— 10 —
Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a fore-cone and an aft-cone. This hyperbola may be called the hyperbola of curvature at P (vide fig. 3). If M be the center of this hyperbola, then we have to deal here with an inter-hyperbola with center M. Let be the sum of the vector MP, then we perceive that the acceleration-vector at P is a vector of magnitude in the direction of MP.
If are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and .
IV.
In order to demonstrate that the assumption of the group for the physical laws does not possibly lead to any contradiction, it is inevitable to undertake a revision of the whole of physics on the basis of the assumptions of this group. The revision has, to a certain extent, already been successfully made in the case of thermodynamics and tadiation,"[1] for electromagnetic phenomena",[2] and finally for Mechanics with the maintenance of the idea of mass.
For the latter area, the question may be asked: if there is a force with the components X, Y, Z (with respect to the space-axes) at a world-point P(x, y, z, t), where the motion-vector is , then how are we to regard this force when the system of reference is changed in any possible manner? Now, certain well-tested theorems about the ponderomotive force in electromagnetic fields exist, where the group is undoubtedly permissible. These theorems lead us to the following simple rule; if the system of reference be changed, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components
where
is the work of the force divided by at the world-point, remains unaltered. This vector is always normal to the motion-vector at P. Such a force-vector belonging to a force at P, may be called a moving force-vector at P.
- ↑ M. Planck, Zur Dynamik bewegter Systeme, Berliner Ber. 1907, p. 542 (also Ann. d. Phys. 26, 1908, p. 1).
- ↑ H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Göttinger Nachr. 1908, p. 53.