A note about the torsion of null curves in the $3$-dimensional Minkowski spacetime and the Schwarzian derivative

TL;DR

This paper establishes a relationship between the torsion of null curves in 3D Minkowski space and the Schwarzian derivative, providing new characterizations of special null curves such as slant helices.

Contribution

It demonstrates that torsion of null curves equals the Schwarzian derivative of a specific function, offering novel insights into their geometric properties and classifications.

Findings

Torsion equals the Schwarzian derivative for null curves in Minkowski space.

Descriptions of slant helices and specific null curves with linear torsion.

Characterization of null curves with torsion proportional to the pseudo-arc parameter.

Abstract

The main topic of this paper is to show that in the 3-dimensional Minkowski spacetime, the torsion of a null curve is equal to the Schwarzian derivative of a certain function appearing in a description of the curve. As applications, we obtain descriptions of the slant helices, and null curves for which the torsion is of the form $\tau=-2\lambda s$, $s$ being the pseudo-arc parameter and $\lambda=\mathop{\rm constant}\not=0$.