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Relativity: The Special and General Theory
Albert Einstein: Relativity
Part I: The Special Theory of Relativity
Minkowski's Four−Dimensional Space
The non−mathematician is seized by a mysterious shuddering when he hears of
"four−dimensional" things, by a feeling not unlike that awakened by thoughts of the occult. And yet
there is no more common−place statement than that the world in which we live is a
four−dimensional space−time continuum.
Space is a three−dimensional continuum. By this we mean that it is possible to describe the
position of a point (at rest) by means of three numbers (co−ordinales) x, y, z, and that there is an
indefinite number of points in the neighbourhood of this one, the position of which can be described
by co−ordinates such as x1, y1, z1, which may be as near as we choose to the respective values of
the co−ordinates x, y, z, of the first point. In virtue of the latter property we speak of a " continuum,"
and owing to the fact that there are three co−ordinates we speak of it as being "
Similarly, the world of physical phenomena which was briefly called " 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 (t1 = 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:
Moreover, according to this equation the time difference ”t1 of two events with respect to K1 does
not in general vanish, even when the time difference ”t1 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