Covariant Newtonian limit of Lorentz space-times

Abstract

The formulation of this limit given by Dautcourt [1] is slightly improved using the notions of Galilei manifold and Newtonian connection. It is then shown under what conditions the conservation equations ▽_μ ^μα = 0 for an arbitrary relativistic continuum have the correct (also covariantly formulated) Newtonian limit. For electromagnetism one obtains a curved space generalization of the electric or magnetic Galileian theory of LeBellac and Lévy-Leblond [4] depending on whether the contravariant or the covariant Maxwell tensor is required to have a regular Galileian limit.