The solution to the quandary posed in the previous post, Limitations of the Lorentz transformation, is to begin with an alternate form of relative motion. In the case of the stantial component this is r´ = r – tc²/v. The relative stantial transformation is then:
r´ = γ (r – tc²/v) and its inverse r = γ (r´ + t´c²/v)
along with a characteristic speed, c, in all reference frames: r = ct, r´ = ct´ implies that
γ² = 1 / (1 – c²/v²) = 1 / (1 – u²/ç²) , which is the Lorentz factor.
The temporal Lorentz transformation is then:
t´ = γ (t – rv/c²) and its inverse t = γ (t´ + r´v/c²)
along with a characteristic speed, c, in all reference frames: t = r/c, t´ = r´/c implies that
γ² = 1 / (1 – v²/c²), which is the Lorentz transformation.
This is the complete Lorentz transformation, which covers all velocities for both relative stantial and temporal components.
Note that the relative stantial component r´ = γ (r – tc²/v) here and the previously used relative stantial component r´ = γ (r – ut), with u instead of v, are equal if u = c²/v. Similarly the relative temporal component t´ = γ (t – rv/c²) and the previously used relative temporal component t´ = γ (t – t/u), with u instead of v, are equal if u = v/c².