Lagrangians for non-relativistic gravity

TL;DR

This paper develops a covariant expansion of Einstein-Hilbert action in powers of 1/c^2, enabling analysis of non-relativistic gravity through a novel parametrization of spacetime geometry.

Contribution

It introduces a 'pre-non-relativistic' parametrization that facilitates the systematic expansion of general relativity in powers of 1/c^2.

Findings

Expanded Einstein-Hilbert action up to next-to-next-to-leading order.

Provided a covariant framework for non-relativistic gravity analysis.

Reformulated general relativity suitable for analytical 1/c^2 expansion.

Abstract

We study the covariant expansion of Einstein-Hilbert action in powers of $1/c^2$ with arbitrary spacetime foliation where $c$ is the speed of light. This is done firstly by suitable parametrization of geometry which is called 'pre-non-relativistic' parametrization. This allows us to rewrite the general relativity in a form suitable for the analytical $1/c^2$ expansion. Consequently, we can study the expansion of Einstein-Hilbert action up to the next-to-next-to-leading order.