chapter  2
2 Pages

Minkowski’s Four-dimensional Space (“World”) [Supplementary to Section 17]

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 = x1

2 + x2 2 + x3

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.”