General form for Pseudo-Newtonian Potentials, imitating Schwarzschild geodesics
Abstract
We propose a new, general form for a pseudo-Newtonian gravitational potential (PNP), expressed as a series of Paczy\'nski-Wiita-like functions with the addition of increasing negative powers of with arbitrary coefficients. We present a procedure for determining these coefficients to construct a custom PNP that replicates key features of Schwarzschild geodesics for a test particle near a black hole. As an example, we construct potentials set to reproduce (I) the presence of an innermost stable circular orbit at the (geometric units), with the correct infall velocity for small deviations (on the geodesic universal infall), (II) the periapsis advance at large distances, and (III) the presence of a marginally bound circular orbit with specific angular momentum , and the periapsis advance of parabolic orbits close to it. We compare the performance of our examples against the Paczy\'nski-Wiita potential and other existing potentials. Finally, we discuss the limitations and advantages of our formulation.
Keywords
Cite
@article{arxiv.2507.15107,
title = {General form for Pseudo-Newtonian Potentials, imitating Schwarzschild geodesics},
author = {Itamar Ben Arosh Arad and Reem Sari},
journal= {arXiv preprint arXiv:2507.15107},
year = {2026}
}
Comments
Post-publication author-updated version. Corrected Eq. (4) to make the coefficients dimensionless, added recent references, and expanded discussion of uses and limitations of the potential