Butterflies on the Stretched Horizon

TL;DR

The paper argues that the ability of Alice to send messages to Bob across the Einstein-Rosen bridge is governed by the time evolution of computational complexity, linking precursor operators to the stretched-horizon geometry via a long-string model. It introduces a concrete criterion distinguishing easy versus hard operators based on how their complexity evolves, showing that successful signaling requires finely tuned future precursors whose complexity increases toward the future, with a concurrent complexity horizon shaping visibility of perturbations. The analysis reframes the AMPSS commutator debates and firewall arguments in terms of complexity growth, chaos, and state dependence, bridging UV/IR dynamics, scrambling, and holographic duality. It further explores very long-time behavior, including recurrences, and discusses implications for evaporating black holes and the robustness of smooth horizons within the ER=EPR framework.

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.

Figures (10)

• Figure 1: ADS and ADS with a unit black hole. The green region indicates the stretched horizon which extends to a distance $l_{ads}$ above the horizon.
• Figure 2: A localized disturbance on a string diffuses.
• Figure 3: The red curve is the outer edge of the stretched horizon. An observer at time $\omega$ looks back to point $\bf{a}.$ This defines a mapping from $\omega$ to a layer at distance $l_a.$
• Figure 4: The thermofield double. The red line is the outer boundary of the stretched horizon. The green lines are surfaces of constant time.
• Figure 5: The lower half of the diagram is fictitious.