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

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We obtain


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.