Complexity and Shock Wave Geometries

TL;DR

This paper refines a conjecture linking the growth of Einstein-Rosen bridges to the complexity of dual quantum states, proposing a specific geometric measure that matches well with shock wave geometries.

Contribution

It introduces a precise geometric definition of complexity as the volume of a maximal surface in ERB, tested across various shock wave geometries.

Findings

Complexity proportional to ERB volume.

Agreement with shock wave geometry calculations.

Supports the geometric complexity conjecture.

Abstract

In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge to the computational complexity of the of the dual quantum state. Our refinement states that the complexity is proportional to the spatial volume of the ERB. More precisely, up to an ambiguous numerical coefficient, we propose that the complexity is the regularized volume of the largest codimension one surface crossing the bridge, divided by $G_N l_{AdS}$. We test this conjecture against a wide variety of spherically symmetric shock wave geometries in different dimensions. We find detailed agreement.