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

TL;DR

The paper shows that gravity-like effects can emerge from quantum constraints by treating time as a relational quantum degree of freedom within the Page-Wootters framework. By introducing a mass-energy coupling term $\dfrac{1}{\Lambda}(\hat{p}_t \otimes \hat{H}_S)$ in a Wheeler-DeWitt-like constraint, it derives an effective Hamiltonian $\hat{H}_{\mathrm{eff}} = \hat{H}_S(\mathbb{I}_S + \hat{H}_S/\Lambda)^{-1}$, yielding both an emergent Newtonian potential between subsystems and gravitational time dilation that matches the Schwarzschild prediction to leading order when $\Lambda^{-1}$ scales with the gravitational coupling $G$ and a characteristic distance. Extending to two subsystems shows a cross-term $-\dfrac{2}{\Lambda} (\hat{H}_A \otimes \hat{H}_B)$ that reproduces Newtonian gravity for $1/\Lambda = G/(2r)$, reinforcing gravity as a consequence of quantum constraints rather than a fundamental force. The framework also suggests renormalization-like features and potential quantum corrections to time dilation in energy superpositions, with implications for clock-based tests of quantum gravity in the weak-field regime. Overall, the work provides a concrete mechanism by which gravitational phenomena can emerge from relational quantum dynamics of time, offering a pathway to integrate gravity into quantum principles and guiding future experimental and theoretical explorations.

Abstract

We study gravitational back-reaction within the Page-Wootters formulation of quantum mechanics by treating time as a quantum degree of freedom. Our model introduces a distinction between global coordinate time, represented as a relational quantum observable, and proper time, measured by internal quantum degrees of freedom of physical systems. By coupling mass-energy with coordinate time through a Wheeler-DeWitt-like constraint, we demonstrate the natural emergence of gravitational time dilation. In the presence of a massive object this agrees with time dilation in a Schwarzchild metric at leading order if the interaction strength is taken to be representative of the gravitational coupling $G$. Additionally, when two particles independently couple to the time coordinate, a Newtonian gravitational interaction arises in the low-energy limit, showing how gravitational potential can emerge from non-interacting quantum systems. Our approach also reveals renormalization features, potentially softening high-energy divergences and suggesting that particles in superposition might introduce quantum corrections to gravitational time dilation.