Topological Photon Sphere

• Topological photon spheres are geometric constructs defined by closed null geodesics around ultracompact objects, characterized by a topologically invariant winding number.
• They employ differential topology and mapping techniques to assess stability and parameter space, linking photon trapping with transitions between black holes and naked singularities.
• This framework extends to engineered photonic systems and quantum optics, offering novel analogues for black hole spectroscopy and practical studies of light trapping.

A topological photon sphere is a geometric and topological structure arising at the locus of closed circular null geodesics—photon orbits—around ultracompact objects such as black holes and horizonless solitonic configurations. Beyond the classical definition via the effective potential, the topological photon sphere is endowed with additional structure: it admits a topological classification, robust under deformations, that encodes global information about spacetime geometry, dynamical trapping, and, in some contexts, optical or quantum field configurations. The topological viewpoint unifies and strengthens the role of photon spheres in gravitational lensing, black hole spectroscopy, and photonic analogue experiments while linking them to indices, winding numbers, and invariants rooted in differential topology and geometric analysis.

1. Geometric and Dynamical Construction

In a static spherically symmetric spacetime, the photon sphere is defined by the condition that the effective potential for null geodesics $V_{\rm eff}(r)$ satisfies

$$V_{\rm eff}(r_{\rm ps}) = 0 \,, \qquad \partial_r V_{\rm eff}(r_{\rm ps}) = 0 \,,$$

where $V_{\rm eff}$ is built from the metric functions, e.g., $$ds^2 = -f(r)\,dt^2 + dr^2/g(r) + h(r)(d\theta^2 + \sin^2\theta\,d\varphi^2)$$, which yields for the photon sphere radius $r_{\rm ps}$: $$\frac{d}{dr}\left(\frac{f(r)}{h(r)}\right)\Big|_{r=r_{\rm ps}} = 0\,,\qquad \text{or equivalently} \qquad f(r) h'(r) - f'(r) h(r) = 0\,.$$ This is the locus at which the closed null (photon) orbit resides. Dynamically, the photon sphere marks the boundary in phase space between incoming trajectories escaping to infinity and those captured by the black hole, as established in both undistorted and perturbed Schwarzschild backgrounds (Shoom, 2017).

If the matter sector is nontrivial—as in, e.g., nonminimally coupled boson stars—the formation and location of the photon sphere depend directly on the interplay between the metric potentials and the matter stress-energy, as quantified by relations such as $(2m)/r = (2/3)[1-4\pi r^2 p]$ at the photon sphere, where $p$ is the radial pressure (Horvat et al., 2013).

2. Topological Classification via Duan’s Mapping Theory

The key conceptual advance of the topological photon sphere is the realization that its existence and stability are encoded in a topological charge computed from a vector field defined over the $(r,\theta)$ plane. Following the approach in (Wei, 2020), one introduces a regular potential $H(r,\theta)$—typically $H(r,\theta) = (1/\sin\theta)\sqrt{f(r)/h(r)}$—and constructs a vector

$$\phi^r = \frac{\partial_r H}{\sqrt{g_{rr}}}\,, \qquad \phi^\theta = \frac{\partial_\theta H}{\sqrt{g_{\theta\theta}}}\,,$$

normalized as $n^a = \phi^a / \|\phi\|$. The vanishing of this vector field ($\phi = 0$) locates the photon sphere.

The topological current is then given by

$$j^\mu = \frac{1}{2\pi} \epsilon^{\mu\nu\rho} \epsilon_{ab} \partial_\nu n^a\, \partial_\rho n^b\,,$$

with nonzero support only at the zeros of $\phi$. The total topological charge, interpreted as a winding number, is

$$Q = \int j^0\,d^2x = \frac{1}{2\pi} \oint_C d\Omega\,,$$

where $C$ encircles the zero and $d\Omega = d\arctan(\phi^2/\phi^1)$. For static spherically symmetric black holes, the standard scenario yields $Q = -1$—the signature of an unstable photon sphere—while horizonless spacetimes (naked singularities) can have $Q = 0$ or, in certain regimes, $Q = +1$ (Wei, 2020, Sadeghi et al., 2023, Sadeghi et al., 10 May 2024). This quantized charge is a topological invariant under smooth deformations of the metric, and is robust under changes in spacetime asymptotics (asymptotically flat, AdS, or dS).

The existence of multiple photon spheres, as in configurations with both stable and unstable circular orbits, results in the global charge tallying the topological difference,

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

a result with geometric proof in (Qiao, 19 Jul 2024).

3. Parameter Space Constraints and Physical Implications

The topological viewpoint provides a sharp tool for classifying the parameter space of gravitational solutions. For any ultra-compact object defined by a metric with parametric freedom (e.g., mass, charge, dark matter coupling, or scalar field content), the criterion $Q=-1$ (unstable photon sphere outside the event horizon) is both necessary and sufficient for “black hole” behavior. The absence of such a photon sphere, or the appearance of compensating stable/unstable pairs with $Q=0$, signals a transition to a naked singularity or horizonless regime (Sadeghi et al., 10 May 2024, Afshar et al., 29 May 2024).

This method enables the computation of permissible parameter intervals by analyzing where the topological charge changes. For examples:

• In non-minimal Einstein-Yang-Mills, a critical value of the coupling sets the boundary between black holes and naked singularities (Sadeghi et al., 10 May 2024).
• In Einstein-Gauss-Bonnet gravity, the interplay between the Gauss–Bonnet coupling and magnetic charge determines the existence of photon spheres and event horizons (Sadeghi et al., 10 May 2024).
• In the presence of perfect fluid dark matter (PFDM), increasing the PFDM parameter can cause the system to lose its event horizon and transition to $Q=0$ (Afshar et al., 29 May 2024).

4. Stability, Distribution, and Topological Index Theorem

Photon spheres can be stable (minimum of the effective potential) or unstable (maximum). Using geometric analysis—especially the intrinsic geodesic curvature $\kappa_g(r)$ and Gaussian curvature $\mathcal{K}(r)$ in the optical metric—the photon sphere’s stability is given by the sign of $\mathcal{K}(r_{\rm ps})$: positive for stable, negative for unstable (Qiao, 19 Jul 2024). The arrangement of photon spheres follows strict alternation in radius, with the relation

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

holding generically for spherically symmetric black holes with suitable asymptotic conditions, as dictated by the Gauss–Bonnet theorem applied to the annular regions in optical geometry.

This topological index connects to the invariance of the winding number in the mapping approach, offering a geometric interpretation of the global photon sphere structure. If the total number is odd, as required by generic smooth metric behavior, the net charge (topological index) is $-1$.

5. Dynamical, Thermodynamic, and Quantum Extensions

The topological photon sphere is not purely a static object. In dynamical or time-dependent spacetimes, e.g., the Vaidya metric with mass accretion, the photon sphere is defined by a global (causal) structure selected by matching null geodesics threading from past to future null infinity. Even here, the topological character is preserved by following the global winding of the vector field $\phi$ (Koga et al., 2022).

Thermodynamic analogues of the topological charge can be constructed using temperature or generalized Helmholtz free energy as effective potentials; their zeroes and critical points in parameter space mirror the classification by photon sphere topology, often correlating with the existence and type of critical phenomena (phase transitions) in black hole systems (Sadeghi et al., 2023).

In quantum and semiclassical settings—particularly for black hole quasinormal modes (QNMs) and the AdS/CFT correspondence—the photon sphere serves as a focal region for wave trapping and mode localization. In high angular momentum limits, effective potentials near the photon sphere approach inverted harmonic oscillator or Pöschl–Teller forms, leading to quantization conditions and spectral patterns controlled by the photon sphere’s location. These features map onto a peculiar subsector of the dual CFT in AdS/CFT, connecting gravitational topology to holographic operator spectra (Hashimoto et al., 2023, Raffaelli, 2021). The geometric/topological nature of these photon spheres underpins the universality of ringdown signals and their role in black hole spectroscopy (Heidmann et al., 2023).

6. Experimental and Optical Analogues

The topological photon sphere concept extends to engineered photonic systems. By constructing optical microcavities with surfaces shaped to reproduce the Fermat metric of a Schwarzschild black hole’s equatorial section, one fabricates an effective potential with a unique maximum (photon sphere) confining quasinormal modes. Lasing at this “optical photon sphere” reveals sharp spatial mode confinement directly analogous to gravitational photon rings. The trapping here is a consequence of spatial curvature, not index contrast, and exhibits topological robustness to perturbations (Xu et al., 2 Jul 2025). Such analogues provide a testbed for astrophysical phenomena and photonic device innovation.

7. Beyond Spacetime: Abstract and Internal “Photon Spheres”

In the context of quantum optics and the representation theory of light, the notion of the “topological photon sphere” may also refer to the structure of the internal state space—generalizations of the Poincaré sphere for polarization to include orbital angular momentum (OAM), color charge analogs, and SU($N$) structure. Here, closed trajectories in the generalized Stokes parameter (or Gell-Mann parameter) space realize nontrivial topological invariants such as Hopf numbers or winding numbers, and experimental manipulation of coherent photons can realize torus, Möbius, or other topologically nontrivial paths—mirroring the geometric and topological structures in physical spacetime (Saito, 2023, Sugic et al., 2021, Tiwari, 2023). The photon's own topological structure (as a singular 4-vector field vortex) with Euclidean group E(2) symmetry also supports the notion of “photon spheres” as a collection of quantized spin-bearing defects (Tiwari, 2023).

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Summary Table: Topological Photon Sphere Properties

Aspect	Metric / Topological Criterion	Physical Consequence
Existence (static spherically sym.)	$f(r)h'(r) - f'(r)h(r) = 0$ at $r = r_{\rm ps}$	Circular null geodesic (photon sphere)
Topological charge (mapping)	$Q = \frac{1}{2\pi} \oint d\Omega$	$Q = -1$ for BH, $Q = 0$ or $+1$ for NS
Stability	$\mathcal{K}(r_{\rm ps}) > 0$ (stable), $< 0$ (unstable)	Determines trapping/escape, alternation
Parameter space classification	$Q$ changes with solution parameters	BH/NS boundary, possible constraints
Dynamics/Analogues	QNM localization at photon sphere; optical analogues in microcavities	Ringdown, black hole spectroscopy, photonics
Internal (abstract) sphere	Winding in Stokes/Gell–Mann space, SU($N$) topology	Encodes photonic degrees of freedom

BH: black hole, NS: naked singularity

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In conclusion, the topological photon sphere provides a unifying framework—spanning global geometry, topology, dynamical stability, quantum field behavior, and even experimental photonics—for characterizing both the existence and the qualitative features of light trapping near ultracompact objects and their analogues. The topological classification, especially the quantized charge associated with the photon sphere, furnishes an invariant crucial for parameter space demarcation, stability analysis, and interpretation of observational and theoretical signatures across classical and quantum domains.