A unified framework for photon and massive particle hypersurfaces in stationary spacetimes

TL;DR

This paper develops a unified geometric framework for understanding photon and massive particle hypersurfaces in stationary spacetimes, linking them to Finsler geometry and providing existence results for particle trajectories.

Contribution

It introduces a unified approach connecting photon and massive particle hypersurfaces via Finsler structures and proves existence and multiplicity of particle solutions.

Findings

Characterization of hypersurfaces as photon or massive particle types

Existence of solutions connecting points to flow lines

Periodic solutions with non-constant projections

Abstract

We revisit the notion of massive particle hypersurfaces and place it within a unified framework alongside photon hypersurfaces in stationary spacetimes. More precisely, for Killing-invariant timelike hypersurfaces $T=\mathbb{R}\times S_0$, where $S_0$ is a smooth embedded surface in a spacelike slice $S$ of the stationary spacetime, we show that $T$ is a photon hypersurface or a massive particle hypersurface if and only if $S_0$ is totally geodesic with respect to certain associated Finsler structures on the slice: a Randers metric governing null geodesics and a Jacobi--Randers metric governing timelike solutions of the Lorentz force equation at fixed energy and charge-to-mass ratio. We also prove existence and multiplicity results for proper-time parametrized solutions of the Lorentz force equation with fixed energy and charge-to-mass ratio, either connecting a point to a flow line of the Killing vector field or having periodic, non-constant projection on $S$.