A Generalized Momentum/Complexity Correspondence

TL;DR

This paper generalizes the Momentum/Complexity correspondence in holographic complexity by incorporating gravitational contributions through the Codazzi equations, validated in a vacuum Einstein pp-wave solution.

Contribution

It introduces a new formula based on the Codazzi equations that accounts for gravitational infall momentum in holographic complexity.

Findings

The generalized formula matches known solutions.

It extends the Momentum/Complexity correspondence to include gravitational effects.

Validated in an exact vacuum Einstein pp-wave solution.

Abstract

Holographic complexity, in the guise of the Complexity = Volume prescription, comes equipped with a natural correspondence between its rate of growth and the average infall momentum of matter in the bulk. This Momentum/Complexity correspondence can be related to an integrated version of the momentum constraint of general relativity. In this paper we propose a generalization, using the full Codazzi equations as a starting point, which successfully accounts for purely gravitational contributions to infall momentum. The proposed formula is explicitly checked in an exact pp-wave solution of the vacuum Einstein equations.