This derivation of the Galilean transformations is similar to that of the Lorentz transformations here.
Since space and time are assumed to be homogeneous, the transformations must be linear. The most general linear relationship is obtained with four constant coefficients: A, B, C, and D:
x′ = Ax − Bt
t′ = Ct − Dx
Without loss of generality, if t = 0, let A = 1, and if x = 0, let C = 1. Then
x′ = x − Bt
t′ = t − Dx
The inverse transformation for the position from frame R′ to frame R should have the same form as the original but with its motion in the opposite direction, as is confirmed by algebra:
x = x′ + Bt′
t = t′ + Dx′
Then B = (x − x′) / t, which is the relative velocity V, and D = (t − t′) / x, which is the relative lenticity W.
x = x′ + Vt′
t = t′ + Wx′
If motion is along the x axis, then t = t′ and W = 0, and if it is along the t axis, then x = x′ and V = 0. In the former case
x’ = x − Vt
t’ = t
and in the latter case
x’ = x
t’ = t − Wx
These are the Galilean transformations.
Revised 2021-06-11.