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Bit threads, Einstein’s equations and bulk locality

Agón, Cesar A., Cáceres, Elena, Pedraza, Juan F.·2021·Journal of High Energy Physics
DOI 10.1007/jhep01(2021)193
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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.

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Figures

Figure 1 A solution to the maximin problem γ A is naturally accompanied by a specific choice of a codimension-one slice Σ on which γ A is a minimal area surface. Such a slice is highly non-unique, however, not all slices that pass through γ A are allowed. Left: in this example Σ is perturbatively close to the null congruence shot out from γ A . In this case the minimal area surface ˜ γ A (orange curve) does not coincide with γ A (red curve). Right: for perturbative excited states, it can be shown that γ A is a minimal area surface on any slice Σ λ that is perturbatively close to Σ 0 . This means that we can pick any of these surfaces, and in particular Σ 0 , to construct relevant bit thread configurations.
Figure 1 A solution to the maximin problem γ A is naturally accompanied by a specific choice of a codimension-one slice Σ on which γ A is a minimal area surface. Such a slice is highly non-unique, however, not all slices that pass through γ A are allowed. Left: in this example Σ is perturbatively close to the null congruence shot out from γ A . In this case the minimal area surface ˜ γ A (orange curve) does not coincide with γ A (red curve). Right: for perturbative excited states, it can be shown that γ A is a minimal area surface on any slice Σ λ that is perturbatively close to Σ 0 . This means that we can pick any of these surfaces, and in particular Σ 0 , to construct relevant bit thread configurations.CC-BY · From the PDF
Figure 2 Contour plot for the magnitude | v | of the geodesic flow given in (3.4), in d = 2 dimensions (i.e., empty AdS 3 ). The contours correspond to the level set surfaces of v , which are all homologous to A and, in particular, include γ A as one of its members. This implies that this vector field v can indeed be used as a seed to generate a good solution v λ in a perturbative excited state.
Figure 2 Contour plot for the magnitude | v | of the geodesic flow given in (3.4), in d = 2 dimensions (i.e., empty AdS 3 ). The contours correspond to the level set surfaces of v , which are all homologous to A and, in particular, include γ A as one of its members. This implies that this vector field v can indeed be used as a seed to generate a good solution v λ in a perturbative excited state.CC-BY · From the PDF
Figure 3 Algorithm to reconstruct the full bulk metric, including its time components δg tt and δg ti . In the left panel we show the components of the metric that can be recovered on a generic boosted slice Σ ′ | β . Combining the data obtained from multiple boosted slices it is possible to recover the full metric on the locus of points in the intersection between the original slice Σ and the new slices Σ ′ , depicted in red. In the right panel we show new slices Σ ′′ | β,σ that are obtained by a simple translation of the original boosted slices Σ ′ . The extra data obtained from the inversion problem in these new slices is sufficient to recover all components of the metric in the full slice Σ .
Figure 3 Algorithm to reconstruct the full bulk metric, including its time components δg tt and δg ti . In the left panel we show the components of the metric that can be recovered on a generic boosted slice Σ ′ | β . Combining the data obtained from multiple boosted slices it is possible to recover the full metric on the locus of points in the intersection between the original slice Σ and the new slices Σ ′ , depicted in red. In the right panel we show new slices Σ ′′ | β,σ that are obtained by a simple translation of the original boosted slices Σ ′ . The extra data obtained from the inversion problem in these new slices is sufficient to recover all components of the metric in the full slice Σ .CC-BY · From the PDF
Figure 4 Perturbed vector field obtained using the unperturbed geodesic construction. Panels (a) and (b) illustrate the vector lines for R = 1 , λ = . 1 , t = . 25 and t = . 5 respectively. Panels (c) and (d) illustrate the corresponding vector norms.
Figure 4 Perturbed vector field obtained using the unperturbed geodesic construction. Panels (a) and (b) illustrate the vector lines for R = 1 , λ = . 1 , t = . 25 and t = . 5 respectively. Panels (c) and (d) illustrate the corresponding vector norms.CC-BY · From the PDF
Figure 5 Perturbed vector field obtained using the level set construction. Panels (a) and (b) illustrate the vector lines for R = 1 , λ = . 02 , t = 0 and t = 1 / 2 respectively. Panels (c) and (d) illustrate the corresponding vector norms.
Figure 5 Perturbed vector field obtained using the level set construction. Panels (a) and (b) illustrate the vector lines for R = 1 , λ = . 02 , t = 0 and t = 1 / 2 respectively. Panels (c) and (d) illustrate the corresponding vector norms.CC-BY · From the PDF
Figure 6 Perturbed vector field obtained using the canonical bit thread construction that relies on the Iyer-Wald formalism. Panels (a) and (b) illustrate the vector lines for R = 1 , λ = . 02 , t = 0 and t = 1 / 2 respectively. Panels (c) and (d) illustrate the corresponding vector norms.
Figure 6 Perturbed vector field obtained using the canonical bit thread construction that relies on the Iyer-Wald formalism. Panels (a) and (b) illustrate the vector lines for R = 1 , λ = . 02 , t = 0 and t = 1 / 2 respectively. Panels (c) and (d) illustrate the corresponding vector norms.CC-BY · From the PDF
Figure 7 Panels (a), (b), (c) and (d) show δv obtained using the level set construction for t = 0 , . 5 , 1 , 1 . 5 respectively. We have taken λ = 0 . 02 , R = 1 .
Figure 7 Panels (a), (b), (c) and (d) show δv obtained using the level set construction for t = 0 , . 5 , 1 , 1 . 5 respectively. We have taken λ = 0 . 02 , R = 1 .CC-BY · From the PDF

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Citations

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Metadata

Authors
Agón, C. A., Cáceres, E., & Pedraza, J. F.
ORCID
Pedraza, Juan F.
Venue
Journal of High Energy Physics
Publisher
Springer Science and Business Media LLC
Published
2021-01-28
Volume
2021
Issue
1
ISSN
1029-8479
DOI
10.1007/jhep01(2021)193
Cited by
34
References
3
Language
en
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