PlanetPhysics/Minkowski's Four Dimensional Space World

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

(Supplementary to Section 17)} From [[../SpecialTheoryOfRelativity/|Relativity: The Special and General Theory]] by Albert Einstein

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

$\begin{array}{ccc}{x}_1& =& x\\ x_2& =& y\\ x_3& =& z\\ x_4& =& \sqrt{-I}\cdot ct\end{array}$

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

${x{\text{'}}_1}^2+{x^{\prime }}_2^2+{x^{\prime }}_3^2+{x^{\prime }}_4^2=x_1^2+x_2^2+x_3^2+x_4^2...\text{(12)}.$

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 $x_4$, 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 ($x{\text{'}}_1,x{\text{'}}_2,x{\text{'}}_3$) with the same origin, then $x{\text{'}}_1,x{\text{'}}_2,x{\text{'}}_3$, are linear homogeneous functions of $x_1,x_2,x_3$ which identically satisfy the equation

${x^{\prime }}_1^2+{x^{\prime }}_2^2+{x^{\prime }}_3^2=x_1^2+x_2^2+x_3^2$

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."

References

This article is derived from the [[../AlbertEinstein/|Einstein]] Reference Archive (marxists.org) 1999, 2002. Einstein Reference Archive which is under the FDL copyright.