world by Minkowski is naturally four dimensional in the space-time
sense. For it is composed of individual events, each of which is
described by four numbers, namely, three space co-ordinates _x, y, z_,
and a time co-ordinate, the time value _t_. The world is in this
sense also a continuum; for to every event there are as many
neighbouring events (realised or at least thinkable) as we care to
choose, the co-ordinates _x1, y1, z1, t1_ of which differ by an
indefinitely small amount from those of the event _x, y, z, t_
originally considered. That we have not been accustomed to regard the
world in this sense as a four-dimensional continuum is due to the fact
that in physics, before the advent of the theory of relativity, time
played a different and more independent role, as compared with the
space coordinates. It is for this reason that we have been in the habit
of treating time as an independent continuum. As a matter of fact,
according to classical mechanics, time is absolute, _i.e._ it is
independent of the position and the condition of motion of the system
of co-ordinates. We see this expressed in the last equation of the
Galileian transformation (_t _ = _t_).
The four-dimensional mode of consideration of the world is natural on
the theory of relativity, since according to this theory time is robbed
of its independence. This is shown by the fourth equation of the
Lorentz transformation:
image030
Moreover, according to this equation the time difference _t _ of two
events with respect to _K _ does not in general vanish, even when the
time difference _t_ of the same events with reference to _K_ vanishes.
Pure space-distance of two events with respect to _K_ results in
time-distance of the same events with respect to _K_. But the
discovery of Minkowski, which was of importance for the formal
development of the theory of relativity, does not lie here. It is to be
found rather in the fact of his recognition that the four-dimensional
space-time continuum of the theory of relativity, in its most essential
formal properties, shows a pronounced relationship to the
three-dimensional continuum of Euclidean geometrical space.[14] In
order to give due prominence to this relationship, however, we must
replace the usual time co-ordinate t by an imaginary magnitude
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