Newton–Cartan, Galileo–Maxwell and Kaluza–Klein

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. (note)