Review on non-relativistic gravity - Hartong, Jelle; Obers, Niels A.; Oling, Gerben (Frontiers in Physics, 2023)
We review the history of Newton–Cartan gravity with an emphasis on recent developments including the covariant, off-shell large speed of light expansion of general relativity. Depending on the matter content, this either leads to a Newton–Cartan geometry with absolute time or to a Newton–Cartan geometry that contains non-relativistic gravitational time dilation effects. The latter shows that non-relativistic gravity includes a strong-coupling regime and goes beyond Newtonian gravity. We start by reviewing early developments in Newton–Cartan geometry, including the covariant description of Newtonian gravity, mainly through the works of Trautman, Dautcourt, Künzle and Ehlers. We then turn to more modern developments, such as the gauging of the Bargmann algebra, and we describe why the latter cannot be used to find an off-shell description of Newton–Cartan gravity. We review recent work on the 1/c expansion of GR and show that this leads to an alternative ‘type II’ notion of Newton–Cartan geometry. Finally, we discuss matter couplings, solutions and odd powers in 1/c, and we conclude with a brief summary of related topics.