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.
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.