Section 26

The Space–time Continuum Of Special Relativity as a Euclidean Continuum

March 12, 2022

We can now use Gaussian coordinates to formulate more exactly the Minkowski spacetime in Section 17. According to Special Relativity (SR), Galileian coordinate systems are preferred for describing a 4D space-time continuum.

The 4 co-ordinates x, y, z, t determine an event or a point of the 4D continuum. 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 SR. These in themselves are nothing but the expression of the universal constancy of the speed of light.

Minkowski found that the Lorentz transformations satisfy the following simple conditions.

Two neighbouring events happen in the 4D continuum with respect to a Galileian reference-body K by the space co-ordinate differences dx, dy, dz and the time-difference dt.

  • The second Galileian system has the corresponding differences for these two events as dx', dy', dz', dt'.
  • These magnitudes always fulfill the condition=
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, expressed as:

ds^2 = dx^2 + dy^2 + dz^2  c^2 dt^2

The magnitude which belongs to two adjacent points of the 4D space-time continuum, has the same value for all selected (Galileian) reference-bodies. If we replace x, y, z, √ (− 1 ct) by x1, x2, x3, x4, we also obtain the result=

ds^2 = dx1^2 + dx2^2 + dx3^2 + dx4^2*

This is independent of the choice of viewpoint. We call the magnitude ds the “distance” apart of the 2 events or 4D points. Thus, if we choose as time-variable the imaginary variable √ (− 1 ct) instead of the real quantity t, we can regard the space-time continuum — in accordance with Special Relativity — as a “Euclidean” 4D continuum.