Logarithmic potential for the gravitational field of Schwarzschild black holes

TL;DR

This paper introduces a logarithmic potential that exactly models particle motion near Schwarzschild black holes, providing a new analytical tool for studying black hole environments and accretion disks.

Contribution

It presents a novel logarithmic gravitational potential that yields exact solutions for particle motion near Schwarzschild black holes, bridging Newtonian and relativistic descriptions.

Findings

Logarithmic potential provides exact solutions for particle trajectories.

Relativistic Bernoulli equation derived in the context of the potential.

Potential improves analytical modeling of black hole accretion phenomena.

Abstract

Approximate gravitational potentials are often used to describe analytically the motion of particles near black holes (BHs), as well as to study the structure of an accretion disk. Such 'pseudo-Newtonian' potentials are used with the flat-metric equations. Here we consider the motion of a free particle near a non-rotating BH in the context of an exact `logarithmic' gravitational potential. We show how the logarithmic potential gives an exact solution for a mechanical problem and present the relativistic Bernoulli equation for the fluid in the Schwarzschild metric.