Evolving laws and cosmological energy

Abstract

We couple the issue of evolution in the laws of physics with that of violations of energy conservation. We define evolution in terms of time variables canonically dual to ``constants'' (such as $Λ$, the Planck mass or the gravitational coupling), mimicking a procedure associated with one formulation of unimodular gravity. We then introduce variability via a dependence of {\it other} fundamental ``constants'' on these clocks. Although this is not needed, sharper results are obtained if this procedure violates local Lorentz invariance, which we define in the spirit of Horava-Lifshitz theories (modifying a $3+1$ split action, so that a Lorentz invariant 4D reassembly is no longer possible). We find that variability in the ``laws of physics'' generically leads to violations of energy conservation if either a matter parameter varies as a function of a gravitational clock, or a gravity parameter depends on a matter clock, with the other combinations sterile. We illustrate this with a variety of clocks (associated with matter, the speed of light, the Ricci scalar, etc) and parameters (mainly the gravitational and matter speed of light, but also the cosmological constant). We can accommodate in this construction (and improve) several early Varying Speed of Light solutions to the flatness and cosmological constant problem, allowing for variability effects related to the spatial curvature and $Λ$ to cause creation of radiation and a Hot Big Bang. But we can also go well beyond, for example modelling signature change by a change in the sign of $c^2$, thereby obtaining a {\it classical} realization of the Hartle-Hawking no-boundary proposal, among other developments.