# A new energy upper bound for AdS black holes inspired by free field theory

## TL;DR

This paper proves a non-perturbative energy upper bound for deformed AdS black holes with flat torus boundaries, inspired by free field theory results, indicating energy increases with boundary deformations.

## Contribution

It establishes a non-perturbative energy upper bound for static deformations of AdS-Schwarzschild black holes with flat boundary metrics, inspired by free field theory.

## Key findings

- Deformed boundary metrics lead to lower energy in the bulk solution.
- The energy of the bulk solution is less than that of the undeformed AdS-Schwarzschild.
- The result is non-perturbative and applies under fixed temperature and area.

## Abstract

We consider the toroidally compactified planar AdS-Schwarzschild solution to 4-dimensional gravity with negative cosmological constant. This has a flat torus conformal boundary metric. We show that if the spatial part of the boundary metric is deformed, keeping it static and the temperature and area fixed, then assuming a static bulk solution exists, its energy is less than that of the AdS-Schwarzschild solution. The proof is non-perturbative in the metric deformation. While we expect the same holds for the free energy for black hole solutions we are so far are not able to prove it. In the context of AdS-CFT this implies a 3-dimensional holographic CFT on a flat spatial torus whose bulk dual is AdS-Schwarzschild has a greater energy than if the spatial geometry is deformed in any way that preserves temperature and area. This work was inspired by previous results in free field theory, where scalars and fermions in 3-dimensions have been shown to energetically disfavour flat space.