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The Geodesics Less Traveled: Nonminimal RT Surfaces and Holographic Scattering

Jacqueline Caminiti, Caroline Lima, Robert C. Myers

TL;DR

This work clarifies when bulk scattering in holography can be diagnosed purely from boundary entanglement. It proves that in pure $AdS_3$ the 2-to-2 converse to the connected wedge theorem holds: a connected entanglement wedge is necessary and sufficient for bulk $2\to2$ scattering, and this extends to a broad class of asymptotically $AdS_3$ vacuum geometries via local AdS$_3$ regions. The analysis, including a shell model with interior AdS$_3$ and exterior conical defects, shows that matter can disrupt this tight relation, with simple counterexamples where nonminimal RT surface inequalities fail to imply scattering. For higher $n$-to-$n$ scattering, the authors show there is no general converse even in pure $AdS_3$: the threshold conditions do not align with RT-phase boundaries except in select cases (e.g., $n=2$, or $n=3$ with special angular arrangements). Overall, the results delineate the limits of the CWT as a diagnostic for bulk scattering and motivate refined boundary-entanglement diagnostics, potentially through focusing arguments or new surface constructs like a proposed $ ilde{u}$, with implications for entwinement and holographic reconstruction.

Abstract

The connected wedge theorem states that in order to have a scattering process in the bulk, it is necessary to have $O(1/G_N)$ mutual information between certain "decision" regions in the boundary theory. While this large mutual information is not generally sufficient to imply scattering, arxiv:2404.15400 showed that for a certain class of geometries, bulk scattering is implied by a certain relation between two (possibly non-minimal) Ryu-Takayanagi surfaces. Here, we show that the 2-to-2 version of the theorem becomes an equivalence in pure AdS$_3$: large mutual information between appropriate boundary subregions is both necessary and sufficient for bulk scattering. This result allows us to extend the findings of arxiv:2404.15400 to a broader class of asymptotically AdS$_3$ spacetimes, which we illustrate with the spinning conical defect geometry. In contrast, we find that matter sources can disrupt this converse relation, and that the $n$-to-$n$ version of the theorem with $n>2$ lacks a converse even in the AdS$_3$ vacuum.

The Geodesics Less Traveled: Nonminimal RT Surfaces and Holographic Scattering

TL;DR

This work clarifies when bulk scattering in holography can be diagnosed purely from boundary entanglement. It proves that in pure the 2-to-2 converse to the connected wedge theorem holds: a connected entanglement wedge is necessary and sufficient for bulk scattering, and this extends to a broad class of asymptotically vacuum geometries via local AdS regions. The analysis, including a shell model with interior AdS and exterior conical defects, shows that matter can disrupt this tight relation, with simple counterexamples where nonminimal RT surface inequalities fail to imply scattering. For higher -to- scattering, the authors show there is no general converse even in pure : the threshold conditions do not align with RT-phase boundaries except in select cases (e.g., , or with special angular arrangements). Overall, the results delineate the limits of the CWT as a diagnostic for bulk scattering and motivate refined boundary-entanglement diagnostics, potentially through focusing arguments or new surface constructs like a proposed , with implications for entwinement and holographic reconstruction.

Abstract

The connected wedge theorem states that in order to have a scattering process in the bulk, it is necessary to have mutual information between certain "decision" regions in the boundary theory. While this large mutual information is not generally sufficient to imply scattering, arxiv:2404.15400 showed that for a certain class of geometries, bulk scattering is implied by a certain relation between two (possibly non-minimal) Ryu-Takayanagi surfaces. Here, we show that the 2-to-2 version of the theorem becomes an equivalence in pure AdS: large mutual information between appropriate boundary subregions is both necessary and sufficient for bulk scattering. This result allows us to extend the findings of arxiv:2404.15400 to a broader class of asymptotically AdS spacetimes, which we illustrate with the spinning conical defect geometry. In contrast, we find that matter sources can disrupt this converse relation, and that the -to- version of the theorem with lacks a converse even in the AdS vacuum.

Paper Structure

This paper contains 19 sections, 2 theorems, 148 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Summary of notation
  3. The connected wedge theorem in pure AdS3
  4. The connected wedge theorem
  5. The converse in pure AdS: 2-to-2 case
  6. The generalized u<d converse
  7. Converse with matter?
  8. Shell geometry
  9. RT surfaces
  10. A simple counterexample to u<d
  11. No converse for n-to-n scattering
  12. The n-to-n CWT
  13. 3-to-3 in pure AdS3
  14. n-to-n in pure AdS3
  15. n-to-n in conical defect background
  16. ...and 4 more sections

Key Result

Theorem 1

Pick four regions ${\hat{\mathcal{C}}}_1$, ${\hat{\mathcal{C}}}_2$, ${\hat{\mathcal{R}}}_1$, ${\hat{\mathcal{R}}}_2$ on the boundary of an asymptotically AdS$_3$ spacetime. From these, define the decision regions Assume that ${\hat{\mathcal{C}}}_i \subseteq \hat{\mathcal{V}}_i$. Assume also that the bulk geometry obeys the null energy condition and the HRRT surface can be found using the maximin

Figures (16)

  • Figure 1: (a) By the CWT, a bulk-only scattering process from $c_1, c_2$ to $r_1, r_2$ requires associated boundary regions (green) to be strongly correlated, as captured by the connectivity of the entanglement wedge. (b) When there is no bulk-only scattering, $\hat{\mathcal{V}}_1$ and $\hat{\mathcal{V}}_2$ need not be strongly correlated. In particular, they can have disconnected minimal-area extremal surfaces. This figure is reproduced from May:2019yxi.
  • Figure 2: The 2-to-2 scattering setup on the boundary cylinder, which was considered in Caminiti:2024ctd. Both input regions, $\hat{\mathcal{V}}_1$ and $\hat{\mathcal{V}}_2$, are on a fixed time slice, have the same size $x$, and their midpoints are separated by an angle $\theta$. The output regions, $\hat{\mathcal{R}}_1$ and $\hat{\mathcal{R}}_2$, are maximized to cover an entire Cauchy slice. Note that the vertical edges on the left and and right are identified.
  • Figure 3: A constant-$t$ slice of the conical defect geometry showing, from left to right, the $d$, $u$, and $o$ candidates for $\mathcal{E}(A\cup B)$. Figure copied from Caminiti:2024ctd.
  • Figure 4: Scattering regions in two-dimensional Minkowski spacetime. While $\hat{\mathcal{R}}_1$ is a finite causal diamond, $\hat{\mathcal{R}}_2$ consists of two semi-infinite causal diamonds.
  • Figure 5: The $u$ entanglement wedge is the codimension-0 bulk subregion bounded by the purple lightsheets. Its topmost ridge is an RT candidate for $\hat{\mathcal{R}}_1$, and by assumption, the minimal RT candidate. The $u$ entanglement wedge contains the scattering region described by the final expression in eq. \ref{['eq:coness']}. To see why, first note that the two future-most purple lightsheets coincide with $\partial J^-(\gamma_{\hat{\mathcal{R}}_1})$. Then, note the scattering region does not extend below the past-most purple lightsheets because the scattering region is cut off by $J^+(c_1) \cap J^+(c_2)$.
  • ...and 11 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2: $n$-to-$n$ CWT