Comparing equivalent gravities: common features and differences

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