Shadow Ringing Effect: Astrophysical Ring Phenomena

• Shadow ringing effect is a family of ring-like, oscillatory, and time-dependent phenomena observed in various astrophysical contexts, defined by different radiative-transfer and dynamical mechanisms.
• It distinguishes features in black-hole imaging by separating the dark shadow from the narrow photon ring and broader lensing ring, with key metrics such as impact parameters around 5.2M.
• Time-dependent variations, including quasinormal mode oscillations and redshift drift, provide actionable diagnostics to constrain accretion rates, gravitational coupling, and models of modified compact objects.

The expression shadow ringing effect is used in several distinct but related ways across contemporary astrophysical and gravitational literature. In black-hole imaging, it can denote the ring and sub-ring structure associated with the shadow boundary, the time-dependent drift of shadow and photon-ring observables induced by cosmological redshift drift, or the oscillatory deformation of the shadow boundary driven by quasinormal modes during ringdown (Gralla et al., 2019, Frion et al., 2021, Pantig, 29 Sep 2025). In studies of modified compact-object environments, the same expression or closely related language is also applied to double photon rings, step-like structures, fractal substructure, or multiple orders of shadow rings produced by shells, chaotic lensing, or torn accretion flows (Li et al., 8 Feb 2026, Wang et al., 2019, Hu et al., 22 Apr 2026). In protoplanetary-disk imaging, it refers to the weak ring or shallow arc that can appear near the edge of a self-shadowed region if a puffed-up inner rim has an extremely sharp vertical edge (Dong, 2015). This suggests that the term is context-dependent rather than uniquely standardized.

1. Terminological scope

Across the cited literature, “shadow ringing” does not denote a single invariant observable. Instead, it labels several families of phenomena that are linked by the appearance of rings, ring-like modulations, or shadow-boundary structure.

Context	Meaning in the literature	Representative papers
Black-hole image morphology	Photon ring, lensing ring, or shadow-boundary substructure	(Gralla et al., 2019, Bronzwaer et al., 2021)
Time-dependent black-hole shadows	Redshift-drift or QNM-driven shadow modulation	(Frion et al., 2021, Pantig, 29 Sep 2025)
Nonstandard compact-object imaging	Double rings, step-like structures, fractal or higher-order shadow rings	(Li et al., 8 Feb 2026, Wang et al., 2019, Hu et al., 22 Apr 2026)
Protoplanetary disks	Weak ring or shallow arc at a shadow edge	(Dong, 2015)

In black-hole studies, the phrase is often attached to phenomena near the critical curve, the photon ring, or the lensing ring, but several papers emphasize that these are not identical objects. In disk-imaging studies outside black-hole physics, the same wording refers not to null-geodesic separatrices or photon-sphere structure, but to radiative-transfer signatures of self-shadowing. The shared vocabulary is therefore morphological rather than strictly dynamical.

2. Shadow, photon ring, and lensing ring in black-hole imaging

A central clarification in the recent black-hole literature is that the black-hole shadow and the photon ring are distinct phenomena. For optically thin and geometrically thick emission regions, the shadow is a sharp-edged dip in brightness coincident with the projection of the unstable-photon region on the observer’s sky, while the photon ring is a thin bright structure produced by rays that travel along extended, horizon-circling paths near the boundary of the unstable-photon region (Bronzwaer et al., 2021). The two key mechanisms are blocking, which produces the sharp-edged darkness, and path-lengthening, which produces brightening near and outside the boundary. The same work states that models exist in which the shadow is visible but the photon ring is not, so the shadow is not reducible to the photon ring (Bronzwaer et al., 2021).

A complementary clarification concerns the distinction between the photon ring and the broader lensing ring. For a Schwarzschild black hole, the critical curve has impact parameter

$b_c = 3\sqrt{3}M \approx 5.2M,$

and the bending angle near the critical curve obeys

$$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$

Because the enhancement is only logarithmic, the photon ring is extremely narrow and its integrated flux is negligible for present observations (Gralla et al., 2019). In the same analysis, the photon ring occupies $5.19M < b < 5.23M$, whereas the lensing ring occupies $5.02M < b < 6.17M$, is centered at a radius $\sim 5\%$ larger than the photon ring, has width $\sim 0.5\text{--}1M$, and can be relatively brighter by a factor of $2\text{--}3$ (Gralla et al., 2019). This reassigns much of the practically visible ring-like structure from the photon ring to the lensing ring.

These distinctions bear directly on interpretation of observed images. In thin- or thick-disk toy models, the characteristic features of the image are dominated by the location and properties of the emitting matter near the black hole rather than by the formal photon ring itself (Gralla et al., 2019). Likewise, for geometrically thick and optically thin emission regions, the shadow is robust and fairly model-independent because blocking and path-lengthening are quite general, whereas the prominence of the photon ring depends on emissivity near the photon sphere (Bronzwaer et al., 2021).

3. Time dependence: cosmological drift and quasinormal modulation

One precise use of the term concerns the slow temporal evolution of the shadow and photon-ring observables in an expanding Universe. The apparent angular radius of a spherically symmetric black-hole shadow at cosmological scales is

$$\alpha_{\text{cosm}} = \frac{3\sqrt{3} \, m}{D_A(z)},$$

with

$$D_A(z) = \frac{c}{1+z} \int_0^{z} \frac{d\tilde{z}}{H(\tilde{z})},$$

and redshift drift

$\frac{dz}{dt} = H_0 (1+z) - H(z).$

Differentiation yields

$$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$0

For the photon-ring diameter $$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$1, the same relative drift appears,

$$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$2

while the drift of the visibility period satisfies

$$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$3

For M87$$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$4, with $$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$5, the effect is of order $$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$6 per day, specifically $$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$7 in the quoted $$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$8CDM estimate (Frion et al., 2021).

The same work notes that non-detection is physically informative. If the black-hole mass varies, then

$$\phi \sim \log \left( \frac{C_\pm}{|b-b_c|} \right), \qquad b \to b_c^\pm .$$9

and if the gravitational coupling varies,

$5.19M < b < 5.23M$0

From Event Horizon Telescope data for M87$5.19M < b < 5.23M$1, the reported constraints are a maximum accretion rate of $5.19M < b < 5.23M$2 per year and $5.19M < b < 5.23M$3 per year (Frion et al., 2021). In this usage, “shadow ringing” is not a bright sub-ring but a temporal drift of shadow size and photon-ring frequency.

A second time-dependent usage arises in linear perturbation theory during black-hole ringdown. For a Schwarzschild background perturbed by a selected quasinormal mode,

$5.19M < b < 5.23M$4

the shadow boundary is modeled as

$5.19M < b < 5.23M$5

with $5.19M < b < 5.23M$6. The derived first-order mapping from $5.19M < b < 5.23M$7 to $5.19M < b < 5.23M$8 is gauge-invariant, and the prediction is that the shadow boundary oscillates coherently at the QNM real frequency $5.19M < b < 5.23M$9 with an exponential damping rate set by $5.02M < b < 6.17M$0, while the azimuthal structure encodes the spherical harmonic content $5.02M < b < 6.17M$1 of the driving QNM (Pantig, 29 Sep 2025). In a related multi-messenger formulation, the same near-photon-sphere geometry determines both shadow observables and eikonal quasinormal frequencies, expressed schematically as

$5.02M < b < 6.17M$2

so “shadow ringing” can also designate the shadow–ringdown correspondence itself (Konoplya et al., 2022).

4. Additional ring structures from nonstandard geometries and accretion flows

A broad class of papers extends the notion of shadow ringing to nonstandard spacetime structure or nonstandard accretion geometry. In slowly rotating Kerr spacetime with a thin shell, the Israel junction construction preserves $5.02M < b < 6.17M$3 and $5.02M < b < 6.17M$4 but not $5.02M < b < 6.17M$5, so the impact parameters $5.02M < b < 6.17M$6 and $5.02M < b < 6.17M$7 are discontinuous at the shell (Li et al., 8 Feb 2026). The resulting images exhibit distinct double photon rings, which can gradually merge into a single ring, and step-like structures produced by abrupt changes in redshift factor and truncated photon regions. The same study states that shadow boundaries and photon rings do not exhibit a one-to-one correspondence (Li et al., 8 Feb 2026).

In a phenomenological torn accretion disk around a Kerr black hole, a radial discontinuity between sub-disks opens a collision-free pathway for backward-traced photons. This generates multiple orders of shadow rings, together with bifurcated shadows, crescent-like structures, and severe erosion of the inner shadow (Hu et al., 22 Apr 2026). The mechanism is explicitly orbital: primary, secondary, and higher-order shadow rings correspond to photons that plunge through the gap after direct passage or after one or more additional orbits. The morphology is governed by the spatial discontinuity between the sub-disks and by the tilt angle of the outer sub-disk (Hu et al., 22 Apr 2026).

In a Schwarzschild black hole surrounded by a Bach-Weyl ring, the external ring makes photon dynamics non-integrable and introduces chaos. The shadow then develops self-similar fractal structures, bright spots, stripes, and eyebrow-shaped features, which the paper attributes to chaotic lensing and to unstable invariant manifolds associated with Lyapunov orbits near fixed points (Wang et al., 2019). Here the “ringing” is fractal rather than periodic.

Other compact-object models produce ring-like variants without invoking time dependence. A Konoplya-Zhidenko rotating naked singularity with a complete unstable photon sphere but no event horizon produces a ring-shaped shadow, interpreted as an infinite number of relativistic Einstein rings rather than a filled dark disk (Wang et al., 2023). In Eddington-inspired Born-Infeld models interpolating between regular black holes and horizonless compact objects, the image can contain a sub-ring structure created by light rays crossing the disk more than once, and in some horizonless cases the extinction of successive sub-rings is less severe than for black holes (Olmo et al., 2023). A Dehnen-type dark matter halo shifts the peak of the received intensity toward a higher impact parameter $5.02M < b < 6.17M$8, moving bright rings outward (Luo et al., 26 May 2025), whereas effective quantum gravity models can shrink the lensing rings and cause the shadow to occupy a larger proportion within the ring (Liu et al., 2024).

5. Self-shadowing in protoplanetary disks

In protoplanetary-disk research, the phrase is used in a different radiative-transfer setting. The relevant geometry is an optically thick disk with a dust-free inner cavity and a vertically puffed-up inner rim at the sublimation radius. The rim blocks stellar photons and casts a shadow over the outer disk. The vertical structure is set by hydrostatic equilibrium,

$5.02M < b < 6.17M$9

and this matters because the sharpness of the rim edge determines whether any ring-like scattered-light feature can appear (Dong, 2015).

For natural HSEQ rims, the transition between the shadowed and flared regions is smooth, with no sharp boundary in density, temperature, or surface brightness. Accordingly, no sharp rings/arcs appear in scattered-light images, and radial profiles remain smooth power laws (Dong, 2015). The paper explicitly states that a puffed-up rim cannot create sharp ring/arc/spiral-arm-like features in the outer disk as have been detected in recent direct NIR imaging.

For artificial sharp rims, the situation changes, but only modestly. The scattered-light radial profile can exhibit a 3-stage broken power law:

• inner shadowed region: $\sim 5\%$0
• transition region: $\sim 5\%$1
• outer flared region: $\sim 5\%$2

Under these extreme assumptions, the transition region can manifest as a marginally bright ring or shallow arc at the shadow edge, but the contrast remains low: the quoted peak-to-gap contrast is $\sim 5\%$3 in unblurred models and drops to almost unity after PSF convolution (Dong, 2015). The same work argues that high-contrast rings or arcs therefore require either a razor-sharp vertical density cutoff or some additional physical cause. The TW Hydrae scattered-light profile is discussed as an observational system with a shallow region that matches the 3-stage form, although the paper states that a natural HSEQ rim would be too smooth to explain it (Dong, 2015).

6. Observational interpretation, limitations, and recurrent misconceptions

Several recurrent misconceptions are addressed explicitly in the literature. The first is that the shadow, photon ring, and brightest observed ring are interchangeable. They are not. The shadow is a sharp-edged dip associated with blocking by the unstable-photon region, the photon ring is a narrow bright structure produced by path-lengthening, and the lensing ring is a broader demagnified image of the back side of the emitting flow (Bronzwaer et al., 2021, Gralla et al., 2019). The second is that a photon ring must dominate the image. In the thin-disk and spherical-accretion models studied for brane-world and quintessence black holes, the direct emission plays the major role, the lensing ring makes a small contribution, and the photon ring makes a negligible contribution (Zeng et al., 2021, Zeng et al., 2020).

Current observational limitations are also emphasized. For M87$\sim 5\%$4, the Event Horizon Telescope achieved an angular resolution of $\sim 5\%$5as, while the expected cosmological shadow drift is of order $\sim 5\%$6 per day; the measured variation in shadow size by EHT pipelines is $\sim 5\%$7 per day, many orders of magnitude larger than the predicted cosmological effect (Frion et al., 2021). In blurred synthetic images for brane-world black holes, EHT-level PSF convolution washes out lensing-ring and photon-ring substructure, and the instrument cannot currently resolve those components (Zeng et al., 2021). Quintessence models make a similar point: even though the locations of photon spheres and ring intervals shift with the quintessence parameter, the direct image remains dominant and higher-order rings remain highly demagnified (Zeng et al., 2020).

At the same time, the literature treats these weak or unresolved effects as diagnostically valuable. Non-detection of shadow drift constrains accretion and possible variation of $\sim 5\%$8 (Frion et al., 2021). Dynamical shadow modulations offer a geometric channel for QNM spectroscopy (Pantig, 29 Sep 2025). Double photon rings, step-like structures, fractal substructure, or multiple shadow rings are proposed as signatures of shells, chaotic multi-source spacetimes, or torn accretion environments rather than of standard equatorial disk imaging (Li et al., 8 Feb 2026, Wang et al., 2019, Hu et al., 22 Apr 2026). In protoplanetary disks, the weakness of the shadow-edge ring is itself the key result, because it limits what self-shadowing by a puffed-up inner rim can explain (Dong, 2015).

In this technical sense, the shadow ringing effect is best understood not as a single observable, but as a family of ring-like, oscillatory, or shadow-edge phenomena whose interpretation depends on the radiative-transfer problem, the spacetime geometry, and the time domain under consideration.