Holographic Complexity in Vaidya Spacetimes I

TL;DR

This paper investigates the evolution of holographic complexity in Vaidya spacetimes using both CV and CA proposals, highlighting the importance of null boundary counterterms and finding late-time growth rates match eternal black holes.

Contribution

It introduces an action principle for null fluids in Vaidya geometries and emphasizes the necessity of boundary counterterms for accurate complexity calculations.

Findings

Late-time complexity growth rate matches eternal black holes.

Null boundary counterterm is essential for proper complexity description.

Both CV and CA proposals yield consistent late-time behavior.

Abstract

We examine holographic complexity in time-dependent Vaidya spacetimes with both the complexity$=$volume (CV) and complexity$=$action (CA) proposals. We focus on the evolution of the holographic complexity for a thin shell of null fluid, which collapses into empty AdS space and forms a (one-sided) black hole. In order to apply the CA approach, we introduce an action principle for the null fluid which sources the Vaidya geometries, and we carefully examine the contribution of the null shell to the action. Further, we find that adding a particular counterterm on the null boundaries of the Wheeler-DeWitt patch is essential if the gravitational action is to properly describe the complexity of the boundary state. For both the CV proposal and the CA proposal (with the extra boundary counterterm), the late time limit of the growth rate of the holographic complexity for the one-sided black hole is precisely the same as that found for an eternal black hole.