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 spatial component this is *r´* = *r – tc²/v*. The relative spatial transformation is then:

*r´* = *γ* (*r – tc²/v*) and its inverse *r* = *γ* (*r´ + t´c²/v*)

along with a standard (characteristic) speed, *c*, in all reference frames: *r = ct, r´ = ct´* implies that

*γ²* = 1 / (1 – *c²/v²*), which is the superluminal Lorentz transformation.

The temporal Lorentz transformation is then:

*t´* = *γ* (*t – rv/c²*) and its inverse *t* = *γ* (*t´ + r´v/c²*)

along with a standard speed, *c*, in all reference frames: *t = r/c, t´ = r´/c* implies that

*γ²* = 1 / (1 – *v²/c²*), which is the subluminal Lorentz transformation.

So we have both subluminal and superluminal Lorentz transformations for both spatial and temporal components. This is the complete Lorentz transformation, which covers all velocities for both relative spatial and temporal components.

Note that the relative spatial component *r´* = *γ* (*r – tc²/v*) here and the previously used relative spatial 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*.