Investigation the geodesic motion of three dimensional rotating black holes

TL;DR

This paper analytically solves the geodesic equations in various three-dimensional rotating black hole spacetimes, classifies possible trajectories, and highlights the advantages of the analytical approach over previous methods.

Contribution

It provides complete analytical solutions of geodesic equations in three-dimensional rotating black holes using elliptic functions, enhancing understanding of particle motion in these spacetimes.

Findings

Analytical solutions expressed in Weierstrass elliptic functions.

Classification of possible particle trajectories around black holes.

Comparison showing benefits of the analytical method over previous approaches.

Abstract

We study the geodesic equations in the space-time of neutral Brans-Dicke Dilaton black hole in three dimensions, BTZ black holes and the 2+1 black hole. We use the process of separation of the Hamilton-Jacobi equation to obtain the constants of motion. The whole analytical solution of the geodesic equations in the space-times of the intended black holes are shown completely. Moreover, the geodesic equations are solved in terms of Weierstrass elliptic functions. Furthermore, with use of the analytical solution and effective potential technique some trajectories around the black holes are classified. Meanwhile, by analytical solution, effective potential and considering the zeroes of underlying polynomials, some possible orbits are plotted. Finally, we compare our results with Cruz {\it et. al.} \cite{Cruz:1994ar} and we indicate the benefits of the analytical method which is applied in this paper.