Notes

*) The general theory of relativity renders it likely that the electrical masses of an electron are held together by gravitational forces.

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[1], y[1], z[1], 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[1], y[1], z[1], t[1] 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: 

eq. 24:  file eq24.gif