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Gravitational Positive Energy Theorems from Information Inequalities

Lashkari, Nima and Lin, Jennifer and Ooguri, Hirosi and Stoica, Bogdan and Van Raamsdonk, Mark (2016) Gravitational Positive Energy Theorems from Information Inequalities. Progress of Theoretical and Experimental Physics, 2016 (12). Art. No. 12C109. ISSN 2050-3911. doi:10.1093/ptep/ptw139.

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In this paper we argue that classical asymptotically anti-de Sitter spacetimes that arise as states in consistent ultraviolet completions of Einstein gravity coupled to matter must satisfy an infinite family of positive energy conditions. To each ball-shaped spatial region B of the boundary spacetime we can associate a bulk spatial region Σ_B between B and the bulk extremal surface B with the same boundary as B. We show that there exists a natural notion of a gravitational energy for every such region that is non-negative, and non-increasing as one makes the region smaller. The results follow from identifying this gravitational energy with a quantum relative entropy in the associated dual conformal field theory state. The positivity and monotonicity properties of the gravitational energy are implied by the positivity and monotonicity of relative entropy, which holds universally in all quantum systems.

Item Type:Article
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URLURL TypeDescription Paper
Lashkari, Nima0000-0003-3446-5933
Ooguri, Hirosi0000-0001-6021-3778
Additional Information:© The Author(s) 2016. Published by Oxford University Press on behalf of the Physical Society of Japan. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. Funded by SCOAP3. Received May 25, 2016; Accepted August 4, 2016; Published December 19, 2016. We thank Xi Dong, Thomas Faulkner, Simon Gentle, Daniel Harlow, Ken Intriligator, Lampros Lamprou, Aitor Lewkowycz, Hong Liu, Juan Maldacena, Travis Maxfield, John McGreevy, Rob Myers, Ingmar Saberi, Jaewon Song, and Edward Witten for discussions. The research of MVR is supported in part by the Natural Sciences and Engineering Research Council of Canada, and by grant 376206 from the Simons Foundation. The research of HO and BS is supported in part by U.S. Department of Energy grant DE-SC0011632 and by Caltech's Walter Burke Institute for Theoretical Physics and Moore Center for Theoretical Cosmology and Physics. The research of HO is also supported in part by the Simons Investigator Award, by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, by JSPS Grant-in-Aid for Scientific Research C-26400240, and by JSPS Grant-in-Aid for Scientific Research on Innovative Areas 15H05895. NL is supported in part by funds provided by the MIT–Skoltech Initiative. JL acknowledges support from the Schmidt Fellowship and the U.S. Department of Energy. We acknowledge the hospitality of the Institute for Advanced Study, where HO was the Director's Visiting Professor in the fall of 2015. HO also acknowledges the hospitality of the Aspen Center for Physics, the Simons Center for Geometry and Physics, and the Center for Mathematical Sciences and Applications and the Center for the Fundamental Laws of Nature at Harvard University, where he is a visiting scholar in the spring 2016. BS thanks MIT, Stanford University, and the Simons Center for Geometry and Physics for their hospitality.
Group:Walter Burke Institute for Theoretical Physics
Funding AgencyGrant Number
Natural Sciences and Engineering Research Council of Canada (NSERC)UNSPECIFIED
Simons Foundation376206
Department of Energy (DOE)DE-SC0011632
Walter Burke Institute for Theoretical Physics, CaltechUNSPECIFIED
Ministry of Education, Culture, Sports, Science and Technology (MEXT)UNSPECIFIED
Japan Society for the Promotion of Science (JSPS)C-26400240
Japan Society for the Promotion of Science (JSPS)15H05895
MIT-Skoltech InitiativeUNSPECIFIED
Schmidt FellowshipUNSPECIFIED
Issue or Number:12
Classification Code:Subject Index: B21, B22, B24, E05, E76
Record Number:CaltechAUTHORS:20161220-092051390
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:72966
Deposited By: Tony Diaz
Deposited On:20 Dec 2016 19:03
Last Modified:11 Nov 2021 05:09

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