In recent post on *Lorentz and dual Lorentz Transformations*, I derived a complete set of Lorentzian transformations:

The Lorentz transformation is

speed: r′ = γ (r − vt) and t′ = γ (t – rv/c²), with γ = (1 – v²/c²)^{–1/2}, or

pace: r′ = γ (r – t/u) and t′ = γ (t – rç²/u), with γ = (1 – ç²/u²)^{–1/2} ,

which applies only if |v| < |c| or |u| > |ç|.

The dual Lorentz transformation is

speed: t′ = *λ* (t − r/v) and r′ = *λ* (r − c² t/v), with *λ* = (1 − c²/v²)^{–1/2}, or

pace: t′ = *λ* (t – ur) and r′ = *λ* (r – tu/ç²), with *λ* = (1 – u²/ç²)^{–1/2},

which applies only if |v| > |c| or |u| < |ç|.

If |v| = |c|, then r′ = r and t′ = t.

Other dimensions were not discussed since the notation didn’t specify whether or not there were other dimensions. Here I note that speed requires that the denominator be a scalar of time, so in that case time is either one-dimensional or resolves itself into one dimension. Similarly, pace requires that the denominator be a scalar of space, so in that case space is either one-dimensional or resolves itself into one dimension. Another way to look at it is that speed allows multidimensional space but not time, whereas pace allows multiple dimensions of time but not of space.

It has been *noted* that tachyons are possible with multiple dimensions of time but not space, which is consistent with the dual Lorentz transformation except that there is also the possibility of space and time resolving themselves into one dimension each. Anyone who travels faster than the characteristic (modal) rate can experience this from their own perspective.

As the Lorentz transformation arises from Minkowski geometry so the dual Lorentz transformation arises from dual Minkowski geometry. *Relativity without tears* (p.23-4) notes that the points of Minkowski geometry correspond to the lines of dual Minkowski geometry (and *vice versa*), and the distance between points in Minkowski geometry correspond to the angle between lines in the dual Minkowski geometry (and *vice versa*). This is consistent with the *previous post*: Linear space and angular time in Minkowski geometry corresponds to angular space and linear time in dual Minkowski geometry.