Review on Non-Relativistic Gravity

TL;DR

This review explores the development of non-relativistic gravity, emphasizing recent covariant off-shell expansions of general relativity that reveal new geometric structures and gravitational effects beyond classical Newtonian theory.

Contribution

It provides a comprehensive overview of Newton-Cartan geometry, recent off-shell large speed of light expansions, and their implications for non-relativistic gravity theories.

Findings

Off-shell large speed of light expansion leads to new Newton-Cartan geometries.

Non-relativistic gravity can include strong field effects beyond Newtonian gravity.

Modern approaches like gauging Bargmann algebra have limitations in covariant formulations.

Abstract

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 expansion either leads to Newton-Cartan geometry with absolute time or to Newton-Cartan geometry with non-relativistic gravitational time dilation effects. The latter shows that non-relativistic gravity includes a strong field 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, K\"unzle 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 covariant description of Newtonian gravity. We review recent work on the $1/c$ expansion of general relativity 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.