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/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.