Entropy as a Clock: Foundations and Parametrizations of Emergent Time

Abstract In canonical quantum gravity, time does not appear as a fundamental coordinate, posing the longstanding problem of how dynamical evolution arises in a fundamentally timeless universe. In this work, we propose that entropy—interpreted as a coarse-grained, monotonically increasing measure of system complexity—can serve as an emergent internal clock. We unify three complementary mechanisms underpinning this idea: (i) the monotonic growth of entanglement entropy under unitary dynamics, (ii) thermal modular flow associated with Kubo-Martin-Schwinger states, and (iii) relational time from the Page–Wootters framework. These mechanisms jointly define a physical arrow and parametrization of time grounded in the informational structure of the quantum state. From this foundation, we derive explicit entropic time laws of the form $$\tau (\Delta S)=\left( \Delta S/\lambda \right) ^{1/\gamma }$$ τ ( Δ S ) = Δ S / λ 1 / γ , showing how the parameters $$(q,N_{0},\lambda ,\gamma )$$ ( q , N 0 , λ , γ ) emerge from microscopic statistical properties such as non-extensive correlations, phase space growth, and entropy production rates. We apply this framework to cosmological epochs, identifying entropy increase across inflation, radiation and matter domination as a natural proxy for internal time progression. This entropic approach provides a unified view linking quantum foundations, thermodynamic irreversibility, and cosmological evolution. We also discuss interpretational subtleties, clarifying how entropic time differs from coordinate time and under what conditions it defines a meaningful temporal structure. We emphasize that the entropic time, $$\tau ,$$ τ , provides an arrow and parametrization of change in relational regimes (Wheeler–DeWitt,Page–Wootters, KMS/modular frameworks), but it is not proposed as a universal bijective substitute for coordinate time, t.