Static spacetimes with a Finsler angular sector

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ϑ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , 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 a Randers sphere of constant positive flag curvature, we obtain exact expressions for the conserved angular charge and 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ϑ</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . We revisit the family of hairy black holes in (Nekouee et al 2025 Class. Quantum Grav. 42 045002), 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 (Ovalle et al 2021 Phys. Dark Univ. 31 100744).