Weyl gauge theories of gravity do not predict a second clock effect

Abstract

We consider Weyl gauge theories of gravity (WGTs), which are invariant both under local Poincar\'e transformations and local changes of scale. Such theories may be interpreted as gauge theories in Minkowski spacetime, but their gravitational interactions are most often reinterpreted geometrically in terms of a Weyl--Cartan spacetime, in which any matter fields then reside. Such a spacetime is a straightforward generalisation of Weyl spacetime to include torsion. As first suggested by Einstein, Weyl spacetime is believed to exhibit a so-called second clock effect, which prevents the existence of experimentally observed sharp spectral lines, since the rates of (atomic) clocks depend on their past history. The prevailing view in the literature is that this rules out WGTs as unphysical. Contrary to this viewpoint, we show that if one adopts the natural covariant derivative identified in the geometric interpretation of WGTs, properly takes into account the scaling dimension of physical quantities, and recognises that Einstein's original objection requires the presence of massive matter fields to represent atoms, observers and clocks, then WGTs do not predict a second clock effect.