Binary black hole shadows, chaotic scattering and the Cantor set

TL;DR

This paper explores the complex, self-similar structure of binary black hole shadows using chaotic scattering theory, symbolic dynamics, and the Cantor set analogy, revealing new insights into null geodesics and shadow morphology.

Contribution

It introduces a novel symbolic dynamics framework for binary black hole shadows, demonstrating their self-similar, Cantor set-like structure and identifying stable null orbits in chaotic regimes.

Findings

Binary black hole shadows exhibit Cantor set-like self-similarity.

Existence of stable, bounded null orbits around equal-mass black holes.

Identification of three types of 1D shadows: regular, Cantor-like, and chaotic.

Abstract

We investigate the qualitative features of binary black hole shadows using the model of two extremally charged black holes in static equilibrium (a Majumdar--Papapetrou solution). Our perspective is that binary spacetimes are natural exemplars of chaotic scattering, because they admit more than one fundamental null orbit, and thus an uncountably-infinite set of perpetual null orbits which generate scattering singularities in initial data. Inspired by the three-disc model, we develop an appropriate symbolic dynamics to describe planar null geodesics on the double black hole spacetime. We show that a one-dimensional (1D) black hole shadow may constructed through an iterative procedure akin to the construction of the Cantor set; thus the 1D shadow is self-similar. Next, we study non-planar rays, to understand how angular momentum affects the existence and properties of the fundamental null orbits. Taking slices through 2D shadows, we observe three types of 1D shadow: regular, Cantor-like, and highly chaotic. The switch from Cantor-like to regular occurs where outer fundamental orbits are forbidden by angular momentum. The highly chaotic part is associated with an unexpected feature: stable and bounded null orbits, which exist around two black holes of equal mass $M$ separated by $a_1 < a < \sqrt{2} a_1$, where $a_1 = 4M/\sqrt{27}$. To show how this possibility arises, we define a certain potential function and classify its stationary points. We conjecture that the highly chaotic parts of the 2D shadow possess the Wada property. Finally, we consider the possibility of following null geodesics through event horizons, and chaos in the maximally-extended spacetime.