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.

Highlights

  • IntroductionGravity and geometry have accompanied each other from the very conception of General

  • Gravity and geometry have accompanied each other from the very conception of GeneralRelativity (GR) brilliantly formulated by Einstein in terms of the spacetime curvature

  • The advent of GR fully ascribed to the non-metricity is materialised in a flat and torsion free geometry [16]

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Summary

Introduction

Gravity and geometry have accompanied each other from the very conception of General. Einstein’s original formulation founded GR on a metric and torsionless spacetime and imputed gravity to the curvature It is, natural to explore, as Einstein did later, if gravity can instead be ascribed to the remaining attributes that a spacetime can have, i.e., to the torsion and to the non-metricity. Natural to explore, as Einstein did later, if gravity can instead be ascribed to the remaining attributes that a spacetime can have, i.e., to the torsion and to the non-metricity In these notes, we will confirm that the very same underlying theory, i.e., GR, can be equivalently described in terms of these three seemingly unrelated elements, knocking into shape a geometrical trinity of gravity. We will illustrate some subtle, conceptual and practical, differences among them

General Relativity
Metric Teleparallelism
Vierbein Formulation
Alternative Theories
Symmetric Teleparallelisms
Symmetric Teleparallel Equivalent of GR
General Quadratic Theory
Matter Couplings
Conclusions
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