Abstract
In this paper, we study the physical significance of the thermodynamic volumes of AdS black holes using the Noether charge formalism of Iyer and Wald. After applying this formalism to study the extended thermodynamics of a few examples, we discuss how the extended thermodynamics interacts with the recent complexity = action proposal of Brown et al. (CA-duality). We, in particular, discover that their proposal for the late time rate of change of complexity has a nice decomposition in terms of thermodynamic quantities reminiscent of the Smarr relation. This decomposition strongly suggests a geometric, and via CA-duality holographic, interpretation for the thermodynamic volume of an AdS black hole. We go on to discuss the role of thermodynamics in complexity = action for a number of black hole solutions, and then point out the possibility of an alternate proposal, which we dub “complexity = volume 2.0”. In this alternate proposal the complexity would be thought of as the spacetime volume of the Wheeler-DeWitt patch. Finally, we provide evidence that, in certain cases, our proposal for complexity is consistent with the Lloyd bound whereas CA-duality is not.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
J.D. Bekenstein, Generalized second law of thermodynamics in black hole physics, Phys. Rev. D 9 (1974) 3292 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S.W. Hawking, Black Holes and Thermodynamics, Phys. Rev. D 13 (1976) 191 [INSPIRE].
D. Kastor, S. Ray and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. Quant. Grav. 26 (2009) 195011 [arXiv:0904.2765] [INSPIRE].
D. Kubiznak and R.B. Mann, Black hole chemistry, Can. J. Phys. 93 (2015) 999 [arXiv:1404.2126] [INSPIRE].
B.P. Dolan, Where is the PdV term in the fist law of black hole thermodynamics?, arXiv:1209.1272 [INSPIRE].
A.M. Frassino, R.B. Mann and J.R. Mureika, Lower-Dimensional Black Hole Chemistry, Phys. Rev. D 92 (2015) 124069 [arXiv:1509.05481] [INSPIRE].
D. Kubiznak, R.B. Mann and M. Teo, Black hole chemistry: thermodynamics with Lambda, Class. Quant. Grav. 34 (2017) 063001 [arXiv:1608.06147] [INSPIRE].
C.V. Johnson, Holographic Heat Engines, Class. Quant. Grav. 31 (2014) 205002 [arXiv:1404.5982] [INSPIRE].
E. Caceres, P.H. Nguyen and J.F. Pedraza, Holographic entanglement entropy and the extended phase structure of STU black holes, JHEP 09 (2015) 184 [arXiv:1507.06069] [INSPIRE].
A. Karch and B. Robinson, Holographic Black Hole Chemistry, JHEP 12 (2015) 073 [arXiv:1510.02472] [INSPIRE].
E. Caceres, P.H. Nguyen and J.F. Pedraza, Holographic entanglement chemistry, arXiv:1605.00595 [INSPIRE].
P.H. Nguyen, An equal area law for holographic entanglement entropy of the AdS-RN black hole, JHEP 12 (2015) 139 [arXiv:1508.01955] [INSPIRE].
D. Kastor, S. Ray and J. Traschen, Extended First Law for Entanglement Entropy in Lovelock Gravity, Entropy 18 (2016) 212 [arXiv:1604.04468] [INSPIRE].
D. Kastor, S. Ray and J. Traschen, Chemical Potential in the First Law for Holographic Entanglement Entropy, JHEP 11 (2014) 120 [arXiv:1409.3521] [INSPIRE].
P. Pradhan, Thermodynamic Products in Extended Phase Space, Int. J. Mod. Phys. D 26 (2016) 1750010 [arXiv:1603.07748] [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].
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].
T. Faulkner, M. Guica, T. Hartman, R.C. Myers and M. Van Raamsdonk, Gravitation from Entanglement in Holographic CFTs, JHEP 03 (2014) 051 [arXiv:1312.7856] [INSPIRE].
J. Lin, M. Marcolli, H. Ooguri and B. Stoica, Locality of Gravitational Systems from Entanglement of Conformal Field Theories, Phys. Rev. Lett. 114 (2015) 221601 [arXiv:1412.1879] [INSPIRE].
N. Lashkari and M. Van Raamsdonk, Canonical Energy is Quantum Fisher Information, JHEP 04 (2016) 153 [arXiv:1508.00897] [INSPIRE].
N. Lashkari, J. Lin, H. Ooguri, B. Stoica and M. Van Raamsdonk, Gravitational Positive Energy Theorems from Information Inequalities, PTEP 2016 (2016) 12C109 [arXiv:1605.01075] [INSPIRE].
B. Mosk, Holographic equivalence between the first law of entanglement entropy and the linearized gravitational equations, Phys. Rev. D 94 (2016) 126001 [arXiv:1608.06292] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].
A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action and black holes, Phys. Rev. D 93 (2016) 086006 [arXiv:1512.04993] [INSPIRE].
M. Alishahiha, Holographic Complexity, Phys. Rev. D 92 (2015) 126009 [arXiv:1509.06614] [INSPIRE].
D. Momeni, M. Faizal, S. Bahamonde and R. Myrzakulov, Holographic complexity for time-dependent backgrounds, Phys. Lett. B 762 (2016) 276 [arXiv:1610.01542] [INSPIRE].
D. Momeni, S.A.H. Mansoori and R. Myrzakulov, Holographic Complexity in Gauge/String Superconductors, Phys. Lett. B 756 (2016) 354 [arXiv:1601.03011] [INSPIRE].
O. Ben-Ami and D. Carmi, On Volumes of Subregions in Holography and Complexity, JHEP 11 (2016) 129 [arXiv:1609.02514] [INSPIRE].
M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett. 115 (2015) 261602 [arXiv:1507.07555] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys. 64 (2016) 24 [arXiv:1403.5695] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [INSPIRE].
M. Cvetič, G.W. Gibbons, D. Kubiznak and C.N. Pope, Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume, Phys. Rev. D 84 (2011) 024037 [arXiv:1012.2888] [INSPIRE].
B.P. Dolan, The compressibility of rotating black holes in D-dimensions, Class. Quant. Grav. 31 (2014) 035022 [arXiv:1308.5403] [INSPIRE].
L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries, Phys. Rev. D 94 (2016) 084046 [arXiv:1609.00207] [INSPIRE].
S. Lloyd, Ultimate physical limits to computation, Nature 406 (2000) 1047.
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
S. Aaronson, The Complexity of Quantum States and Transformations: From Quantum Money to Black Holes, arXiv:1607.05256 [INSPIRE].
M.R. Dowling and M.A. Nielsen, The geometry of quantum computation, quant-ph/0701004.
N. Margolus and L.B. Levitin, The Maximum speed of dynamical evolution, Physica D 120 (1998) 188 [quant-ph/9710043] [INSPIRE].
J.D. Bekenstein, A Universal Upper Bound on the Entropy to Energy Ratio for Bounded Systems, Phys. Rev. D 23 (1981) 287 [INSPIRE].
J.D. Bekenstein, How does the entropy/information bound work?, Found. Phys. 35 (2005) 1805 [quant-ph/0404042] [INSPIRE].
Y. Sekino and L. Susskind, Fast Scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang and R.-H. Peng, Action growth for AdS black holes, JHEP 09 (2016) 161 [arXiv:1606.08307] [INSPIRE].
H.-S. Liu and H. Lü, Thermodynamics of Lifshitz Black Holes, JHEP 12 (2014) 071 [arXiv:1410.6181] [INSPIRE].
W.G. Brenna, R.B. Mann and M. Park, Mass and Thermodynamic Volume in Lifshitz Spacetimes, Phys. Rev. D 92 (2015) 044015 [arXiv:1505.06331] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish and J. Sully, Integral Geometry and Holography, JHEP 10 (2015) 175 [arXiv:1505.05515] [INSPIRE].
B. Swingle, Entanglement Renormalization and Holography, Phys. Rev. D 86 (2012) 065007 [arXiv:0905.1317] [INSPIRE].
B. Czech, P. Hayden, N. Lashkari and B. Swingle, The Information Theoretic Interpretation of the Length of a Curve, JHEP 06 (2015) 157 [arXiv:1410.1540] [INSPIRE].
B. Czech et al., Tensor network quotient takes the vacuum to the thermal state, Phys. Rev. B 94 (2016) 085101 [arXiv:1510.07637] [INSPIRE].
L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole complementarity, Phys. Rev. D 48 (1993) 3743 [hep-th/9306069] [INSPIRE].
G. ’t Hooft, On the Quantum Structure of a Black Hole, Nucl. Phys. B 256 (1985) 727 [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: 1610.02038
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
Couch, J., Fischler, W. & Nguyen, P.H. Noether charge, black hole volume, and complexity. J. High Energ. Phys. 2017, 119 (2017). https://doi.org/10.1007/JHEP03(2017)119
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2017)119