3 States of Equilibrium
We derive all the transformations based on the observation of a given quantity of an ideal gas, which is enclosed in a (variable) volume Vj. If such transformations exist, which are applicable to all systems, then they have to be applicable to an enclosed mole of an ideal gas too !
If such a gas is thermally isolated (i.e., it neither absorbs nor radiates energy to its environment), then for statistical reasons it will very quickly reach an equilibrium state. It is essential that this statement is also true from the perspective of a fast-moving observer:
( P m constant m and m v2 mconstant ) ==> P' j= fP (jv2j) · P mmconstant
The relative velocity is indeed constant as assumed in the STR, and therefore P’ jhas to be constant too. If, for example, pressure, volume and temperature are constant in the rest frame, then the same will hold for P' j, V' jand T' j. Furthermore, this is true regardless of the exact formulation of the corresponding transformations !
But if P'j, V' jund T' jare likewise constant, this means nothing other than that the gas is also in a state of equilibrium for the fast-moving observer. The presence of a state of equilibrium is not 'relative'. It is a fact, given for all frames of inertia - or for none.