# Analytic solutions of the geodesic equation for Reissner-Nordstr\"om-(anti-)de Sitter black holes surrounded by different kinds of regular and exotic matter fields

## TL;DR

This paper derives exact analytical solutions for particle and light trajectories in Reissner-Nordström-(anti-)de Sitter black hole spacetimes with various matter fields, revealing all possible orbit types.

## Contribution

It provides the first complete analytical solutions of geodesic equations in these complex black hole backgrounds using elliptic and hyperelliptic functions.

## Key findings

- Complete classification of possible orbits.
- Analytical expressions for geodesics in terms of elliptic and hyperelliptic functions.
- Insights into particle and light motion in exotic matter environments.

## Abstract

The purpose of this study is the derivation of the equation of motion for particles and light in the spacetime of Reissner-Nordstr\"{o}m-(anti-)de Sitter black holes in the background of different kinds of regular and exotic matter fields. The complete analytical solutions of the geodesic equations are given in terms of the elliptic Weierstra{\ss} $\wp$-function and the hyperelliptic Kleinian $\sigma$-function. Finally after analyzing the geodesic motion of test particles and light using parametric diagrams and effective potentials, we present a list of all possible orbits.