# The geodesics structure of Schwarzschild electromagnetic black hole

## TL;DR

This paper analyzes how an electromagnetic field alters the geodesic structure of a Schwarzschild black hole, affecting particle trajectories, horizons, and innermost stable circular orbits with exact solutions.

## Contribution

It provides the first exact analytical solutions for geodesics in a Schwarzschild electromagnetic black hole, revealing modifications to horizons and ISCO radii.

## Key findings

- Radial geodesics are solved exactly for both photons and particles.
- Electromagnetic field causes the horizon to shrink and delays particle infall.
- No stable photon circular orbits exist; stable and unstable orbits for massive particles are identified.

## Abstract

The geodesic equations are considered in static mass imbedded in a uniform electromagnetic field. Due to electromagnetic field horizon shrinks and geodesics are modified. By analyzing the behavior of the effective potentials for the massless and massive particle we study the radial and circular trajectories. Radial geodesics for both photons and particles are solved exactly. It is shown that a particle falls toward the horizon in a finite proper time slows down so that the particle reaches the singularity in longer time than Schwarzschild case. Timelike and null circular geodesics are investigated. We have shown that, there is no stable circular orbits for photons, however stable and unstable second kind orbits are exists for massive particle. An exact analytical solution for the innermost circular orbits (ISCO) has been obtained. It has been shown that the radius of ISCO shrinks due to the presence of electromagnetic field.