Page:Zur Dynamik bewegter Systeme.djvu/13

so that the volume is diminished by the ratio ${\sqrt {1-{\frac {v^{2}}{c^{2}}}}}:1$ for any body, will adopt the same temperature and pressure. Therefore, if we know for a single body the change of temperature set forth by such a process, then we know the change for any arbitrary body in nature.

Now, especially for a black cavity radiation we have for q₁=0, q₂=v according to (2)

$S_{1}={\frac {4aT_{1}^{3}V_{1}}{3}},\quad S_{2}={\frac {4ac^{4}T_{3}^{2}V_{2}}{3(c^{2}-v^{2})^{2}}}$,

consequently, since under the condition S₁ = S₂ and $V_{2}=V_{1}\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$,

$T_{2}=T_{1}\cdot {\sqrt {1-{\frac {v^{2}}{c^{2}}}}}$

and by (4):

p₁ = p₂,

that is, the common final pressure is equal to the common initial pressure. The last two relations are thus generally valid for any arbitrary body subjected to that process.

It also follows that we can replace the volume condition (16) of § 5 by the simpler condition, in which the final pressure p₂ is equal to the initial pressure p₁. Then we can say: By a reversible adiabatic isobaric (i.e. p = const.) acceleration (in an arbitrary way) of any body from velocity 0 up to velocity v, both the volume and the temperature of the body is diminished in the ratio ${\sqrt {1-{\frac {v^{2}}{c^{2}}}:1}}$. In this theorem, of course, the direction of the velocity v is negligible. Therefore, the same theorem is valid even if we substitute the arbitrarily oriented velocity q instead of velocity v directed into the x-axis.

§ 7.

The last theorem makes it possible to express in a very general way the relation between the values of the temperature and the pressure of an arbitrarily moving body for the two reference frames used by us. We imagine that a moving body with arbitrarily directed velocity is given. The magnitude of velocity for the unprimed frame is q, and q' for the primed frame. If, from the given condition, the body is brought in a reversible, adiabatic and isobaric way to rest for the unprimed