Unimodular-like times, evolution and Brans–Dicke gravity

In unimodular-like theories, the constants of nature are demoted from pre-given parameters to phase space variables. Their canonical duals provide physical time variables. We investigate how this interacts with an alternative approach to varying constants, where they are replaced by dynamical scalar fields. Specifically, we investigate the Brans–Dicke theory of gravity and its interaction with clocks dual to the cosmological constant, the Planck mass, etc. We crucially distinguish between the different role of Newton’s G in this process, leading to the possibility of local Lorentz invariance violation. A large number of possible theories emerge, for example where the Brans–Dicke coupling, $\omega$, depends on unimodular-like times (in a generalization of scalar-tensor theories), or even become the dual variable to unimodular-like clocks ticking variations in other demoted constants, such as the cosmological constant. We scan the space of possible theories and select those most interesting regarding the joint variations of the Brans–Dicke $\omega$ and other parameters, (such as the cosmological constant); and also regarding their energy conservation violation properties. This ground work is meant to provide the formalism for further developments, namely regarding cosmology, black holes and the cosmological constant problem.