# What magnitude is absolute: temperature or entropy?

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1. Jun 8, 2015

### ORF

Hello

In relativity, what magnitude is absolute*: temperature or entropy?

*absolute = equal for all observers (= a Lorentz scalar)

Thank you for your time :)

Greetings!

2. Jun 8, 2015

Staff Emeritus
There is no consistent relativistic definition of temperature. Classically, there are several different definitions of temperature, but they all give the same answers. Relativistically, they do not.

3. Jun 8, 2015

### ORF

Hello

Thanks for answering so quickly! :)

Is there a consistent relativistic definition of entropy? (just by curiosity... why there is no consistent relativistic definition of temperature? I suppose because of the non-simultaneous interactions, but I haven't the idea totally clear so I prefer asking.)

On the other hand, I thought the temperature can be infered from the Maxwell-Jüttner distribution:
http://en.wikipedia.org/wiki/Maxwell–Jüttner_distribution

Greetings!

4. Jun 8, 2015

5. Jun 8, 2015

### vanhees71

For a point of view from statistical mechanics (kinetic theory), see

http://fias.uni-frankfurt.de/~hees/publ/kolkata.pdf

It's all covariant. There's a big mess in the older literature from the time where the math of relativity wasn't fully understood. I don't know, how Vanadium comes to his statement in #50. I'd like to see a convincing argument, why it shouldn't be possible to define temperature uniquely in relativistic theory.

To my understanding it's a scalar (field) as you see in my manuscript, and that's the definition of temperature used in the heavy-ion community and, as far as I know, also by the relativistic astro-physics community in its generalization to the General Theory of Relativity.

6. Jun 8, 2015

### pervect

Staff Emeritus
I'd agree with the observations that the literature is a bit of a mess, and it's hard to tell from it if a consensus view even exists.

My personal point of view is that entropy is a simpler concept than temperature, entropy being basically a scalar density. You can regard is as the log of the number of states using the POV of statistical thermodynamics.

Be warned that my views of thermo are basically based on reading one paper that I found persuasive, rather than any in-depth research. Said paper is http://arxiv.org/abs/physics/0505004.

This paper itself admits that there is/was a bit of controversy on the issue, and mentions some of the competing ideas that I never really looked into deeply.

To oversimplify a lot, we start with the point of view that entropy, being the log of the number of states, is a world scalar. More precisely, entropy per unit volume is a scalar density. Going on from this starting point, the traditional idea is that entropy, work (energy) and temperature are related by the equation $\Delta S = \Delta Q / T$. But in special relativity, energy is one component of a 4-vector, so our first step in expressing the above law covariaintly replaces $\Delta Q$ with a 4 vector, which represents the exchange of energy-momentum between systems rather than the exchange of a scalar energy. Since we want to map a change in energy-momentum to a change in a scalar value, we need another 4-vector which replaces inverse temperature in the equation. Thus inverse temperature becomes not a scalar, but a 4-vector.

There is at least one issue that remains to be fixed, though. This is the fact that energy-momentum is only a 4-vector for an isolated system, and we often wish to treat non-isolated systems in thermodynamics. We can still treat the entropy per unit volume as a scalar density, but the energy/momentum per unit volume is best treated via the stress-energy tensor. We still keep the concept of inverse temperature as a 4-vector, and we represent the energy/momentum per unit volume via the stress energy tensor.

On a more practical note, all the textbook treatmens I have just choose some particular frame, and make a point of doing the thermodynamics in a non-covariant manner, rather than worry about the issue of how to do it covariantly. I'd say this is probably the easiest approach to communicate.

7. Jun 12, 2015

### vanhees71

Do you understand Eq. (2) in this paper? The final results nevertheless seem to be correct to me. I've to check more carefully. The point is to define all thermodynamic quantities in the local restframe of the heat bath (as Nakamura restricts himself to equilibrium, that's good enough) and write everything covariantly. It's indeed true that everything is clearly resolved by just taking into account the four-flow $u^{\mu}(t,\vec{x})$ of the medium.

For the point of view from a kinetic-theory perspective, see

http://fias.uni-frankfurt.de/~hees/publ/kolkata.pdf

It's all standard. Usually the papers by Israel are correct (van Kampen seems to have overcomplicated many things but also almost always got it right). It was new to me that already Fermi got it right in 1923 :-)). At the moment Springer link seems to be down, however :-((.