# Appendix II. Minkowski’s Four-Dimensional Space (“World”) [Supplementary to Section XVII]

Relativity: The Special and General TheoryAlbert 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

$x_1 = x$ $x_2 = y$ $x_3 = z$ $x_4 = \sqrt{-I} \cdot ct$

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

$\tag{12} {x'_1}^2 + {x'}_2^2 + {x'}_3^2 + {x'}_4^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2$

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.”

Appendix III. The Experimental Confirmation of the General Theory of Relativity

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