  We study Kaluza-Klein reduction in Newton-Cartan gravity. In particular we
show that dimensional reduction and the nonrelativistic limit commute. The
resulting theory contains Galilean electromagnetism and a nonrelativistic
scalar. It provides the first example of back-reacted couplings of scalar and
vector matter to Newton-Cartan gravity. This back-reaction is interesting as it
sources the spatial Ricci curvature, providing an example where nonrelativistic
gravity is more than just a Newtonian potential.
