Many Grains, One Flow: A Zeroth Law of Time and the Free Fall of Heat

We propose a zeroth law of time: good clocks keep step, whatever they are made with and without being directly coupled to each other. No exception is known within the accuracy reached. Thermometers agree because they have exchanged heat; clocks agree although they have exchanged nothing. They are linked only indirectly, through the web of physical laws, which are mutually induced and mutually constraining. We state the law within a picture in which classical time is made with records: each record-making event, a time grain, costs at least kBln 2 of entropy per bit over a full cycle. The result is classical time, a coarse-grained quantity accumulated from many grains and shared by all good clocks. Keeping step has a price: two stochastic clocks agree to a relative precision ε only after producing at least 8kB/ε² of entropy together. Where the law meets the zeroth law of thermodynamics, it gains a test: near equilibrium, thermal clocks at different heights read one common time exactly when heat falls like all other energy. An erasure–Szilard cycle shows that energy conservation and local position invariance enforce this, and free-fall data imply that their readings agree to within about one percent of the gravitational redshift between them. Many grains, one flow.