Page:On the Conception of the Current of Energy.djvu/1

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On the the conception of the current of energy.

By M. Laue. (Communicated by Prof. J. D. van der Waals).

The law of the inertia of the energy, which with perfect generality brings the momentum per unit of volume ${\displaystyle {\mathfrak {g}}}$ in connection with the energy current ${\displaystyle {\mathfrak {S}}}$ according to the formula

${\displaystyle {\mathfrak {g}}={\frac {\mathfrak {S}}{c^{2}}}}$

has again drawn the attention to the conception of the current of energy, which at the time was discussed with vivid interest in relation to Poynting's theorem. The author has given a rule for the transformation of the density of the energy current ${\displaystyle {\mathfrak {S}}}$. This rule states that in every department of physics a tensor of stress ${\displaystyle p}$ exists, which with the three components of the vector ${\displaystyle {\tfrac {i}{c}}{\mathfrak {S}}}$ and the density of the energy ${\displaystyle W}$ taken negatively forms the components of a symmetrical "world tensor" ${\displaystyle T}$, i. e. we shall have

${\displaystyle {\begin{array}{llll}T_{jk}=p_{jk}&&\mathrm {if} \ j,k&=x,y,z\\\\T_{jl}={\frac {i}{c}}{\mathfrak {S}}&&{\mathsf {if}}\ j&=x,y,z\ \mathrm {and} \ (l=ict)\\\\T_{ll}=-W\end{array}}}$

In Electrodynamics the tensor ${\displaystyle p}$ represents the Maxwell stresses, in mechanics it is closely connected with the elastic stresses.

Now the conception of the current of energy has been formed in analogy to the conception of the current of a fluid. If we denote the density of the fluid by ${\displaystyle \varrho }$, its velocity by ${\displaystyle {\mathfrak {q}}}$, then the density of the current is of course ${\displaystyle \varrho {\mathfrak {q}}}$. In a recent paper[1] van der Waals Jr. transfers this relation to the energy current, and so he arrives at the conception of velocity of the motion of the energy, which is connected with the energy current ${\displaystyle {\mathfrak {S}}}$ and the energy density ${\displaystyle W}$ according to the relation

 ${\displaystyle {\mathfrak {w}}={\frac {\mathfrak {S}}{W}}}$ (1)

This velocity appears to him even to be the more lucid conception, from which the conception of the energy current must be deduced. And in the final remark of his paper[2] he expresses a doubt whether the above quoted transformation formula for the density of the energy

1. Van der Waals Jr. Proc. Amsterdam. 1911. 239.
2. Van der Waals Jr. p. 253 last paragraph. The note on this page is undoubtedly the consequence of an oversight, for in formula XXVIII I have explicitly equated to zero die divergence of the sum of all the world tensors as van der Waals wishes.