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 instantaneous. 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/*γ*² = 1 − *β*²; and *γ* = 1/*γ* + *β*²*γ*.

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 − *β*² = 1/*γ*². Also 1/*γ* + *β*²*γ* = *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^{−1}ML = M):

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

This can be better expressed with the transpose (G^{T}) such that G^{T}JG = L:

Here is the *right* para-Galilean transformation (F^{†}) such that GF^{†} = L:

This is expressed with the transpose (G^{T}) such that GJ^{−1}G^{T} = L:

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

From G^{T}JG = L we see that the Galilean transformation (G) represents one-way light as instantaneous, with a turn-around transformation (J). The round trip with turn-around comprises the Lorentz transformation (L).

*Last updated April 23, 2020*.