The Schr\"odinger-Newton equations beyond Newton

TL;DR

This paper derives a covariant low-velocity limit of gravitoelectromagnetic equations, coupling them with the Schr"odinger equation to extend the Schr"odinger-Newton model, revealing relativistic corrections and generalizations to many particles.

Contribution

It introduces a minimal, covariant extension of the Schr"odinger-Newton equations incorporating gravitoelectric and gravitomagnetic effects, with relativistic corrections and a many-particle generalization.

Findings

Relativistic gravitomagnetic correction scales with system mass over Planck mass.

Gravitomagnetic effects reinforce Newtonian attraction.

Extended model includes many-particle systems via Wigner function.

Abstract

The scope of this paper is twofold. First, we derive rigorously a low-velocity and Galilei-covariant limit of the gravitoelectromagnetic (GEM) equations. Subsequently, these reduced GEM equations are coupled to the Schr\"odinger equation with gravitoelectric and gravitomagnetic potentials. The resulting extended Schr\"odinger-Newton equations constitute a minimal model where the three fundamental constants of nature ($G$, $\hbar$, and $c$) appear naturally. We show that the relativistic correction coming from the gravitomagnetic potential scales as the ratio of the mass of the system to the Planck mass, and that it reinforces the standard Newtonian (gravitoelectric) attraction. The theory is further generalized to many particles through a Wigner function approach.