13   The Transformation of Total Entropy following Planck

Planck [1] wrote in his 1907 paper

"We want to prove that the entropy of the body under consideration in relation to the primed system has the same value as for the unprimed system. You could generally base this proof on the close relationship  between entropy and probability, whose values cannot possibly depend on the choice of the reference system, however, here we prefer to take a more direct approach, independent from the introduction of the concept of probability.
We consider the body taken from a state in which it is at rest with respect to unprimed reference system and is brought by any reversible adiabatic process to a second state in which it is at rest for the primed reference system. If we denote the entropy of the body for the unprimed system in the initial state as S1, and that for the final state with S2, then because of reversibility and adiabaticity S1 j= S2 . But also for the primed reference system, the process is reversible and adiabatic, so we have also:  S1' j= S2' .
 Now if S1' is not equal to S1, but rather S1' j> S1 this would mean that: The entropy of the body in the reference system, which is undergoing motion, is greater than that for the reference system, in which it is at rest. Then, according to this proposition it must also be that S2 j> S2' , for in the second case of the body is at rest in the primed reference system, and is concidered to be in motion for the unprimed reference system. But these inequalities contradict the two equations established above. Nor can it be that S1' j< S1 ; consequently S1' j= jS1 , and in general:

S' nS

that is, the entropy of a body does not depend on the choice of the reference system."

Is Planck's argument watertight? Even the savvy Pauli accepts it in his famous encyclopedia article [3] on relativity theory (§46, p.694). However, it suffers from a mistake in logic called petitio principii (i.e., begging the question): what one wants to show is already used in the proof !

Planck must bring a body initially at rest (first state) in the unprimed system to the velocity v jof the primed system so that in this second state it is at rest in that system. So in the unprimed system one already measures the entropy S2 jon the fast-moving body, while reciprocally in the primed system one measures S1' jon a fast-moving body. Planck simply assumes that S2 jwill have the same value as S1 , but this is exactly what would have to be shown. It could also be that jS2 j= jS1 · √ , as it would precisely be, if entropies of fast-moving systems transform! This would then give the following, also contradiction-free set of measurements:

S2 jS1 · √    ,  S2' jS1 ,    S1' jS2' · √    ,  S2 jS1'

For reasons of symmetry, the two formulas without root factor must apply. The set of all four formulas is consistent - with or without the root factors!

So Planck simply assumes that entropy is invariant. From this 'result', he then correctly concludes that the transformation of the temperature isj
T' j= T · √ .

In section 15 we will present the argument against Planck, Einstein and Pauli, along the lines of Avramov. But first we again consider the second law of thermodynamics, this time in the formulation of entropy.