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

Erasmo Caponio, Anna valeria Germinario, Antonio Masiello

TL;DR

This work develops a unified Finsler-geometric framework to characterize photon and massive-particle hypersurfaces in stationary spacetimes. By reducing null and timelike Lorentz-force dynamics to Randers and Jacobi--Randers metrics on a spacelike slice, it shows that a Killing-invariant hypersurface T = \mathbb{R} × S_0 is a photon surface or a massive-particle surface exactly when S_0 is totally geodesic for the respective Finsler metric; this yields a coordinate-free, intrinsic criterion for confinement of light and charged massive particles. The paper also derives a master equation linking energy, curvature, and field data, and proves existence and multiplicity results for proper-time Lorentz-force trajectories with fixed ε and ρ, including trajectories connecting a point to a flow line or exhibiting nontrivial periodic projections, with a Jacobi--Randers reduction guiding the variational construction. Overall, it provides a cohesive geometric bridge between photon trapping and massive-particle confinement in stationary spacetimes, enabling practical criteria and variational methods to analyze shadows, particle dynamics, and orbit patterns near compact objects via Finsler geometry.

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 $S$.

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

TL;DR

This work develops a unified Finsler-geometric framework to characterize photon and massive-particle hypersurfaces in stationary spacetimes. By reducing null and timelike Lorentz-force dynamics to Randers and Jacobi--Randers metrics on a spacelike slice, it shows that a Killing-invariant hypersurface T = \mathbb{R} × S_0 is a photon surface or a massive-particle surface exactly when S_0 is totally geodesic for the respective Finsler metric; this yields a coordinate-free, intrinsic criterion for confinement of light and charged massive particles. The paper also derives a master equation linking energy, curvature, and field data, and proves existence and multiplicity results for proper-time Lorentz-force trajectories with fixed ε and ρ, including trajectories connecting a point to a flow line or exhibiting nontrivial periodic projections, with a Jacobi--Randers reduction guiding the variational construction. Overall, it provides a cohesive geometric bridge between photon trapping and massive-particle confinement in stationary spacetimes, enabling practical criteria and variational methods to analyze shadows, particle dynamics, and orbit patterns near compact objects via Finsler geometry.

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 , where is a smooth embedded surface in a spacelike slice of the stationary spacetime, we show that is a photon hypersurface or a massive particle hypersurface if and only if 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 .
Paper Structure (12 sections, 15 theorems, 120 equations)

This paper contains 12 sections, 15 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Stationary spacetimes: standard form and Fermat principles
  3. Photon surfaces as totally geodesic surfaces
  4. Characterization of photon surfaces
  5. Massive particle surfaces at fixed charge-to-mass ratio and energy
  6. Extrinsic characterization of $(\rho,\varepsilon)$-MPS
  7. On some weak umbilicity notions
  8. MPS master equation
  9. Reduction of the Lorentz force equation to a non-relativistic electromagnetic system
  10. Massive particle surfaces as totally geodesic surfaces
  11. Solutions to the Lorentz force equation connecting a point to a line
  12. Jacobi-Randers metrics for non-relativistic electromagnetic systems

Key Result

Theorem 2.1

Let $p=(t_0,x_0)$ and $q=(t_1,x_1)$ with $t_1>t_0$. A future-pointing lightlike curve, $\gamma\colon [0,1]\to M$, $\gamma(s)= (t(s),x(s))$, from $p$ to the timelike line $\mathbb{R}\xspace\times\{x_1\}$ is a geodesic of $(M,g)$ if and only if $x$ is a pregeodesic of $F^+$ joining $x_0$ to $x_1$, par Analogously, if $t_1<t_0$, then for past--pointing lightlike curves one has to replace $F^+$ with $

Theorems & Definitions (43)

  • Theorem 2.1: Fermat principle cjm
  • Definition 3.1
  • Definition 3.2: Killing-invariant hypersurface
  • Remark 3.3: Reduction under the standard stationary splitting
  • Definition 3.4
  • Theorem 3.5: $\partial_t$-invariant photon surfaces
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.2
  • ...and 33 more