Motion of particles on a $z=2$ Lifshitz black hole in 3+1 dimensions

TL;DR

This paper analyzes the geodesic motion of particles around a specific Lifshitz black hole in 3+1 dimensions, providing analytical solutions and showing that confined orbits are not possible in this spacetime.

Contribution

It offers the first analytical solutions for particle trajectories in a $z=2$ Lifshitz black hole and confirms the absence of confined orbits, aligning with recent findings for similar black holes.

Findings

Analytical expressions for particle trajectories using elliptic functions

Confined orbits are not possible in this Lifshitz black hole spacetime

Results agree with recent literature on Lifshitz black holes

Abstract

We study the geodesic structure of a $z=2$ Lifshitz black hole in 3+1 spacetime dimensions that is an exact solution to the Einstein-scalar-Maxwell theory. We investigate the motion of massless and massive particles in this background using the standard Lagrangian procedure. Analytical expressions are obtained for radial and angular motions of the test particles, where the polar trajectories are given in terms of the $\wp$ - Weierstrass elliptic function. It is shown that confined orbits are not allowed on this spacetime, this result agrees with the obtained recently in the literature for other Lifshitz black holes.