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What is the current consensus on how temperature transforms in relativity?...Around 1905 Planck and Einstein concluded that a moving body would appear colder than its proper temperature. In 1963 Ott asserted that that a moving body would appear hotter than its proper temperature. A few years later Landsberg concluded temperature was a Lorentz invariant.

Here is another simple thought experiment. Consider two very long rectangular objects A and B, that are moving relative to each other, while remaining in thermal contact along their longest sides. In a particular reference frame (S), A and B are moving at the same speed wrt S, and have equal temperatures.

If we take the Ott view that temperature increases with relative motion, then in the rest frame of object A, the temperature of object B is greater than that of A and heat flows from B to A.

The converse would then be true in the rest frame of B, so that heat flows in the opposite direction from A to B. This would appear to be a physical contradiction to what is measured in the rest frame of A.

A similar contradiction appears if we imagine temperature gets colder with relative motion.

If temperature is not a Lorentz invariant, it would appear that one of the consequences is that entropy is not a Lorentz invariant either.

Another approach is to consider phase transitions, (as in the pressure thread). If temperature is not Lorentz invariant, then the absolute temperature of the triple point of water for example, would be an observer dependent quantity.