Abstract
Holographic complexity, in the guise of the Complexity = Volume prescription, comes equipped with a natural correspondence between its rate of growth and the average infall momentum of matter in the bulk. This Momentum/Complexity correspondence can be related to an integrated version of the momentum constraint of general relativity. In this paper we propose a generalization, using the full Codazzi equations as a starting point, which successfully accounts for purely gravitational contributions to infall momentum. The proposed formula is explicitly checked in an exact pp-wave solution of the vacuum Einstein equations.
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References
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
E. P. Verlinde, On the Origin of Gravity and the Laws of Newton, JHEP 04 (2011) 029 [arXiv:1001.0785] [INSPIRE].
L. Susskind, Why do Things Fall?, arXiv:1802.01198 [INSPIRE].
A. R. Brown, H. Gharibyan, A. Streicher, L. Susskind, L. Thorlacius and Y. Zhao, Falling Toward Charged Black Holes, Phys. Rev. D 98 (2018) 126016 [arXiv:1804.04156] [INSPIRE].
L. Susskind, Complexity and Newton’s Laws, Front. in Phys. 8 (2020) 262 [arXiv:1904.12819] [INSPIRE].
J. M. Magán, Black holes, complexity and quantum chaos, JHEP 09 (2018) 043 [arXiv:1805.05839] [INSPIRE].
H. W. Lin, J. Maldacena and Y. Zhao, Symmetries Near the Horizon, JHEP 08 (2019) 049 [arXiv:1904.12820] [INSPIRE].
A. Mousatov, Operator Size for Holographic Field Theories, arXiv:1911.05089 [INSPIRE].
J. L. F. Barbón, J. Martín-García and M. Sasieta, Momentum/Complexity Duality and the Black Hole Interior, JHEP 07 (2020) 169 [arXiv:1912.05996] [INSPIRE].
J. L. F. Barbón, J. Martin-Garcia and M. Sasieta, Proof of a Momentum/Complexity Correspondence, Phys. Rev. D 102 (2020) 101901 [arXiv:2006.06607] [INSPIRE].
L. Susskind and Y. Zhao, Complexity and Momentum, JHEP 03 (2020) 239 [arXiv:2006.03019] [INSPIRE].
S. Chapman, H. Marrochio and R. C. Myers, Holographic complexity in Vaidya spacetimes. Part I, JHEP 06 (2018) 046 [arXiv:1804.07410] [INSPIRE].
S. Chapman, H. Marrochio and R. C. Myers, Holographic complexity in Vaidya spacetimes. Part II, JHEP 06 (2018) 114 [arXiv:1805.07262] [INSPIRE].
L. Susskind and Y. Zhao, Switchbacks and the Bridge to Nowhere, arXiv:1408.2823 [INSPIRE].
L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690] [INSPIRE].
L. Susskind, Three Lectures on Complexity and Black Holes, SpringerBriefs in Physics, Springer, (2018), DOI [arXiv:1810.11563] [INSPIRE].
D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90 (2014) 126007 [arXiv:1406.2678] [INSPIRE].
D. A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [arXiv:1409.8180] [INSPIRE].
J. Couch, S. Eccles, T. Jacobson and P. Nguyen, Holographic Complexity and Volume, JHEP 11 (2018) 044 [arXiv:1807.02186] [INSPIRE].
E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, Cambridge University Press (2009), [DOI] [INSPIRE].
C. W. Misner, K. Thorne and J. Wheeler, Gravitation, W.H. Freeman, San Francisco U.S.A. (1973).
H. Bondi, F. Pirani and I. Robinson, Gravitational waves in general relativity III. Exact plane waves, Proc. Roy. Soc. Lond. A 251 (1959) 519.
J. Ehlers and W. Kundt, Exact solutions of the Gravitational Field Equations, in L. Witten ed., Gravitation: An Introduction to Current Research, John Wiley & Sons Inc., London U.K. (1962), pp. 49–101.
A. Einstein, Der energiesatz in der allgemeinen relativitätstheorie, in Albert Einstein: Akademie-Vorträge: Sitzungsberichte der Preußischen Akademie der Wissenschaften 1914–1932 1 (2006) 154.
L. Landau and E. Lifschitz, Course of Theoretical Physics. Volume II: The Classical Theory of Fields, Pergamon Press, Oxford U.K. (1975).
L. F. Abbott and S. Deser, Stability of Gravity with a Cosmological Constant, Nucl. Phys. B 195 (1982) 76 [INSPIRE].
M. A. G. Bonilla and J. Senovilla, M. M., Some Properties of the Bel and Bel-Robinson Tensors, Gen. Rel. Grav. 29 (1997) 91 [INSPIRE].
J. M. M. Senovilla, Superenergy tensors, Class. Quant. Grav. 17 (2000) 2799 [gr-qc/9906087] [INSPIRE].
M. A. G. Bonilla and J. M. M. Senovilla, Very Simple Proof of the Causal Propagation of Gravity in Vacuum, Phys. Rev. Lett. 78 (1997) 783 [INSPIRE].
J. L. F. Barbón and E. Rabinovici, Holographic complexity and spacetime singularities, JHEP 01 (2016) 084 [arXiv:1509.09291] [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].
P. Caputa and J. M. Magan, Quantum Computation as Gravity, Phys. Rev. Lett. 122 (2019) 231302 [arXiv:1807.04422] [INSPIRE].
A. Belin, A. Lewkowycz and G. Sárosi, Complexity and the bulk volume, a new York time story, JHEP 03 (2019) 044 [arXiv:1811.03097] [INSPIRE].
M. Flory and N. Miekley, Complexity change under conformal transformations in AdS3/CFT2, JHEP 05 (2019) 003 [arXiv:1806.08376] [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].
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].
J. Couch, W. Fischler and P. H. Nguyen, Noether charge, black hole volume, and complexity, JHEP 03 (2017) 119 [arXiv:1610.02038] [INSPIRE].
P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi and K. Watanabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett. 119 (2017) 071602 [arXiv:1703.00456] [INSPIRE].
D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman, A Universal Operator Growth Hypothesis, Phys. Rev. X 9 (2019) 041017 [arXiv:1812.08657] [INSPIRE].
J. L. F. Barbón, E. Rabinovici, R. Shir and R. Sinha, On The Evolution Of Operator Complexity Beyond Scrambling, JHEP 10 (2019) 264 [arXiv:1907.05393] [INSPIRE].
E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner, Operator complexity: a journey to the edge of Krylov space, arXiv:2009.01862 [INSPIRE].
S.-K. Jian, B. Swingle and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT gravity, JHEP 03 (2021) 014 [arXiv:2008.12274] [INSPIRE].
A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan and J. Simón, First Law of Holographic Complexity, Phys. Rev. Lett. 123 (2019) 081601 [arXiv:1903.04511] [INSPIRE].
A. Bernamonti, F. Galli, J. Hernandez, R. C. Myers, S.-M. Ruan and J. Simón, Aspects of The First Law of Complexity, arXiv:2002.05779 [INSPIRE].
A. R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018) 086015 [arXiv:1701.01107] [INSPIRE].
H. Jeffreys, On isotropic tensors, Math. Proc. Cambridge Phil. Soc. 73 (1973) 173.
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Barbón, J.L.F., Martín-García, J. & Sasieta, M. A generalized Momentum/Complexity correspondence. J. High Energ. Phys. 2021, 250 (2021). https://doi.org/10.1007/JHEP04(2021)250
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DOI: https://doi.org/10.1007/JHEP04(2021)250