Albert Einstein > Relativity: The Special and General Theory > Appendix 2.

Albert Einstein (1879–1955).  Relativity: The Special and General Theory.  1920.

Appendix II

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


WE can characterise the Lorentz transformation still more simply if we introduce the imaginary

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

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

  That is, by the afore-mentioned choice of “co-ordinates” (11a) is transformed into this equation.   2
  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.   3
  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.”   4
  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 x1, x2, x3, which identically satisfy the equation

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 imaginary time co-ordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”


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