Emergence of Gravitational Potential and Time Dilation from Non-interacting Systems Coupled to a Global Quantum Clock

Abstract: We study gravitational back-reaction within relational time formulations of quantum mechanics by considering two versions of time: a time coordinate, modelled as a global quantum degree of freedom, and the proper time of a given physical system, modelled via an internal degree of freedom serving as a local quantum "clock". We show that interactions between coordinate time and mass-energy in a global Wheeler-DeWitt-like constraint lead to gravitational time dilation. In the presence of a massive object this agrees with time dilation in a Schwarzchild metric at leading order in $G$. Furthermore, if two particles couple independently to the time coordinate we show that Newtonian gravitational interaction between those particles emerges in the low energy limit. We also observe features of renormalization of high energy divergences.