Albert Einstein (1879–1955). Relativity: The Special and General Theory. 1920.

**Appendix II**

# Appendix II. Minkowski’s Four-Dimensional Space (“World”)

[SUPPLEMENTARY TO SECTION XVII]

W

*ct*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:

*a*) is transformed into this equation.

We see from (12) that the imaginary time co-ordinate 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 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.”