Radial Shadow Terms: Theory & Applications

• Radial shadow terms are radius-dependent quantities that distinguish visible from hidden regions in black-hole imaging and contour-based analysis.
• They are derived from effective potentials and photon-sphere conditions, separating escaping from captured photons in various spacetimes.
• These terms also extend to observer-plane expansions using multipole coefficients to quantify shadow distortion and noncircular features.

Radial shadow terms are the radius-dependent quantities that determine a shadow boundary, a shadow-aware visibility factor, or a shadow-induced dynamical response. In the black-hole literature, the phrase is used most naturally for the radial null-geodesic potential, the photon-sphere or spherical-orbit conditions, and the critical impact-parameter relations that separate escaping from captured photons. In contour-based image analysis, the same idea appears as a radial representation of the observed boundary itself, while in shadow-aware rendering and sensing it appears as radial return bounds, transmittance-like visibility factors, or shadow-modified pressure forcing. The term is therefore not universal, but the common structure is a radial quantity that divides visible from hidden trajectories or illuminated from shadowed regions (Mandal et al., 2022, Abdujabbarov et al., 2015, Derksen et al., 2021, Boisguezennec et al., 24 Jun 2026).

1. Core meaning in static and spherically symmetric shadow theory

For static, spherically symmetric black-hole geometries of Schwarzschild type,

$ds^2=f(r)\,dt^2-f(r)^{-1}dr^2-r^2 d\Omega^2_{d-2},$

the radial shadow problem reduces to null motion in the effective potential

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$

The photon sphere radius $r_{ph}$ is fixed by

$r_{ph}f'(r_{ph})-2f(r_{ph})=0,$

and the shadow radius for an observer at infinity is

$$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$

In this setting, the radial shadow terms are precisely $f(r)$, $V_{\rm eff}(r)$, $r_{ph}$, and the critical impact parameter $b_{cr}$ (Paithankar et al., 2023).

The five-dimensional Reissner–Nordström anti-de Sitter case makes this structure explicit through a radial polynomial,

$\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$

together with the critical relations

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$0

or equivalently

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$1

For the specific RN-AdS$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$2 metric, the photon-sphere equation becomes

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$3

Because the spacetime is static and spherically symmetric, the shadow is a perfect dark circle, and the paper reports that the shadow radius decreases with charge while increasing with the plasma parameter (Mandal et al., 2022).

A perturbative variant of the same logic appears for asymptotically flat metrics expanded as

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$4

with $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$5. The photon sphere is written as

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$6

so the radial correction terms $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$7 shift both $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$8 and the shadow size already at leading order (Vertogradov et al., 2024). In a related generic quartic ansatz,

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$9

the photon sphere is determined by

$r_{ph}$0

and the shadow radius is

$r_{ph}$1

This formulation makes the inverse-power radial terms $r_{ph}$2, $r_{ph}$3, $r_{ph}$4, and $r_{ph}$5 the direct shadow-defining quantities (Mafuz et al., 2023).

The same pattern persists in the Kalb–Ramond black hole coupled to nonlinear electrodynamics, where

$r_{ph}$6

and the photon sphere is selected by

$r_{ph}$7

The paper writes the explicit photon-sphere equation as

$r_{ph}$8

with shadow radius

$r_{ph}$9

Here the magnetic monopole charge $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$0, the Lorentz-violating hair $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$1, and the exponent-setting parameter $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$2 are all radial shadow terms in the strict sense that they enter $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$3, $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$4, and hence $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$5 and $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$6 (Ahmed et al., 15 May 2026).

2. Rotating spacetimes and radial photon potentials

In stationary axisymmetric spacetimes, radial shadow terms are usually encoded in a Kerr-like radial potential $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$7 together with the conditions

$r_{ph}f'(r_{ph})-2f(r_{ph})=0,$8

which determine critical impact parameters $r_{ph}f'(r_{ph})-2f(r_{ph})=0,$9 and $$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$0. In the Kerr–Sen dilaton–axion black hole,

$$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$1

The explicit critical parameters $$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$2 and $$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$3 follow from these radial conditions, and for an equatorial observer the shadow contour is

$$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$4

The radial charge deformation enters through the replacement $$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$5, so the charge parameter changes the shadow through the radial metric function itself (Dastan et al., 2016).

The same formal structure appears in rotating regular black holes. For Hayward and Bardeen,

$$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$6

with $$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$7 or $$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$8. For the rotating Ayon-Beato-Garcia metric in the near-equatorial approximation,

$$r_{sh}=b_{cr}=\frac{r_{ph}}{\sqrt{f(r_{ph})}}=\frac{2\sqrt{f(r_{ph})}}{f'(r_{ph})}.$$9

In all three cases, the decisive radial shadow terms are $f(r)$0, $f(r)$1, $f(r)$2, $f(r)$3, and the resulting $f(r)$4. The reported qualitative trend is that increasing $f(r)$5, $f(r)$6, or $f(r)$7 decreases the shadow size and increases the distortion parameter (Abdujabbarov et al., 2016).

Rotating anisotropic-matter geometries make the radial deformation even more explicit. One construction uses

$f(r)$8

while another writes

$f(r)$9

In both cases, the critical impact parameters depend on $V_{\rm eff}(r)$0, $V_{\rm eff}(r)$1, or on $V_{\rm eff}(r)$2 and $V_{\rm eff}(r)$3, so the anisotropic matter term $V_{\rm eff}(r)$4 or $V_{\rm eff}(r)$5 is itself a radial shadow term (Badía et al., 2021, Lee et al., 2021).

A Lorentz-violating example is the Einstein-bumblebee Kerr-like solution, with

$V_{\rm eff}(r)$6

and radial potential

$V_{\rm eff}(r)$7

The paper’s central radial conclusion is that the unstable equatorial circular orbit radius $V_{\rm eff}(r)$8 decreases for $V_{\rm eff}(r)$9 and increases for $r_{ph}$0, so the Lorentz-breaking parameter modifies the shadow by shifting the radial turning point of the null potential (Ding et al., 2019).

3. Generalized radial criteria: dynamics, media, and formation conditions

Not all radial shadow terms are static. In the shadow of a collapsing star in a regular Hayward spacetime, the relevant radial quantities are the stellar surface radius $r_{ph}$1, the observer radius $r_{ph}$2, the turning-point radius $r_{ph}$3, the limiting photon-sphere radius $r_{ph}$4, and the horizon radii $r_{ph}$5. The edge ray is determined by the null turning-point condition

$r_{ph}$6

and the observed shadow angle for a static observer is written as

$r_{ph}$7

Here the shadow forms dynamically because $r_{ph}$8 evolves from the stellar radius toward the limiting unstable orbit $r_{ph}$9 during collapse (Nunez et al., 2023).

Media introduce additional radial terms by modifying the null potential. In the RN-AdS$b_{cr}$0 plasma extension, the vacuum radial polynomial becomes

$b_{cr}$1

while in Kerr–Sen and in rotating regular black holes the plasma contribution appears as an extra term proportional to $b_{cr}$2 or $b_{cr}$3. For the common choice

$b_{cr}$4

the refractive index contributes an explicitly radial correction, and the papers report that plasma changes the shadow size through the modified critical orbit conditions (Mandal et al., 2022, Dastan et al., 2016, Abdujabbarov et al., 2016).

A stronger conceptual generalization is that a shadow need not require a conventional photon sphere or photon shell. One paper argues that a shadow forms if the effective potential of null geodesics has a positive finite upper bound and includes a region where photons are trapped or scattered. In that framework, the apparent boundary is controlled by the critical impact parameter and by the global radial structure of $b_{cr}$5, whereas the bright ring is tied more specifically to lingering near unstable photon orbits. This directly separates the existence of a shadow from the existence of a photon sphere in the strict Kerr sense (Bambhaniya et al., 7 Sep 2025).

A further reinterpretation appears in the relation between black-hole shadows and radial linear uniformly accelerated trajectories. For Schwarzschild-type metrics and boundary data $b_{cr}$6, the turning-point bound $b_{cr}$7 and acceleration bound $b_{cr}$8 satisfy

$b_{cr}$9

In the more general class $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$0, the relation becomes

$\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$1

This identifies shadow quantities with radial bounds from timelike accelerated motion rather than with null geodesics alone (Paithankar et al., 2023).

4. Radial descriptions on the observer plane

Once the shadow boundary is projected to an image plane, radial shadow terms can be defined geometrically rather than geodesically. A coordinate-independent formalism treats the observed shadow as a closed radial curve $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$2 around an effective center extracted from the curve itself. The boundary is then expanded as

$\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$3

In this representation, $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$4 is the dominant circular radius, $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$5 controls the leading asymmetry or denting, and higher $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$6 encode progressively finer noncircular structure. The paper reports essentially exponential convergence of the coefficients: for Kerr shadows, $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$7 terms are already at the $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$8 level and $\mathcal R(r)=r^4E^2-(L^2+\mathcal C)r^2 f(r),$9 around $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$00, while for Bardeen and Kerr–Taub–NUT the convergence is faster. The same formalism supports several coordinate-independent distortion parameters and remains robust under direct perturbations of the coefficients with $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$01 over $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$02 realizations (Abdujabbarov et al., 2015).

A more phenomenological radial description appears in the Schwarzschild black hole with halo containing quadrupolar and octopolar terms. That paper does not introduce an analytic $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$03, but it does isolate low-order observer-plane quantities that act like radial coefficients: the width $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$04, height $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$05, oblateness $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$06, center shift $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$07, and distortion parameter $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$08. The quadrupole term changes the even-parity horizontal extent of the contour, while the octopole term produces an odd-parity vertical shift and asymmetry. In that sense, the quadrupole and octopole behave like low-order radial deformation modes of the observed shadow (Wang et al., 2021).

For a general class of integrable stationary axisymmetric spacetimes written in terms of four free radial functions $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$09, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$10, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$11, and $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$12, the polar-observer shadow size is determined solely by $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$13. The spherical photon orbit satisfies

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$14

and the shadow radius is

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$15

High-order photon-ring spacing depends on the radial instability combination involving $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$16, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$17, and $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$18, while remaining independent of $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$19 and $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$20. This isolates a particularly sharp notion of radial shadow terms: $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$21 sets the shadow size, while $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$22 enters only through ring-spacing instability (Salehi et al., 2023).

5. Related usages outside black-hole imaging

Outside general relativity, the phrase is usually absent, but closely related quantities appear. In “Shadow Neural Radiance Fields,” the closest explicit shadow term is the local solar visibility field

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$23

described as the ratio of incoming solar light with respect to the diffuse sky light and loosely interpreted as the visibility of the directional light source. It enters the illumination model

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$24

and is regularized against volumetric transmittance

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$25

The paper explicitly notes that there is no inverse-square radial attenuation term and no normal-light cosine factor; the shadowing variable is directional and local rather than a literal radial potential (Derksen et al., 2021).

In close-proximity FMCW radar geometry reconstruction, the closest radial shadow terms are the chassis-shadow opening angle

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$26

the inner and outer radial bounds $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$27 and $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$28, and their ratio

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$29

The inclination of a nearby slender vertical object is recovered from a closed-form piecewise mapping $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$30. Here the shadow is not a dark silhouette cast by gravity but an occluded near-field region generated by the vehicle chassis, and the decisive radial quantities are the return boundaries of the segmented target (Boisguezennec et al., 24 Jun 2026).

In transition disks, the paper does not define a formal radial shadow term, but the shadow is imposed through the irradiation field

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$31

The temperature drop in the shadow lowers the pressure and acts as an asymmetric driving force at the cavity edge, producing spirals with zero pattern speed. The pitch angle is

$V_{\rm eff}(r)=\frac{f(r)}{r^2}.$32

quoted as $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$33 if $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$34, and the transport level is $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$35. A plausible implication is that, in this hydrodynamic setting, the effective radial shadow term is the shadow-induced perturbation of the radial pressure force rather than a boundary curve or photon potential (Zhang et al., 2024).

6. Conceptual synthesis and recurrent misconceptions

A useful synthesis is that radial shadow terms are not a single invariant object but a family of radius-dependent structures selected by the modeling framework. In static spherical black-hole problems they are usually $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$36, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$37, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$38, and $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$39; in rotating spacetimes they become $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$40, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$41, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$42, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$43, and $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$44; in contour analysis they are the coefficients $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$45 of a radial expansion; and in sensing or rendering they may instead be visibility fields, transmittance factors, or radial return boundaries. This suggests a taxonomy rather than a unique definition (Mafuz et al., 2023, Abdujabbarov et al., 2015, Derksen et al., 2021).

One common misconception is that a shadow boundary is always synonymous with a photon sphere. Several of the black-hole papers do use unstable circular or spherical null orbits as the defining radial criterion, but the generalized effective-potential analysis shows that a shadow can form whenever the null effective potential has a positive finite upper bound and contains a region where photons are trapped or scattered. A related misconception is to identify the dark shadow boundary with the bright ring; one paper explicitly distinguishes the apparent boundary from the bright ring on the observer’s screen (Bambhaniya et al., 7 Sep 2025).

Another recurring point is that not every radial function in a metric is equally observable. In the general integrable axisymmetric family, the horizon-defining function $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$46 determines the polar shadow size, while $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$47, which controls the ergosphere, does not enter that observable. In the generic inverse-power spherical family, shadow measurements constrain the coefficients appearing in $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$48, and in the KR–NED model the EHT-compatible region is expressed directly in terms of the parameters that enter $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$49. This suggests that shadow observations probe specific radial combinations rather than the full metric indiscriminately (Salehi et al., 2023, Mafuz et al., 2023, Ahmed et al., 15 May 2026).

The broad literature therefore treats radial shadow terms as the mathematically minimal radius-dependent quantities that fix a critical boundary. Whether written as $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$50, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$51, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$52, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$53, $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$54, or $V_{\rm eff}(r)=\frac{f(r)}{r^2}.$55, they encode the same structural operation: the selection of limiting trajectories or return boundaries that separate shadowed from unshadowed, captured from escaping, or occluded from visible configurations.