Chaos and fractals of the black hole photon ring

Abstract

The photon ring of a Kerr black hole decomposes into a self-similar hierarchy of subrings. Here, we show that this self-similar structure persists in phase space. Moreover, near the photon shell of bound photon orbits, dynamics are controlled by a Lyapunov exponent $γ$, whose role we highlight by computing the first-return map for light rays close to an unstably bound orbit. Despite an exponential $e^γ$ sensitivity to initial conditions, nearly bound rays do not exhibit chaotic behavior. However, as the background spacetime is deformed away from the Kerr geometry, chaos sets in, with its first onset most visible near strongly resonant bound orbits in the photon shell. We display two animations: one illustrating the emergence of chaos near the photon shell, which results in a fractal phase-space structure, and another exhibiting how this chaotic, fractal, self-similar structure is encoded in the first-return map.

Figures (8)

• Figure 1: The fates of three nearby photons, separated by small differences in initial radius.
• Figure 2: Schematic illustration of the first-return map on the Poincaré section $\mathcal{S}$. A base point $p\in\mathcal{S}$ (green dot, left) returns to $\mathcal{S}$ after an affine time $T(p)$ under the Hamiltonian flow (large green orbit). A small circular bundle of nearby initial conditions $\mathcal{C}$ (solid red circle) is co-evolved for the same amount of time. At time $T(p)$, some points have not yet returned to $\mathcal{S}$ (dashed red segment), while others have already crossed it. Projecting the evolved bundle back onto $\mathcal{S}$ yields the image ellipse $\tilde{\mathcal{C}}$ (gray ellipse, right), which represents the action of the differential of the return map $dF_p$ on infinitesimal deviations (blue vectors). Because the flow is symplectic, the first-return map is area-preserving.
• Figure 3: Triptych of the equatorial Poincaré section $\mathcal{S}$ for Kerr null geodesics near the unstable spherical photon orbit $(r, p_r) = (r_c, 0)$, illustrating the photon ring in phase space. The boxes mark the region enlarged in the next panel; from left to right, each panel zooms in by a factor of $10$. The diagonal boundary is the separatrix associated with the hyperbolic fixed point. The blue, green, red, and purple bands correspond to the $n=0$, $1$, $2$, and $3$ photon rings, respectively, while the saturation encodes the fractional part of $n$. The $n=0$ and $n=1$ rings do not yet lie in the universal large-$n$ regime, which begins around $n\gtrsim2$. In that regime, the rings exhibit a simple self-similar structure governed by the Lyapunov exponent of the bound orbit.
• Figure 4: Triptych of the equatorial Poincaré section $\mathcal{S}$ for Kerr null geodesics near the unstable spherical photon orbit $(r, p_r) = (r_c, 0)$. Pixels are colored by the half-orbit count $n$, with blue, black, red, and orange corresponding to $n=0$, $1$, $2$, and $3$, respectively. The dashed curves are contours of the energy-rescaled Carter constant $\eta = Q/E^2$, which is preserved under the first-return map. At sufficiently small scales, the phase-space structure approaches the scale-invariant normal form of a hyperbolic fixed point, making the local self-similarity manifest.
• Figure 5: Linearized first-return dynamics near the hyperbolic fixed point associated with the unstable spherical photon orbit. Left: a small circle of initial conditions on the Poincaré section $\mathcal{S}$, centered at $(r, p_r) = (r_c, 0)$, and its image under the first-return map $F$ (shown as an ellipse), overlaid on the exit basin (green = escape, blue = capture). Right: schematic action of the differential $dF_p$ on infinitesimal deviations: a circular bundle is mapped to an ellipse. Deviations along the eigendirections $\vec{e}_1$ and $\vec{e}_2$ of $dF_p$ are stretched and contracted by factors $e^{\pm 2\gamma}$ along the unstable and stable directions, respectively. Symplecticity implies that the enclosed area is preserved under $dF_p$ before any visual rescaling introduced for legibility.