iSoul In the beginning is reality.

Galilean decompositions of the Lorentz transform

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:

\begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix} = \begin{pmatrix} \gamma ct \\ x/\gamma \end{pmatrix} = \begin{pmatrix} c{t}' \\ {x}' \end{pmatrix}

The gamma transformation is conjugate to the Lorentz boost (Λ), i.e., GTΓG = Λ:

or

\begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} \begin{pmatrix} \gamma & 0 \\ 0 & 1/\gamma \end{pmatrix} \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{pmatrix}

or

\begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1/\gamma & 0 \\ 0 & \gamma \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -\beta & 1 \end{pmatrix} = \begin{pmatrix} 1 & -\beta \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1/\gamma & -\beta\gamma \\ 0 & \gamma \end{pmatrix} = \begin{pmatrix} \gamma & -\beta \gamma \\ -\beta \gamma & \gamma \end{pmatrix}

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 − β²)−1/2:

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

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