The complete set of solutions of the geodesic equations in the space-time of a Schwarzschild black hole pierced by a cosmic string

TL;DR

This paper provides a comprehensive analytical solution to the geodesic equations in a Schwarzschild black hole space-time pierced by a cosmic string, exploring particle trajectories, perihelion shifts, and light deflection.

Contribution

It presents the complete set of analytical solutions for geodesic equations in this specific space-time, including classifications based on energy, angular momentum, and deficit angle.

Findings

Derived analytical solutions for massive and massless particles.

Calculated perihelion shift for bound orbits.

Discussed light deflection and Newtonian limit.

Abstract

We study the geodesic equations in the space-time of a Schwarzschild black hole pierced by an infinitely thin cosmic string and give the complete set of analytical solutions of these equations for massive and massless particles, respectively. The solutions of the geodesic equations can be classified according to the particle's energy and angular momentum, the ratio between the component of the angular momentum aligned with the axis of the string and the total angular momentum, the deficit angle of the space-time and as well the horizon radius (or mass) of the black hole. For bound orbits of massive test particles we calculate the perihelion shift, we discuss light deflection and comment on the Newtonian limit.