In order to demonstrate that the assumption of the group G_{_c_} 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 G_{_c_} 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
_mc²_ _dt_/_d_τ = _mc²_ /√(1 - _v²_/_c²_)
_i.e._, if we deduct from this the additive constant _mc²_, we obtain the expression ½ _mv²_ of Newtonian-mechanics up to magnitudes of _the order of_ 1/_c²_. Hence it appears that _the energy_ depends _upon the system of reference_. But since the _t_-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting case c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.[33]
From the very beginning, we can establish the ratio between the units of time and space in such a manner, that the velocity of light becomes unity. If we now write √-1 _t_ = _l_, in the place of _l_, then the differential expression
_d_τ² = -(_dx²_ + _dy²_ + _dz²_ + _dl²_),
becomes symmetrical in (_x_, _y_, _r_, _l_); this symmetry then enters into each law, which does not contradict the _world-postulate_. We can clothe the “essential nature of this postulate in the mystical, but mathematically significant formula
3·10⁵ _km_ = √-1 Sec.
