Chapter IX: 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

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in place of _t_, as time-variable. If, in accordance with this, we insert

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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_12 + _x_22 + _x_32 + _x_42 (12).

That is, by the afore-mentioned choice of “coordinates,” (11_a_) [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_12 + _x_22 + _x_32

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.”