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.
