Chaos and fractals of the black hole photon ring

TL;DR

This paper investigates the self-similar and fractal structures of photon rings around Kerr black holes, analyzing their phase space dynamics, stability, and the onset of chaos when spacetime is deformed from Kerr geometry.

Contribution

It reveals that the self-similar hierarchy of photon ring substructures persists in phase space and identifies how chaos emerges with spacetime deformations from Kerr black holes.

Findings

Photon rings exhibit self-similar, fractal structures in phase space.

Chaos appears near the photon shell when spacetime deviates from Kerr geometry.

The first-return map encodes the fractal structure of photon trajectories.

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 $\gamma$, whose role we highlight by computing the first-return map for light rays close to an unstably bound orbit. Despite an exponential $e^\gamma$ 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.