Entropy, extremality, euclidean variations, and the equations of motion

• Published in: The journal of high energy physics Vol. 2018; no. 1; pp. 1 - 33
• Main Authors: Dong, Xi, Lewkowycz, Aitor
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2018; Springer Nature B.V; Springer Berlin; SpringerOpen
• Subjects: AdS-CFT Correspondence; Black Holes in String Theory; Classical and Quantum Gravitation; Classical Theories of Gravity; Computation; Elementary Particles; Entropy; Equations of motion; Gauge-gravity correspondence; Gravitation; High energy physics; Mathematical analysis; Operators (mathematics); Physics; Physics and Astronomy; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Quantum entanglement; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Black Holes in String Theory; AdS-CFT Correspondence; Gauge-gravity correspondence; Classical Theories of Gravity
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP01(2018)081

A bstract We study the Euclidean gravitational path integral computing the Rényi entropy and analyze its behavior under small variations. We argue that, in Einstein gravity, the extremality condition can be understood from the variational principle at the level of the action, without having to solve explicitly the equations of motion. This set-up is then generalized to arbitrary theories of gravity, where we show that the respective entanglement entropy functional needs to be extremized. We also extend this result to all orders in Newton’s constant G N , providing a derivation of quantum extremality. Understanding quantum extremality for mixtures of states provides a generalization of the dual of the boundary modular Hamiltonian which is given by the bulk modular Hamiltonian plus the area operator, evaluated on the so-called modular extremal surface. This gives a bulk prescription for computing the relative entropies to all orders in G N . We also comment on how these ideas can be used to derive an integrated version of the equations of motion, linearized around arbitrary states.