Non-relativistic limit of gravity theories in the first order formalism

TL;DR

This paper explores the non-relativistic limit of four-dimensional gravity theories in the first order formalism, analyzing Einstein-Hilbert and Mardones-Zanelli actions, revealing differences in field equations and conditions for Newtonian time.

Contribution

It provides a detailed analysis of the non-relativistic limit of gravity in the first order formalism, especially highlighting the role of the boost connection in the Mardones-Zanelli action.

Findings

The Einstein-Hilbert limit yields vacuum and matter configurations at leading order.

The Mardones-Zanelli action's non-relativistic limit fully determines the field equations.

The cosmological constant must vanish in the non-relativistic Mardones-Zanelli action.

Abstract

We consider the non-relativistic limit of gravity in four dimensions in the first order formalism. First, we revisit the case of the Einstein-Hilbert action and formally discuss some geometrical configurations in vacuum and in the presence of matter at leading order. Second, we consider the more general Mardones-Zanelli action and its non-relativistic limit. The field equations and some interesting geometries, in vacuum and in the presence of matter, are formally obtained. Remarkably, in contrast to the Einstein-Hilbert limit, the set of field equations is fully determined because the boost connection appears in the action and field equations. It is found that the cosmological constant must disappear in the non-relativistic Mardones-Zanelli action at leading order. The conditions for Newtonian absolute time be acceptable are also discussed. It turns out that Newtonian absolute time can be safely implemented with reasonable conditions.