Exact solutions to the geodesic equations of linear dilaton black holes

TL;DR

This paper derives exact analytical solutions for the geodesic equations in 4D linear dilaton black hole spacetimes, revealing the mathematical structure of test particle motion using elliptic functions.

Contribution

It provides the first explicit solutions to the geodesic equations in LDBH spacetime, including solutions expressed via Weierstrass elliptic functions.

Findings

Exact solutions for radial and angular geodesics obtained.

Radial trajectories expressed in terms of WeierstrassP-function.

Insights into particle motion in non-asymptotically flat black hole backgrounds.

Abstract

In this paper, we analyze the geodesics of the 4-dimensional ($4D$) linear dilaton black hole (LDBH) spacetime, which is an exact solution to the Einstein-Maxwell-Dilaton (EMD) theory. LDBHs have non-asymptotically flat (NAF) geometry, and their Hawking radiation is an isothermal process. The geodesics motions of the test particles are studied via the standard Lagrangian method. After obtaining the Euler-Lagrange (EL) equations, we show that exact analytical solutions to the radial and angular geodesic equations can be obtained. In particular, it is shown that one of the possible solutions for the radial trajectories can be given in terms of the WeierstrassP-function ($\wp$-function), which is an elliptic-type special function.