Connes–Rovelli thermal time conjecture for gravitational clock rates

Let $g_{ik}(x)$ be a spacetime metric, let $dt=dx^0$ be coordinate time, and consider an ideal clock fixed at spatial position $x^i$, $i=1,2,3$. Its proper time satisfies

$$d\tau^2=|g_{00}|(x)dt^2.$$

Let $H_T$ be the Tomita, or modular, Hamiltonian and let $\Delta^{is}=\exp(isH_T)$ be the local modular group. Thermal-time conjecture. The microscopic cause of the rate dependence of a clock in a gravitational field is its interaction with the deformed quantum fluctuation spectrum induced by $H_T$; moreover, the local time at $x$ is given by the evolution of the local modular group, $\Delta^{is}=\exp(isH_T)$. This connects the thermal time concept of Connes–Rovelli with gravitational time dilation and proposes a microscopic quantum explanation for the dependence of clock rates on the gravitational field. The paper presents this as a conjectural physical interpretation rather than a mathematical result.