Ringed Black Hole Structures

• Ringed black hole structures are configurations exhibiting concentric photon rings and subrings, arising from unstable null geodesics near the event horizon.
• They result from a synthesis of strong gravitational lensing, general-relativistic hydrodynamics, and complex accretion disk phenomena that test Kerr geometry and the no-hair theorem.
• Observational studies using VLBI and multi-frequency imaging decode ring morphology to constrain black hole mass, spin, and potential exotic compact object scenarios.

A ringed black hole structure is defined as a black hole or exotic compact object surrounded by multiple discrete rings or annuli in its image, most notably the photon ring and higher-order subrings generated by the strong-field lensing of light, occasionally accompanied by real matter structures, such as toroidal accretion flows, and complemented by gravitational lensing effects of thin shells or other ultra-compact configurations. The finest ring features in black hole imagery are determined by a confluence of spacetime geometry, general-relativistic hydrodynamics, invariant phase-space structures, strong lensing, and radiative transfer in the vicinity of the event horizon.

1. Geometry and Formation of Photon Rings

In the Kerr spacetime, the photon shell comprises unstable, bound null geodesics at radii $r_{\rm ph}^\pm$, given by

$$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$

for mass $M$ and spin parameter $a$ with $|a| \leq M$ (Lupsasca et al., 2024). Photons emitted from the critical curve in the observer's sky asymptote to these bound orbits, executing multiple half-orbits before escaping. The image of a black hole's surroundings thus naturally develops a sequence of concentric rings:

• The $n=1$ photon ring, corresponding to trajectories experiencing a single half-orbit near $r_{\rm ph}$, is ultra-narrow and nearly independent of the spatial distribution of emissivity, acting as a direct probe of the Kerr geometry (Gralla et al., 2019).
• Higher-order subrings $(n\geq2)$, generated by photons that complete additional half-orbits, are narrower by successive factors of $\sim e^{-2\pi/\lambda}$, where $\lambda$ is the Lyapunov exponent controlling the instability of the photon orbit (Grover et al., 2017, Kapec et al., 2022).

In models where the emission is sharply peaked at radius $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$0, the nth ring appears at impact parameter $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$1 with $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$2 for Schwarzschild (Wang, 2020).

2. Ringed Accretion Disks: Matter Structures and Instabilities

Ringed accretion disks generalize single-torus ("Polish doughnut") models in Kerr geometry to collections of perfect-fluid tori ("rings"), each with distinct constant specific angular momentum $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$3 and equipotential parameter $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$4. The equilibrium configuration for each torus is determined by the Boyer condition: $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$5 with $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$6 defining equipressure surfaces (Pugliese et al., 2015, Pugliese et al., 2016). Instability develops at cusp radii where

$$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$7

and equipotential $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$8 triggers accretion or jet formation (Pugliese et al., 2016).

The maximum number of rings, and hence instability points, is regulated by spin $$r_{\rm ph}^\pm = 2M\bigg\{1 + \cos\left[\frac{2}{3}\arccos\left(\pm\frac{a}{M}\right)\right]\bigg\}$$9 and rotation sense: corotating sequences may admit an infinite number in principle (only the innermost accrete), while mixed counterrotating sequences are restricted by angular momentum constraints and separation conditions (Pugliese et al., 2015). Perturbations in $M$0 or $M$1 yield coupled radial and vertical oscillations in the rings, underpinning quasi-periodic oscillation (QPO) phenomenology observed in AGN and X-ray binaries (Pugliese et al., 2015, Pugliese et al., 2017).

3. Phase-Space Structures and Subring Self-Similarity

The shadow edge and ring structure in black hole images map directly to invariant phase-space manifolds emanating from unstable fixed points (light rings) of the null Hamiltonian. Each saddle × center fixed point spawns a Lyapunov family of periodic orbits: $M$2 with corresponding stable and unstable manifolds forming "tubes," onto which all ring-generating trajectories asymptote (Grover et al., 2017). The $M$3th subring is an image of the intersection of the $M$4-winding unstable manifold with the observer's sky. Phase-space methods thus naturally unify the geometric-optics approach with the formation of self-similar, exponentially demagnified subrings, whose spacing reflects fundamental Lyapunov exponents of the strong-field geometry (Kapec et al., 2022).

4. Exotic Ringed Structures: Multiple Photon Spheres and Novel Compact Objects

Configurations with more than one unstable photon sphere (e.g., certain regular black holes, wormholes, Schwarzschild stars with thin shells, or horizonless gravastars) generate "rings within rings" in their images:

• Spherically symmetric metrics with two distinct critical curves $M$5 produce both inner and outer photon rings (Guerrero et al., 2022, Giribet et al., 2023).
• For horizonless regular solutions (e.g., rotating Bardeen, Hayward, nonsingular metrics), closed photon rings exist in well-defined parametric ranges even in the absence of an event horizon. The circularity deviation observable $M$6 remains within empirical bounds for values $M$7 (Kumar et al., 2020).
• In both transparent and reflective gravastar models, additional rings appear due to transmission or reflection effects, offering direct constraints on horizon reflectivity, e.g., a non-detection at $M$8 pins the reflectivity $M$9 (Wang, 2020).
• Eddington-inspired Born-Infeld (EiBI) compact objects (regular black holes and traversable wormholes) modify the diameter, width, and relative subring intensity in systematic ways, with wormholes potentially presenting a richer hierarchy of rings (Olmo et al., 2023).

5. Observational Properties and Astrophysical Diagnostics

VLBI and other horizon-scale imaging studies, such as EHT and future BHEX, aim to resolve the fine structure of these ringed morphologies:

• For M87*, the ring diameter at 1.3 mm is $a$0 ($a$1), with a thicker and larger ($a$2) ring at 3.5 mm due to absorption in the accretion flow (Lu et al., 2023). The outer ring traces the "photon ring," while the greater width reflects optical depth effects in the hot RIAF plasma.
• The shape and diameter of the photon ring encode mass ($a$3), spin ($a$4), and inclination ($a$5); deviations from Kerr-circular predictions can be pinpointed via parametric "circlipse" fits and mapped to tests of the no-hair theorem (Lupsasca et al., 2024, Johannsen, 2015).
• The $a$6 photon ring is always visible; higher-order rings can be suppressed by absorption, particularly below 300 GHz (2206.12066). At sufficiently high frequencies, clear multi-ring "wedding-cake" morphologies appear.
• Lensing rings, typically $a$7 larger in radius than the photon ring and with widths of $a$8–$a$9, can impart a pronounced local brightness (factor $|a| \leq M$0–$|a| \leq M$1) atop the direct emission, but contribute only $|a| \leq M$2 of total flux (Gralla et al., 2019).
• Sub-annular features, separated by $|a| \leq M$3–$|a| \leq M$4as for SMBHs, may be observed in the presence of matter shells or double photon spheres, providing clear evidence for non-standard compact objects (Giribet et al., 2023, Guerrero et al., 2022).

6. Higher-Dimensional and Topological Ring Structures

In $|a| \leq M$5, Myers–Perry black holes exhibit ultraspinning instabilities leading to horizon "ripples," bifurcations to lumpy black holes, and ultimately merger to black ring solutions of $|a| \leq M$6 topology (Dias et al., 2014). Black ring branches, their mass–spin relations, and curvature invariants precisely match blackfold predictions including finite-size terms, with $|a| \leq M$7 agreement in $|a| \leq M$8 (Dias et al., 2014). These phenomena provide a broader context for ringed horizon topologies beyond four dimensions.

7. Physical Implications and Future Prospects

Ringed black hole structures encode both the gravitational field's deepest strong-field regime and the dynamics of surrounding matter. The photon ring and its substructure directly probe the near-horizon geometry, cisterns of spacetime multipolar deviations, and fundamental features such as horizon reflectivity and quantum resonance spectra (Kapec et al., 2022). Multiple material rings in accretion disks—arising from complex feeding or merger histories—manifest in time-variable QPOs, spectral line splitting, and plausible jet multiplicities (Pugliese et al., 2015, Pugliese et al., 2017).

Future space-VLBI and multi-wavelength campaigns, especially at frequencies $|a| \leq M$9 GHz, are essential to resolve sub-$n=1$0as ring features and test both the predictions of general relativity and the existence of exotic compact object alternatives. Coordinated modeling of ring morphology, dynamics, radiative transfer, and phase-space structures remains the critical path forward for extracting robust physical inferences from ringed black hole observations.