Gauge invariant approach to nonmetricity theories and the second clock effect

In this paper we discuss on recent attempts aimed at demonstrating that, contrary to well-known results, the second clock effect (SCE) does not take place in generalized Weyl spaces -- spaces with arbitrary nonmetricity -- denoted here as $W_4$ spaces. These attempts include Weyl gauge theories of gravity, as well as the symmetric teleparallel theories (STTs). Our approach to this issue is based on the adoption of Weyl gauge symmetry (WGS) which is a manifest symmetry of the basic laws of Weyl geometry. We shall consistently adapt mathematical and geometrical quantities and concepts so that the resulting geometrical framework be gauge invariant. This issue is of special relevance for the fate of nonmetricity theories, including a class of the STTs which is being intensively applied in the cosmological framework. As we shall show, if realize that WGS is a manifest symmetry of generalized Weyl spaces $W_4$, and identify physical vectors and tensors with corresponding hypothetical vectors and tensors living in $W_4$, neither the Weyl gauge theories nor the nonmetricity theories are free of the SCE, unless Weyl integrable geometry (WIG) spaces are considered.