There are several insights in the previous post *Subluminal and superluminal Lorentz transformations* to explore here.

Case 1 begins with *r´ = r – vt* or *t´ = t – rv/c²*. The equation for *r´* comes straight out of the Galilei transformation with the equation for *t´* allowed to change. So the ghost of Galilei lives on in the Lorentz transformation.

What if we began with the Galilei transformation for *t´*? Then *t´ = γt* and *t´ = r´/c* leads to *r´ = γct* so the reference frames are simply proportional. Space and time are equivalent. This would be the case if space and time were both scalars, essentially one dimensional. That is the case if *v = c*.

Look again at Case 1:

*r´ = r – vt* ⇔ *r´/c* = *r/c* – *t* (*v/c*) = *t´* = *t* (1 – *v/c*) and

*t´ = t – rv/c²* ⇔ *ct´* = *ct* – *r* (*v/c*) = *r´* = *r* (1 – *v/c*),

which shows the parallelism between the two beginnings for the subluminal Lorentz transformation.

Look again at Case 2:

*r´ = r – tc²/v* ⇔ *r´/c* = *r/c* – *t* (*c/v*) = *t´* = *t* (1 – *c/v*) and

*t´ = t – r/v* ⇔ *ct´* = *ct* – *r* (*c/v*) = *r´* = *r* (1 – *c/v*),

which shows the parallelism between the two beginnings for the superluminal Lorentz transformation. It also shows that the superluminal Lorentz transformation may be derived from a form of the Galilei transformation. So much depends on pre-Einstein mechanics, which is called non-relativistic although it includes Galilei relativity.

What is the difference between (*v/c*) and (*c/v*)? Both are dimensionless. In the first case *v* is denominated in units of *c* and in the second case *c* is denominated in units of *v*. They are just slightly different perspectives, which lead to the two main parts of the complete Lorentz transformation.