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 $$\tau (\Delta S)=\left( \Delta S/\lambda \right) ^{1/\gamma }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mfenced> <mml:mi>Δ</mml:mi> <mml:mi>S</mml:mi> <mml:mo>/</mml:mo> <mml:mi>λ</mml:mi> </mml:mfenced> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>γ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , showing how the parameters $$(q,N_{0},\lambda ,\gamma )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>γ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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, $$\tau ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> 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.