# Galilean decompositions of the Lorentz transformation

The background for this post is here.

For space with time (3+1):

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

$\begin{pmatrix}&space;\gamma&space;&&space;0&space;\\&space;0&space;&&space;1/\gamma&space;\end{pmatrix}&space;\begin{pmatrix}&space;t&space;\\&space;x&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;\gamma&space;t&space;\\&space;x/\gamma&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;t'&space;\\&space;x'&space;\end{pmatrix}$

Use vector (t  x)T. The gamma transformation is conjugate to the Lorentz boost (matrix Λ) by the Galilean transformations (G, GT), i.e., GTΓG = Λ:

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

or

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

This expands to

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

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

For time with space (1+3):

Use vector (x  t)T. The dual gamma transformation (matrix Γ) expresses the time dilation of clocks and length contraction of rods with a relative pace:

$\begin{pmatrix}&space;\gamma&space;&&space;0&space;\\&space;0&space;&&space;1/\gamma&space;\end{pmatrix}&space;\begin{pmatrix}&space;x&space;\\&space;t&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;\gamma&space;x&space;\\&space;t/\gamma&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;x'&space;\\&space;t'&space;\end{pmatrix}$

The dual gamma transformation is conjugate to the dual Lorentz boost (matrix Λ) by the dual Galilean transformations (GT, G), i.e., GTΓG = Λ:

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

or

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

Note that superluminal motion is excluded in both space-time and time-space.