Relativity
The Special and General Theory

two neighboring events, the relative position of which in the four-dimensional continuum is given with respect to a Galilean reference-body K by the space co-ordinate differences dx, dy, dz and the time-difference dt. With reference to a second Galilean system we shall suppose that the corresponding differences for these two events are dx', dy', dz', dt'. Then these magnitudes always fulfill the condition^[1]

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,

√

-1

·ct, by x₁, x₂, x₃, x₄, we also obtain the result that

ds2 = dx12 + dx22 + dx32 + dx42.

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

√

-1

·ct, instead of the real quantity t, we can regard the space-time continuum -- in accordance with the special theory of relativity -- as a "Euclidean" four-dimensional continuum, a result which follows from the considerations of the preceding section.

[1] 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).