## The Space-Time Continuum of the Special Theory of Relativity Considered as a Euclidean Continuum

We are now in a position to formulate more exactly the idea of Minkowski, which was only vaguely indicated in Section 17. In accordance with the special theory of relativity, certain coordinate systems are given preference for the description of the four-dimensional, space-time continuum. We called these “Galileian co-ordinate systems.” For these systems, the four co-ordinates x, y, z, t, which determine an event or-in other words-a point of the four-dimensional continuum, are deﬁned physically in a simple manner, as set forth in detail in the ﬁrst part of this book. For the transition from one Galileian system to another, which is moving uniformly with reference to the ﬁrst, the equations of the Lorentz transformation are valid. These last form the basis for the derivation of deductions from the special theory of relativity, and in themselves they are nothing more

condition1

dx2 + dy2 + dz2 − c2dt2 = dx′2 + dy′2 + dz′2 − c2dt′2

The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude

ds2 = dx2 + dy2 + dz2 − c2dt2,

which belongs to two adjacent points of the four-dimensional space-time continuum, has the same value for all selected (Galileian) reference-bodies. If we replace x, y, Z, − 1ct, by x1, x2, x3, x4, we also obtain the result that

ds2 = dx1 2 + dx2

2 + dx3 2 + dx4

is independent of the choice of the body of reference. We call the magnitude ds the “distance” apart of the two events or four-dimensional points.