Relativity: The Special and General Theory · Albert Einstein

Appendix II: Minkowski’s Four-Dimensional Space (“World”)

35 of 37 · 2 min of reading

We can characterise the Lorentz transformation still more simply if we introduce the imaginary StartRoot negative 1 EndRoot dot c t in place of t , as time-variable. If, in accordance with this, we insert x 1 equals x x 2 equals y x 3 equals z x 4 equals StartRoot negative 1 EndRoot dot c t, and similarly for the accented system K′ , then the condition which is identically satisfied by the transformation can be expressed thus: (12) x prime 1 squared plus x prime 2 squared plus x prime 3 squared plus x prime 4 squared equals x 1 squared plus x 2 squared plus x 3 squared plus x 4 squared. (12) That is, by the afore-mentioned choice of “coordinates,” ( 11 a ) 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 prime 1, x prime 2, x prime 3 ) with the same origin, then x prime 1, x prime 2, x prime 3 , are linear homogeneous functions of x 1, x 2, x 3 which identically satisfy the equation x prime 1 squared plus x prime 2 squared plus x prime 3 squared equals x 1 squared plus x 2 squared plus x 3 squared

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.”

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End of part 35 of 37