Non-relativistic gravity theories in four spacetime dimensions

TL;DR

This paper develops a non-relativistic gravity theory in four dimensions using a geometrical approach, deriving a Newtonian gravity action with connections to MOND and extending to generalized algebras.

Contribution

It introduces a novel non-relativistic gravity framework in four dimensions based on the MacDowell-Mansouri formulation, linking to Newtonian and MOND theories.

Findings

Derivation of a Newtonian gravity action from a non-relativistic algebra.

Demonstration that the theory includes the Poisson equation with a cosmological constant.

Extension to a generalized Newton-Hooke algebra for broader applicability.

Abstract

In this work we present a non-relativistic gravity theory defined in four spacetime dimensions using the MacDowell-Mansouri geometrical formulation. We obtain a Newtonian gravity action which is constructed from the curvature of a Newton-Hooke version of the so-called Newtonian algebra. We show that the non-relativistic gravity theory presented here contains the Poisson equation in presence of a cosmological constant. Moreover we make contact with the Modified Newtonian Dynamics (MOND) approach for gravity by considering a particular ansatz for a given gauge field. We extend our results to a generalized non-relativistic MacDowell-Mansouri gravity theory by considering a generalized Newton-Hooke algebra.