Bit threads, Einstein’s equations and bulk locality

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