Page:On the Conception of the Current of Energy.djvu/2

 current follows from the transformation formula for the density of the energy $$W$$ and for its velocity $$\mathfrak{w}$$. He assumes there, if I understand him rightly, that the addition theorem of applies to $$\mathfrak{w}$$ as well as to the velocity of a material point.

This, however, is not the case. For if we start from the transformation for $$\mathfrak{S}$$ and $$W$$, we find quite a different law for the transformation formula for $$\mathfrak{w}$$. It is the question if an objection to that transformation can he derived from this fact.

To me this seems not to be the case. The claim that the addition theorem should apply presupposes that for energy as for matter we can distinguish individually the particles of which it consists. Only on this supposition can the paths of a particle relative to two differently moving coordinate systems be possibly compared with one another, which then leads to the addition theorem of. This assumption, however, does certainly not hold, for the transformation formula for $$W$$, i. e. the equation

$W=\frac{W'+\beta^{2}p'_{xx}+2\frac{v}{c^{2}}\mathfrak{S}'_{x}}{1-\beta^{2}}$|undefined

shows, that energy can also then be present in the accentuated system, when in the unaccentuated system no energy of the same kind is to be found.

It is true that in the electromagnetic field in vacuo this case cannot occur. But it can occur for the elastic energy of a body subjected to a tension which is equal in all directions.

If the body rests relatively to the accentuated system, then we have

$\mathfrak{S}'=0,\ p'_{xx}<0,\ W'>0$

and if the body is body little compressible:

$p'_{xx}>>W'$

We shall then have $$W=0$$ if the relative velocity of translation of file two systems reaches the not very large value

$v=c\sqrt{-\frac{p'_{xx}}{W'}}$|undefined

If $$v$$ increases to a still higher value, $$W$$ will even become negative. In such a case it is certainly impossible to compare the motion of a particle of energy when evaluated with the aid of the two systems.

Perhaps the objection may be raised against this consideration that in the last equation the tensor transformation has been used, whereas its applicability is just to be proved. Therefore I will adduce an instance which shows independently of every special theory, that the