Stable Photon Sphere: Theory & Implications

• Stable photon spheres are regions in spacetime where photons follow closed, stable orbits due to a positive second derivative of the effective potential.
• They typically arise in exotic gravitational scenarios, such as hairy black holes or modified gravity theories, often involving violations of standard energy conditions.
• Their presence affects gravitational lensing, shadow formation, and quasinormal mode behavior, making them key targets for strong-field tests of general relativity.

A stable photon sphere is a geometric structure in spacetime where photons (massless particles) can execute circular, closed, or more generally bound null orbits, such that small radial perturbations do not cause the photon to leave the orbit but instead are restored or remain bounded. Stable photon spheres are an extension of the classical photon sphere concept, which is central to gravitational lensing and black hole shadow formation, but are less common and often associated with matter distributions or exotic gravitational configurations. Their existence, distribution, and physical implications are subjects of significant focus in recent theoretical work, particularly in the context of strong-field tests of gravity, compact astrophysical objects with “hair,” exotic matter, and alternative gravitational theories.

1. Mathematical Characterization and Existence Conditions

Stable photon spheres are defined by the presence of stable closed null geodesics—circular light orbits at fixed radius—such that the effective potential $V_{\rm eff}(r)$ governing null radial motion satisfies

$$V_{\rm eff}(r_{\rm sps}) = 0, \qquad V'_{\rm eff}(r_{\rm sps}) = 0, \qquad V''_{\rm eff}(r_{\rm sps}) > 0$$

where $r_{\rm sps}$ is the radius of the photon sphere and the sign of the second derivative indicates stability. In spherically symmetric, static spacetimes with metric

$$ds^2 = -A(r)\, dt^2 + B(r)\, dr^2 + C(r)\, d\Omega^2,$$

the photon sphere condition is often written

$$\frac{d}{dr} \left(\frac{A(r)}{C(r)}\right)\bigg|_{r_{\rm ps}} = 0$$

with the stability determined by the second derivative. For matter-supported black holes, an explicit existence criterion for stable photon spheres is

$$\rho(r_{\rm sps}) + p_T(r_{\rm sps}) > \frac{1}{8\pi r_{\rm sps}^2},$$

where $\rho$ is the energy density and $p_T$ the tangential pressure at $r_{\rm sps}$ (Song et al., 27 Aug 2025).

Stable photon spheres can appear in the vicinity of black holes with nonstandard matter content (“hair”), in regular or horizonless compact objects, or in certain scalar-tensor and Horndeski theories where energy conditions may be violated. Under additional assumptions, such as monotonic decrease of $m(r)/r^3$ outside the horizon, a universal upper bound can be established: $r_{\rm sps} < 6 M$ where $M$ is the ADM mass (Song et al., 27 Aug 2025). This bound is independent of the specific solution and provides a sharp constraint on their possible occurrence.

2. Geometric, Topological, and Optical Frameworks

The existence, stability, and distribution of photon spheres are elegantly formulated in terms of intrinsic and extrinsic curvature properties in “optical geometry”—the two-dimensional Riemannian geometry in which null geodesics map to spatial geodesics. The photon sphere corresponds to a closed curve (or surface) where the geodesic curvature $\kappa_g$ vanishes: $\kappa_g(r_{\rm ps}) = 0.$ Stability is determined by the Gaussian curvature $\mathscr{K}$ at $r_{\rm ps}$: $$\mathscr{K}(r_{\rm ps}) > 0 \Rightarrow \text{stable}; \qquad \mathscr{K}(r_{\rm ps}) < 0 \Rightarrow \text{unstable}$$ (Qiao, 19 Jul 2024). The Gauss–Bonnet theorem constrains the possible arrangements, forcing stable and unstable photon spheres to alternate, and leading to the topological relation

$n_{\rm stable} - n_{\rm unstable} = -1$

for asymptotically regular black holes (Qiao, 19 Jul 2024). This framework applies universally to static, spherically symmetric (and, more generally, suitably regular) spacetimes.

An alternative topological classification invokes the sum of winding numbers (total topological charge, TTC) associated with zeros of a vector field constructed from the regular potential $H(r, \theta)$. TTC $= -1$ signals a single unstable photon sphere (usual black holes), while TTC $= 0$ or $+1$ marks the coexistence of an unstable and a stable photon sphere, characteristic of naked singularities or horizonless ultracompact objects (Afshar et al., 29 May 2024).

3. Physical Manifestation and Examples across Theories

Stable photon spheres are rare in standard black hole spacetimes satisfying the usual energy conditions and regularity. Their presence requires exotic matter content, nonminimal couplings, or modifications to gravity:

• In minimally coupled Schwarzschild, Reissner–Nordström, and Kerr–Newman black holes, the photon sphere outside the event horizon is always unstable.
• In extremal Reissner–Nordström black holes, a stable photon sphere coincides with the event horizon itself; in Kerr–Newman(-AdS), the stability of the horizon photon orbit depends subtly on the charge and rotation parameter (Khoo et al., 2016, Tang et al., 2017).
• Hairy black holes with nonminimally coupled scalar fields, e.g., the Einstein–Maxwell–scalar model, can have one (or more) stable photon spheres (“anti-photon spheres”) coexisting with unstable ones (Guo et al., 2021). In these cases, the effective potential possesses both maxima and minima, with stable photon orbits at the minima.
• In Weyl conformal gravity (Mannheim–Kazanas metric), both stable and unstable photon spheres exist; loading the stable photon sphere with a shell of null matter preserves its area and, at threshold, generates an extremal horizon with a near-horizon AdS$_2 \times$S$^2$ geometry independent of cosmological curvature (Kusano et al., 21 Oct 2025).
• In regular black holes with an asymptotically Minkowski core, the photon sphere structure becomes multi-valued with stable and unstable branches depending on model parameters (Berry et al., 2020).
• In Konoplya–Zhidenko naked singularity spacetimes and in Kerr naked singularities, stable spherical photon orbits are permitted, producing complex observational signatures (Charbulák et al., 2018, Wang et al., 2023).

4. Astrophysical and Observational Implications

The existence and properties of stable photon spheres have several astrophysical consequences:

• Black hole shadows: The classical, unstable photon sphere dictates the shadow edge. Stable photon spheres can, in principle, trap photons, but their impact on shadow formation is more subtle and often negligible for direct shadow features (Kudo et al., 2022, Wang et al., 2023).
• Gravitational lensing: Unstable photon spheres lead to divergent (logarithmic) light deflection, producing relativistic images and characteristic lensing signatures. For stable photon spheres, the deflection is nondivergent and characterized instead by a finite (arcsine-like) angular shift, provided a photon can approach the sphere only up to a finite “gap” distance (Kudo et al., 2022). This leads to milder lensing signatures.
• Quasinormal modes and linear stability: Stable photon spheres are associated with long-lived, slowly decaying quasinormal modes (QNMs) due to tunneling effects across potential barriers. Their existence can generate late-time “tails” in gravitational wave signals or even indicate linear instability of the background spacetime, as the perturbation energy accumulates near the stable orbit (Guo et al., 2021).
• Accretion disk structure: In systems where stable photon orbits intersect or approach the equatorial plane, reflected or “echoed” illumination of the disk is possible, affecting the dynamics and radiative properties of accretion flows. This is especially pronounced in Kerr naked singularity spacetimes (Charbulák et al., 2018).
• Wormhole stability classification: For thin-shell wormholes constructed by gluing spacetimes along surfaces coinciding with photon spheres, the stability of the throat under perturbations is set purely by photon sphere stability: throats at stable (anti-photon) spheres are stable, while those at unstable spheres are unstable, independent of the barotropic fluid’s sound speed (Tsukamoto et al., 2023).

5. Boundaries, Universal Constraints, and Topological Distribution

The radius of any stable photon sphere outside a static, asymptotically flat black hole is universally constrained by

$r_{\rm sps} < 6M.$

This bound obtains when the external matter satisfies the weak energy condition, a nonpositive trace energy condition, and monotonic mass–radius behavior. The result is independent of the detailed black hole solution and extends to broad classes of “hairy” or matter-surrounded black holes (Song et al., 27 Aug 2025). The distribution of photon spheres—by geometric arguments in optical geometry—enforces that stable and unstable photon spheres must alternate, and that $n_{\rm stable} - n_{\rm unstable} = -1$ holds in physically sensible black hole solutions (Qiao, 19 Jul 2024). This topological alternation is robust across asymptotic behaviors (flat, de Sitter, anti-de Sitter).

6. Stability, Dynamical Impacts, and Quantum Implications

The mathematical formulation of stability, applicable to both photon spheres and more general photon surfaces, is given in terms of the curvature contraction

$$R_{acbd}\,k^a n^c k^b n^d > 0\ \text{(stable)}, \quad < 0\ \text{(unstable)}$$

with $k^a$ tangent to the null geodesic and $n^c$ the normal to the surface (Koga et al., 2019). Alternatively, the normal derivative of the second fundamental form on the surface controls the response to perturbations.

Stable photon spheres can trap perturbations (including gravitational waves and null matter), potentially leading to secular instabilities or the formation of near-horizon AdS$_2\times$S$^2$ throats in certain models (Kusano et al., 21 Oct 2025). Near the photon sphere, the field equations exhibit enhanced algebraic and conformal structure (hidden SL(2,ℝ) symmetry), which affects quasinormal spectra and connections to chaos bounds and holography (Raffaelli, 2021). This makes the stable photon sphere a preferred location for exploring links between strong gravity, quantum chaos, and gauge/gravity duality.

7. Limitations and Model Dependencies

Stable photon spheres are strongly model-dependent and are absent in standard vacuum black hole solutions of general relativity. Their existence typically requires either matter external to the black hole (e.g., scalar hair), modified gravity, or violation of energy conditions. In spacetimes obeying the usual (e.g., strong) energy conditions, there is typically a unique unstable photon sphere outside the event horizon, with stable spheres appearing only for “exotic” matter content or horizonless/ultracompact objects (Cvetic et al., 2016, Afshar et al., 29 May 2024). Moreover, their observational impact is often subleading compared to unstable spheres, except in configurations where stable photon spheres support long-lived modes or induce characteristic resonances in gravitational wave or electromagnetic signals.

Table: Characteristic Aspects of Stable Photon Spheres

Criterion / Feature	Stable Photon Sphere (SPS)	Unstable Photon Sphere (UPS)
Stability	$V''_{\rm eff} > 0$	$V''_{\rm eff} < 0$
Typical Occurrence	Nonvacuum, exotic, or hairy spacetimes	Standard black holes (outside $r_H$)
Lensing Effect	Finite, arcsine-like angle (“gap”)	Divergent (logarithmic), multiple images
Shadow formation	Minimal impact on edge	Sets edge of shadow
Trap null/geometric waves	Yes; traps, induces long-lived QNMs	No; photons quickly escape/trapped

Summary

Stable photon spheres are geometric loci of closed null geodesics with restoring behavior under perturbation, arising in black holes with external matter, certain scalar–tensor theories, or exotic compact objects. Their existence and properties are governed by precise curvature, topological, and energy condition criteria, and they have major implications for gravitational lensing, stability analysis, and astrophysical observation. The maximal allowed radius obeys a universal bound $r_{\rm sps} < 6M$ for asymptotically flat black holes with “reasonable” matter, and stable and unstable photon spheres must alternate in radius, obeying $n_{\rm stable} - n_{\rm unstable} = -1$. While not present in standard vacuum black holes, stable photon spheres represent a well-defined theoretical possibility in a range of physically motivated and alternative gravitational systems, providing a target for future observational and theoretical studies (Song et al., 27 Aug 2025, Afshar et al., 29 May 2024, Qiao, 19 Jul 2024, Kusano et al., 21 Oct 2025, Kudo et al., 2022, Guo et al., 2021, Charbulák et al., 2018, Koga et al., 2019).