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

Abstract

Abstract: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.

Traversable wormholes via a double trace deformation

“…More generally, it was proposed in [18] (and proven in [17]) that, at general orders in 1/N , one should consider the entropy outside the quantum extremal surface, obtained by extremizing the total generalized entropy S gen = A/4G N +S bulk . When calculating the O(1) piece of the entropy, these two prescriptions agree on the value of the entropy but [16,18] argued that the location of the quantum extremal surface is also physically important, because it provides a natural boundary for how much of the bulk can be reconstructed from the CFT state on a single boundary.…”

One-loop universality of holographic codes

“…One should thus understand [19] to rely on having a bulk effective action valid at locally-measured energies below some bulk cutoff scale Λ. In particular, the operator L R is determined by applying the Lewkowycz-Maldacena procedure [29] to this effective action and so also depends on Λ; see [25][26][27][28] for treatments of higher derivative corrections. We will discuss this procedure in more detail in section 3, but for now we note that, although dynamical fluctuations below the cutoff Λ contribute to the expectation value of L R in (2.1), the procedure determining the form of L R is entirely classical and makes no reference to these fluctuations.…”

Marginally trapped surfaces and AdS/CFT

“…We expect the divergence to be in part determined by τ a , though it must be contracted with some one index object that contains information about the boundary region A. A natural candidate is the trace of the extrinsic curvature of ∂A, 26) where the i index runs over the boundary indices, the b index runs over the 2 vectors orthogonal to our boundary subregion (and contained in the boundary), and n is the normal to ∂A that points outwardly away from A. The only nonzero component is…”

“…This would presumably involve replacing the area of each leaf with the generalized entropy as in [9,23,25,26]. However, it is unclear just how the bulk entanglement term should be defined for bulk gravitons.…”

“…However, it is unclear just how the bulk entanglement term should be defined for bulk gravitons. While the arguments of [25] and [26] can be used to define this entanglement across an HRT surface, at least at present there is no general understanding of how to define such entanglement across a JHEP02(2018)049 general bulk surface -or even a general one that is marginally trapped. The issue is a classic one associated with the failure of the linearized graviton action to be gauge invariant on off-shell backgrounds such as those that would naturally be used in attempting to define this entanglement using the replica trick.…”