# One and two-way transformations

The transformation of Galileo is a one-way transformation, i.e., it uses only the one-way speed of light, which for simplicity is assumed to be infinite. The transformation of Lorentz is a the two-way transformation, which uses the universal two-way speed of light. The following approach defines two different one-way transformations, which combine to equal the two-way Lorentz transformation. Note that β = v/c; 1/γ² = 1 − β²; and γ = 1/γ + β²γ.

Galilean transformation:

Dual Galilean transformation:

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

Lorentz transformation (boost): .

General Lorentz boost (see here):

with   and k = 1/c for the Lorentz boost.

General dual Lorentz boost:

with 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 one-way Galilean transformation (G):

The inverse Galilean transformation (G-1):

The two-way Lorentz transformation (L):

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

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

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

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

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

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

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

Last updated April 23, 2020.