Relativity (1931)/Appendix 2

by Albert Einstein

Layout 2

 

APPENDIX II

MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”) [Supplementary to Section XVII]

WE can characterise the Lorentz transformation still more simply if we introduce the imaginary ${\sqrt {-1}}.ct$ in place of $t$, as time-variable. If, in accordance with this, we insert

${\begin{aligned}x_{1}&=x\\x_{2}&=y\\x_{3}&=z\\x_{4}&={\sqrt {-1}}.ct\end{aligned}}$

and similarly for the accented system $K'$, then the condition which is identically satisfied by the transformation can be expressed thus:

$x_{1}'^{2}+x_{2}'^{2}+x_{3}'^{2}+x_{4}'^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}$

(12).

That is, by the afore-mentioned choice of “coordinates” (11a) is transformed into this equation. We see from (12) that the imaginary time coordinate $x_{4}$, 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

$x_{1}'^{2}+x_{2}'^{2}+x_{3}'^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$

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 imaginary time co-ordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”