# The Geometrical Trinity of Gravity

## TL;DR

This paper explores three equivalent geometric formulations of gravity—curvature, torsion, and non-metricity—in flat spacetime, highlighting their conceptual differences and discussing possible extensions of these theories.

## Contribution

It presents a comprehensive comparison of the three formulations of General Relativity and discusses their extensions, emphasizing their equivalence and distinct geometric interpretations.

## Key findings

- Three equivalent formulations of General Relativity are detailed.
- Each formulation attributes gravity to a different geometric property.
- Extensions of these formulations are discussed.

## Abstract

The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the spacetime curvature, which is now the standard interpretation of gravity. However, this is not the only possibility. In these notes we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory. Based on these three alternative formulations of General Relativity, we then discuss some extensions.