Pseudo-Newtonian equations for evolution of particles and fluids in stationary space-times

TL;DR

This paper develops a general pseudo-Newtonian formalism for simulating particles, light, and fluids in stationary space-times, enabling effective Newtonian-based modeling of black-hole related astrophysical processes.

Contribution

It introduces a comprehensive pseudo-Newtonian framework applicable to stationary space-times, including explicit fluid equations in Schwarzschild space-time, and assesses its accuracy and limitations.

Findings

Formulas are simplest for non-rotating black holes (Schwarzschild)

Quantitative errors are minimal for low binding energy motions

Provides ready-to-use fluid equations in Schwarzschild coordinates

Abstract

Pseudo-Newtonian potentials are a tool often used in theoretical astrophysics to capture some key features of a black-hole space-time in a Newtonian framework. As a result, one can use Newtonian numerical codes, and Newtonian formalism in general, in an effective description of important astrophysical processes such as accretion onto black holes. In this paper we develop a general pseudo-Newtonian formalism which pertains to the motion of particles, light, and fluids in stationary space-times. In return, we are able to assess the applicability of the pseudo-Newtonian scheme. The simplest and most elegant formulas are obtained in space-times without gravitomagnetic effects, such as the Schwarzschild rather than the Kerr space-time; the quantitative errors are smallest for motion with low binding energy. Included is a ready-to-use set of fluid equations in Schwarzschild space-time in Cartesian and radial coordinates.