## 36   Summary for the Sections 1 to 35

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

• temperature T n( and temperature changes )
• pressure  P n( and pressure changes )
• particle number  N j and amount of substance  n j=j N / NA
• thermodynamic efficiency
• heat capacity ratio   kappa  =  CP / CV

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

• volume  V n( and volume changes )
• the Boltzmann constant  k nand the universal gas constant  R
• average velocity of the particles relative to the center of mass
• heat quantity  Q n( and their changes )
• internal energy  U n( and their changes )
• volume work  P · V n( and their changes )
• enthalpy H n( and their changes )
• entropy S  ( and their changes )
• molar heat capacity  CV  and CP
• molrelated melting- and evaporation energies

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

• total energy  E  jof the gas
• particle density

There is no simple transformation in the sense of section 2 for the following process variable:

• peformed work  jW

Taking into account these transformations, all the essential relationships of thermodynamics persist !
The following relationships are invariant:

• the first law of thermodynamics, thus mmmjmmE j= jQ j+ ∆W
• the second law of thermodynamics, thus   mnni ∆S jj0       in a closed system
• for ideal gases       mmmm CP j–  CV j=j R
• for ideal gases mmm      m P · V j= n · R · T j= jN · k · T
• for the entropy mmmmmm dS · T j=j dQ mm and mm Sj=j k · ln(Ω)

For non-relativistic ideal gases we further have the invariant relationships

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 a transformation exists for  T , and in addition we have chosen a definition of temperature within the limiting conditions of
k' · T' n=   k · T · √    so that temperature is relativistically invariant. Most of the results were established even before this decision ( see section 33 ).

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

Avramov, however, made the better choice !