# Speed of time : temperature tells thermal time/proper time (Smerlak Rovelli)

1. May 18, 2010

### marcus

"Speed of time": temperature tells thermal time/proper time (Smerlak Rovelli)

IMHO this is an elegant 4-page gem. Check it out.

http://arxiv.org/abs/1005.2985
Thermal time and the Tolman-Ehrenfest effect: temperature as the "speed of time"
Carlo Rovelli, Matteo Smerlak
(Submitted on 17 May 2010)
"The thermal time hypothesis has been introduced as a possible basis for a fully general-relativistic thermodynamics. Here we use the notion of thermal time to study thermal equilibrium on stationary spacetimes. Notably, we show that the Tolman-Ehrenfest effect (the variation of temperature in space so that $$T\sqrt{g_{00}}$$ remains constant) can be reappraised as a manifestation of this fact: at thermal equilibrium, temperature is locally the rate of flow of thermal time with respect to proper time - pictorially, 'the speed of (thermal) time'. Our derivation of the Tolman-Ehrenfest effect makes no reference to the physical mechanisms underlying thermalization, thus illustrating the import of the notion of thermal time."

2. May 18, 2010

### marcus

Re: "Speed of time": temperature tells thermal time/proper time (Smerlak Rovelli)

A nice fact from 1930 is that at equilibrium a vertical column of fluid standing in a gravitational field must be warmer at the bottom.

(see the discussion on page 1 right after equation (2), and see the simple one-paragraph explanation on page 4, in the brief Appendix A.)

The appendix shows that energy "falls". If there are two systems at different gravitational potential and some energy is transferred from the higher to the lower. The amount added to the lower will be greater than the amount deducted from the higher. The energy increased by "falling". This applies to the transfer of heat.
When the two are in equilibrium so that total entropy is unchanged, this example provides a simple proof of the Tolman-Ehrenfest effect. At equilibrium, temperature cannot be constant in space. Instead, what is constant is the temperature multiplied by sqrt(g00).

They recall the concept of thermal time developed by Rovelli in 1993, and further elaborated by Alain Connes and Rovelli in 1994
http://arxiv.org/abs/gr-qc/9406019
and they discover that the temperature T is, in fact, the ratio of thermal time to proper time.

Amusingly this can make one think of biological maturation. Biological time can also pass more quickly when it is warm. It was observed among the Inuit, when they lived in igloo ice houses which were normally kept very warm, that the young people reached puberty sooner---as also was seen in the tropics. In agriculture the concept of "degree day" is used to predict how soon the crops will ripen. Warmer weather makes things go faster. This is certainly not the Tolman-Ehrenfest effect! But it serves to help remember the direction of the effect.

Last edited: May 18, 2010
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