The 1+3-Newton-Cartan system and Newton-Cartan cosmology

We perform a covariant 1+3 split of the Newton-Cartan equations. The resulting 3-dimensional system of equations, called \textit{the 1+3-Newton-Cartan equations}, is structurally equivalent to the 1+3-Einstein equations. In particular it features the momentum constraint, and a choice of adapted coordinates corresponds to a choice of shift vector. We show that these equations reduce to the classical Newton equations without the need for special Galilean coordinates. The solutions to the 1+3-Newton-Cartan equations are equivalent to the solutions of the classical Newton equations if space is assumed to be compact or if fall-off conditions at infinity are assumed. We then show that space expansion arises as a fundamental field in Newton-Cartan theory, and not by construction as in the classical formulation of Newtonian cosmology. We recover the Buchert-Ehlers theorem for the general expansion law in Newtonian cosmology.