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 XVII. In accordance with the special theory of relativity, certain co-ordinate 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 than the expression of the universal validity of the law of transmission of light for all Galileian systems of reference.
Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us consider two neighbouring events, the relative position of which in the four-dimensional continuum is given with respect to a Galileian reference-body _K_ by the space co-ordinate differences _dx, dy, dz_ and the time-difference _dt_. With reference to a second Galileian system we shall suppose that the corresponding differences for these two events are _dx′, dy′, dz′, dt′_. Then these magnitudes always fulfill the condition.[21]
[21] Cf. Appendixes I and II. The relations which are derived there for the co-ordinates themselves are valid also for co-ordinate _differences_, and thus also for co-ordinate differentials (indefinitely small differences).
_dx_2 + _dy_2 + _dz_2 – _c_2_dt_2 = _dx′_2 + _dy′_2 + _dz′_2 – _c_2_dt′_2.
The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude
_ds_2 = _dx_2 + _dy_2 + _dz_2 – _c_2 _dt_2,
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_,
image034
by _x_1, _x_2, _x_3, _x_4, we also obtain the result that
_ds_2 = _dx_12 + _dx_22 + _dx_32 + _dx_42.
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.
Thus, if we choose as time-variable the imaginary variable
image035
instead of the real quantity _t_, we can regard the space-time contintium—accordance with the special theory of relativity—as a “Euclidean” four-dimensional continuum, a result which follows from the considerations of the preceding section.
