# Relativity: The Special and General Theory

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-ordinates) 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 " 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 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 Δt1 of two events with respect

to K1 does not in general vanish, even when the time difference Δt1 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.1) In order to give due prominence to this relationship,