Comparing equivalent gravities: common features and differences

Abstract

We discuss equivalent representations of gravity in the framework of metric-affine geometries pointing out basic concepts from where these theories stem out. In particular, we take into account tetrads and spin connection to describe the so called Geometric Trinity of Gravity. Specifically, we consider General Relativity, constructed upon the metric tensor and based on the curvature R; Teleparallel Equivalent of General Relativity, formulated in terms of torsion T and relying on tetrads and spin connection; Symmetric Teleparallel Equivalent of General Relativity, built up on non-metricity Q, constructed from metric tensor and affine connection. General Relativity is formulated as a geometric theory of gravity based on metric, whereas teleparallel approaches configure as gauge theories, where gauge choices permit not only to simplify calculations, but also to give deep insight into the basic concepts of gravitational field. In particular, we point out how foundation principles of General Relativity (i.e. the Equivalence Principle and the General Covariance) can be seen from the teleparallel point of view. These theories are dynamically equivalent and this feature can be demonstrated under three different standards: (1) the variational method; (2) the field equations; (3) the solutions. Regarding the second point, we provide a procedure starting from the (generalised) second Bianchi identity and then deriving the field equations. Referring to the third point, we compare spherically symmetric solutions in vacuum recovering the Schwarzschild metric and the Birkhoff theorem in all the approaches. It is worth stressing that, in extending the approaches to f(R), f(T), and f(Q) gravities respectively, the dynamical equivalence is lost opening the discussion on the different number of degrees of freedom intervening in the various representations of gravitational theories.