  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.
