The structure of the Newtonian limit

✍ Scribed by Juan A. Navarro Gonzalez; Juan B. Sancho de Salas

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 217 KB
• Volume: 44
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(02)00116-x

✦ Synopsis

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 2 ε 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.