Analytical solutions of the geodesic equation in the spacetime of a black hole in f(R) gravity

TL;DR

This paper derives exact analytical solutions for test particle trajectories in a black hole spacetime within f(R) gravity, using advanced mathematical functions, and characterizes the orbit types based on physical parameters.

Contribution

It provides the first complete set of analytic solutions to the geodesic equations in f(R) black hole spacetimes, expanding understanding of particle motion in modified gravity.

Findings

Solutions expressed via Weierstrass elliptic functions and Kleinian sigma functions

Classification of orbit types based on energy, angular momentum, and cosmological constant

Analytic framework applicable to various parameter regimes

Abstract

We consider the motion of test particles in the spacetime of a black hole in f(R) gravity. The complete set of analytic solutions of the geodesic equation in the spacetime of this black hole are presented. The geodesic equations are solved in terms of Weierstrass elliptic functions and derivatives of Kleinian sigma functions. The different types of the resulting orbits are characterized in terms of the conserved energy and angular momentum as well as the cosmological constant $\Lambda$ and the real constant $\beta$.