Relativity: The Special and General Theory by Albert Einstein - HTML preview

Appendix II: Minkowski's Four-Dimensional Space ("World")(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

x₁ = x x₂ = y x₃ = z x₄ =

and similarly for the accented system K¹, then the condition which is identically satisfied by the transformation can be expressed thus :

 

x
1
'
2 2 2 3
² + x₂'² + x₃'² + x₄'² = x₁ + x₂ + x² + x₄ (12).

 

That is, by the afore-mentioned choice of " coordinates," (11a) [see the end of Appendix I] is transformed into this equation.

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.

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."

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 coordinate 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

x'
2 2
1
+ x'
2
+ x'
² = x² + x² + x²
3 1 2 3

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."