Relativity: The Special and General Theory

We can characterise the Lorentz transformation still more simply if we introduce the imaginary

in place of t, as time-variable. If, in accordance with this, we insert

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 “coordinates,” (11a) [see the end of Appendix II] 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 (x′₁, x′₂, x′₃) with the same origin, then x′₁, x′₂, x′₃, 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 an imaginary time coordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”