Rational Orbits around Charged Black Holes

TL;DR

This paper introduces a classification system for eccentric orbits around charged black holes using rational numbers, aiding in approximating complex orbits and distinguishing charged from uncharged spacetimes.

Contribution

It develops a taxonomy based on rational numbers for classifying orbits in Reissner-Nordström spacetime, applicable to both charged and uncharged particles, and offers a method to differentiate spacetimes via orbital dynamics.

Findings

Orbit taxonomy based on rational numbers enables orbit approximation.

Comparison of orbits distinguishes charged and uncharged black hole spacetimes.

Method reduces computational complexity in gravitational wave calculations.

Abstract

We show that all eccentric timelike orbits in Reissner-Nordstr\"{o}m spacetime can be classified using a taxonomy that draws upon an isomorphism between periodic orbits and the set of rational numbers. By virtue of the fact that the rationals are dense, the taxonomy can be used to approximate aperiodic orbits with periodic orbits. This may help reduce computational overhead for calculations in gravitational wave astronomy. Our dynamical systems approach enables us to study orbits for both charged and uncharged particles in spite of the fact that charged particle orbits around a charged black hole do not admit a simple one-dimensional effective potential description. Finally, we show that comparing periodic orbits in the RN and Schwarzschild geometries enables us to distinguish charged and uncharged spacetimes by looking only at the orbital dynamics.