Bargmann structures and Newton-Cartan theory

Abstract

It is shown that Newton-Cartan theory of gravitation can best be formulated on a five-dimensional extended space-time carrying a Lorentz metric together with a null parallel vector field. The corresponding geometry associated with the Bargmann group (nontrivially extended Galilei group) viewed as a subgroup of the affine de Sitter group AO(4,1) is thoroughly investigated. This new global formalism allows one to recast classical particle dynamics and the Schr\"odinger equation into a purely covariant form. The Newton-Cartan field equations are readily derived from Einstein's Lagrangian on the space-time extension.