Dual Euclidean transformations

Dual Euclidean transformations are required to transform six dimensions of length and duration: one Euclidean transformation for length space with time and one Euclidean transformation for duration space with distance. The two Euclidean transformations are:

x′ = xvt and z′ = zws

where x and x′ are length space vectors, t is the time parameter, z and z′ are duration space vectors, and s is the distance parameter. The two transformations are related by the rod-clock rate:

x = cz and s = ct

where c is the rod-clock speed. The two Euclidean transformations are related by the rod-clock speed:

x′ = xvt = cz′ = czvs/c = czβs

where β = v/c. We then have

z′ = zvs/c² = zws

where w = v/c². That is, cw = v/c = β.

Note there is some similarity with the Lorentz transformations without the gamma factor.