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Static spacetimes with a Finsler angular sector

Erasmo Caponio

TL;DR

The paper develops static Lorentz--Finsler spacetimes with a Randers angular sector of constant positive flag curvature to analyze photon-sphere physics and black-hole solutions. It derives a model-independent photon-sphere condition $r_C\nu'(r_C)=2$ and, for Randers angular sectors, explicit expressions for the axial charge and critical impact parameter, along with a Finslerian Sagnac delay. The work also shows how Finslerian non-reversibility affects orbit equations and clock-based tests, and critically revisits recent hairy-black-hole results, showing their equivalence to earlier Ovalle constructions after appropriate rescalings. Collectively, the results illuminate how direction-dependent angular geometry modifies null geodesics while leaving the radial sector to GR-like dynamics, and they provide a concrete diagnostic to distinguish static Lorentz--Finsler from stationary Lorentzian spacetimes via timelike clocks.

Abstract

We consider static spacetimes in spherical coordinates whose angular sector is described by a Finsler metric rather than the standard round metric on $S^2$. Our first contribution is kinematical: maintaining arbitrary lapse and radial factors $e^{ν(r)}$, $e^{\vartheta(r)}$, and relying solely on Killing symmetries and the null constraint, we derive model--independent relations for circular photon orbits and the effective dynamics. By specializing the angular sector to Randers sphere of constant positive flag curvature, we obtain exact expressions for the conserved angular charge, the critical impact parameter and we quantify a Finslerian Sagnac--type effect. Our second contribution is dynamical: we examine the field equations used in the literature to determine $(e^ν,e^{\vartheta})$. We revisit the family of hairy black holes in \cite{Nekouee2025}, demonstrating that the analysis therein neglects crucial non-reversible Finsler features. Furthermore, we show that the solutions presented as new reproduce previously known results in \cite{Ovalle2021}.

Static spacetimes with a Finsler angular sector

TL;DR

The paper develops static Lorentz--Finsler spacetimes with a Randers angular sector of constant positive flag curvature to analyze photon-sphere physics and black-hole solutions. It derives a model-independent photon-sphere condition and, for Randers angular sectors, explicit expressions for the axial charge and critical impact parameter, along with a Finslerian Sagnac delay. The work also shows how Finslerian non-reversibility affects orbit equations and clock-based tests, and critically revisits recent hairy-black-hole results, showing their equivalence to earlier Ovalle constructions after appropriate rescalings. Collectively, the results illuminate how direction-dependent angular geometry modifies null geodesics while leaving the radial sector to GR-like dynamics, and they provide a concrete diagnostic to distinguish static Lorentz--Finsler from stationary Lorentzian spacetimes via timelike clocks.

Abstract

We consider static spacetimes in spherical coordinates whose angular sector is described by a Finsler metric rather than the standard round metric on . Our first contribution is kinematical: maintaining arbitrary lapse and radial factors , , and relying solely on Killing symmetries and the null constraint, we derive model--independent relations for circular photon orbits and the effective dynamics. By specializing the angular sector to Randers sphere of constant positive flag curvature, we obtain exact expressions for the conserved angular charge, the critical impact parameter and we quantify a Finslerian Sagnac--type effect. Our second contribution is dynamical: we examine the field equations used in the literature to determine . We revisit the family of hairy black holes in \cite{Nekouee2025}, demonstrating that the analysis therein neglects crucial non-reversible Finsler features. Furthermore, we show that the solutions presented as new reproduce previously known results in \cite{Ovalle2021}.

Paper Structure

This paper contains 13 sections, 4 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Lorentz--Finsler Randers framework
  3. Randers angular sector
  4. Circular geodesics, critical impact parameter and a Sagnac--type effect
  5. Photon sphere
  6. Axial Noether charge and critical impact parameter
  7. Orbit equation
  8. A Finslerian Sagnac--type effect
  9. Distinguishing static Finsler from stationary Lorentzian spacetimes under shared optical data
  10. Finslerian field equations and their simplification
  11. The vacuum equations
  12. The Finslerian hairy black hole solutions
  13. Conclusions

Key Result

Proposition 2.2

Let $\overline F$ be a Randers metric on $S^2$ of constant positive flag curvature $K$. Then, locally up to isometry, there exist a Killing vector field $X$ of the round metric on $S^2$ and a constant $\varepsilon\in [0,\sqrt K)$, such that in spherical coordinates $(\theta,\phi)$ adapted to the $SO

Theorems & Definitions (11)

  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3: Symmetries of Randers spheres with constant flag curvature
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 3.3: Closed geodesics on $(S^2,\overline F)$
  • Proposition 3.4: Equatorial closed geodesic for constant flag curvature Randers spheres
  • proof
  • Corollary 3.5: Equatorial light ring
  • ...and 1 more