Page:Relativity (1931).djvu/167

 $$x_4$$ 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 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 $$x_1$$, $$x_2$$, $$x_3$$, which identically satisfy the equation

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.”