# Static spacetimes with a Finsler angular sector

## TL;DR

This paper explores static spacetimes with Finslerian angular sectors, deriving model-independent photon orbit relations, exact Finslerian effects, and revisiting black hole solutions to highlight Finsler features.

## Contribution

It introduces a kinematical framework for Finsler angular sectors in static spacetimes and critically analyzes existing black hole solutions for Finsler properties.

## Key findings

- Derived model-independent relations for photon orbits.
- Obtained exact Finslerian conserved quantities and effects.
- Revealed overlooked Finsler features in existing black hole solutions.

## 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 $e^{\nu(r)}$, $e^{\vartheta(r)}$, 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 $(e^{\nu},e^{\vartheta})$. 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}.