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
x₁ = x, x₂ = y, x₃ = z, x₄ = √(-1)·ct,
and similarly for the accented system K ', then the condition which is identically satisfied by the transformation can be expressed thus:
x₁'² + x₂'² + x₃'² + x₄'² = x₁² + x₂² + x₃² + x₄². (12).
That is, by the afore-mentioned choice of "co-ordinates," (11a) is transformed into this equation.
We see from (12) that the imaginary time co-ordinate x₄ enters into the condition of transformation in exactly the same way as the space co-ordinates x₁ , x₂ , x₃ . It is due to this fact that, according to the theory of relativity, the "time" x₄ enters into natural laws in the same form as the space co-ordinates x₁ , x₂ , x₃ .
A four-dimensional continuum described by the "co-ordinates" x₁ , x₂ , x₃ , x₄ , was called "world" by Minkowski, who also termed a point-event a "world-point." From a "happening" in three-dimensional 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 ( x₁ ', x₂ ', x₃ ') with the same origin, then x₁ ', x₂ ', x₃ ', are linear homogeneous functions of x₁ , x₂ , x₃ , which identically satisfy the equation
x₁'² + x₂'² + x₃'² = x₁² + x₂² + x₃².
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."