Extended Geometric Trinity of Gravity

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. We discuss the straightforward extension of General Relativity, the $f(R)$ gravity, where $f(R)$ is an arbitrary function of the Ricci scalar. With this paradigm in mind, we take into account $f(T)$ and $f(Q)$ extensions showing that they are not equivalent to $f(R)$. Dynamical equivalence is achieved if boundary terms are considered, that is $f(T-\tilde{B})$ and $f(Q-B)$ theories. The latter are the extensions of Teleparallel Equivalent of General Relativity and Symmetric Teleparallel of General Relativity, respectively. We obtain that $f(R)$, $f(T-\tilde{B})$, and $f(Q-B)$ form the Extended Geometric Trinity of Gravity. The aim is to show that also if dynamics are equivalent, foundations of theories of gravity can be very different.