Non-relativistic expansion of the Einstein-Hilbert Lagrangian

TL;DR

The work provides a systematic $1/c^2$ expansion to derive a non-relativistic gravity Lagrangian from the Einstein–Hilbert action, casting the problem in a covariant, off-shell Newton–Cartan framework. It presents a general expansion scheme for fields and Lagrangians, yielding LO, NLO, and NNLO sectors where NNLO reproduces LO dynamics and encodes second-variation structure. The metric is expanded within twistless torsional Newton–Cartan geometry, with Milne boosts ensuring invariance of key quantities, and the resulting non-relativistic gravity (NRG) Lagrangian recovers Newtonian gravity and its extensions, including gravitational time dilation, under the theory’s gauge symmetries. This formalism provides a covariant path to connect relativistic and non-relativistic gravity and to explore higher-order approximations akin to post-Newtonian expansions.

Abstract

We present a systematic technique to expand the Einstein-Hilbert Lagrangian in inverse powers of the speed of light squared. The corresponding result for the non-relativistic gravity Lagrangian is given up to next-to-next-to-leading order. The techniques are universal and can be used to expand any Lagrangian theory whose fields are a function of a given parameter.