Is temperature an observer-invariant concept in relativity?

[Naty1]

Page 52 of motionmountain (Learning Materials) has the following which I believe bears on a thread here in Physics Forums in the last week..can't find it now and I don't recall all the posts... I'm wondering if the following clarifies or rather is inconsistent with that thread discussion ...I'd appreciate anyone who can identify that thread (posted here in relativity) and also any comments regarding the following quote. (Someone posted in that earlier thread that the KE was measured relative to the center of mass...I posted a reference to DrGreg..regarding his insight in another thread regarding length contraction/compression heat/work... )

(my boldface)

The literature on temperature is confusing. Albert Einstein and Wolfgang Pauli agreed
on the following result: the temperature T seen by an observer moving with speed v is
related to the temperature T0 measured by the observer at rest with respect to the heat
bath via
T = T0 (1 − v²/c². ) ^-1/2

A moving observer thus always measures lower values than a resting one.
In 1908, Max Planck used this expression, together with the corresponding transformation
for heat, to deduce that the entropy is invariant under Lorentz transformations.
Being the discoverer of the Boltzmann constant k, Planck proved in this way that the
constant is a relativistic invariant.
Not all researchers agree on the expression. Others maintain that T and T0 should
be interchanged in the temperature transformation. Also, powers other than the simple
square root have been proposed.The origin of these discrepancies Ref. 56 is simple: temperature is only defined for equilibrium situations, i.e., for baths. But a bath for one observer is not a bath for the other. For low speeds, a moving observer sees a situation that is almost a heat bath; but at higher speeds the issue becomes tricky. Temperature is deduced from the speed of matter particles, such as atoms or molecules. For moving observers, there is no good way to measure temperature. The naively measured temperature value even depends on the energy range of matter particles that is measured! In short, thermal equilibrium is not an observer-invariant concept.Therefore, no temperature transformation formula is correct. (With certain additional assumptions, Planck’s expression does seem to hold, however.) In fact, there are not even any experimental observations that
would allow such a formula to be checked. Realizing such a measurement is a challenge
for future experimenters – but not for relativity itself.

 

[Dmitry67]

Hm, interesting question
So 2 objects are flying close to each other at v
They exchange the heat radiation
The temperature equlibrium can be defined based on the Orthogonal doppler effect
Based on the formula here: http://en.wikipedia.org/wiki/Transverse_Doppler_effect
formula for T is correct

 

[Dmitry67]

Hmmm... I am confused...
So there are 2 infinite plates heated at T1 and T2 exchange the black body radiation moving at v (to each other) . At what T1 and T2 they are in equilibrium?

So for each plate other plate is COLDER...

 

[Naty1]

Dimitry...Note: I just updated the Lorentz factor exponent in my original post from an incorrect "1/2" to the correct "-1/2".