A General Discussion on Photon Spheres in Different Categories of Spacetimes

Photon spheres have attracted considerable interest in the studies of black holes and other astrophysical objects. For different categories of spacetimes (or gravitational sources), the existence of photon spheres and their distributions are dramatically influenced by the geometric and topological properties of spacetimes and characteristics of the corresponding gravitational fields. In this work, we carry out a geometric analysis on photon spheres for different categories of spacetime (including black hole spacetime, ultra-compact object’s spacetime, regular spacetime, and naked singularity spacetime). Some universal properties and conclusions are obtained for these spacetimes. We mostly focus on the existence of photon spheres, the total number of photon spheres $n = n_{\text{stable}} + n_{\text{unstable}}$, the subtraction of stable photon sphere and unstable photon sphere $w = n_{\text{stable}} - n_{\text{unstable}}$ in different categories of spacetimes. These conclusions are derived solely from geometric properties of optical geometry of spacetimes, irrelevant to the specific spacetime metric forms. Besides, our results successfully recover some important theorems on photon spheres proposed in recent years.

💡 Research Summary

This paper presents a comprehensive geometric analysis of photon spheres (and light rings) in several broad classes of static, spherically symmetric spacetimes. By mapping the original four‑dimensional Lorentzian manifold onto a two‑dimensional optical geometry via the null condition (ds^{2}=0), the authors identify two intrinsic curvature quantities that fully determine the existence and stability of photon spheres: the geodesic curvature (\kappa_{g}(r)) and the Gaussian curvature (K(r)) of the optical surface. The photon‑sphere condition (\kappa_{g}(r_{\rm ph})=0) is shown to be mathematically equivalent to the familiar effective‑potential condition (dV_{\rm eff}/dr=0). Stability follows from the sign of the Gaussian curvature: (K<0) signals an unstable sphere (no conjugate points), while (K>0) indicates a stable sphere (conjugate points present). This relationship is a direct consequence of the Cartan–Hadamard theorem and holds independently of the explicit metric functions (f(r)) and (g(r)).

The authors then apply this framework to four representative categories of spacetimes: (i) black‑hole spacetimes (Schwarzschild, Reissner‑Nordström, Kerr, etc.), (ii) ultra‑compact objects (UCOs) with matter interiors, (iii) regular spacetimes that are free of curvature singularities but may possess horizons, and (iv) naked‑singularity spacetimes lacking horizons. For each class they examine the asymptotic behavior of (\kappa_{g}(r)) near the relevant boundaries (event horizon, the origin, and spatial infinity) and determine how many zero‑crossings of (\kappa_{g}) can occur. The key results are:

• Black holes: (\kappa_{g}) changes sign exactly once outside the horizon, guaranteeing a single photon sphere. The associated Gaussian curvature is negative, so the sphere is unstable. Consequently the “topological charge” (w=n_{\rm stable}-n_{\rm unstable}) equals (-1) for any static black hole, reproducing recent theorems based on (\phi)-mapping topological currents.

• Ultra‑compact objects: The presence of matter allows (\kappa_{g}) to oscillate, producing zero, one, or multiple photon spheres. Both stable and unstable spheres may coexist, leading to (w) values of 0, (\pm1), or even (\pm2) depending on the interior equation of state. This explains the diverse shadow and lensing signatures predicted for exotic compact objects.

• Regular spacetimes: When the central region is regular, (\kappa_{g}) may remain finite at (r\to0). Depending on the specific regular metric (e.g., Bardeen or Hayward), one can obtain no photon sphere, a single unstable sphere, or a pair consisting of one stable and one unstable sphere, giving (w=0) or (+1).

• Naked singularities: Without a horizon, the behavior of (\kappa_{g}) near the singularity dominates. In many models the curvature drives (\kappa_{g}) to stay of one sign, eliminating the unstable sphere that is typical of black holes. Stable spheres can appear, yielding non‑negative (w) values. This aligns with earlier studies showing the possible disappearance of light rings in certain naked‑singularity geometries.

All these findings are shown to be fully consistent with the effective‑potential approach and with the topological charge formalism previously introduced by Cunha, Wei, and collaborators. The geometric method, however, requires only the sign of (\kappa_{g}) and (K) and thus works without explicit knowledge of the metric functions, making it a powerful diagnostic for new gravity models, numerical relativity outputs, or phenomenological spacetimes.

The paper also discusses observational implications. Since photon spheres determine the edge of black‑hole shadows, the structure of strong‑lensing rings, and the frequencies of quasi‑normal modes, the classification of (n) and (w) provides a direct link between measurable signatures and the underlying spacetime topology. For instance, multiple photon spheres in a UCO could produce concentric Einstein rings, while a stable photon sphere in a regular spacetime could support long‑lived photon shells, potentially observable as persistent electromagnetic or gravitational‑wave echoes.

In summary, the authors establish a universal, metric‑independent framework for analyzing photon spheres across a wide spectrum of spacetimes. They derive robust, category‑specific constraints on the total number of photon spheres and on the topological charge (w), confirming known theorems for black holes and extending them to ultra‑compact, regular, and naked‑singularity geometries. The work opens the door to rapid assessment of photon‑sphere properties in novel theoretical models and suggests concrete avenues for testing these predictions with upcoming high‑resolution imaging and gravitational‑wave observations.