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# Two-way transformations

The transformations associated with Galileo and Lorentz are two of several transformations between frames of reference. A one-way transformation assumes that the one-way speed of light is a constant, whereas a two-way transformation does not. Here are the available transformations:

Galilean transformation:  ${x}'&space;\mapsto&space;x-vt;\;&space;\;&space;{t}'&space;\mapsto&space;t.$

Dual Galilean transformation:  ${x}'&space;\mapsto&space;x;\;&space;\;&space;{t}'&space;\mapsto&space;t-wx.$

These could be combined with a selection factor κ of zero or one:

${x}'&space;\mapsto&space;x&space;-&space;\epsilon&space;vt;\;&space;\;&space;{t}'&space;\mapsto&space;t-(1-\epsilon&space;)wx.$

Lorentz transformation (boost): ${x}'&space;\mapsto&space;\gamma&space;(x-vt);\;&space;\;&space;{t}'&space;\mapsto&space;\gamma&space;(t-vx/c^{2})$.

General Lorentz boost (see here): ${x}'&space;\mapsto&space;\gamma&space;(x-vt);\;&space;\;&space;{t}'&space;\mapsto&space;\gamma(t-k^{2}vx)$

with $\gamma&space;=\left&space;(1-\frac{v^{2}}{c^{2}}&space;\right&space;)^{-1}$  and k = 1/c for the Lorentz boost.

General dual Lorentz boost:  ${x}'&space;\mapsto&space;\gamma_{2}&space;(x-kwt);\;&space;\;&space;{t}'&space;\mapsto&space;\gamma_{2}&space;(t-wx)$

with $\gamma_{2}&space;=\left(1-\frac{w^{2}}{k^{2}}&space;\right)^{-1}$and k = 1/c.

The following approach defines two different one-way transformations, which combine to equal the one-way Lorentz transformation. Note that β = v/c and 1 − β² = 1/γ². Also 1/γ + β²γ = γ.

The two-way Galilean transformation (G):

$\begin{pmatrix}&space;{x}'\\&space;{t}'&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;1&space;&&space;-v&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;x&space;\\&space;t&space;\end{pmatrix}&space;=\begin{pmatrix}&space;x-vt\\&space;t&space;\end{pmatrix}$

$\begin{pmatrix}&space;{x}'\\&space;{ct}'&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;1&space;&&space;-\beta&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;x&space;\\&space;ct&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;x-\beta&space;ct&space;\\&space;ct&space;\end{pmatrix}$

The inverse Galilean transformation (G-1):

$\begin{pmatrix}&space;{x}'\\&space;{t}'&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;1&space;&&space;v&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;x&space;\\&space;t&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;x+vt&space;\\&space;t&space;\end{pmatrix}$

$\begin{pmatrix}&space;{x}'\\&space;{ct}'&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;1&space;&&space;\beta&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;x&space;\\&space;ct&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;x+\beta&space;ct&space;\\&space;ct&space;\end{pmatrix}$

The one-way Lorentz transformation (L):

$\begin{pmatrix}&space;{x}'\\&space;{t}'&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;\gamma&space;&&space;-\gamma&space;v&space;\\&space;-\gamma&space;v/c^{2}&space;&&space;\gamma\end{pmatrix}&space;\begin{pmatrix}&space;x&space;\\&space;t&space;\end{pmatrix}&space;=&space;\begin{pmatrix}\gamma&space;(x-vt)\\&space;\gamma&space;(t-vx/c^{2})&space;\end{pmatrix}$

$\begin{pmatrix}&space;{x}'\\&space;{ct}'&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;\gamma&space;&&space;-\beta&space;\gamma&space;\\&space;-\beta&space;\gamma&space;&&space;\gamma&space;\end{pmatrix}&space;\begin{pmatrix}&space;x&space;\\&space;ct&space;\end{pmatrix}&space;=&space;\gamma&space;\begin{pmatrix}x-\beta&space;ct\\&space;ct-\beta&space;x&space;\end{pmatrix}$

Note the metric (M) is compatible with the Lorentz transformation (L−1ML = M):

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

The (left) para-Galilean transformation (F = LG−1) such that FG = L:

$\begin{pmatrix}&space;\gamma&space;&&space;0&space;\\&space;-\gamma&space;v/c^{2}&space;&&space;1/\gamma&space;\end{pmatrix}&space;\begin{pmatrix}&space;1&space;&&space;-v&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;\gamma&space;&&space;-\gamma&space;v&space;\\&space;-\gamma&space;v/c^{2}&space;&&space;\gamma&space;\end{pmatrix}$

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

This can be better expressed with the transpose (GT) such that GTJG = L:

$\begin{pmatrix}&space;1&space;&&space;0&space;\\&space;-v/c^{2}&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;\gamma&space;&&space;0&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;1&space;&&space;0&space;\\&space;0&space;&&space;1/\gamma&space;\end{pmatrix}&space;\begin{pmatrix}&space;1&space;&&space;-v&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;=\begin{pmatrix}\gamma&space;&&space;0&space;\\&space;-\gamma&space;v/c^{2}&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;1&space;&&space;-v&space;\\&space;0&space;&&space;1/\gamma&space;\end{pmatrix}$

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

Here is the right para-Galilean transformation (F) such that GF = L:

$\begin{pmatrix}&space;1&space;&&space;-v&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;1/\gamma&space;&&space;0&space;\\&space;-\gamma&space;v/c^{2}&space;&&space;\gamma&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;\gamma&space;&&space;-\gamma&space;v&space;\\&space;-\gamma&space;v/c^{2}&space;&&space;\gamma&space;\end{pmatrix}$

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

This is expressed with the transpose (GT) such that GJ−1GT = L:

$\begin{pmatrix}&space;1&space;&&space;-v&space;\\&space;0&space;&&space;1&space;\end{pmatrix}&space;\begin{pmatrix}&space;1/\gamma&space;&&space;0&space;\\&space;0&space;&&space;\gamma&space;\end{pmatrix}&space;\begin{pmatrix}&space;1&space;&&space;0&space;\\&space;-v/c^{2}&space;&&space;1&space;\end{pmatrix}&space;=&space;\begin{pmatrix}&space;\gamma&space;&&space;-\gamma&space;v&space;\\&space;-\gamma&space;v/c^{2}&space;&&space;\gamma&space;\end{pmatrix}$

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

Either way, the two-way Galilean transformation is compatible with the one-way Lorentz transformation. The matrices J and L are congruent.

From GTJG = L we see that the Galilean transformation (G) represents one-way light as instantaneous, with a finite turn-around speed (J). The round trip with turn-around comprises the Lorentz transformation (L).

Last updated April 23, 2020.