iSoul In the beginning is reality

Complete Lorentz group

The complete Lorentz transformation may be written as

r′ = γ (r − ct(v/c)), ct′ = γ (ct – rv/c), and γ = (1 – v2/c2)–1/2,

which applies only if |v| < |c|, and

r′ = γ (rct(c/v)), ct′ = γ (ct − r(c/v)), and γ = (1 − c2/v2)–1/2,

which applies only if |v| > |c|. If |v| = |c|, then r′ = r and t′ = t.

In order to express this more easily, define β and γ as follows:

β =

  1. v/c if |v| < |c|
  2. c/v if |v| > |c|
  3. 0 if |v| = |c|

Based on this define γ = 1 / √(1 – β²) for all v. Then the complete Lorentz transformation may be expressed as

r′ = − ctβγ and ct′ = ctγ – rβ,

which may be displayed in matrix form (imagination required) as:

( r′ )  = ( γ    −βγ ) ( r )

( ct′ ) = ( −βγ    γ ) ( ct )

This is formally identical to the Lorentz transformation, which forms a multiplicative group, and so the complete Lorentz transformation forms a group as well.

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