On the low-energy limit of stationary and axisymmetric solutions in General Relativity

TL;DR

This paper investigates the low-energy limit of stationary, axisymmetric solutions in General Relativity with dust, revealing persistent differences from Newtonian gravity due to strong gravitomagnetic effects, and demonstrating how GR can explain Tully-Fisher-like relations.

Contribution

It introduces the concept of strong gravitomagnetism in the low-energy limit, providing a systematic way to solve the equations and showing how GR accounts for phenomena like the Tully-Fisher relation.

Findings

Differences between GR and Newtonian gravity persist in the weak-field, slow-motion regime.

Analytical solutions for specific cases and the vacuum case are provided.

GR can naturally explain a Tully-Fisher-like relation.

Abstract

We study the low-energy limit of General Relativity in the presence of stationarity and axial symmetry, coupled to dust. Specifically, we demonstrate that differences between the dynamics of General Relativity and those of Newtonian gravity persist even in the weak-field and slow-motion regime. Notably, these differences are driven by dragging terms that are not necessarily small, as is typically the case in the well-known gravitomagnetic limit. To highlight this distinction, we use the concept of strong gravitomagnetism that we introduced in previous works. We provide a pedagogical discussion of how these discrepancies arise and outline a systematic procedure to solve the equations of motion for such systems. Furthermore, we present analytical results for specific cases and also give the general solution for the vacuum case. A particularly notable result is our demonstration of how General Relativity can naturally account for a Tully-Fisher-like relation.