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

Massive particle surfaces (MPS), introduced by Kobialko, Bogush and Gal'tsov, are timelike hypersurfaces confining the trajectories of charged massive particles with fixed Killing energy and charge-to-mass ratio, in analogy with the photon surfaces of Claudel, Virbhadra and Ellis. We study both classes in standard stationary spacetimes within a unified Finslerian framework. For Killing-invariant hypersurfaces $\mathbb{R}\times S_0$, with $S_0$ a hypersurface of a spacelike slice $S$, we prove that photon hypersurfaces and MPS are exactly those for which $S_0$ is totally geodesic with respect to the Fermat metric and to a Jacobi--Randers metric on $S$, respectively. For MPS we also give a gauge-invariant version, in terms of a Riemannian Jacobi metric and a tangency condition on the reduced magnetic field. For arbitrary timelike hypersurfaces, we give a coordinate-free formulation of the partial umbilicity characterization of MPS and, in dimension at least four and under a natural energy condition, we characterize when an MPS is quasi-umbilical in the sense of B.-Y. Chen. In the Som--Raychaudhuri spacetime of a rigidly rotating charged dust we exhibit MPS which are neither photon hypersurfaces nor quasi-umbilical. Finally, under completeness assumptions on the Jacobi--Randers metric, we prove existence and multiplicity results for proper-time parametrized solutions of the Lorentz force equation connecting a point to a flow line of the Killing field and, under compactness assumptions, existence of solutions with periodic spatial projection, possibly confined to an MPS.