Towards a Reconstruction of General Bulk Metrics

TL;DR

The paper presents a covariant framework to reconstruct the bulk spacetime metric, up to a conformal factor, from boundary data via light-cone cuts associated with bulk points. It shows how the full conformal metric can be recovered locally from the geometry of these cuts and provides a field-theory route to determine the cuts through bulk-point singularities in boundary correlators, at least for points in the causal wedge. The authors also discuss extensions to obtain the conformal factor, behavior outside the causal wedge, and implications for subregion duality, while noting limitations in shadow regions near black holes. Overall, the work offers a novel, causality-based route to bulk reconstruction that is covariant and dimension-agnostic, complementing entanglement-based approaches.

Abstract

We prove that the metric of a general holographic spacetime can be reconstructed (up to an overall conformal factor) from distinguished spatial slices - "light-cone cuts" - of the conformal boundary. Our prescription is covariant and applies to bulk points in causal contact with the boundary. Furthermore, we describe a procedure for determining the light-cone cuts corresponding to bulk points in the causal wedge of the boundary in terms of the divergences of correlators in the dual field theory. Possible extensions for determining the conformal factor and including the cuts of points outside of the causal wedge are discussed. We also comment on implications for subregion/subregion duality.

Figures (10)

• Figure 1: The intersection of the lightcone (up to caustics) of a bulk point $p$ with the asymptotic boundary defines the past and future cuts of $p$, $C^{\pm}(p)$. The cuts are complete spatial slices of the asymptotic boundary.
• Figure 2: The point $p$ lies inside the event horizon (dotted line) of a collapsing star. The boundary of the future of $p$ never intersects the asymptotic boundary. $p$ has only a past cut. More generally, the interior of an event horizon lies at least partly inside the boundary's domain of influence, but (by definition) not within the boundary's causal wedge.
• Figure 3: Cuts corresponding to a null bulk geodesic. The cuts are all tangent at the point $x$ at which the null geodesic reaches the boundary.
• Figure 4: (a): $\partial J^{-}(p)$ will generally have caustics and some isolated $C^{0}$ points on the cut $C(p)$. At any regular point $x$, there is a null achronal geodesic $\gamma$ from $p$ all the way to $C(p)$. (b): In the space of cuts ${\cal M}$, a point $P$ corresponds to a cut $C(p)$; the null curve $\gamma$ of $\partial J^{-}(p)$ corresponds to a null curve $\gamma$, where points $Q$ on $\gamma$ are cuts $C(q)$ which are tangent to $C(p)$ at $x$.
• Figure 5: A Landau diagram of a bulk-point singularity in a 7-point function: $z_1$, $z_2$, and the $x_{i}$ are all null-separated from a bulk point $y$ so that high energy test particles from $z_1, z_2$ scatter at $y$, conserving energy and momentum. To find the past cut of $y$, we vary $z_1, z_2$ in a spatial direction while keeping the 7-point function singular.

Theorems & Definitions (6)

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