Periodic orbits of neutral test particles in Reissner-Nordstr\"{o}m naked singularities

TL;DR

This paper explores the classification and properties of periodic orbits of neutral test particles around Reissner-Nordström naked singularities, revealing complex distribution patterns and physical characteristics of hypothetical circular orbits.

Contribution

It provides a detailed mapping of periodic orbits in parameter space and analyzes their distribution and properties in the context of Reissner-Nordström naked singularities.

Findings

Periodic orbits can exist in both regions around the singularity.

Sets of periodic orbits can share the same energy and angular momentum.

Different regions in parameter space require distinct analytical solutions.

Abstract

We conduct studies on Levin's taxonomy of periodic orbits for neutral test particles around a Reissner-Nordstr\"{o}m naked singularity. It was known that naked singularities could harbor two distinct regions of time-like bound orbits and thus we expect periodic orbits to appear in both regions. It is possible for a pair of periodic orbits from both regions to possess the exact same angular momentum $L$ and energy $E$ values. We chart the sets of periodic orbits in $(L,E)$-parameter space and highlight the general distribution pattern of these sets for three possible scenarios. Regions within $(L,E)$-space can be partitioned into multiple domains $\mathcal{D}_k$ based on the roots configuration of the quartic polynomial $P(u)$ where $u$ is the inverse radial coordinate. Consequently, each domain and interestingly enough, portions of certain periodic orbits sets that lie in different $\mathcal{D}_k$ require different analytical solutions to plot the resulting orbit. Furthermore, we uncover physical properties of some hypothetical circular orbits residing in the inner region from analysing the $(L,E)$-space.