# A Is there a Thermal State definition inside Relativity Theory?

#### DaTario

Summary
If hard balls start colliding in such a way that most of them have velocities near c, is it possible to speak of a thermal state in the Maxwell- Bolstzmann sense?
Hi All,

Considering a set of many many small hard balls which start colliding inside a box. The velocities of these balls being mostly greater than c/2. Is it possible, in this case, to speak of convergence to a thermal state in the same sense of ordinary thermodynamics (i.e., using Maxwell-Bolzmann distribution)?
Are there any relativistic corrections for different observers?

Best Regards,
DaTario

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#### vanhees71

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For an intro to relativistic kinetic theory, see

#### DaTario

Thanks to you both.

So it seems to be correct to say that there exists, in the relativistic regime of a system with many many neutral particles, a stationary state (or perhaps an asymptotic state) which can be called the system's thermal state (having thus a given temperature T associated with it). Is it?

Comment: I have read partially these materials and it have become clear that there exists a probability density for $\gamma$, which depends on the velocity. So the answer to my question above seems to be yes. I am just searching for a confirmation.

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Yes.

#### DaTario

Thank you, Orodruin.

In this respect, considering a reference frame S' with velocity v (near c) wrt the lab's reference frame S, when an observer in S' "sees" a typical gas sample whose volume is static in S and have temperature T also in S, does this S' observer measure the same temperature T (although a drift velocity v must be taken out)?

Best wishes

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#### vanhees71

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That's a very good question, and the issue of the behavior of the thermodynamic quantities under Poincare transformations has been a decade-long debate between eminent physicists. Planck was among the first who worked on it and gave one partially consistent picture. A better solution has been found by Ott.

Nowadays we work with manifestly covariant quantities, i.e., with tensors in Minkowski space (including scalars and vectors and the corresponding fields of course; in QT the proper orthochronous Poincare group is substituted by its covering group which means to substitute $\mathrm{SL}(2,\mathbb{C})$ for the proper orthochronous Lorentz group $\mathrm{SO}(1,3)^{\uparrow}$, and then also spinors of various kinds are added to the theoretical toolkit).

Today we define (sic!) quantities like temperature, chemical potential, internal energy density, entropy density,... as scalars. As with mass one simply takes over the definitions from non-relativistic physics in the local restframe of the medium. For details see

van Kampen, N. G.: Relativistic thermodynamics of moving systems , Phys. Rev. 173(1), 295, 1968

Everything, of course, derives also from (quantum) statistics and thus (quantum) many-body theory and coarsegraining of the microscopic dynamics to macroscopic observables. In this process the one-particle phase-space distribution function plays an important role (e.g., in the Boltzmann transport equation), and accordingly to the above strategy the phase-space distribution function is a scalar field. For details, see

#### DaTario

Thank you, Vanhees71!

"Is there a Thermal State definition inside Relativity Theory?"

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