PlanetPhysics/Minkowski's Four Dimensional Space World

Minkowski's Four-Dimensional Space (``World")

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(Supplementary to Section 17)} From Relativity: The Special and General Theory by Albert Einstein

We can characterise the Lorentz transformation still more simply if we introduce the imaginary   in place of  , as time-variable. If, in accordance with this, we insert

 

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

 

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  , enters into the condition of transformation in exactly the same way as the space co-ordinates  . It is due to this fact that, according to the theory of relativity, the "time"  , enters into natural laws in the same form as the space co ordinates  .

A four-dimensional continuum described by the ``co-ordinates"  , 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 ( ) with the same origin, then  , are linear homogeneous functions of   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 an imaginary time coordinate); the Lorentz transformation corresponds to a "rotation" of the co-ordinate system in the four-dimensional ``world."

References

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This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. Einstein Reference Archive which is under the FDL copyright.