Geometrical aspects of gravitational theory: equivalent representations of gravitation

In this work we present an analysis of different non-riemannian geometrical approaches to the gravitation that possess equivalent physical content to Einstein’s gravitational theory. At a first moment, we review mathematical and physical aspects of the geometries and how each of them have distinct concepts and point-of-view (but equivalent) on regard to the gravitational theory. Among the many approaches to the gravitational interaction, in particular, we discuss the riemannian geometry and the general theory of relativity (in terms of the curvature tensor R), the Weitzenböck geometry and the teleparallel theory (in terms of the torsion tensor T), at last Weyl’s geometry and the symmetric teleparallel theory (in terms of the non-metricity tensor Q). Next, we establish that they are physically equivalent by showing that they have the same field equations, obtained from the variational principle. Finally, in order to illustrate some aspects of the symmetric teleparallel theory, we analyze the cosmology in terms of the theory f(Q) and its contributions to the Friedmann equations, and also how it can contribute to phenomena at early universe (high curvature regime) and at late-time cosmology (low curvature regime).

Keywords
Modified Gravity; Weitzenböck geometry; Weyl geometry; Teleparallel theory; Symmetric Teleparallel theory