Bit threads, Einstein's equations and bulk locality

TL;DR

This paper introduces a new method for bulk metric reconstruction in holography using bit threads, emphasizing locality and differential form formalism, with explicit solutions for spherical regions and recursive perturbation techniques.

Contribution

It develops a novel approach to metric reconstruction based on bit threads, leveraging differential forms and the Iyer-Wald formalism for improved locality and explicit inversion methods.

Findings

Canonical thread configurations encode metric information locally.

Explicit inverse operators are derived for spherical regions.

Recursive perturbation inversion is feasible at higher orders.

Abstract

In the context of holography, entanglement entropy can be studied either by i) extremal surfaces or ii) bit threads, i.e., divergenceless vector fields with a norm bound set by the Planck length. In this paper we develop a new method for metric reconstruction based on the latter approach and show the advantages over existing ones. We start by studying general linear perturbations around the vacuum state. Generic thread configurations turn out to encode the information about the metric in a highly nonlocal way, however, we show that for boundary regions with a local modular Hamiltonian there is always a canonical choice for the perturbed thread configurations that exploits bulk locality. To do so, we express the bit thread formalism in terms of differential forms so that it becomes manifestly background independent. We show that the Iyer-Wald formalism provides a natural candidate for a canonical local perturbation, which can be used to recast the problem of metric reconstruction in terms of the inversion of a particular linear differential operator. We examine in detail the inversion problem for the case of spherical regions and give explicit expressions for the inverse operator in this case. Going beyond linear order, we argue that the operator that must be inverted naturally increases in order. However, the inversion can be done recursively at different orders in the perturbation. Finally, we comment on an alternative way of reconstructing the metric non-perturbatively by phrasing the inversion problem as a particular optimization problem.