Relational Time as a Stochastic Variable in ADM Gravity

The absence of a fundamental time parameter in canonical quantum gravity is motivating the search for internal clocks by which evolution is defined relationally. While formal solutions are provided classical deparametrization schemes they often rely on semiclassical limits or fixed foliations that break general covariance. In this work, a canonical framework is constructed where time emerges dynamically as a stochastic degree of freedom, identified with a massless scalar field whose conserved momentum current defines a relational foliation. Unlike semiclassical stochastic gravity approaches where noise is introduced externally, here the stochasticity arises from coarse-graining of unresolved transverse-traceless graviton modes, leading to an intrinsic, dynamically coupled stochastic clock field. This leads to a diffusion-like broadening of the wavefunctional across neighboring clock slices, offering a novel stochastic phenomenology of the Wheeler–DeWitt equation. The resulting evolution remains consistent with hypersurface-deformation algebra in an ensemble sense, while introducing an intrinsic probabilistic nature to relational time. This work thus establishes the theoretical foundation for future applications in quantum cosmology and the emergence of classicality from quantum gravitational systems.