Geometric Approach to Light Rings in Axially Symmetric Spacetimes

TL;DR

This work extends a geometric framework for circular photon orbits from spherically symmetric spacetimes to stationary, axially symmetric spacetimes by employing Randers–Finsler optical geometry. Light rings are located by the vanishing geodesic curvature $\kappa_{g}^{(F)}=0$, while their stability is governed by the intrinsic flag curvature $\mathcal{K}^{(F)}_{\text{flag}}$, with positive (negative) flag curvature signaling stable (unstable) rings. The analysis yields exact equivalence with the conventional effective-potential approach, including the angular-velocity condition, extremum criteria, and second-derivative stability, and it reproduces known LR radii in Kerr and Kerr–Newman spacetimes. The framework is metric-agnostic for any stationary, axisymmetric spacetime and naturally reduces to Gaussian-curvature results in the slow-rotation limit, offering a robust geometric tool for exploring photon orbits, spacetime topology, and related observational signatures.

Abstract

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. The development of analytical methods for these circular orbits has also drawn extensive attention. 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 give an extension of this geometric approach from spherically symmetric spacetimes to axially symmetric rotational spacetimes. In such a geometric approach, light rings in the equatorial plane are determined through the intrinsic curvatures in the optical geometry of Lorentz spacetime, which gives rise to a Randers-Finsler geometry for axially symmetric spacetimes. Specifically, light rings can be precisely determined by the condition of vanishing geodesic curvature, and the stability of light rings is classified through 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 completely equivalent results with those derived from the conventional approach (based on the effective potential of photons).

Figures (5)

• Figure 1: The research backgrounds and motivations of our present work.
• Figure 2: Illustration of the nontrivial connections between the stability of circular photon orbits and the existence of conjugate points in optical geometry. (a) The left panel illustrates the photon beams perturbed from an unstable circular photon orbit at a given point $p$. The perturbed photons would inevitably move away from this unstable circular orbit (escape to infinity or fall into the event horizon produced by black holes), and it is not possible to find another point $q$ conjugate to $p$ in this unstable circular photon orbit. (b) The right panel illustrates the photon beams perturbed from a stable circular photon orbit. In such cases, the perturbed photons may travel along other bound photon orbits near this stable circular photon orbit. There are conjugate points in this stable circular photon orbit (it is easy to observe that $p$ and $q_{1}$ are conjugate points, meanwhile, $q_{1}$ and $q_{2}$ are also conjugate points).
• Figure 3: Illustration of the embedding of a curved surface $S$ (which is a 2-dimensional Riemannian geometry) into a 3-dimensional background Euclidean space. The $\gamma=\gamma(s)$ is a continuous curve residing on this surface $S$, and $p$ is an arbitrary point on this curve. The $\{ \boldsymbol{e_{1}},\boldsymbol{e_{2}},\boldsymbol{e_{3}}\}$ compose an orthonormal frame field in 3-dimensional Euclidean space such that $\boldsymbol{e_{1}}=\boldsymbol{T}$ is the unit tangent vector of this curve, $\boldsymbol{e_{2}}=\boldsymbol{V}$ is a unit vector in the tangent space of surface $S$ that is orthogonal to $\boldsymbol{T}$, and $\boldsymbol{e_{3}}=\boldsymbol{N}$ is the unit normal vector of surface $S$ at point $p$. The tangent plane of the curved surface $S$ at point $p$ is spanned by frame vectors $\boldsymbol{e_{1}}$ and $\boldsymbol{e_{2}}$.
• Figure 4: This figure illustrates the left-handed frame $\{ e_{1}^{(\alpha)}, e_{2}^{(\alpha)}, e_{3}^{(\alpha)} \} = \{ T^{\alpha}, T^{\alpha}\times N^{\alpha}, N^{\alpha} \}$ used in the geodesic curvature formula in (\ref{['geodesic curvature alpha']}). The handedness (left-handed or right-handed) of the frame $\{ e_{1}^{(\alpha)}, e_{2}^{(\alpha)}, e_{3}^{(\alpha)} \}$ or $\{ \boldsymbol{e_{1}}, \boldsymbol{e_{2}}, \boldsymbol{e_{3}} \}$ depends on the choice of normal vector orientation. Here we follow the convention given by Ono et. al in reference Asida2017, where the unit normal vector $N^{\alpha}$ is assumed in the upward direction. This is different from the right-handed frame presented in figure \ref{['figure Embedding']}, where the normal vector $\boldsymbol{e_{2}} = \boldsymbol{N}$ is oriented downward. However, the vanishing of Finslerian geodesic curvature $\kappa_{g}^{(\alpha)} + \kappa_{\beta} = 0$ is independent of the choice of left-handed and right-handed frames. Furthermore, for any photon orbit parametrized by arc-length parameter $l$ in the Riemannian geometry $dl^{2}=\alpha_{ij}dx^{i}dx^{j}$, the tangent vector $T^{(\alpha)}=\frac{dx}{dl}$ always has a constant unit norm. In this case, the"covariant acceleration" vector $a^{(\alpha)} = \frac{DT^{(\alpha)}}{dl}$ of photons is automatically orthogonal to the vector $T^{(\alpha)}$ in the tangent space.
• Figure 5: Illustration of the ingredients in the definition of flag curvature, including the base point $x$ in the Finsler manifold, as well as the flagpole $y$ and transverse edge $V$ in the tangent space $T_{x}M$.