Canonical Clock Sectors and Relational Frame Equivalence in Brans--Dicke Theory

Abstract

We investigate the equivalence of the Jordan and Einstein frames of the Brans--Dicke theory before and after relational deparametrization of the constrained Hamiltonian system. Although the two conformal formulations are equivalent at the covariant level and related by a canonical transformation on the standard ADM phase space, their equivalence after reduction with respect to an internal clock is nontrivial. Using the Brans--Dicke scalar as a relational clock, we show that the apparent discrepancy between the reduced Hamiltonians does not indicate a physical inequivalence of the two frames. Instead, it arises from a mismatch in the canonical embedding of the clock sector prior to reduction. While the scalar configuration variable is preserved under the conformal transformation, its conjugate momentum is shifted by a contribution involving the gravitational momentum trace. Consequently, relational dynamics are determined not by the clock variable $T$ alone, but by the complete canonical clock pair $(T,P_T)$. We construct a frame-adapted canonical chart in which the clock sector is consistently transformed prior to deparameterization. The resulting reduced Hamiltonian coincides with that of the Einstein frame, restoring the equivalence of the reduced relational dynamics. Our results identify the canonical clock sector as the fundamental structure governing relational evolution in Brans--Dicke theory and provide a general framework for understanding frame dependence in scalar--tensor gravity and reduced phase-space quantization.