The Tortoise and the Hare: A Causality Puzzle in AdS/CFT

TL;DR

This paper investigates a causality puzzle in AdS/CFT, demonstrating that the finite size of encoding regions prevents paradoxes and introduces a tortoise coordinate to measure nonlocality, with implications for bulk null geodesics.

Contribution

It proves that finite encoding regions in AdS/CFT prevent causality violations and introduces a tortoise coordinate to quantify nonlocality in the encoding.

Findings

Finite encoding regions prevent causality paradoxes.

A tortoise coordinate measures nonlocality of encoding.

Critical encoding avoids causality violation in null geodesics.

Abstract

We pose and resolve a holographic puzzle regarding an apparent violation of causality in AdS/CFT. If a point in the bulk of $AdS$ moves at the speed of light, the boundary subregion that encodes it may need to move superluminally to keep up. With $AdS_3$ as our main example, we prove that the finite extent of the encoding regions prevents a paradox. We show that the length of the minimal-size encoding interval gives rise to a tortoise coordinate on $\mathrm{AdS}$ that measures the nonlocality of the encoding. We use this coordinate to explore circular and radial motion in the bulk before passing to the analysis of bulk null geodesics. For these null geodesics, there is always a critical encoding where the possible violation of causality is barely avoided. We show that in any other encoding, the possible violation is subcritical.