Sara Cesare

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 $$f(T-\tilde{B})$$ f ( T - B ~ ) and $$f(Q-B)$$ 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), $$f(T-\tilde{B})$$ f ( T - B ~ ) , and $$f(Q-B)$$ 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.

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.