Extended geometric trinity of gravity

Abstract 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 $$f(T-\tilde{B})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>-</mml:mo> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$f(Q-B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>-</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> theories. The latter are the extensions of Teleparallel Equivalent of General Relativity and Symmetric Teleparallel of General Relativity, respectively. We obtain that f ( R ), $$f(T-\tilde{B})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>-</mml:mo> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and $$f(Q-B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>-</mml:mo> <mml:mi>B</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.