Nonmetricity theories and aspects of gauge symmetry

In this paper we discuss on the phenomenological viability of nonmetricity theories of gravity which are based in the class of generalized Weyl spacetimes -- denoted by $W_4$ -- where arbitrary nonmetricity is allowed. This class of geometry includes the so called teleparallel spaces $Z_4$, which are the geometric basement of the symmetric teleparallel theories (STTs). The guiding principle in our discussion is Weyl gauge symmetry (WGS), which is a manifest symmetry of $W_4$ spaces. Here we derive the master equation that drives the gauge invariant variations of the length of vectors during parallel transport in $W_4$. This is the mathematical basis of the second clock effect (SCE). We are able to give qualitative and quantitative estimates for the SCE, as well as for the perihelion shift, in the coincident gauge of $Z_4$ space. We conclude that generalized Weyl spaces do not represent phenomenologically viable descriptions of Nature due to the SCE and, also to their predictions for the perihelion shift. All of the present results are based in the assumption of: (i) a gauge invariant parallel transport law and (ii) a consistency hypothesis which enables identifying hypothetical vectors and tensors defined in $W_4$, with related physical vectors and tensors arising in the given gravitational theory. Our discussion is mostly geometrical without relying on specific theories of gravity, unless it is absolutely necessary.