Non-relativistic Limits of General Relativity

We discuss non-relativistic limits of general relativity. In particular, we define a special fine-tuned non-relativistic limit, inspired by string theory, where the Einstein-Hilbert action has been supplemented by the kinetic term of a one-form gauge field. Taking the limit, a crucial cancellation takes place, in an expansion of the action in terms of powers of the velocity of light, between a leading divergence coming from the spin-connection squared term and another infinity that originates from the kinetic term of the one-form gauge field such that the finite invariant non-relativistic gravity action is given by the next sub-leading term. This non-relativistic action allows an underlying torsional Newton-Cartan geometry as opposed to the zero torsion Newton-Cartan geometry that follows from a more standard limit of General Relativity but it lacks the Poisson equation for the Newton potential. We will mention extensions of the model to include this Poisson equation.

Automated summary

In this article, the non-relativistic limits of general relativity are discussed, focusing on a special finely tuned limit inspired by string theory. This limit involves adding the kinetic term of a one-form gauge field to the Einstein-Hilbert action, resulting in a finite invariant non-relativistic gravity action. This action allows for an underlying torsional Newton-Cartan geometry, but lacks the Poisson equation for the Newton potential. Extensions to include this equation are mentioned for further research.