Critical Curves, Shadows, and Photon Rings

A black-hole image can contain several nested ideas that are often called a “shadow”: a metric-defined critical curve on the observer’s screen, a capture silhouette under a specified illumination, higher-order images accumulating near the curve, and an emission-dependent depression after finite resolution and scattering. They are not interchangeable. This page isolates the geometric benchmark, then restores emission and instrumental layers. In particular, a reconstructed dark region is not a direct image of an event horizon.

Helpful background. Black-Hole Geometry and Mechanics supplies null geodesics and observer tetrads; Relativistic Matter and Compact Objects supplies emitting and absorbing plasma; Black-Hole Astrophysics supplies horizon-scale source context; and Computational Relativity supplies ray tracing and synthetic-observation checks.

The Schwarzschild critical impact parameter

Work first in a stationary Schwarzschild spacetime, in geometric optics, with a distant static observer. In the equatorial plane,

$$ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\phi^2, \qquad f(r)=1-\frac{2GM}{r}.$$

For an affinely parameterized null geodesic, stationarity and spherical symmetry give

$$E=f\dot t, \qquad L=r^2\dot\phi, \qquad b=\frac{L}{E},$$

where a dot denotes differentiation with respect to the affine parameter and $b$ is the impact parameter. The null condition becomes

$$\dot r^2=E^2-\frac{fL^2}{r^2} =E^2\left(1-b^2\frac{f}{r^2}\right).$$

The boundary between capture and escape is the ray that asymptotes to an unstable circular null orbit. Its radius makes $f/r^2$ stationary:

$$\frac{d}{dr}\left(\frac{1}{r^2}-\frac{2GM}{r^3}\right)=0 \quad\Longrightarrow\quad r_{\rm ph}=3GM.$$

At the circular orbit, $\dot r=0$, hence

$$b_c^2=\frac{r_{\rm ph}^2}{f(r_{\rm ph})}=27G^2M^2, \qquad b_c=3\sqrt{3}\,GM.$$

For a distant observer at distance $D$, the small angular radius is $\theta_c\simeq b_c/D$. This is a geometric result, independent of emissivity. Perlick and Tsupko derive observer-sky shadow constructions and their assumptions in Perlick and Tsupko 2022, Open PDF, §§ II–III, especially Eq. (8).

In Kerr spacetime, there is a photon region of unstable spherical null orbits rather than one photon sphere. Constants of motion map that region through a declared observer tetrad to a generally displaced, noncircular screen curve that depends on $a_*=J/(GM^2)$ and inclination. The departure from circularity can become substantial near extremal spin viewed close to the equatorial plane. “Prograde side” and “left side” are meaningless until the screen basis, observer location, ray direction, and spin orientation are stated. The observer construction and limiting shapes are developed in Perlick and Tsupko 2022, Open PDF, § IV.

Four features that must be separated

The comparison axes are defining operation, dependence on the spacetime, dependence on the emitter, and sensitivity to the instrument.

Feature	Defining operation	What changes it
Critical curve	Screen directions whose rays asymptote to unstable bound null motion	Spacetime, observer position and tetrad; dispersive propagation if vacuum geometric optics is abandoned
Capture silhouette or shadow	Directions that receive no light under a specified source illumination	Illumination geometry, opacity, foreground emission, and the convention used for “shadow”
Photon subrings	Successive highly lensed images made by rays executing additional half-orbits near the photon region	Source intersections and opacity set brightness; geometry sets their accumulation curve and asymptotic scaling
Central brightness depression	Low reconstructed intensity inside a bright annulus	Emissivity, absorption, redshift, beaming, scattering, time averaging, Fourier coverage, reconstruction, and beam

The critical curve is a property of a propagation problem. A subring is an image of emitting matter. A brightness depression is an intensity feature. Gralla, Holz, and Wald make these distinctions explicit and show why the traditional backlit “shadow” need not describe near-hole emission in Gralla et al. 2019, §§ I–III.

The terminology also depends on image order. A direct image reaches the observer without a large extra winding. A lensing ring contains a strongly bent secondary image of the source. Photon subrings correspond to still higher orders and accumulate exponentially close to the critical curve. Counting equatorial-plane crossings can be useful for a thin disk, but it is not a universal definition for a thick, time-dependent emitter.

One critical curve, two different depressions

Consider two optically thin, spherically symmetric teaching emissivities in the same Schwarzschild metric. Model A emits with $j_\nu\propto r^{-2}$ down to $r=2.2GM$. Model B has the same outer emissivity but an inner cavity,

$$j_{\nu,B}(r)=j_{\nu,A}(r)H(r-6GM),$$

where $H$ is a step function. Both models have $b_c=3\sqrt{3}GM$ because neither changes the metric or observer. Yet Model B removes direct emission from the inner flow and can produce a wider, deeper central depression. Conversely, adding optically thick foreground emission can fill the central region even though the critical curve remains present.

The example specifies a qualitative emissivity change, not a computed image: a numerical depression width also requires the emitter four-velocity, emitted spectrum, outer boundary, and a definition of that width. Its point is that “dark diameter equals critical-curve diameter” is not a geometric identity. Gravitational redshift, Doppler beaming, path length through emitting material, and absorption further change intensity. Convolution with a finite beam mixes bright annular emission into neighboring pixels. The EHT geometric models therefore fit an emission ring with an interior depression and calibrate its relation to a gravitational scale using model families; Paper VI does not equate a reconstructed pixel boundary with the horizon Event Horizon Telescope Collaboration 2019, §§ 3–5.

Subrings and interferometric scales

Near an unstable photon orbit, neighboring rays separate exponentially. Successive image orders therefore approach the critical curve schematically as

$$\lvert b_n-b_c\rvert\propto e^{-\gamma n},$$

where $\gamma$ is set by the orbit’s instability and the definition of image order. Their widths and integrated fluxes shrink correspondingly, although optically thin path length can enhance surface brightness close to the curve. Johnson and collaborators show that this hierarchy produces a sequence of long-baseline visibility regimes with approximately periodic amplitude structure in Johnson et al. 2020, main text and Figs. 1–3.

That asymptotic signature is not the same as resolving the broad ring in present images. Finite bandwidth, interstellar scattering, optical depth, source thickness, variability, and sparse baseline coverage can erase or mix subrings. A stable broad annulus can be strong evidence for lensed near-hole emission without being a resolved $n\geq2$ photon subring.

Checks and model boundary

A reliable calculation should reverse the ray direction and recover the same screen boundary, verify $b_c$ against the analytic result, and demonstrate convergence under affine-step and screen-resolution refinement. It should map local screen directions with an orthonormal tetrad, not identify coordinate components with viewing angles. Any executable ray tracer must preserve the separation between this geometric critical structure and the emitting model.

The benchmark assumes vacuum geometric optics and a stationary spacetime. Frequency-dependent plasma refraction, scattering, polarization transport, finite optical depth, and source evolution are additional forward-model layers. A claim about an observed ring must carry those layers as well as angular scale, distance, inclination, calibration, and likelihood conventions.

Common pitfalls

Calling the photon sphere a visible surface. It is a set of unstable null orbits, not material and not the horizon. What becomes bright is emitting matter mapped along near-critical rays.

Treating every annulus as a photon ring. A broad direct or secondary image can be annular. Image order and visibility-domain tests are needed before assigning an asymptotic subring.

Solved diagnostic: critical angular diameter

Suppose a source has gravitational angular scale $GM/D=3.8\,\mu\mathrm{as}$. Find the Schwarzschild critical-curve diameter. Does agreement of a measured broad-ring diameter with this number prove that the ring is a resolved photon subring?

Solution

The critical angular radius is $3\sqrt{3}GM/D$, so

$$d_c=2\theta_c=6\sqrt{3}\frac{GM}{D} =6\sqrt{3}(3.8\,\mu\mathrm{as})=39.5\,\mu\mathrm{as}.$$

The numerical agreement is a useful scale check, but it does not identify the observed emission order. A broad ring’s diameter also depends on where the plasma emits, optical depth, redshift and beaming, scattering, Fourier coverage, and reconstruction. Establishing a photon subring requires the predicted narrow hierarchy or its visibility signature, while interpreting a central depression requires an explicit illumination and transfer model.

References

• Event Horizon Telescope Collaboration (2019). “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole.” The Astrophysical Journal Letters 875, L6. DOI; Open PDF.
• Gralla, S. E., Holz, D. E., and Wald, R. M. (2019). “Black Hole Shadows, Photon Rings, and Lensing Rings.” Physical Review D 100, 024018. DOI; Open PDF.
• Johnson, M. D., Lupsasca, A., Strominger, A., et al. (2020). “Universal Interferometric Signatures of a Black Hole’s Photon Ring.” Science Advances 6, eaaz1310. DOI; Open PDF.
• Perlick, V., and Tsupko, O. Y. (2022). “Calculating Black Hole Shadows: Review of Analytical Studies.” Physics Reports 947, 1–39. DOI; Open PDF.