Relativity: The Special and General Theory/Appendix II

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←Appendix I Relativity: The Special and General Theory by Albert Einstein, translated by Robert W. Lawson
Appendix II - Minkowski's Four-Dimensional Space ("World")		 Appendix III→


[edit] Appendix II - Minkowski's Four-Dimensional Space ("World")

(SUPPLEMENTARY TO SECTION 17)


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{align} x_1 & = x \\ x_2 & = y \\ x_3 & = z \\ x_4 & = \sqrt{-1}ct \end{align}

and similarly for the accented system K1, then the condition which is identically satisfied by the transformation can be expressed thus :

x_1\mbox{ }'^2 + x_2\mbox{ }'^2 + x_3\mbox{ }'^2 + x_4\mbox{ }'^2 = x_1^{\mbox{ }2} + x_2^{\mbox{ }2} + x_3^{\mbox{ }2} + x_4^{\mbox{ }2} (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 x4, enters into the condition of transformation in exactly the same way as the space co-ordinates x1, x2, x3. It is due to this fact that, according to the theory of relativity, the "time" x4, enters into natural laws in the same form as the space co ordinates x1, x2, x3.

A four-dimensional continuum described by the "co-ordinates" x1, x2, x3, x4, 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 (x1 ', x2 ', x3 ') with the same origin, then x1 ', x2 ', x3 ', are linear homogeneous functions of x1, x2, x3 which identically satisfy the equation

x_1\mbox{ }'^2 + x_2\mbox{ }'^2 + x_3\mbox{ }'^2 = x_1^{\mbox{ }2} + x_2^{\mbox{ }2} + x_3^{\mbox{ }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 fourdimensional "world."

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