Scattering map for two black holes

TL;DR

This paper models light trajectories around two distant Schwarzschild black holes using a simplified 2D map, revealing chaotic behavior and fractal basin boundaries that depend on the separation distance.

Contribution

It introduces a novel approximation reducing the complex dynamics to a 2D map and analyzes the chaotic properties and fractal structures of light trajectories.

Findings

The map exhibits chaos with fractal basin boundaries.

The basin boundary approaches a Cantor set as separation increases.

The fractal dimension decreases logarithmically with distance.

Abstract

We study the motion of light in the gravitational field of two Schwarzschild black holes, making the approximation that they are far apart, so that the motion of light rays in the neighborhood of one black hole can be considered to be the result of the action of each black hole separately. Using this approximation, the dynamics is reduced to a 2-dimensional map, which we study both numerically and analytically. The map is found to be chaotic, with a fractal basin boundary separating the possible outcomes of the orbits (escape or falling into one of the black holes). In the limit of large separation distances, the basin boundary becomes a self-similar Cantor set, and we find that the box-counting dimension decays slowly with the separation distance, following a logarithmic decay law.