Motion and Trajectories of Particles Around Three-Dimensional Black Holes

TL;DR

This paper investigates the motion of relativistic particles around three-dimensional black holes using Hamilton-Jacobi formalism, revealing that particles are inevitably drawn into the black hole and stable orbits are not possible.

Contribution

It extends the Hamilton-Jacobi approach to three-dimensional black holes and provides exact solutions for particle trajectories at extreme angular momentum values.

Findings

Particles are trapped regardless of energy and angular momentum.

Matter always falls into the black hole center.

Exact solutions found for specific angular momentum conditions.

Abstract

The motion of relativistic particles around three dimensional black holes following the Hamilton-Jacobi formalism is studied. It follows that the Hamilton-Jacobi equation can be separated and reduced to quadratures in analogy with the four dimensional case. It is shown that: a) particles are trapped by the black hole independently of their energy and angular momentum, b) matter alway falls to the centre of the black hole and cannot understake a motion with stables orbits as in four dimensions. For the extreme values of the angular momentum of the black hole, we were able to find exact solutions of the equations of motion and trajectories of a test particle.