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′ = x − vt and z′ = z − ws
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′ = x − vt = cz′ = cz − vs/c = cz − βs
where β = v/c. We then have
z′ = z − vs/c² = z − ws
where w = v/c². That is, cw = v/c = β.
Note there is some similarity with the Lorentz transformations without the gamma factor.