Page:Bemerkungen zum Prinzip der Aktion und Reaktion.djvu/3

From the point of view described we can generally denote the principle of equality of action and reaction as the "theorem of inertia of energy".

But we can go a step further. As the constancy of energy implies the concept of energy flux, the constancy of momentum also implies the concept of "flux of the momentum", or briefly spoken: the "momentum flux". As the momentum located in a particular area can only change due to external effects, i.e. according to the theory of direct action only by processes at the surface of that space, the amount of change in unit time is a surface integral which can be described as the total momentum flux in the interior of that space. A major difference, however, to the energy flux is the fact that the energy is a scalar, yet the momentum is a vector. Therefore, the incoming energy in that space is expressed by a single surface integral, and the energy flux is a vector. However, the momentum flux into that space is expressed by three surface integrals, corresponding to the three components of momentum, and the momentum flux in one place is a tensor triple, in the notation of W. Voigt^[1], which is characterized by six components

To get an idea of the significance of these tensor triples, we first consider the mechanical momentum and its corresponding mechanical momentum flux. The total momentum flux into the interior of a room, i.e. the increase in momentum in the interior per unit time, is equal to the resulting mechanical force which acts on the entire mass located within that space. Consequently, the momentum flux through a surface element is nothing else than the mechanical pressure on the surface element, and its components have the form:

$X_{n}=X_{x}\cos(nx)+X_{y}\cos(ny)+X_{z}\cos(nz)$

$Y_{n}=Y_{x}\cos(nx)+Y_{y}\cos(ny)+Y_{z}\cos(nz)$

$Z_{n}=Z_{x}\cos(nx)+Z_{y}\cos(ny)+Z_{z}\cos(nz)$,

where n denotes the inner normal of the surface element. $X_{x},\ Y_{y},\ Z_{z},\ X_{y}=Y_{x},\ Y_{z}=Z_{y},\ Z_{x}=X_{z}$ are the six components of the tensor triples representing the momentum flux.

The same is true for the electromagnetic momentum flux in a vacuum. The components of this tensor triple are nothing else than the known Maxwell stresses. Its integration over a closed surface gives the total momentum flux into the interior, and thus the increase of all mechanical and electromagnetic momentum contained within the enclosed space. It is remarkable, how by this theorem the physical meaning of the Maxwell stresses is increased for the stationary aether theory. Because, as a force of pressure these stresses make not really sense in this theory, since we may not attribute a meaning to a force acting on something absolutely immovable.^[2] The fact that the Maxwell stresses, although they were officially abolished, have maintained themselves within the theory of a stationary aether, because they often proved so convenient as mathematical tools for certain calculations, could already suggest the idea, that they play some special physical role by which they are legitimate also for the stationary aether.

It is tempting to transfer the concept of momentum flux to the gravitational field, where, apart from the fatal sign, there are given a remarkable number of analogies; but a more detailed discussion of this problem would take too far at this place.

Discussion.

Minkowski: The theorems on momentum can be derived directly from the energy theorem in my point of view. Namely, the energy equation in Lorentz's theory depends on the reference frame for space and time. If we write the energy theorem for each possible reference frame, we have some equations, and in those the theorems on momentum are included.

Planck: Certainly. But I consider the independence from the reference frame not as a solid physical result, but more as a hypothesis, which I consider as promising but not to be established yet. However, we just need to consider whether these relationships actually exist in nature. This we can only learn by experiment, and hopefully the time is not far off when we experience it.

1. See M. Abraham, Enzyklopädie d. math. Wiss. IV, 14, p. 28
2. Vgl. H. A. Lorentz, Versuch einer Theorie der elektrischen und optischen Erscheinungen, p. 28. Leiden 1895.