*) The general theory of relativity renders it likely that the
electrical masses of an electron are held together by gravitational
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 x, y, z, 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 " three-dimensional."
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 x, y, z, t 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 =
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:
eq. 24: file eq24.gif
Moreover, according to this equation the time difference Dt1 of two
events with respect to K1 does not in general vanish, even when the
time difference Dt1 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.* In order to give due prominence to this relationship,
however, we must replace the usual time co-ordinate t by an imaginary
magnitude eq. 25 proportional to it. Under these conditions, the
natural laws satisfying the demands of the (special) theory of
relativity assume mathematical forms, in which the time co-ordinate
plays exactly the same role as the three space co-ordinates. Formally,
these four co-ordinates correspond exactly to the three space
co-ordinates in Euclidean geometry. It must be clear even to the
non-mathematician that, as a consequence of this purely formal
addition to our knowledge, the theory perforce gained clearness in no
These inadequate remarks can give the reader only a vague notion of
the important idea contributed by Minkowski. Without it the general
theory of relativity, of which the fundamental ideas are developed in
the following pages, would perhaps have got no farther than its long
clothes. Minkowski's work is doubtless difficult of access to anyone
inexperienced in mathematics, but since it is not necessary to have a
very exact grasp of this work in order to understand the fundamental
ideas of either the special or the general theory of relativity, I
shall leave it here at present, and revert to it only towards the end
of Part 2.
*) Cf. the somewhat more detailed discussion in Appendix II.
THE GENERAL THEORY OF RELATIVITY