Extended Gravity
Abstract
We shall show equivalence between Palatini-f() theories and Brans-Dicke (BD) theories at the level of action principles in generic dimension with generic matter coupling. We do that by introducing the Helmholtz Lagrangian associated to Palatini-f() theory and then performing frame transformations in order to recover Einstein frame and Brans-Dicke frame. This clarifies the relation among different formulations and the transformations among different frames. Additionally, it defines a formulation a l\'a Palatini for the Brans-Dicke theory which is dynamically equivalent to metric BD (unlike the standard Palatini-formulation of metric BD theory which are not dynamically equivalent). In conclusion we discuss interpretation of extended theories of gravitation and perspectives.
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