The background for this post is the previous one, *here*.

The *gamma transformation* (Γ) expresses the time dilation of clocks and length contraction of rods with a relative speed:

The gamma transformation is conjugate to the Lorentz boost (Λ), i.e., G^{T}ΓG = Λ:

or

The matrix second from the right represents the Tangherlini transformation (or inertial synchronized Tangherlini transformation).

Note that for time-space (1+3) there is a different decomposition, with *k* = pace of light, *β*^{∗} = *w*/*k* = *c*/*v* and *γ*^{∗} = (1 − *β ^{∗}²*)

^{−1/2}:

Harmonic algebra changes *β*^{∗} and *γ*^{∗} to space-time (3+1): H(*β*^{∗}) = *β* and H(*γ*^{∗}) = 1/*γ*.