Large and small speed of light expansion of general relativity

The quest for a consistent theory of quantum gravity is one of the most important outstanding problems in theoretical physics. In the landscape of physical theories, quantum gravity sits at the corner where all the physical constants (speed of light, Newton’s and Planck’s constant) are finite. A region that is often overlooked is the nonrelativistic gravity regime. Contrary to common lore, it has become clear that the theory of nonrelativistic gravity is much richer than was so far appreciated, containing much more than just Newtonian gravity. I will review recent developments in the large speed of light expansion of GR and its coupling to matter, and present its geometric formulation in terms of a novel version of Newton-Cartan geometry. I will also discuss recent insights into the complementary case when the speed of light goes to zero, which is known as the Carroll or ultra-local expansion of gravity, in which case the underlying geometric structure is Carroll geometry. Finally, I will briefly comment on approaches towards embedding such limits in string theory and holography.