Geometric Approach to Light Rings in Axially Symmetric Spacetimes
Circular photon orbits have become an attractive topic in recent years. They play extremely important roles in black hole shadows, gravitational lensings, quasi-normal modes, and spacetime topological properties. In our recent work, \href{https://doi.org/10.1103/PhysRevD.106.L021501}{Phys. Rev. D \textbf{106}, L021501 (2022)}, a geometric approach to circular photon orbits was proposed for spherically symmetric spacetimes. In the present study, we extend this geometric approach from spherically symmetric spacetimes to axially symmetric spacetimes. In this geometric approach, light rings in the equatorial plane are determined by the intrinsic curvatures in the optical geometry of Lorentz spacetime, which gives rise to a Randers-Finsler geometry in axially symmetric cases. Specifically, light rings can be precisely determined by the vanishing of geodesic curvature, and the stability of light rings is classified using the intrinsic flag curvature in Randers-Finsler optical geometry. This geometric approach presented in this work is generally applicable to any stationary and axially symmetric spacetime, without imposing any restriction on the spacetime metric forms. Furthermore, we provide a rigorous demonstration to show that our geometric approach yields results that are completely equivalent to those derived from the conventional approach (based on the effective potential of photons).