Ehrenfest–Tolman effect explained

\xi , the temperature T satisfies instead the Tolman-Ehrenfest relation: T||\xi|| = constant , where ||\xi|| = \sqrt{g _{ab} \xi ^{a\xi} ^{b}} is the norm of the timelike Killing vector field.

This relationship leads to the concept of thermal time which has been considered as a possible basis for a fully general-relativistic thermodynamics. It has been shown that the Tolman–Ehrenfest effect can be derived by applying the equivalence principle to the concept that temperature is the rate of thermal time with respect to proper time.^[1]

References

1. 1005.2985 . 10.1088/0264-9381/28/7/075007 . Thermal time and Tolman–Ehrenfest effect: 'temperature as the speed of time' . 2011 . Rovelli . Carlo . Smerlak . Matteo . 119250231 . Classical and Quantum Gravity . 28 . 7 . 075007 . 2011CQGra..28g5007R .