Energy Level Diagrams for Black Hole Orbits

TL;DR

This paper develops a framework for understanding black hole orbits through energy level diagrams and rational number classifications, drawing an analogy to atomic physics and applicable to astrophysical systems.

Contribution

It introduces a novel method to construct energy level diagrams for black hole orbits using a rational number correspondence, including a closed form for quantifying zoom-whirl behavior.

Findings

Constructed periodic tables of black hole orbits.

Derived a closed form expression for rational q.

Linked orbit classifications to observable astrophysical phenomena.

Abstract

A spinning black hole with a much smaller black hole companion forms a fundamental gravitational system, like a colossal classical analog to an atom. In an appealing if imperfect analogy to atomic physics, this gravitational atom can be understood through a discrete spectrum of periodic orbits. Exploiting a correspondence between the set of periodic orbits and the set of rational numbers, we are able to construct periodic tables of orbits and energy level diagrams of the accessible states around black holes. We also present a closed form expression for the rational q, thereby quantifying zoom-whirl behavior in terms of spin, energy, and angular momentum. The black hole atom is not just a theoretical construct, but corresponds to extant astrophysical systems detectable by future gravitational wave observatories.