Albert Einstein: Relativity

Appendix

(supplementary to section 17)

We can characterise the Lorentz transformation still more simply if we introduce the imaginary 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} =

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} (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 fourdimensional " world."

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