Proof of a momentum/complexity correspondence

J L F Barbón,J Martín-García,M Sasieta

doi:10.1103/physrevd.102.101901

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.

Highlights

Quantum complexity has been identified as a key notion in the development of the holographic dictionary for its promise to offer a peek into the interior of black holes [1]
Volume prescription, as a result of the momentum constraint in general relativity (GR). This PVC correspondence is based on two ingredients that were advanced in the thin-shell analysis of
[10]: the use of maximal-volume hypersurfaces as the time foliation to measure the momentum, and a particular choice of momentum component along the extremal surfaces, determined by an appropriate “infall field” CΣ

Summary

INTRODUCTION

Quantum complexity has been identified as a key notion in the development of the holographic dictionary for its promise to offer a peek into the interior of black holes [1]. Explicit momentum/complexity (PC) relations for states or operators have been described for low-dimensional models [5,6,7,8] and thin-shell approximations in higherdimensional models (cf for example [9,10,11]) In all these situations, the relevant radial momentum arises as some canonical momentum in an effective 1 þ 1 dimensional effective Lagrangian, and the detailed form of the PC correspondence depends to some extent on the coordinates chosen and the dynamical assumptions on the nature of the shells (massive dust, null dust, branes, etc.). In this paper we confirm this expectation, showing that the content of (1) is essentially the momentum constraint of general relativity (GR)

PVC FROM THE MOMENTUM CONSTRAINT

Exact PVC

A Newtonian interpretation

Obstructions

CONCLUSIONS AND OUTLOOK