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

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 $S$.

💡 Research Summary

The paper develops a unified geometric framework for photon surfaces and massive‑particle surfaces in stationary spacetimes, employing Finsler geometry to relate the intrinsic geometry of hypersurfaces to the dynamics of null and timelike (charged) trajectories.
First, the authors recall the standard stationary splitting of a spacetime ((M,g)) with a complete timelike Killing field (K). In adapted coordinates ((t,x)) the metric takes the form
(g=-\beta(x)(dt-\omega_x)^2+h_0(x)),
where (\beta>0), (\omega) is a one‑form, and (h_0) is a Riemannian metric on the spatial slice (S).
Using the Fermat principle, any future‑pointing null geodesic projects onto a pre‑geodesic of a Randers metric on (S):
(F_{\pm}(x,y)=\sqrt{h_0(y,y)}\pm\omega(y)).
Theorem 3.5 shows that a Killing‑invariant timelike hypersurface (T=\mathbb R\times S_0) is a photon surface (equivalently, totally umbilic) if and only if the spatial submanifold (S_0) is totally geodesic for both Randers metrics (F_{+}) and (F_{-}). Thus the classical umbilicity condition is recast as a Finsler‑geodesic condition.
Next the paper introduces a stationary electromagnetic potential (A) with field (F=dA) and fixes a charge‑to‑mass ratio (\rho=q/m). Charged particles obey the Lorentz force equation
(\nabla_{\dot\gamma}\dot\gamma=-\rho,\iota_{\dot\gamma}F^{\sharp}).
Because (A) is stationary, the quantity
(\varepsilon=-g(K,\dot\gamma)-\rho A(K))
is conserved along any solution. The “kinetic part” of the energy is (E_k=\varepsilon+\rho A(K)). By parametrising trajectories with proper time and fixing ((\rho,\varepsilon)), the authors rewrite the Lorentz‑force dynamics as a non‑relativistic Lagrangian system of Tonelli type. This system admits a Jacobi‑Randers metric (\tilde F) whose geodesics, after re‑parametrisation, coincide with the spatial projections of the original charged trajectories. Explicitly,
(\tilde F(x,y)=\sqrt{(E_k^2-\kappa^2)h_0(y,y)}+\rho,\omega(y)+\text{electric potential term}),
where (\kappa) encodes the projection of the Killing field onto the hypersurface.
Definition 4.4 introduces ((\rho,\varepsilon))-massive particle surfaces (((\rho,\varepsilon))-MPS): a timelike hypersurface (T) such that every admissible initial velocity (unit timelike, satisfying the energy condition) generates a Lorentz‑force trajectory that remains in (T). Theorem 4.6 provides an extrinsic characterization: for every admissible vector (u) tangent to (T), the second fundamental form satisfies (\Pi(u,u)=\rho,F(u,n)), where (n) is the unit normal of (T). This condition is equivalent to the tensorial identity (\Pi=H\kappa\otimes\kappa+\rho E_k F), with (H) the (\kappa)-orthogonal mean curvature and (\kappa) the projection of (K) onto (T).
Section 5 constructs the Jacobi‑Randers metric associated with a given ((\rho,\varepsilon)) and proves the main result (Theorem 5.5): a Killing‑invariant hypersurface (T=\mathbb R\times S_0) is a ((\rho,\varepsilon))-MPS if and only if the spatial slice (S_0) is totally geodesic for the Jacobi‑Randers metric (\tilde F). This unifies photon and massive‑particle surfaces under a single “total geodesicity” criterion in two different Finsler structures.
The authors then apply variational methods to obtain existence and multiplicity results for proper‑time parametrised solutions of the Lorentz‑force equation with fixed ((\rho,\varepsilon)). Theorem 5.7 guarantees a trajectory joining any prescribed point to a Killing flow line, while Theorem 5.10 (and Corollary 5.10) shows that on a compact spatial slice, for sufficiently large energy levels, there exists at least one non‑constant periodic trajectory whose spatial projection is closed. These results rely on the completeness of the Jacobi‑Randers metric and on standard critical point theory for Tonelli Lagrangians.
The appendix supplies a rigorous derivation of the correspondence between fixed‑energy solutions of the non‑relativistic electromagnetic system and geodesics of the Jacobi‑Randers metric, clarifying the relationship with earlier works by Weinstein, Routh reduction, and recent studies on charged particle dynamics.
Overall, the paper achieves a conceptual synthesis: photon surfaces (null geodesics) and massive‑particle surfaces (timelike, possibly charged, geodesics) are both characterised by the total geodesicity of an associated Finsler metric on the spatial slice. This perspective not only streamlines the geometric understanding of trapping surfaces in black‑hole spacetimes but also provides a robust analytical toolbox for proving existence of bound orbits for charged particles, with potential implications for astrophysical modeling of plasma dynamics near compact objects.