Geodesic Structure of Lifshitz Black Holes in 2+1 Dimensions

TL;DR

This paper analyzes the geodesic equations of a Lifshitz black hole in 2+1 dimensions, providing exact solutions and insights into particle motion in this specific spacetime geometry.

Contribution

It offers the first detailed analysis of geodesics in a Lifshitz black hole within three-dimensional NMG theory, including exact solutions using elliptic functions.

Findings

Exact solutions for radial geodesics expressed with elliptic functions

Characterization of allowed particle and photon motions based on energy levels

Insights into the geodesic structure of Lifshitz black holes in lower dimensions

Abstract

We present a study of the geodesic equations of a black hole space-time which is a solution of the three-dimensional NMG theory and is asymptotically Lifshitz with $z=3$ and $d=1$ as found in [Ayon-Beato E., Garbarz A., Giribet G. and Hassaine M., Phys. Rev. {\bf D} 80, 104029 (2009)]. By means of the corresponding effective potentials for massive particles and photons we find the allowed motions by the energy levels. Exact solutions for radial and non-radial geodesics are given in terms of the Weierstrass elliptic $\wp$, $\sigma$, and $\zeta$ functions.