# Butterfly Velocity Bound and Reverse Isoperimetric Inequality

## TL;DR

This paper derives a universal formula for butterfly velocities in AdS black holes, links thermodynamical volume to the reverse isoperimetric inequality, and shows the inequality constrains butterfly speeds under energy conditions.

## Contribution

It introduces a universal expression for butterfly velocities and a geometrical formula for thermodynamical volume, connecting these to the reverse isoperimetric inequality in AdS black holes.

## Key findings

- Butterfly velocities follow a universal formula involving thermodynamic quantities.
- The thermodynamical volume has a near-horizon geometrical expression.
- The reverse isoperimetric inequality is derived from the null-energy condition.

## Abstract

We study the butterfly effect of the AdS planar black holes in the framework of Einstein's general relativity. We find that the butterfly velocities can be expressed by a universal formula $v_{\rm B}^2 = TS/(2V_{\rm th} P)$. In doing so, we come upon a near-horizon geometrical formula for the thermodynamical volume $V_{\rm th}$. We verify the volume formula by examining a variety of AdS black holes. We also show that the volume formula implies that the conjectured reverse isoperimetric inequality follows straightforwardly from the null-energy condition, for static AdS black holes. The inequality is thus related to an upper bound of the butterfly velocities.