Energy Versus Internal Energy and a Lorentz Boost of an Ideal Gas

 We argue that in thermodynamics and statistical mechanics, energy and internal energy are not equivalent. It seems that internal energy is associated with a  rest frame of an ideal gas system, even though a Lorentz boost does not discriminate between the two. In particular, internal energy appears in the Sackur-Tetrode formula for entropy and entropy is  linked to a physical counting of states. Thus, a special energy, namely internal energy, is associated with the Sackur-Tetrode formula. 

   We suggest that a problem arises when one Lorentz boosts an ideal gas because energy —> g(v) energy, where g(v)=1/sqrt(1-vv/cc), but g(v) internal energy is no longer strictly internal energy.  It is a combination of internal energy and collective motion energy.Thus, it seems that an equation which applies to internal energy should not really be used with a new energy which is not strictly speaking internal. For example, in (1),  P dV  is taken to transform as energy i.e. g(v) energy, dV as dV/g(v) so P transforms as g(v)g(v) P. We argue, however, that P, like internal energy, is based on special conditions and that objects like g(v)g(v) P and g(v) T, where T is temperature, no longer represent purely statistical objects. Furthermore, if one transforms variables such as internal energy, V and rest m, one finds that the entropy value of the Sackur-Tetrode entropy divided by k (Boltzmann constant) changes which should not happen if it represents ln(number of states). This seems to lead to the conclusion that thermodynamic quantities like T, entropy etc apply to a system with bias removed (if possible). A moving frame is a biased frame and so we suggest that one perform thermodynamic/statistical mechanical equations in the rest frame.