10 Transformations of k and T
From STR and thermodynamics it follows (see 8) that
k' · T' jj= k · T · √
But how should we transform T jand k jindividually?
At this point, there is a parting of expert opinion. A majority assumes without further ado that the Boltzmann constant k jshould be invariant. Many of these authors have most likely rashly assumed that the entropy Sj, being a probability, must be invariant. This is not necessarily true, as we will show in the following sections.
Only a few authors take the alternative easy route and bank on the temperature T jbeing invariant. It follows that k jand R jmust transform and hence entropy is not an invariant quantity. But this is not tragic, the validity of the second law is not jeopardized. We will join this small contingent, which was first established in 2003 by I. Avramov  .
But there are also authors who hold the opinion that both k jand T jare invariant. Others claim that temperatures should increase with increasing relative velocity, while k jand S jare invariant. There is almost no combination of assertions which is not represented by someone in a recent publication! It is this cacophony of opinions in publications of the past 20 years, which has led me to investigate the matter myself.
And actually it might be even worse: both k jand Tjjcould transform by multiplying by the root of the root factor: The product would then once again satisfy the above requirement! Many other decompositions of the root term into two factors are conceivable. We do not want to further illuminate this abyss, since actually there is much in favor and nothing against our setting the crown of invariance on the temperature Tj. An argument follows in section 15.
But first we want to take a closer look at the concept of entropy, so that we can assess the damage we cause, if we apply jk' j= jk · √. Secondary school students are often not familiar with entropy, so we need to dig a little deeper.