In order to demonstrate that the assumption of the group Gc for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of “Thermodynamics and Radiation,”[[30]] for “Electromagnetic phenomena”,[[31]] and finally for “Mechanics with the maintenance of the idea of mass.”
For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (x, y, z, t), where the velocity-vector is ([.x], [.y], [.z], [.t]), then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group Gc is undoubtedly permissible. These theorems lead us to the following simple rule; if the system of reference be changed in any way, 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
[.t]X, [.t]Y, [.t]Z, [.t]T,
where T = 1/c² ([.x]/[.t] X + [.y]/[.t] Y + [.z]/[.t] Z) = 1/c²
(the rate of
which work is done at the world-point), remains unaltered.
This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a moving force-vector at P.
Now the world-line passing through P will be described by a substantial point with the constant mechanical mass m. Let us call m-times the velocity-vector at P as the impulse-vector, and m-times the acceleration-vector at P as the force-vector of motion, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector[[32]]:
The force-vector of motion is equal to the moving force-vector.
This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the “Energy-law.” Accordingly c²-times the component of the impulse-vector in the direction of the t-axis is to be defined as the kinetic-energy of the point-mass. The expression for this is