and discuss results in section VI.

What is the physical frame?
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Some considerations are in order at this point. The conformal transformations from the Jordan to the Einstein frame are geometrical maps allowing to set many features of scalar-tensor gravity, $f(R)$ gravity and, in general, any modified theory of gravity. Taking into account both the Jordan and the Einstein conformal frames (infinite conformal frames can be chosen assuming a suitable conformal factor), the question is whether the frames are infinitely many *physically* equivalent or only mathematically related. In other words, the problem is whether the physical information contained in the theory is “preserved” or not by the conformal transformations. In other words, one has the metric $g_{ij} $ and its conformally related one $\bar{g}_{ij}$: the question is what is the “physical metric”, *i.e.*, the metric from which curvature, geometry, and physical effects have to be calculated and compared with experiments and observations [@mercadante].

More precisely, every Killing and homothetic vector is also a symmetry of the energy-momentum tensor. The latter is also a [*metric*]{} of spacetime because a “metric” can be considered any second order tensor. Therefore, up to the homothetic group, metric, energy-momentum tensor, and Ricci tensor have 