The Non-Relativistic Geometric Trinity of Gravity

TL;DR

This paper extends the geometric trinity of gravity to the non-relativistic regime by deriving a non-metric theory equivalent to Newtonian gravity and its teleparallel counterpart, unifying different formulations of gravity in the non-relativistic limit.

Contribution

It introduces the non-relativistic analogue of STEGR and demonstrates its equivalence to Newton--Cartan theory and its teleparallel version, completing the non-relativistic geometric trinity.

Findings

Derived the non-relativistic limit of STEGR.

Established equivalence with Newton--Cartan theory.

Unified non-relativistic gravity formulations.

Abstract

The geometric trinity of gravity comprises three distinct formulations of general relativity: (i) the standard formulation describing gravity in terms of spacetime curvature, (ii) the teleparallel equivalent of general relativity describing gravity in terms of spacetime torsion, and (iii) the symmetric teleparallel equivalent of general relativity (STEGR) describing gravity in terms of spacetime non-metricity. In this article, we complete a geometric trinity of non-relativistic gravity, by (a) taking the non-relativistic limit of STEGR to determine its non-relativistic analogue, and (b) demonstrating that this non-metric theory is equivalent to the Newton--Cartan theory and its teleparallel equivalent, i.e., the curvature and the torsion based non-relativistic theories that are both geometrised versions of classical Newtonian gravity.