This section describes the shell of unstable bound photon orbits surrounding a black hole, its lensed photon ring image, the photon subrings labeled by half-orbit number, and the angle-dependent Lyapunov exponents that govern the subring brightness ratio asymmetry. Previous treatments of these structures include (8) and (10–13).

A closely related formula appears in ( 14 ). Hence, the nearly bound geodesic will typically cross the equatorial plane a number of times of order n ≈ 1 γ ln | δ r n δ r 0 | (9)until δr n ≫ δr 0 , when the geodesic is well separated from the bound orbit and it shoots off to infinity (or crosses the event horizon if δr 0 < 0). These Lyapunov exponents are central and potentially observable quantities that characterize the geometry of the Kerr photon shell.

The bound geodesics are unstable in the sense that, if perturbed slightly, they either fall into the black hole or escape to infinity where they can reach a telescope. The observed photon ring image arises from photons traveling on such “nearly bound” geodesics. Consider two geodesics, one of which is bound, with the other initially differing only by an infinitesimal radial separation δr 0 . The equation of geodesic deviation shows that, after n half-orbits between θ ± , their separation grows to δ r n = e γ n δ r 0 (7)

The inner circular equatorial orbit at r − γ is prograde, while the outer one at r + γ is retrograde: ℓ ( r ∓ γ ) ≷ 0 . The overall direction of the orbits reverses at the intermediate value r 0 γ for which 𝓁 vanishes. At that radius, [θ − , θ + ] equals [0, π], and the orbits can pass over the poles.

We will refer to one such complete oscillation (e. g., from θ − back to itself) as one orbit, since the photon typically returns to a point near, but not identical to (since the azimuthal angle ϕ also shifts), its initial position.

Every point in the equatorial annulus r − γ ≤ r ≤ r + γ , θ = π/2, has a unique bound orbit passing through it. On the boundaries r = r ± γ , the orbits reside entirely in the equatorial plane. At generic points, on the other hand, they oscillate in the θ direction between polar angles θ ± = arccos ( ∓ u + ) (3)where u ± = r a 2 ( r − M ) 2 [ − r 3 + 3 M 2 r − 2 a 2 M ± 2 M Δ ( 2 r 3 − 3 Mr 2 + a 2 M ) ] (4)

( A ) Cross section of the photon shell in the (r, θ) plane in Boyer-Lindquist coordinates. The black hole spin is a/M = 0.94, directed vertically, and the color varies with r. The intersection of an observer’s line of sight with the photon shell boundaries at r = r ± γ determines the visible subregion of the photon shell. ( B to D ) Photon ring on the screen of an observer at varying inclinations θ obs relative to the spin axis, whose projection onto the plane perpendicular to the line of sight is depicted by the (left-pointing) arrow. The center of the photon ring has a displacement from the origin that increases with spin. The color coding on the ring denotes the matching radius on the shell from which the photon emanated. The photon shell r − γ ≤ r ≤ r + γ is only visible in its entirety to the edge-on (θ obs = 90 ∘ ) observer. The face-on (θ obs = 0 ∘ ) observer only receives photons from the white r = r 0 γ orbit. The θ obs = 17 ∘ observer sees the portion of the shell delineated by the dashed lines.

Photon shell. The photon shell, illustrated in Fig. 2 , is the region of a black hole spacetime containing bound null geodesics or “bound orbits” that neither escape to infinity nor fall across the event horizon. For Schwarzschild, the photon shell is the two-dimensional sphere at r = 3M and any θ, ϕ, and t. For Kerr, this two-dimensional sphere fattens to a three-dimensional spherical shell. It is best described using Boyer-Lindquist coordinates, in which the metric of a Kerr black hole of mass M and angular momentum J = aM (with 0 ≤ a ≤ M) is d s 2 = − Δ Σ ( d t − a sin 2 θ d ϕ ) 2 + Σ Δ d r 2 + Σ d θ 2 + sin 2 θ Σ [ ( r 2 + a 2 ) d ϕ − a d t ] 2 (1A) Δ = r 2 − 2 Mr + a 2 , Σ = r 2 + a 2 cos 2 θ (1B)

Photon ring and subrings. The photon ring is the image on the observer screen produced by photons on nearly bound geodesics (7). In the limit in which the photons become fully bound, it may be shown that their images approach a closed curve C γ given by ρ = D − 1 a 2 ( cos 2 θ obs − u + u − ) + ℓ 2 (10A) φ ρ = arccos ( − ℓ ρ D sin θ obs ) (10B)where (ρ, φ ρ ) are dimensionless polar coordinates on the observer screen, while (D, θ obs ) denote the observer’s distance and inclination from the Kerr spin axis, respectively. We can view C γ as parameterized by the shell radius r − γ ≤ r ≤ r + γ from which the photon originated. For each value of r, Eq. 10B has two solutions for φ ρ in the range 0 ≤ φ ρ ≤ 2π, so each radius in the photon shell appears at two positions on C γ . A notable consequence of Eqs. 10A and 10B is that for θ obs ≠ 0, both 𝓁 and ρ, and hence φ ρ , are functions only of r, θ obs , and D. Hence, a measurement at a specific angle φ ρ along the ring probes a specific radius r of the Kerr geometry and not, as might have been expected, a specific angle around the black hole!

Astrophysically observed photon intensities I ring (ρ, φ ρ ) at the screen can be computed by backward ray tracing. One follows the null geodesics from the observer screen back into the Kerr spacetime, integrating the Doppler-shifted strength J of matter sources along the geodesic, with attenuation factors accounting for the optical depth. Scattering effects are negligible because the expected plasma frequency and electron gyroradius are in the megahertz range, several orders of magnitude below the observing frequencies that we consider. For the images in this paper, we used ipole (15). A light ray aimed exactly at the curve C γ is captured by the photon shell and (unstably) orbits the black hole forever. Those aimed inside C γ fall into the black hole, while those aimed outside escape to infinity. Therefore, C γ is the edge of the black hole shadow.

If we shoot a light ray very near, a distance δρ from the shadow edge at ρ c , it will circle many times through the emission region before falling into the black hole or escaping to infinity. The affine length of the ray and its number of half-orbits accordingly diverge as δρ → 0 n ≈ − 1 γ ln | δ ρ ρ c | (11)

This follows from Eq. 9 together with a computed relation between δρ and δr 0 . For optically thin matter distributions, Eq. 11 implies a mild divergence in the observed ring intensity I ring ∼ n as the shadow edge is approached, since a light ray that completes n half-orbits through the emission region can collect ∼n times more photons along its path. The photon ring is then the bump in the photon intensity containing this logarithmic divergence at the shadow edge. Although the divergence is cut off by a finite optical depth, this notable feature remains visually prominent in many ray-traced images of GRMHD simulations, as in Fig. 1.

The photon ring can be subdivided into subrings arising from photons that have completed n half-orbits between their source and the screen. This definition for the photon ring agrees with that in (16) but differs from the later usage in (9) and (10) by the inclusion of the n = 1 and 2 contributions. These low n contributions fully account for the thin ring image visible in Fig. 1. To orbit at least n/2 times around the black hole, the photon must be aimed within an exponentially narrowing window δ ρ n ρ c ≈ e − γ n (12)around the shadow edge. Hence, the subrings occupy a sequence of exponentially nested intervals centered around C γ .

Each subring consists of photons lensed toward the observer screen after having been collected by the photon shell from anywhere in the universe. Hence, in an idealized setting with no absorption, each subring contains a separate, exponentially demagnified image of the entire universe, with each subsequent subring capturing the visible universe at an earlier time. Together, the set of subrings are akin to the frames of a movie, capturing the history of the visible universe as seen from the black hole. In an astrophysical setting, these images are dominated by the luminous matter around the black hole. For a black hole surrounded by a uniform distribution extending over the poles, the contributions made by each subring to the total intensity profile cannot be told apart, and the individual subrings cannot be distinguished on the image. However, for a realistic disk or jet with emission peaked in a conical region, the subrings are visibly distinct: the nth subring is approximately a smooth peak of width e−γn. Summing these smooth peaks, like layers in a tiered wedding cake (see Fig. 3), reproduces the leading logarithmic divergence in the intensity (Eq. 11).

Fig. 3 Image cross sections of a photon ring and its subrings. (A) Brightness cross sections for the time-averaged GRMHD image shown in Fig. 1. The blue/red curves show cross sections perpendicular/parallel to the projected spin axis. (B and C) Decomposition of the left perpendicular peak and the right parallel peak into subrings indexed by the number n of photon half-orbits executed between turning points (Eq. 3) in the polar motion. Similar results are also seen in image cross sections of simple geometrical models (10).

The photons comprising successive subrings for the same angle φ ρ traverse essentially the same orbits and hence encounter the same matter distribution around the black hole. Apart from source variations on the time scale of an orbit, intensities of the nth and (n + 1)th subring differ only because they correspond to windows whose widths δρ n and δρ n + 1 differ by a factor of e−γ. Hence, for large enough n, the intensities are related by I ring n + 1 ( ρ c + δ ρ , φ ρ ) ≈ I ring n ( ρ c + e γ δ ρ , φ ρ ) (13)

We therefore find the angle-dependent subring flux ratio F ring n + 1 F ring n ≈ e − γ (14)

Equations 13 and 14 are matter-independent predictions for the photon ring structure that involve only general relativity. The prediction holds only for “large enough” n: At small n, there are nonuniversal matter-dependent effects from photons that do not traverse exactly the same region around the black hole. Insight into when n is large enough might be obtained from GRMHD simulations.