ling, P. Liu, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic superconductor on Q-lattice, JHEP 02 (2015) 059 [arXiv:1410.6761] [INSPIRE].
M. Baggioli and O. Pujolàs, Electron-phonon interactions, metal-insulator transitions and holographic massive gravity, Phys. Rev. Lett. 114 (2015) 251602 [arXiv:1411.1003] [INSPIRE].
L. Cheng, X.-H. Ge and Z.-Y. Sun, Thermoelectric DC conductivities with momentum dissipation from higher derivative gravity, JHEP 04 (2015) 135 [arXiv:1411.5452] [INSPIRE].
A. Donos and J.P. Gauntlett, Navier-Stokes equations on black hole horizons and DC thermoelectric conductivity, Phys. Rev. D 92 (2015) 121901 [arXiv:1506.01360] [INSPIRE].
E. Banks, A. Donos and J.P. Gauntlett, Thermoelectric DC conductivities and Stokes flows on black hole horizons, JHEP 10 (2015) 103 [arXiv:1507.00234] [INSPIRE].
A. Donos, J.P. Gauntlett, T. Griffin and L. Melgar, DC conductivity of magnetised holographic matter, JHEP 01 (2016) 113 [arXiv:1511.00713] [INSPIRE].
X.-H. Ge, Y. Tian, S.-Y. Wu and S.-F. Wu, Hyperscaling violating black hole solutions and Magneto-thermoelectric DC conductivities in holography, Phys. Rev. D 96 (2017) 046015 [arXiv:1606.05959] [INSPIRE].
X.-H. Ge, Y. Tian, S.-Y. Wu, S.-F. Wu and S.-F. Wu, Linear and quadratic in temperature resistivity from holography, JHEP 11 (2016) 128 [arXiv:1606.07905] [INSPIRE].
S. Cremonini, H.-S. Liu, H. Lü and C.N. Pope, DC conductivities from non-relativistic scaling geometries with momentum dissipation, JHEP 04 (2017) 009 [arXiv:1608.04394] [INSPIRE].
A. Donos, J.P. Gauntlett, T. Griffin and L. Melgar, DC conductivity and higher derivative gravity, Class. Quant. Grav. 34 (2017) 135015 [arXiv:1701.01389] [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].
M. Baggioli and W.-J. Li, Diffusivities bounds and chaos in holographic Horndeski theories, JHEP 07 (2017) 055 [arXiv:1705.01766] [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].
Y.-Z. Li, H.-S. Liu and H. Lü, Quasi-Topological Ricci Polynomial Gravities, arXiv:1708.07198 [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].
D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys. 12 (1971) 498 [INSPIRE].
G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].
J.W. York, Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
N. Deruelle, M. Sasaki, Y. Sendouda and D. Yamauchi, Hamiltonian formulation of f(Riemann) theories of gravity, Prog. Theor. Phys. 123 (2010) 169 [arXiv:0908.0679] [INSPIRE].
R.C. Myers, Higher derivative gravity, surface terms and string theory, Phys. Rev. D 36 (1987) 392 [INSPIRE].
J.T. Liu and W.A. Sabra, Hamilton-Jacobi counterterms for Einstein-Gauss-Bonnet gravity, Class. Quant. Grav. 27 (2010) 175014 [arXiv:0807.1256] [INSPIRE].
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].
C. Teitelboim and J. Zanelli, Dimensionally continued topological gravitation theory in Hamiltonian form, Class. Quant. Grav. 4 (1987) L125 [INSPIRE].
A. Cisterna and C. Erices, Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario, Phys. Rev. D 89 (2014) 084038 [arXiv:1401.4479] [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].
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].
X.-H. Feng, H.-S. Liu, H. Lü and C.N. Pope, Thermodynamics of charged black holes in Einstein-Horndeski-Maxwell theory, Phys. Rev. D 93 (2016) 044030 [arXiv:1512.02659] [INSPIRE].
E. Caceres, R. Mohan and P.H. Nguyen, On holographic entanglement entropy of Horndeski black holes, arXiv:1707.06322 [INSPIRE].
X.-H. Feng, H.-S. Liu, W.-T. Lu and H. Lü, Horndeski gravity and the violation of reverse isoperimetric inequality, arXiv:1705.08970 [INSPIRE].
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Liu, HS., Lü, H. & Pope, C. Holographic heat current as Noether current. J. High Energ. Phys. 2017, 146 (2017). https://doi.org/10.1007/JHEP09(2017)146
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DOI: https://doi.org/10.1007/JHEP09(2017)146