The Toll of the Tolman Effect: On the Status of Classical Temperature in General Relativity

The Tolman effect is well-known in relativistic cosmology but rarely discussed outside it. That is surprising because the effect -- that systems extended over a varying gravitational potential exhibit temperature gradients while in thermal equilibrium -- conflicts with ordinary classical thermodynamics. In this paper we try to better understand this effect from a foundational perspective. We make five claims. First, as Tolman knew, it was Einstein who first discovered the effect, and furthermore, Einstein's derivation helps us appreciate how robust it is. Second, the standard interpretation of the effect in terms of 'local temperature' leads to the breakdown of much of classical thermodynamics. Third, one can rescue thermodynamics by using Einstein's preferred interpretation in terms of the 'wahre Temperatur' -- what we'll call global temperature -- but it too has some costs. Fourth, the effect is perhaps best understood in terms of clocks as opposed to energy loss. Fifth, inspired by a proposal of Einstein's elsewhere, we sketch an interpretation of the effect in terms of a third novel temperature, which we call the 'wahre-local temperature'. On this view, temperature -- and thermodynamics -- is defined only in relation to local clocks. In sum, we view the fragmentation of temperature in thermodynamics as a natural and expected result of the fragmentation of time in general relativity.