Proof of a momentum/complexity correspondence

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.

A generalized Momentum/Complexity correspondence

“…with C µ ≡ e µ a C a . As shown in [10], a sufficient condition for this to happen is for Σ t to be topologically and conformally trivial. This includes single-boundary states with either spherical symmetry or in (2 + 1) dimensions (cf.…”

“…For sufficiently localized bulk states, this ansatz can be used to solve the box boundary condition (2.22) when we place the box at infinity in an asymptotically flat or AdS spacetime. As explained in [10] for the AdS case, the solution involves a C-field with asymptotic behavior…”

“…This paper is organized as follows. In section 2 we review the PVC correspondence introduced in [10] and explain its proposed generalization. In section 3 we test our proposal by an explicit calculation in the case of an exact gravitational wave background.…”

“…A recurrent idea since the early days of holography is a somewhat implicit relation between notions of 'complexity' of quantum states and the degree of 'gravitational clumpling' in the bulk [1,2]. Recently, these intuitions came gradually into focus in low-dimensional models [3][4][5][6][7][8] and other AdS/CFT examples [9][10][11][12][13], leading to momentum/complexity relations of the generic form dC dt ∼ P infall , (1.1) equating the rate of complexity growth with a suitable 'infall' momentum measuring the rate of gravitational clumping of matter in the bulk. The claimed ability to see 'deep in the bulk' is one of the prime motivations for the introduction of quantum complexity in general discussions of holography.…”

“…It was suggested in [9] that a fairly general form of the correspondence should exist, beyond the specific dynamical details of the various shell models. This expectation was born out by the results in [10], showing that an equation of the form (1.1) is always present, as a consequence of the momentum constraint of general relativity (GR), provided the complexity is defined with the Complexity=Volume prescription [14][15][16][17][18]. More precisely, if C is defined as an extremal codimension-one volume, one finds…”