The *complete Lorentz transformation* may be written as

*r′* = *γ* (*r − ct*(*v/c*)), c*t′* = *γ* (c*t – rv/c*), and *γ* = (1 – *v*^{2}/*c*^{2})^{–1/2},

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

*r′* = *γ* (*r* − *c**t*(*c/v*)), c*t′* = *γ* (c*t − r*(*c/v*)), and *γ* = (1 − *c*^{2}/*v*^{2})^{–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 – *β*²) for all *v*. Then the complete Lorentz transformation may be expressed as

*r′* = *rγ** − ct βγ and* c

*t′*= c

*tγ – r*,

*β*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.