if we introduce a new momentum in addition to the mechanical momentum only known so far, i.e. the electromagnetic momentum. Abraham has made this even more plausible, by bringing the conservation of momentum in comparison with the conservation of energy. Similarly, as the energy principle is violated when we don't consider the electromagnetic energy, and fulfilled when we introduce this type of energy, the reaction principle is violated if we only consider the mechanical momentum, and is satisfied on the other hand, as soon as we have considered the electromagnetic momentum.
However, this certainly indisputable comparison doesn't affect a still substantial difference. As to the energy we know already a number of different forms: the kinetic energy, gravitation, the elastic deformation energy, heat, chemical energy, and it is therefore not a principally new feature when to these different forms, the electromagnetic energy is attached as another form. Conversely, until now only a single form was known as momentum: mechanical momentum. While the energy from the very outset represented a universal physical expression, the momentum was so far a special mechanical expression, the reaction principle a special mechanical theorem, and therefore the extension recognized as necessary was perceived as a revolution of principle, by which the previous relatively simple and uniform concept of momentum becomes considerably more complicated in nature.
Is it not possible, even from the standpoint of general dynamics, to form the definition of momentum (although it now includes both the mechanical and the electromagnetic form) in the same uniform way as it was earlier done in mechanics? A positive answer to this question would certainly lead to an advance in the understanding of the true meaning of the reaction principle.
In fact, such a uniform definition of momentum seems to be possible and feasible, at least if we also admit of Einstein's theory of relativity.[1] However, it must now be emphasized that this theory is regarded nowadays by no means assured. But as their deviations from other eligible theories are only limited to very small terms, one may say however, that they can be considered as correct just up to those differences, and to that extent, therefore, the following considerations keep in all circumstances a certain importance.
In the theory of relativity, the momentum can now be generally attributed to that vector which expresses the energy flux, not only the Poynting electromagnetic energy flux, but the energy flux in general. From the standpoint of direct action theory all types of energy can change their location within space only by continuous propagation, not by changes in the form of leaps. Therefore, the energy principle generally requires that the change of total energy contained in a certain space is like a surface integral, namely, the algebraic sum of the entire incoming energy through the surface of that space. The flux can be submitted by radiation, such as the Poynting vector, by conduction as with pressure or with pushes and with heat conduction, and by convection as with the admission of ponderable atoms or electrons through the considered surface. In any case, the entire energy flux at every point in space, with respect to unit surface and unit time, is a certain finite vector, and the ratio of this vector by the square of the speed of light c is in general the momentum in relation to unit volume.
Take for example a ponderable fluid moving with velocity q under the pressure p. Through a surface element df of a stationary area perpendicularly directed to q, there is a flow of energy through conduction and through convection during the time dt. The conducted energy is the mechanical work: p · df · qdt. The convected energy is: df · ε · qdt, where ε denotes the energy density. Accordingly, by that definition the momentum of unit volume is:
.
Comparing this expression with the usual mechanical momentum kq, where k is the density of the liquid, we find:
,
a known relationship of the theory of relativity[2]. {{nop]]
- ↑ See especially F. Hasenöhrl (Sitzungsbericht de. Akad. d. Wiss. zu Wien vom 31. Oktober 1907, S. 1400), although he doesn't directly proceed from the theory of relativity, but as far as I can see he get quite the same results.
- ↑ See for example M. Planck, Ann. d. Phys. (4), 25, 27, 1908. equation (48).
