[Paper Review] Analytical solutions of the geodesic equations in the spacetime of a rotating charged black hole in $f(R)$ gravity

Saheb Soroushfar, Reza Saffari+3 more|arXiv (Cornell University)|May 29, 2016

Black Holes and Theoretical Physics3 citations

TL;DR

This paper derives analytical solutions to the geodesic equations for test particles and light rays in the spacetime of a rotating, charged black hole within $f(R)$ gravity. Using special functions like Weierstrass elliptic and Kleinian sigma functions, it classifies orbit types through effective potentials and parametric diagrams, offering exact solutions and dynamical insights beyond general relativity.

ABSTRACT

We study the geodesic equations in the space time of a rotating charged black hole in $f(R)$ gravity. We derive the equations of motion for test particles and light rays and present their solutions in terms of the Weierstrass $\wp$, $\zeta$ and $\sigma$ functions as well as the Kleinian $\sigma$ function. With the help of parametric diagrams and effective potentials we analyze the geodesic motion and classify the possible orbit types.

Motivation & Objective

• To extend the understanding of geodesic motion in modified gravity by studying rotating, charged black holes in $f(R)$ gravity.
• To derive exact analytical solutions for timelike and null geodesics in this spacetime.
• To classify possible orbit types using effective potentials and parametric diagrams.
• To provide a systematic framework for analyzing particle and light ray trajectories beyond general relativity.

Proposed method

• Derivation of the geodesic equations from the metric of a rotating, charged black hole in $f(R)$ gravity.
• Solution of the equations of motion using Weierstrass $\wp$, $\zeta$, $\sigma$ functions and the Kleinian $\sigma$ function.
• Construction of effective potentials to analyze the nature of geodesic motion.
• Use of parametric diagrams to visualize and classify orbital types such as bounded, unbounded, and plunging orbits.
• Application of special functions to express time and spatial coordinates as functions of affine parameters.
• Comparison of dynamical behavior with general relativity through qualitative analysis of orbit structures.

Experimental results

Research questions

• RQ1 How do the geodesic equations for test particles and light rays behave in the spacetime of a rotating, charged black hole in $f(R)$ gravity?
• RQ2 What analytical forms can the solutions of these geodesic equations take in terms of special functions?
• RQ3 Which types of orbits—bounded, unbounded, or plunging—are possible under $f(R)$ gravity for such black holes?
• RQ4 How do the effective potentials in $f(R)$ gravity differ from those in general relativity, and what do they reveal about orbital stability?
• RQ5 What role do the Weierstrass and Kleinian functions play in parameterizing the full trajectory of particles and photons?

Key findings

• The geodesic equations for both massive particles and photons are solved analytically using Weierstrass $\wp$, $\zeta$, $\sigma$ functions and the Kleinian $\sigma$ function.
• The solutions express the time and spatial coordinates as explicit functions of an affine parameter, enabling full trajectory reconstruction.
• Parametric diagrams reveal distinct orbit types, including bounded, unbounded, and plunging trajectories, depending on energy and angular momentum.
• Effective potentials demonstrate that $f(R)$ gravity modifies orbital dynamics compared to general relativity, particularly in the stability and shape of bound orbits.
• The use of special functions allows for a complete and exact description of geodesic motion without numerical approximation.
• The classification of orbits is consistent across different parameter regimes, showing robustness of the analytical framework.