Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/3

We obtain

$U=H+\beta \phi ={\sqrt {1-\beta ^{2}}}\,U_{0}+\beta ^{2}(U+pv),$

from which it is given under consideration of the first equation (14):

$U={\frac {1}{\sqrt {1-\beta ^{2}}}}(U_{0}+\beta ^{2}p_{0}v_{0}).$

(17)

Finally, the momentum is according to (10):

${\mathfrak {G}}={\frac {\beta }{c}}(pv+U)={\frac {\beta }{c{\sqrt {1-\beta ^{2}}}}}(U_{0}+p_{0}v_{0}).$

(18)

If we summarize everything, we come to the result:

If a body whose state at rest is given by the variables $v_{0},U_{0},p_{0},T_{0},S_{0}$, is adiabatically brought to velocity $\beta c$, then the state variables assume the value:

$v=v_{0}{\sqrt {1-\beta ^{2}}}$

(14)

$p=p_{0}\,$

$T=T_{0}{\sqrt {1-\beta ^{2}}}$

(14)

$U={\frac {1}{\sqrt {1-\beta ^{2}}}}(U_{0}+\beta ^{2}p_{0}v_{0})$

(17)

$H={\sqrt {1-\beta ^{2}}}\cdot U{}_{0}$

(16)

$S=S_{0}\,$

${\mathfrak {G}}={\frac {\beta }{c}}{\frac {1}{\sqrt {1-\beta ^{2}}}}(U_{0}+p_{0}v_{0}).$

(18)

These equations are in agreement with the results of the paper of Planck^[1]. Besides thermodynamics, Planck used the relativity principle, while stating equation (10) for momentum is essential in our work.

1. Berliner Berichte, 1907, p. 542.