Minkowski’s Four — Dimensional Space (world)

We can characterise the Lorentz transformation still more simply if we introduce the imaginary −1 ⋅ ct in place of t, as time-variable. If, in accordance with this, we insert

x 1 = x
x 2 = y
x 3 = z
x 4 = − 1 ⋅ ct

and similarly for the accented system K’, then the condition which is identically satisfied by the transformation can be expressed thus:

x 1 ' 2 + x 2 ' 2 + x 3 ' 2 + x 4 ' 2 = x 1 2 + x 2 2 + x 3 2 + x 4 2 . (12).

That is, by the afore-mentioned choice of “coordinates” (11a) is transformed into this equation.

We see from (12) that the imaginary time coordinate x 4 enters into the condition of transformation in exactly the same way as the space co-ordinates x 1 , x 2 , x 3.

It is due to this fact that, according to the theory of relativity, the “time” x 4 enters into natural laws in the same form as the space co-ordinates x 1 , x 2 , x 3.

A four-dimensional continuum described by the “co-ordinates” x 1 , x 2 , x 3 , x 4 , was called “world” by Minkowski, who also termed a point-event a “world-point.”

From a “happening” in 3D space, physics becomes, as it were, an “existence” in the four-dimensional “world.”

This four-dimensional “world” bears a close similarity to the three-dimensional “space” of (Euclidean) analytical geometry. If we introduce into the latter a new Cartesian co-ordinate system (x’ 1 , x’ 2 , x’ 3 ) with the same origin, then

x’ 1 , x’ 2 , x’ 3 , are linear homogeneous functions of x 1 , x 2 , x 3 , which identically satisfy the equation x 1 ’ 2 + x 2 ’ 2 + x 3 ’ 2 = x 1 2 + x 2 2 + x 3 2 .

The analogy with (12) is a complete one. We can regard Minkowski’s “world” in a formal manner as a four-dimensional Euclidean space (with imaginary time co-ordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”