Abstract
We specialize the Wald formalism to derive the thermodynamical first law for static black holes with spherical/torus/hyperbolic symmetries in a variety of supergravities or supergravity-inspired theories involving multiple scalars and vectors. We apply the formula to study the first law of a general class of Lifshitz black holes. We analyse the first law of three exact Lifshitz black holes and the results fit the general pattern. In one example, the first law is TdS + ΦdQ = 0 where (Φ, Q) are the electric potential and charge of the Maxwell field. The unusual vanishing of mass in this specific solution demonstrates that super-extremal charged black holes can exist in asymptotic Lifshitz spacetimes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large- \( \mathcal{N} \) limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [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].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].
M. Taylor, Non-relativistic holography, arXiv:0812.0530 [INSPIRE].
S.S. Gubser and A. Nellore, Ground states of holographic superconductors, Phys. Rev. D 80 (2009) 105007 [arXiv:0908.1972] [INSPIRE].
S.A. Hartnoll and A. Tavanfar, Electron stars for holographic metallic criticality, Phys. Rev. D 83 (2011) 046003 [arXiv:1008.2828] [INSPIRE].
U.H. Danielsson and L. Thorlacius, Black holes in asymptotically Lifshitz spacetime, JHEP 03 (2009) 070 [arXiv:0812.5088] [INSPIRE].
R.B. Mann, Lifshitz topological black holes, JHEP 06 (2009) 075 [arXiv:0905.1136] [INSPIRE].
S.A. Hartnoll, J. Polchinski, E. Silverstein and D. Tong, Towards strange metallic holography, JHEP 04 (2010) 120 [arXiv:0912.1061] [INSPIRE].
K. Balasubramanian and K. Narayan, Lifshitz spacetimes from AdS null and cosmological solutions, JHEP 08 (2010) 014 [arXiv:1005.3291] [INSPIRE].
A. Donos and J.P. Gauntlett, Lifshitz solutions of D = 10 and D = 11 supergravity, JHEP 12 (2010) 002 [arXiv:1008.2062] [INSPIRE].
H. Singh, Special limits and non-relativistic solutions, JHEP 12 (2010) 061 [arXiv:1009.0651] [INSPIRE].
R. Gregory, S.L. Parameswaran, G. Tasinato and I. Zavala, Lifshitz solutions in supergravity and string theory, JHEP 12 (2010) 047 [arXiv:1009.3445] [INSPIRE].
A. Donos, J.P. Gauntlett, N. Kim and O. Varela, Wrapped M5-branes, consistent truncations and AdS/CMT, JHEP 12 (2010) 003 [arXiv:1009.3805] [INSPIRE].
D. Cassani and A.F. Faedo, Constructing Lifshitz solutions from AdS, JHEP 05 (2011) 013 [arXiv:1102.5344] [INSPIRE].
N. Halmagyi, M. Petrini and A. Zaffaroni, Non-relativistic solutions of N = 2 gauged supergravity, JHEP 08 (2011) 041 [arXiv:1102.5740] [INSPIRE].
H. Singh, Lifshitz/Schródinger Dp-branes and dynamical exponents, JHEP 07 (2012) 082 [arXiv:1202.6533] [INSPIRE].
H. Lü, Y. Pang, C.N. Pope and J.F. Vazquez-Poritz, AdS and Lifshitz black holes in conformal and Einstein-Weyl gravities, Phys. Rev. D 86 (2012) 044011 [arXiv:1204.1062] [INSPIRE].
D. Mallayev, J.F. Vazquez-Poritz and Z. Zhang, Marginal deformations of non-relativistic field theories, Phys. Rev. D 90 (2014) 106007 [arXiv:1309.3257] [INSPIRE].
H.-S. Liu and H. Lü, Lifshitz and Schrödinger vacua, superstar resolution in gauged maximal supergravities, JHEP 02 (2014) 122 [arXiv:1310.8348] [INSPIRE].
F.-W. Shu, K. Lin, A. Wang and Q. Wu, Lifshitz spacetimes, solitons and generalized BTZ black holes in quantum gravity at a Lifshitz point, JHEP 04 (2014) 056 [arXiv:1403.0946] [INSPIRE].
K. Lin, F.-W. Shu, A. Wang and Q. Wu, High-dimensional Lifshitz-type spacetimes, universal horizons and black holes in Hořava-Lifshitz gravity, arXiv:1404.3413 [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].
S. Gao, The first law of black hole mechanics in Einstein-Maxwell and Einstein-Yang-Mills theories, Phys. Rev. D 68 (2003) 044016 [gr-qc/0304094] [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, 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].
H. Lü, Y. Pang and C.N. Pope, AdS dyonic black hole and its thermodynamics, JHEP 11 (2013) 033 [arXiv:1307.6243] [INSPIRE].
D.-W. Pang, On charged Lifshitz black holes, JHEP 01 (2010) 116 [arXiv:0911.2777] [INSPIRE].
K. Balasubramanian and J. McGreevy, An analytic Lifshitz black hole, Phys. Rev. D 80 (2009) 104039 [arXiv:0909.0263] [INSPIRE].
Y. Gim, W. Kim and S.-H. Yi, The first law of thermodynamics in Lifshitz black holes revisited, JHEP 07 (2014) 002 [arXiv:1403.4704] [INSPIRE].
C. Charmousis, B. Gouteraux, B.S. Kim, E. Kiritsis and R. Meyer, Effective holographic theories for low-temperature condensed matter systems, JHEP 11 (2010) 151 [arXiv:1005.4690] [INSPIRE].
N. Iizuka, N. Kundu, P. Narayan and S.P. Trivedi, Holographic Fermi and non-Fermi liquids with transitions in dilaton gravity, JHEP 01 (2012) 094 [arXiv:1105.1162] [INSPIRE].
J. Tarrio and S. Vandoren, Black holes and black branes in Lifshitz spacetimes, JHEP 09 (2011) 017 [arXiv:1105.6335] [INSPIRE].
B. Gouteraux and E. Kiritsis, Generalized holographic quantum criticality at finite density, JHEP 12 (2011) 036 [arXiv:1107.2116] [INSPIRE].
L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B 85 (2012) 035121 [arXiv:1112.0573] [INSPIRE].
M. Alishahiha, E. O Colgain and H. Yavartanoo, Charged black branes with hyperscaling violating factor, JHEP 11 (2012) 137 [arXiv:1209.3946] [INSPIRE].
Z. Fan, Holographic fermions in asymptotically scaling geometries with hyperscaling violation, Phys. Rev. D 88 (2013) 026018 [arXiv:1303.6053] [INSPIRE].
Z. Fan, Dynamic Mott gap from holographic fermions in geometries with hyperscaling violation, JHEP 08 (2013) 119 [arXiv:1305.1151] [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: 1410.6181
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
Liu, HS., Lü, H. Thermodynamics of Lifshitz black holes. J. High Energ. Phys. 2014, 71 (2014). https://doi.org/10.1007/JHEP12(2014)071
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2014)071