Divergent Reflections around the Photon Sphere of a Black Hole

TL;DR

This paper analyzes the behavior of light trajectories around black holes, revealing exponential patterns in how images and light paths cluster near the photon sphere, with extensions to rotating Kerr black holes.

Contribution

It introduces a formalism describing light trajectories near black holes using exponential functions, generalizes findings to Kerr black holes, and explains image formation near the photon sphere.

Findings

Exponential clustering of light paths near the photon sphere.

Generalization of exponential behavior to Kerr black holes with spin dependence.

No logarithmic divergence for prograde trajectories in extremely rotating Kerr black holes.

Abstract

From any location outside the event horizon of a black hole there are an infinite number of trajectories for light to an observer. Each of these paths differ in the number of orbits revolved around the black hole and in their proximity to the last photon orbit. With simple numerical and a perturbed analytical solution to the null-geodesic equation of the Schwarzschild black hole we will reaffirm how each additional orbit is a factor $e^{2 \pi}$ closer to the black hole's optical edge. Consequently, the surface of the black hole and any background light will be mirrored infinitely in exponentially thinner slices around the last photon orbit. Furthermore, the introduced formalism proves how the entire trajectories of light in the strong field limit is prescribed by a diverging and a converging exponential. Lastly, the existence of the exponential family is generalized to the equatorial plane of the Kerr black hole with the exponentials dependence on spin derived. Thereby, proving that the distance between subsequent images increases and decreases for respectively retrograde and prograde images. In the limit of an extremely rotating Kerr black hole no logarithmic divergence exists for prograde trajectories.