First Law of Thermodynamics from Mechanics of Photon Absorption in a Moving Frame Part II

In Part I, we considered a photon being absorbed by a gas at rest and argued that essentially all of the photon energy, dmo, is converted into thermal energy because the photon momentum is small compared to the momentum of the system after absorption, i.e. the system hardly moves.We then analyzed the system as seen by someone watching a moving gas and separated the change in energy into an energy change portion involving no change in momentum of the system, (1-v) g(v) dmo, and  another, a change strictly due to a change in momentum, v g(v) dmo, where g(v)=1/sqrt(1-vv/cc). 

    In this note, we argue that the zero momentum piece should be further decomposed into a heat term and an internal work piece. In other words, (1-v) g(v) dmo is not entirely heat.  We argue that there are two types of work in the problem: an internal one by the gas associated with a decrease in velocity if momentum is held constant, and an increase in momentum due to the absorption of the photon. These two work pieces cancel, leaving g(v) dmo as the increase in energy. This is linked to an increase in internal energy, but is enhanced by g(v) because this internal energy is moving. Thus, there is consistency between what is seen in the lab and moving frames. 

   This suggests that the definition 1/T = dS/dE  holding collective parameters constant is more complicated than it appears. In the case of a boost, S is the same in the rest and moving frames, but energy changes. Thus, it seems that one wants an energy change not associated with a collective parameter when calculating temperature. A boost is associated with the collective parameter velocity of the moving system. Such an interpretation leads to the dS/dE as being evaluated in a rest frame with further collective parameters (such as volume) held constant in that frame, we suggest. Even though T is calculated in the rest frame, heat energy receives the usual g(v) factor from a boost.