Action Principle for Newtonian Gravity

TL;DR

This paper derives an action principle for Newtonian gravity using a novel Newton--Cartan geometry, incorporating effects like gravitational time dilation, based on a covariant expansion of general relativity.

Contribution

It introduces a new geometric framework and action principle for Newtonian gravity that includes relativistic effects such as gravitational time dilation.

Findings

Derivation of an action principle with Poisson equation as equations of motion.

Development of a new Newton--Cartan geometry based on a modified symmetry algebra.

Extension of Newtonian gravity to include gravitational time dilation effects.

Abstract

We derive an action whose equations of motion contain the Poisson equation of Newtonian gravity. The construction requires a new notion of Newton--Cartan geometry based on an underlying symmetry algebra that differs from the usual Bargmann algebra. This geometry naturally arises in a covariant $1/c$ expansion of general relativity with $c$ the speed of light. By truncating this expansion at subleading order we obtain the field content and transformation rules of the fields that appear in the action of Newtonian gravity. The equations of motion generalize Newtonian gravity by allowing for the effect of gravitational time dilation due to strong gravitational fields.