Some examples of the behaviour of conformal geodesics

TL;DR

This paper explores the behavior of conformal geodesics in curved spacetimes, examining whether they can spiral into smaller regions, and provides specific examples and conditions where such behavior is excluded.

Contribution

It offers concrete examples and conditions under which conformal geodesics do not spiral, including a reduction of equations in a particular non-conformally flat metric.

Findings

Certain conformal geodesic behaviors are ruled out in specific settings

Reduction of conformal-geodesic equations to quadratures in a non-conformally flat metric

Examples illustrating possible and impossible behaviors of conformal geodesics

Abstract

With the aid of concrete examples, we consider the question of whether, in the presence of conformal curvature, a conformal geodesic can become trapped in smaller and smaller sets, or phrased informally: are spirals possible? We do not arrive at a definitive answer, but we are able to find situations where this behaviour is ruled out, including a reduction of the conformal-geodesic equations to quadratures in a specific non-conformally flat metric.