Complexity Growth in Flat Spacetimes

TL;DR

This paper investigates the rate of complexity growth in holographic duals of flat spacetimes using the complexity equals action proposal, revealing dimension-dependent behaviors and bounds.

Contribution

It extends the complexity equals action framework to asymptotically flat spacetimes, deriving explicit late-time growth rates and their relation to Lloyd's bound.

Findings

For dimensions > 3, complexity growth approaches Lloyd's bound from above.

In 3D, the growth rate is constant and differs from Lloyd's bound by a logarithmic term.

The results match the flat-space limit of AdS complexity formulas.

Abstract

We use the complexity equals action proposal to calculate the rate of complexity growth for field theories that are the holographic duals of asymptotically flat spacetimes. To this aim, we evaluate the on-shell action of asymptotically flat spacetime on the Wheeler-DeWitt patch. This results in the same expression as can be found by taking the flat-space limit from the corresponding formula related to the asymptotically AdS spacetimes. For the bulk dimensions that are greater than three, the rate of complexity growth at late times approaches from above to Lloyd's bound. However, for the three-dimensional bulks, this rate is a constant and differs from Lloyd's bound by a logarithmic term.