What follows are four derivations of the Lorentz transformation from the complete Galilei (Galilean) transformations in space with time (3+1) and time with space (1+3). Their intersection is linear space and time (1+1), which is the focus of the derivations. The other dimensions may be reached by rotations in space or time.

I. Space with Time (3+1)

Consider two inertial frames of reference *O* and *O′*, assuming *O* to be at rest while *O′* is moving with velocity *v* with respect to *O* in the positive *x*-direction. The origins of *O* and *O′* initially coincide with each other. A light signal is emitted from the common origin and travels as a spherical wave front. Consider a point *P* on a spherical wavefront at a distance *x* and *x′* from the origins of *O* and *O′* respectively. According to the second postulate of the special theory of relativity the speed of light, *c*, is the same in both frames, so for the point *P*:

*x = ct*, and *x′ = ct′*.

A. Time velocity

Define velocity *v* as the time velocity *v*_{t} = *ds*/*dt*. Consider the standard Galilean transformation of *ct* and *x* with a factor *γ*, which is to be determined and may depend on *β*, where *β = v/c*:

*x′* = γ(*x − vt*) = γ(*x − βct*) = *γx*(1 − *β*).

The inverse transformation is the same except that the sign of *β* is reversed:

*x* = *γ*(*x′ + vt′*) = γ(*x + βct*) = *γx′*(1 + *β*).