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 (r − ct(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 =
- v/c if |v| < |c|
- c/v if |v| > |c|
- 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′ = rγ − ctβγ and ct′ = ctγ – rβ,
which may be displayed in matrix form as:
This is formally identical to the Lorentz transformation, which forms a multiplicative group, and so the complete Lorentz transformation forms a group as well.