Page:Zur Dynamik bewegter Systeme.djvu/7

Furthermore, the momentum^[1]

$G={\frac {4q\epsilon V}{3c^{2}+q^{2}}}$.

Substituting these values into the expression of A, and the values of A and E into the equation for dS, the latter is as follows:

$dS={\frac {d(\epsilon V)-qd\left({\frac {4q\epsilon V}{3c^{2}+q^{2}}}\right)+{\frac {c^{2}-q^{2}}{3c^{2}+q^{2}}}\epsilon dV}{T}}$.

The condition that this expression forms a complete differential of the three independent variables q, V and T (bearing in mind that ε only depends on q and T, not on V) gives as a necessary consequence the relations:

$\epsilon ={\frac {ac^{4}}{3}}\cdot {\frac {3c^{2}+q^{2}}{(c^{2}-q^{2})^{3}}}T^{4}$

(1)

and

$S={\frac {4ac^{4}}{3}}\cdot {\frac {T^{3}V}{(c^{2}-q^{2})^{2}}}$

(2)

where the constant a is determined by the fact that ε goes over to aT⁴ for q = 0, which is in accordance with the Stefan-Boltzmann radiation law.

With these values we obtain for the energy E, the pressure p and the momentum G of the moving cavity radiation as functions of the independent variables q, V and T, the following expressions:

$E={\frac {ac^{4}}{3}}\cdot {\frac {3c^{2}+q^{2}}{(c^{2}-q^{2})^{3}}}T^{4}V$

(3)

$p={\frac {ac^{4}}{3}}\cdot {\frac {T^{4}}{(c^{2}-q^{2})^{2}}}$

(4)

$G={\frac {4ac^{4}q}{3}}\cdot {\frac {T^{4}V}{(c^{2}-q^{2})^{3}}}$

(5)

So, for example, if we impart some acceleration to the cavity radiation, while its volume V is kept constant and no heat is supplied from outside so that also the entropy S remains constant, the temperature T of the radiation is decreased by (2) in the ratio

1. According to K. Von Mosengeil, l.c. equation (24*) it is namely:
   $G={\frac {16\pi q}{3c^{3}}}{\frac {K\left({\frac {\pi }{2}},q\right)}{\left(1-{\frac {q^{2}}{c^{2}}}\right)^{3}}}$

   where according to equation (25*):

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