Page:Zur Dynamik bewegter Systeme.djvu/14

reference frame, then its volume has grown from V' to ${\frac {V}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}$, its temperature from T' to ${\frac {T}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}$. However, if from the given condition the body is brought in a reversible, adiabatic and isobaric way to rest for the primed reference frame, then its volume has grown from V to ${\frac {V'}{\sqrt {1-{\frac {q'^{2}}{c^{2}}}}}}$, its temperature from T to ${\frac {T'}{\sqrt {1-{\frac {q'^{2}}{c^{2}}}}}}$. However, the state of rest achieved in this way in the unprimed system is in all respects identical to the previously obtained state of rest in the primed system. For the conditions, under which the theorem of § 5 is valid, are all satisfied here when we think that the body (at rest for the unprimed system) is brought from the initially given state in a reversible, adiabatic and isobaric way to rest for the primed system. Consequently:

$p=p',\ {\frac {V}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}={\frac {V'}{\sqrt {1-{\frac {q'^{2}}{c^{2}}}}}}$ and ${\frac {T}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}={\frac {T'}{\sqrt {1-{\frac {q'^{2}}{c^{2}}}}}}$, or:

${\frac {V'}{V}}={\frac {T'}{T}}={\sqrt {\frac {c^{2}-q'^{2}}{c^{2}-q{}^{2}}}},\ p'=p,\ S'=S$

(17)

as a generally valid relation between the primed and unprimed the variables.

§ 8.

Now we are mainly concerned with the comparison of the values of the kinetic potential in both reference frames. For this purpose we first write the differential equations (7) in accordance with the principle of relativity for the primed system:

${\frac {\partial H'}{\partial V'}}=p',\ {\frac {\partial H'}{\partial T'}}=S'$

(18)

These two equations give with respect to the equations (7) and the relations (17):

${\frac {\partial }{\partial V}}\left(H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}\right)={\frac {\partial H}{\partial V}},\quad {\frac {\partial }{\partial T}}\left(H'{\sqrt {\frac {c^{2}-q^{2}}{c^{2}-q'^{2}}}}\right)={\frac {\partial H}{\partial T}}$.

(19)