Relativistic thermodynamics problem

[bernhard.rothenstein]

Authors (Tolmann, Arzelies) consider that proper temperature T(0) and non-proper one T are related in the I inertial reference frame by
T=T(0)/(1-uu/cc)^1/2 (1) whereas in I' they are related by T'=T(0)/(1-u'u'/cc)^1/2 (2). Expressing the right side of (1) as a function of u' via the addition law of relativistic velocities we obtain
T=T'(1+Vu'/cc)/(1-VxV/cc)^1/2. (3)
Do you see some physics behind (3). Has u'T'/cc a physical meaning eventually via the Boltzmann constant?
Thanks

 

[pervect]

The question of how to treat relativistic thermodynamics comes up from time to time, but usually doesn't get much of an answer. My thinking on the topic is based on the arguments presented in http://arxiv.org/abs/physics/0505004

which argues for inverse temperature as a 4-vector and which I personally find convincing. Unfortunately, I have *not* read all the relevant literature in this field (relativistic thermodynamics) thus it's quite possible that as a result of this I'm missing some of the fine or not-so-fine points.

Here are a couple of quotes from the above paper:

Let us generalize the above statement to relativity. Heat is a form of energy in non-relativistic thermodynamics, where the energy and momentum
are distinct quantities. In relativity, however, the energy and momentum are
components of one physical entity, energy-momentum four vector namely, and thus cannot be treated independently. Therefore we must treat the energy-momentum exchange between the objects, not energy alone.

Consequently, the inverse temperature must have four components, , corresponding to each component of energy-momentum four vector.

Apparently there has been some amount of controversy in the field, the authors state:

Theory of relativistic thermodynamics has a long and controversial history
(see, e.g., [1] and references therein). The controversy seems to have been
settled more or less by the end of 1960s [2], however, papers are still being
published to this date (e.g., [3]).

Thus the early papers such as those of Tollman may not be representative of the current thinking.

Unfortunately, it is not particularly clear to me how one could go about demonstrating whether or not the above paper is representative of current thinking either - all I can say is that I find the arguments presented in this paper convincing.

 

[Entropia]

for sharing

The expression (3) is a result of the addition law of relativistic velocities, which states that velocities in different reference frames are not simply added, but rather must be calculated using a more complex formula. This formula takes into account the effects of special relativity, such as time dilation and length contraction.

In this case, the authors are considering the relationship between proper and non-proper temperatures in different inertial reference frames. The equation (3) shows that the temperature in one reference frame is related to the temperature in another reference frame by a factor of (1+Vu'/cc)/(1-VxV/cc)^1/2. This factor takes into account the relative velocities between the two frames, as well as the effects of special relativity.

The physical meaning behind this equation is that temperature is not an absolute quantity, but rather is dependent on the reference frame in which it is measured. This is a consequence of the fact that time and space are relative in special relativity.

Additionally, the term u'T'/cc in the equation has a physical meaning as it represents the ratio of the velocity of the particles (u') to the speed of light (c), multiplied by the temperature (T'). This quantity can be related to the Boltzmann constant, which is a fundamental constant in thermodynamics that relates temperature to the kinetic energy of particles. This shows that even in the context of relativistic thermodynamics, the Boltzmann constant maintains its physical significance.