The structure of the Newtonian limit

Abstract

We consider a smooth one-parameter family of four-dimensional manifolds X ε , ε≥0, each one endowed with a covariant metric g ε . It is assumed that g ε is a Lorentz metric for each ε>0, i.e., the signature of g ε is (+,−,−,−) for ε>0, while the limit metric g 0 on X 0 is assumed to be degenerated of rank 1, i.e., the signature of g 0 is (+,0,0,0). We characterize when the limit manifold X 0 inherits the geometric structure of a Newtonian gravitation. The limit manifold X 0 is a Newtonian gravitation if and only if there exist the limits of the Levi-Civita connection ∇ ε , the curvature operator R ε and the contravariant Einstein tensor G ε 2 as ε→0. Moreover, the existence of these limits is characterized in terms of the Taylor expansion of the family { g ε } with respect to the parameter ε.