This is a revision to the derivation of the superluminal Lorentz transformation, as presented previously such as *here*. The problem with the previous derivation is that it did not distinguish between speed and pace, velocity and celerity.

Consider Case 2 again.

This case begins with: *r´ = r – tc²/v* or *t´* = *t – r/v*. These can be written as: *r´ = r – ct *(*c/v*) or c*t´* = c*t – r *(*c/v*). If *β = c/v*, then the equations are: *r´ = r – ctβ* or c*t´* = c*t – rβ*.

Note that *v* ≠ 0 in these equations since it is in the denominator. But the restriction should be *c* ≠ 0 instead. More precisely, *β* should be a ratio of paces, not speeds. That is, the rate of change should be measured as the change in time per change in space (length or distance).

If we notate the pace by an underline, then *β* should be defined as *v/c*. That allows the definition of *β* as follows:

*β* = *v/c* if |*v*| < |*c*|, in the spatial direction of *v*,

*β* = *v/c* = *c/v* if |*v*| > |*c*|, i.e., |*v*| < |*c*|, in the temporal direction of *v* (for emphasis this could be notated as *β*),

*β* = 0 if |*v*| = |*c*|, without direction.

Then for the superluminal Lorentz transformation: *γ²* = 1 / (1 – *v²/c²*) = 1 / (1 – *β²*),

This matters even more with the vector version of *β*, i.e., **β**:

**β** = **v***/c* if |**v**| < |*c*|, with the spatial direction of **v**,

**β** = **v***/c* if |**v**| > |*c*|, with the temporal direction of **v** (for emphasis this could be notated as **β**),

**β** = **0** if |**v**| = |*c*|, without direction.