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enclosed in absolutely reflecting moving walls, whose volume V may be chosen so great that the influence of the mass of the walls is too small to be considered. All changes happening in that system we consider as reversible, that is, they happen so slowly that there is in every moment a stationary state. Then the state of the system is completely determined by the speed q, which can be an arbitrarily large fraction of the speed of light c, the volume V and the temperature T. For an infinitesimal change of state, the change in energy E of the radiation is according to the first law of thermodynamics:

dE = A + Q.

where A is the mechanical work applied from the outside to the radiation, Q is the heat supplied from the outside: and after the second law, S is the change in entropy of the radiation:

$dS={\frac {Q}{T}}={\frac {dE-A}{T}}$.

By aid of the last equation we want to calculate the properties of the radiation in their dependence on the independent variables q, V and T. The energy of the radiation is:

E = ε · V,

where ε is the spatial energy density, which depends only on q and T. Moreover, the external work A shall be additively composed of the translation work and the compression work. The former is equal to the product of velocity q and the increase of momentum G, the latter is equal to the product of pressure p and the decrease of volume V, thus:

A = qdG - pdV.

Now the pressure is^[1]

$p={\frac {c^{2}-q^{2}}{3c^{2}+q^{2}}}\cdot \epsilon$.

1. Kurd Von Mosengeil, Ann. d. Phys. (4) 22, S. 867, 1907, gives (based on a formula for the pressure of a single ray on a moving mirror by M. Abraham (Elektromagnetische Theorie der Strahlung. Leipzig, B. G. Teubner 1905, p. 351) the equation (42):
   $p={\frac {4\pi }{3c}}K(0)\left(1-{\frac {q^{2}}{c^{2}}}\right)^{\frac {2}{3}}$

   and as equation (44):

   $\epsilon ={\frac {4\pi }{c}}K(0){\frac {1+{\frac {q^{2}}{3c^{2}}}}{\left(1-{\frac {q^{2}}{c^{2}}}\right)^{\frac {1}{3}}}}$

   Combined, both equations give the above relation, which is by the way generally valid, not only for adiabatic processes.