Now the question may be asked,—what circumstances lead us to these
changed views about time and space, are they not in contradiction with
observed phenomena, do they finally guarantee us advantages for the
description of natural phenomena?
Before we enter into the discussion, a very important point must be
noticed. Suppose we have individualised time and space in any manner;
then a world-line parallel to the _t_-axis will correspond to a
stationary point; a world-line inclined to the _t_-axis will correspond
to a point moving uniformly; and a world-curve will correspond to a
point moving in any manner. Let us now picture to our mind the
world-line passing through any world point _x_, _y_, _z_, _t_; now if we
find the world-line parallel to the radius vector OA′ of the
hyperboloidal sheet, then we can introduce OA′ as a new time-axis, and
then according to the new conceptions of time and space the substance
will appear to be at rest in the world point concerned. We shall now
introduce this fundamental axiom:—
_The substance existing at any world point can always be conceived to be
at rest, if we establish our time and space suitably._ The axiom denotes
that in a world-point, the expression
_c²__dt²_ - _dx²_ - _dy²_ - _dz²_
shall always be positive or what is equivalent to the same thing, every
velocity V should be smaller than _c_. _c_ shall therefore be the upper
limit for all substantial velocities and herein lies a deep significance
for the quantity _c_. At the first impression, the axiom seems to be
rather unsatisfactory. It is to be remembered that only a modified
mechanics will occur, in which the square root of this differential
combination takes the place of time, so that cases in which the velocity
is greater than _c_ will play no part, something like imaginary
coordinates in geometry.
The _impulse_ and real cause of inducement _for the assumption of the
group-transformation G_{c}_ is the fact that the differential equation
for the propagation of light in vacant space possesses the
group-transformation G_{_c_}. On the other hand, the idea of rigid
bodies has any sense only in a system mechanics with the group
G_{infinity}. Now if we have an optics with G_{_c_}, and on the other
hand if there are rigid bodies, it is easy to see that a _t_-direction
can be defined by the two hyperboloidal shells common to the groups
G_{∞}, and G_{_c_}, which has got the further consequence, that by means
of suitable rigid instruments in the laboratory, we can perceive a
change in natural phenomena, in case of different orientations, with
regard to the direction of progressive motion of the earth. But all
efforts directed towards this object, and even the celebrated
interference-experiment of Michelson have given negative results. In
order to supply an explanation for this result, H. A. Lorentz formed a
hypothesis which practically amounts to an invariance of optics for the
group G_{_c_}. According to Lorentz every substance shall suffer a
contraction