# Entropy, Extremality, Euclidean Variations, and the Equations of Motion

## TL;DR

This paper explores the variational principles behind entanglement entropy in gravity theories, generalizing extremality conditions to quantum regimes and providing a bulk method to compute relative entropies.

## Contribution

It demonstrates that extremality conditions for entanglement entropy can be derived from the variational principle without solving equations of motion, extending to quantum extremality and general gravity theories.

## Key findings

- Extremality condition derived from variational principle in Einstein gravity.
- Generalization of entanglement entropy extremization to arbitrary gravity theories.
- Bulk prescription for computing relative entropies to all orders in Newton's constant.

## Abstract

We study the Euclidean gravitational path integral computing the Renyi 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.