Butterflies on the Stretched Horizon

TL;DR

This paper explores how perturbations on one side of an Einstein-Rosen bridge can send messages to the other side, linking the difficulty of operators to the complexity growth at the black hole's stretched horizon.

Contribution

It proposes that the distinction between easy and hard operators is related to the evolution of computational complexity at the stretched horizon, offering new insights into black hole information transfer.

Findings

Complexity growth correlates with message-sending capability.

AMPSS commutator is more related to butterfly effects than firewalls.

Difficulty of operators is tied to their complexity evolution over time.

Abstract

In this paper I return to the question of what kind of perturbations on Alice's side of an Einstein-Rosen bridge can send messages to Bob as he enters the horizon at the other end. By definition "easy" operators do not activate messages and "hard" operators do, but there are no clear criteria to identify the difference between easy and hard. In this paper I argue that the difference is related to the time evolution of a certain measure of computational complexity, associated with the stretched horizon of Alice's black hole. The arguments suggest that the AMPSS commutator argument is more connected with butterflies than with firewalls.