Extended Geometric Trinity of Gravity

Understand

Extensions of equivalent representations of gravity are discussed in the metric-affine framework.

• First, we focus on: (i) General Relativity, based upon the metric tensor whose dynamics is given by the Ricci curvature scalar $R$; (ii) the Teleparallel Equivalent of General Relativity, based on tetrads and spin connection whose dynamics is given by the torsion scalar $T$; (iii) the Symmetric Teleparallel Equivalent of General Relativity, formulated with respect to both the metric tensor and the affine connection and characterized by the non-metric scalar $Q$ with the role of gravitational field.
• They represent the so-called Geometric Trinity of Gravity, because, even if based on different frameworks and different dynamical variables, such as curvature, torsion, and non-metricity, they express the same gravitational dynamics.
• Starting from this framework, we construct their extensions with the aim to study possible equivalence.