Frenet-Serret formalism for null world lines

TL;DR

This paper extends the Frenet-Serret formalism to null world lines in general spacetimes, clarifies Fermi-Walker transport for these trajectories, and applies the framework to null circular orbits in Kerr spacetime, providing new insights and tools.

Contribution

It introduces a novel extension of the Frenet-Serret formalism to null trajectories using Newman-Penrose formalism and explores null geodesics in Kerr spacetime.

Findings

Recovered and clarified properties of null trajectories

Analyzed null circular orbits in Kerr spacetime

Provided methods for integrating parallel transport along null geodesics

Abstract

The Frenet-Serret curve analysis is extended from nonnull to null trajectories in a generic spacetime using the Newman-Penrose formalism, recovering old results which are not well known and clarifying the associated Fermi-Walker transport which has been left largely unexplored in the literature. This machinery is then used to discuss null circular orbits in stationary axisymmetric spacetimes using the Kerr spacetime as a concrete example, and to integrate the equations of parallel transport along null geodesics in any spacetime.