Subregion Action and Complexity

TL;DR

This paper investigates the on-shell action of black brane solutions in Einstein gravity across various spacetime subregions, revealing properties like subadditivity and introducing the concept of mutual complexity, with implications for holographic complexity measures.

Contribution

It introduces the analysis of subregion on-shell actions for black branes, proposes a new mutual complexity measure, and explores the role of boundary joints in the action's behavior.

Findings

On-shell action obeys subadditivity.

Joint points significantly influence the action's behavior.

Spacelike boundaries lead to linear time growth.

Abstract

We evaluate finite part of the on-shell action for black brane solutions of Einstein gravity on different subregions of spacetime enclosed by null boundaries. These subregions include the intersection of WDW patch with past/future interior and left/right exterior for a two sided black brane. Identifying the on-shell action on the exterior regions with subregion complexity one finds that it obeys subadditivity condition. This gives an insight to define a new quantity named mutual complexity. We will also consider certain subregion that is a part of spacetime which could be causally connected to an operator localized behind/outside the horizon. Taking into account all terms needed to have a diffeomorphism invariant action with a well-defined variational principle, one observes that the main contribution that results to a nontrivial behavior of the on-shell action comes from joint points where two lightlike boundaries (including horizon) intersect. A spacelike boundary gives rise to a linear time growth, while we have a classical contribution due to a timelike boundary that is given by the free energy.