On the Newtonian Limit of the Weyl Tensor

TL;DR

This paper clarifies how the electric and magnetic parts of the Weyl tensor behave in the Newtonian limit, identifying the minimal set of equations needed for this approximation in cosmology.

Contribution

It provides a precise definition of the Newtonian limits of the Weyl tensor's electric and magnetic parts and discusses the necessary equations for this limit.

Findings

Defined Newtonian limits of Weyl tensor components

Identified subset of equations for Newtonian approximation

Clarified relation between relativistic and Newtonian frameworks

Abstract

In this note we wish to complement some recent work in the cosmological literature concerning the Weyl conformal curvature tensor and its parts. In particular, we shall give a clear-cut definition of the Newtonian limits of electric and magnetic parts of the Weyl tensor. We also discuss that only a subset of the relativistic equations is needed to obtain a closed system of equations in the Newtonian limit.