## 18   Summary of the First Part

If one defines temperature as a fundamental unit of physics, as in section 15 of this paper, then the following values are relativistically invariant:

• temperature T j( and temperature changes )
• pressure  P j( and pressure changes )
• particle number  N j and amount of substance  n j=j N / NA
• thermodynamic efficiency

The following values are transformed by multiplying by the root factor:

• volume  V j( and volume changes )
• the Boltzmann constant  k nand the universal gas constant  R
• entropy S j( and their changes )

The following value is transformed by dividing by the root factor:

• density of particles

Taking into account these transformations, the essential relationships of thermodynamics remain valid.
The following relationships are form invariant:

• the second law of thermodynamics    mmmm ∆S jj 0 mm in a closed system
• for ideal gases mmmm P · V j= jn · R · T j= jN · k · T
• the entropy is mmmym Sj =j k · ln(Ω)

It should be noted that we have not assumed the validity of these relationships for a fast-moving observer in order to derive the transformation rules! We only assumed that such transformations exist, and in addition we have defined temperature within the limits of  k' · T' n=   k · T · √   so that it is relativistically invariant.

Planck, Einstein, Hasenöhrl and von Mosengeil made no errors (unlike many others) in the derivation of their results. They just decided that jS' j= S should apply, and then they derived their other results in a logically correct way.

Avramov, however, made the better choice!