Minkowski's Four-Dimensional Space (``World")

(Supplementary to Section 17)} From Relativity: The Special and General Theory by Albert Einstein

We can characterise the Lorentz transformation still more simply if we introduce the imaginary ${\displaystyle {\sqrt {-I}}\cdot ct}$ in place of ${\displaystyle t}$, as time-variable. If, in accordance with this, we insert

${\displaystyle {\begin{matrix}x_{1}&=&x\\x_{2}&=&y\\x_{3}&=&z\\x_{4}&=&{\sqrt {-I}}\cdot ct\end{matrix}}}$

and similarly for the accented system ${\displaystyle K^{1}}$, then the condition which is identically satisfied by the transformation can be expressed thus:

${\displaystyle {x'_{1}}^{2}+{x'}_{2}^{2}+{x'}_{3}^{2}+{x'}_{4}^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}\quad .\quad .\quad .\quad {\mbox{(12)}}.}$

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 ${\displaystyle x_{4}}$, enters into the condition of transformation in exactly the same way as the space co-ordinates ${\displaystyle x_{1},x_{2},x_{3}}$. It is due to this fact that, according to the theory of relativity, the "time" ${\displaystyle x_{4}}$, enters into natural laws in the same form as the space co ordinates ${\displaystyle x_{1},x_{2},x_{3}}$.

A four-dimensional continuum described by the ``co-ordinates" ${\displaystyle x_{1},x_{2},x_{3},x_{4}}$, 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 (${\displaystyle x'_{1},x'_{2},x'_{3}}$) with the same origin, then ${\displaystyle x'_{1},x'_{2},x'_{3}}$, are linear homogeneous functions of ${\displaystyle x_{1},x_{2},x_{3}}$ which identically satisfy the equation

${\displaystyle {x'}_{1}^{2}+{x'}_{2}^{2}+{x'}_{3}^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$

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."