Complexity/Action duality of shock wave geometry in a massive gravity theory

TL;DR

This paper studies how shock wave geometries in massive gravity theories influence holographic complexity, revealing that graviton mass affects local shock wave dynamics but not global action growth, and extends black hole thermodynamics.

Contribution

It demonstrates the duality between complexity and action in massive gravity, showing the impact of graviton mass on local shock wave behavior and thermodynamics.

Findings

Global shock wave action growth unaffected by graviton mass.

Local shock wave complexity proportional to butterfly velocity, which is reduced in massive gravity.

Extended black hole thermodynamics with a valid Smarr formula.

Abstract

On the holographic complexity dual to the bulk action, we investigate the action growth for a shock wave geometry in a massive gravity theory within the Wheeler-De Witt (WDW) patch at the late time limit. For a global shock wave, the graviton mass does not affect the action growth in the bulk, i.e. the complexity on the boundary, showing that the action growth (complexity) is the same for both the Einstein gravity and the massive gravity. Nevertheless, for a local shock wave that depends on transverse coordinates, the action growth (complexity) is proportional to the butterfly velocity for the two gravity theories, but the butterfly velocity of the massive gravity theory is smaller than that of the Einstein gravity theory, indicating that the action growth (complexity) of the massive gravity is depressed by the graviton mass. In addition, we extend the black hole thermodynamics of the massive gravity and obtain the right Smarr formula.