Proof of a Momentum/Complexity Correspondence

TL;DR

This paper establishes a general Momentum/Complexity correspondence within holographic complexity, linking matter momentum flux to complexity variation, and demonstrates its exactness in specific gravitational solutions.

Contribution

It formalizes a general Momentum/Complexity correspondence based on Einstein's equations, extending previous work to broader classes of solutions and clarifying the role of gravitational radiation and topology.

Findings

Exact PC correspondence for 2+1 dimensional solutions

Extension to spherically symmetric solutions in arbitrary dimensions

Identification of topological obstructions like wormholes

Abstract

We show that the holographic Complexity = Volume proposal satisfies a very general notion of Momentum/Complexity correspondence (PC), based on the Momentum Constraint of General Relativity. It relates the rate of complexity variation with an appropriate matter momentum flux through spacelike extremal surfaces. This formalizes the intuitive idea that `gravitational clumpling' of matter increases complexity, and the required notion of `infall momentum' is shown to have a Newtonian avatar which expresses this idea. The proposed form of the PC correspondence is found to be exact for any solution of Einstein's equations in 2+1 dimensions, and any spherically symmetric solution in arbitrary dimensions, generalizing all previous calculations using spherical thin shells. Gravitational radiation enters through a correction which does not have a straightforward interpretation as a PC correspondence. Other obstructions to an exact PC duality have a topological origin and arise in the presence of wormholes.