Stable Photon Spheres: Geometry & Implications

• Stable photon spheres are spatial hypersurfaces in curved spacetimes where circular null geodesics remain stable under radial perturbations.
• The analysis employs potential methods and topological charge, linking energy conditions and curvature to establish existence and stability criteria.
• Their unique signatures in gravitational lensing, quasinormal modes, and accretion dynamics offer practical insights for distinguishing exotic compact objects.

A stable photon sphere (often abbreviated SPS) is a spatial hypersurface in a Lorentzian manifold—typically a static, spherically symmetric or axisymmetric spacetime—on which circular null geodesics exist and are linearly stable with respect to radial perturbations. Standard unstable photon spheres are textbook features of black hole solutions (e.g., at radius $r=3M$ for Schwarzschild), but stable photon spheres require specific matter configurations, extremal horizon structure, or more intricate energy-momentum distributions. The existence, topological arrangement, and universal bounds on SPS have become a central topic in the interface of mathematical relativity, black hole optics, and astrophysical phenomenology.

1. Geodesic, Curvature, and Potential Characterizations

In static spherically symmetric spacetimes with line element

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

circular photon orbits (photon spheres) occur at radii where the radial potential for null geodesics $(d^2r/d\lambda^2) = -V'_{\mathrm{eff}}(r)$ is extremized:

• Existence: $V_{\mathrm{eff}}(r_{\mathrm{ps}}) = 0$, $V'_{\mathrm{eff}}(r_{\mathrm{ps}}) = 0$.
• Stability: $V''_{\mathrm{eff}}(r_{\mathrm{ps}})<0$ (unstable), $V''_{\mathrm{eff}}(r_{\mathrm{ps}})>0$ (stable).

Equivalently, using the optical geometry, a photon sphere is a circle in the 2D optical metric whose geodesic curvature vanishes:

$\kappa_g(r_{\mathrm{ps}}) = 0.$

The stability is encoded in the sign of the Gaussian curvature $\mathcal{K}$ at the photon sphere; stable if $\mathcal{K}(r_{\mathrm{ps}}) > 0$, unstable if negative (Qiao, 19 Jul 2024).

In matter-filled horizons (including "hairy" black holes and boson stars), a general existence condition is (with $m(r)$ mass function, $p$ radial pressure):

$$\frac{2m(r_{\mathrm{ps}})}{r_{\mathrm{ps}}} = \frac{2}{3}[1 - 4\pi r_{\mathrm{ps}}^2 p(r_{\mathrm{ps}})].$$

The stability condition for the outermost photon sphere in these spacetimes is typically,

$\rho + p_T > \frac{1}{8\pi r_{\mathrm{ps}}^2},$

where $\rho$ is energy density and $p_T$ tangential pressure (Song et al., 27 Aug 2025).

For more general static warped product or less symmetric backgrounds, the photon surface condition reduces to extrema of a pseudopotential $V(r) = f(r)/r^2$:

• Existence: $dV/dr = 0$,
• Stability: $d^2V/dr^2 > 0$ (minimum—SPS), $<0$ (maximum—unstable), with separability ensured by a rank-2 Killing tensor (Koga et al., 2020).

For horizon photon orbits in extremal black holes, analyticity of the metric coefficient $f(r)$ at the horizon $r_H$ yields: the photon orbit at $r_H$ is stable if the first nonzero Taylor coefficient beyond first order is even and positive (Khoo et al., 2016, Tang et al., 2017).

2. Topological Distribution and Alternation Laws

The distribution of SPS and unstable photon spheres in spherically symmetric black hole spacetimes is highly constrained. By geometric analysis of optical curvature and employing the Gauss–Bonnet theorem, it is established that:

• The number of photon spheres $n_{\text{ps}}$ is odd and their stabilities alternate: each unstable is sandwiched between two stable ones, and vice versa.
• The key topological relation is

$n_{\text{stable}} - n_{\text{unstable}} = -1,$

i.e., there is always one more unstable than stable photon sphere (Qiao, 19 Jul 2024).

This alternation is tied to the sign changes of the geodesic and Gaussian curvatures in the optical metric, regardless of the detailed asymptotics (flat, dS, AdS) or the inclusion of external fields.

An alternative topological approach attributes an integer "topological charge" to each photon sphere via currents constructed from the gradient of the potential; a total charge $Q_{\text{TTC}}=-1$ indicates a black hole (single unstable photon sphere), while $Q_{\text{TTC}}=0$ or $+1$ is associated with naked singularities or horizonless ultra-compact objects featuring an SPS (Afshar et al., 29 May 2024).

3. Existence Conditions and Universal Upper Bounds

For static, spherically symmetric, asymptotically flat black holes with external matter fields (e.g., hairy black holes), an SPS can occur outside the event horizon if the matter satisfies:

• The weak energy condition ($\rho\geq 0$, $\rho+p\geq 0$)
• Non-positive trace ($T=-\rho+p+2p_T\leq 0$)
• Monotonically decreasing mass-radius ratio ($d/dr[m(r)/r^3] \leq 0$).

Under these conditions, the radius $r_{\mathrm{sps}}$ of any SPS is subject to a universal upper bound:

$r_{\mathrm{sps}} < 6M,$

where $M$ is the ADM mass (Song et al., 27 Aug 2025). This result is independent of the specific solution and applies equally to boson stars, black holes with matter hair, and various regular models. It complements existing lower bounds for unstable photon spheres ($r_{\mathrm{ph,in}}\leq 3M$ established by Hod and others).

For horizon photon spheres in extremal electrovacua, existence and stability are primarily governed by the balance of angular momentum and charge: small $a$ (rotation) and large $Q$ (charge) favor stability at the horizon (Khoo et al., 2016, Tang et al., 2017).

4. Physical and Astrophysical Implications

The presence of an SPS has several important consequences:

• Lensing: SPS do not generally define the shadow edge (in contrast to unstable photon spheres), and the deflection angle near an SPS is non-divergent—reaching a finite limit described by an arcsine law. Thus, strong lensing featuring 'relativistic images' and infinite photon rings does not occur near an SPS. Spacetimes with an outermost SPS cannot be asymptotically flat (Kudo et al., 2022).
• Quasinormal Modes and Instabilities: Scalar or gravitational quasinormal modes localized near an SPS are extremely long-lived in the eikonal limit (their damping times scale exponentially with multipole index $\ell$), implying that perturbations may linger and potentially lead to non-linear instabilities due to energy trapping (Guo et al., 2021).
• Accretion Disk and Aschenbach Effect: SPS locations set upper limits for regions where velocity profiles of accretion disks can exhibit non-monotonic behavior (the Aschenbach effect), which may be probed in high-energy astrophysical observations (Song et al., 27 Aug 2025).
• Test of Exotic Compact Objects: Multiple photon spheres (including SPS) or the detection of lensing signatures lacking a shadow edge can indicate the presence of horizonless compact objects—such as boson stars, regular black holes, or wormholes—rather than standard black holes (Horvat et al., 2013, Tangphati et al., 2023, Berry et al., 2020).

5. Model-Specific Manifestations and Constraints

• Boson Stars: Nonminimally coupled boson stars generically allow both inner (stable) and outer (unstable) photon spheres when exceeding certain critical configurations (e.g., high central scalar field amplitude and strong non-minimal coupling). Their lensing properties result in sharply enhanced central photon flux and possible observable differences from black holes (Horvat et al., 2013).
• Dyonic Black Holes/Quasi-Topological Electromagnetism: The inclusion of non-minimal electromagnetic terms can yield configurations with up to three photon spheres, of which the innermost or intermediate can be stable. These cases highlight that SPS can arise even with strong energy condition compliance in suitably engineered matter sectors (Liu et al., 2019).
• Kerr Naked Singularities: For spacetimes with $a > M$ (Kerr naked singularities), a rich structure of stable and unstable spherical photon orbits exists, with energetic and dynamical features not possible in black hole spacetimes (Charbulák et al., 2018).
• Binary Black Holes/Di-holes: Parameter regimes in Majumdar–Papapetrou or Reissner–Nordström di-hole systems can exhibit both stable and unstable photon rings, leading to novel lensing phenomena including double sets of relativistic images and chaotic transition in photon orbit dynamics (Patil et al., 2016, Dolan et al., 2016).

6. Mathematical Structures and Generalizations

The stability of a photon sphere or photon surface (in the sense of Claudel–Virbhadra–Ellis) can always be characterized locally by the sign of a Riemann or Weyl curvature contraction:

$$\mathcal{A}_k = - X^2 R_{\alpha c \beta d} k^\alpha n^c k^\beta n^d.$$

Stable if $>0$, unstable if $<0$ (Koga et al., 2019). For (warped) less symmetric spacetimes, existence and stability of photon surfaces reduce to analysis of a one-dimensional pseudopotential, with separability locally ensured by the presence of a Killing tensor (Koga et al., 2020).

Global distribution statements, including alternation, are underpinned by topological arguments and index theorems applied to the geometry of the optical metric, ensuring that these results are robust across a wide class of gravitational models, including asymptotically de Sitter/anti-de Sitter and regular spacetimes (Qiao, 19 Jul 2024, Afshar et al., 29 May 2024).

7. Observational, Theoretical, and Classification Contexts

Observational constraints arising from the Event Horizon Telescope and gravitational-wave ringdowns can, in principle, probe the presence and distribution of multiple photon spheres, especially if an SPS-related phenomenon (e.g., lack of shadow edge, abnormal photon ring, or anomalously long-lived perturbations) is detected.

From a model classification perspective, the effective potential/topological charge method enables rapid distinction between true black holes (with unstable photon spheres and $Q_{\mathrm{TTC}} = -1$) and naked singularities or ultra-compact objects (with minima in $V_{\mathrm{eff}}$ supporting an SPS and $Q_{\mathrm{TTC}} \neq -1$) (Afshar et al., 29 May 2024).

In summary, stable photon spheres are exceptional, highly-constrained geometric objects whose existence requires delicate balancing of matter and spacetime structure. Their distribution is governed by universal topological laws, their spatial locations are subject to strict bounds, and their presence yields unique signatures in gravitational lensing, dynamical stability, and the spectrum of quasinormal modes. A full classification now relies on curvature, topology, and energy conditions, connecting the physics of black hole optics with the broader mathematical theory of light propagation and global spacetime geometry.