Generalized Newtonian description of particle motion in spherically symmetric spacetimes

TL;DR

This paper introduces a generalized Newtonian framework for particle motion in spherically symmetric spacetimes, accurately capturing key orbital features and applicable to various black hole models with high precision.

Contribution

It provides a new low-energy limit approach that reproduces key relativistic orbital properties in a simple Newtonian form for diverse spherically symmetric spacetimes.

Findings

Reproduces marginally stable, bound, and photon orbits exactly.

Achieves better than 10% accuracy in orbital and epicyclic frequencies.

Applicable to multiple black hole spacetimes with simple equations of motion.

Abstract

We present a generalized Newtonian description of particle dynamics valid for any spherically symmetric, static black hole spacetime. This approach is derived from the geodesic motion of test particles in the low-energy limit. It reproduces exactly the location of the marginally stable, marginally bound, and photon circular orbits; the radial dependence of the energy and angular momentum of circular orbits; parabolic motion; pericentre shift; and the spatial projection of general trajectories. As explicit examples of the new prescription, we apply it to the Schwarzschild, Schwarzschild-de Sitter, Reissner-Nordstr\"om, Ay\'on Beato-Garc\'ia, and Kehagias-Sfetsos spacetimes. In all of these examples, the orbital and epicyclic frequencies are reproduced to better than 10%. The resulting equations of motion can be implemented easily and efficiently within existing Newtonian frameworks.