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.
