PlanetPhysics/Minkowski's Four Dimensional Space World

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 $$\sqrt{-I} \cdot ct$$ in place of $$t$$, as time-variable. If, in accordance with this, we insert $$\begin{matrix} x_1 & = & x \\ x_2 & = & y \\ x_3 & = & z \\ x_4 & = & \sqrt{-I} \cdot ct \end{matrix}$$ and similarly for the accented system $$K^1$$, then the condition which is identically satisfied by the transformation can be expressed thus:


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

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 $$x_4$$, 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 an imaginary time coordinate); the Lorentz transformation corresponds to a "rotation" of the co-ordinate system in the four-dimensional ``world."