iSoul In the beginning is reality

# Galilei doesn’t lead to Lorentz

I haven’t mentioned this before because I have a solution to it but there is a problem with deriving the Lorentz transformation from the Galilei transformation. If one uses the spatial Galilei (Galilean) transformation, the gamma factor leads to the Lorentz transformation. But if one uses the temporal Galilei (Galilean) transformation, the gamma factor does not lead to the Lorentz transformation.

As usual, the standard transformation of reference frames begins with two frames in uniform relative motion along one axis (usually called x). Here we take the spatial axis to be the r-axis, which parallels the spatial axis of motion.

The two frames are differentiated by primed and unprimed letters. They coincide at time t = 0 and their relative speed is v. In the Galilei (Galilean) transformation, there is a universal time that is available to all reference frames.

The Galilei (Galilean) transformation is: r′ = rtv and t′ = t.

To derive the Lorentz transformation one includes the gamma factor with the first equation. Why not include the gamma factor with the second equation? Because it won’t work.

For the transformation t′ = t, the reverse transformation is simply t = t′.

If we include the gamma factor as with these equations, similar to the usual derivations, and follow the usual procedure to combine the Galilei (Galilean) transformation and its reverse, the result is:

t′ = γt and t = γt′.

As before, multiply these together and solve for γ: tt′ = γ²tt′.

Divide out tt′: 1 = γ².

And so γ = ±1, which is not the gamma of the Lorentz transformation.

So the Galilei (Galilean) transformation is not sufficient to derive the Lorentz without qualification or modification. There should be (1) a qualification that the temporal transformation cannot be used to derive the Lorentz transformation, and (2) an expansion to include the co-Galilei (co-Galilean) transformation, in which the temporal transformation leads to the co-Lorentz transformation. For details, see here and here.