Spinning probes and helices in AdS$_3$

TL;DR

This paper explores extremal curves related to a torsion-dependent functional in AdS3, revealing their connection to spinning probes and helices, with implications for entanglement entropy and elastic structures.

Contribution

It derives shape equations for extremal curves in general spaces and characterizes their solutions as helices, especially in AdS3, linking geometric motions to spinning probes.

Findings

Solutions correspond to Mathisson's helical motions.

Lancret ratio relates to observer speeds.

Rich family of helices in AdS3 discovered.

Abstract

We study extremal curves associated with a functional which is linear in the curve's torsion. The functional in question is known to capture the properties of entanglement entropy for two-dimensional conformal field theories with chiral anomalies and has potential applications in elucidating the equilibrium shape of elastic linear structures. We derive the equations that determine the shape of its extremal curves in general ambient spaces in terms of geometric quantities. We show that the solutions to these shape equations correspond to a three-dimensional version of Mathisson's helical motions for the centers of mass of spinning probes. Thereafter, we focus on the case of maximally symmetric spaces, where solutions correspond to cylindrical helices and find that the Lancret ratio of these equals the relative speed between the Mathisson-Pirani and the Tulczyjew-Dixon observers. Finally, we construct all possible helical motions in three-dimensional manifolds with constant negative curvature. In particular, we discover a rich space of helices in AdS$_3$ which we explore in detail.