Page:Zur Dynamik bewegter Systeme.djvu/11

Furthermore,^[1]

${\sqrt {\frac {c^{2}-q'^{2}}{c^{2}-q^{2}}}}={\frac {c{\sqrt {c^{2}-v^{2}}}}{c^{2}-v{\dot {x}}}}={\frac {c^{2}+v{\dot {x}}'}{c{\sqrt {c^{2}-v^{2}}}}}={\frac {V'}{V}}={\frac {dt}{dt'}}.$

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We want to prove now that the entropy of the considered body has with respect to the primed system the same value as with respect to the unprimed system. We could found this prove, more generally, on the close connection of entropy with probability, whose quantity can impossibly depend on the choice of the reference frame; however, here we prefer a more direct way, completely independent of the introduction of the concept of probability.

We think of a body brought from a state at rest in the unprimed reference frame, into a second state by any reversible adiabatic process, so that it is at rest in the primed reference frame. If we denote the entropy of the body for the unprimed frame in the initial state by S₂, in the final state by S₂, then because of reversibility and adiabasy S₁ = S₂. But also for the primed reference frame the process is reversible and adiabatic, so we also have: $S'_{1}=S'_{2}$.

If S'₁ would not be equal to S₁, but $S'_{1}>S_{1}'$, then this would mean: the entropy of the body for that reference frame for which it is in motion, is greater than for that reference frame for which it is at rest. Then according to this proposition it should be $S_{2}>S'_{2}$ as well; for in the second condition the body rests in the primed reference frame, while it is in motion for the unprimed reference frame. However, these two inequalities contradict the equations stated above. Nor can $S'_{1}<S_{1}$; hence $S'_{1}=S_{1}$, and in general:

$S'=S,$

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that is, the entropy of the body does not depend on the choice of the reference frame.

§ 5.

Hence it follows the important conclusion: If a body (which in the initial state is at rest in the unprimed system) is brought in any

1. All these relations are also valid for a non-uniformly moving medium in which the velocity continuously varies in magnitude and direction from point to point. In this case, V can be understood as any infinitesimal volume element.