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* = *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*).

Multiply these two equations to get

*xx′* = *γ*²*xx′*(1 − *β*²).

Divide out *xx′* to get

*γ*² = 1/(1 − *β*²),

or *γ* = 1/√(1 − *β*²).

B. Space velocity

Define velocity *v* as the space velocity *v*_{s} = (*dt*/*ds*)^{−1}. Consider the transposed standard Galilean transformation of *ct* = *x* with a factor *γ*, which is to be determined and may depend on *β*, where *β = v/c*:

*ct′* = *γ*(c*t − βx*) = *γct*(1 − *β*).

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

*ct* = *γ*(c*t′ + βx′*) = *γct′*(1 + *β*).

Multiply these two equations to get

*c*²*tt′* = *γ*²c²*tt′*(1* − β*²).

Divide out c²*tt′* to get

*γ*² = 1/(1 − *β*²),

or *γ* = 1/√(1 − *β*²).

Therefore, the Lorentz transformation for space with time is

*ct′ = γ(ct − βx) *and *x′* = γ(*x − βct*) with *γ* = 1/√(1 − *β*²).

II. Time with Space (1+3)

Consider two inertial frames of reference *O* and *O′*, assuming *O* to be at rest while *O′* is moving with lenticity *w* with respect to *O* in the positive *t*-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 time-point *Q* on a spherical wavefront at a distime *t* and *t′* from the origins of *O* and *O′* respectively. According to the second postulate of the special theory of relativity the pace of light, *k*, is the same in both frames, so for the time-point *Q*:

*t* = *kx*, and *t′* = *kx′*.

A. Space lenticity

Define lenticity *w* as the space lenticity *w*_{s} = *dt*/*ds*. Consider the dual standard Galilean transformation of *t*/*k* = *x* with a factor *λ*, which is to be determined, which may depend on *β* = *k*/*w*:

*kx’ = t′* = *γ*(*x* − *t/w*) = *γ*(*x* − *βt*/*k*) = *λt*(1 − *β*).

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

*t* = *λ*(*x* + *t’*/*w*) = *λ*(*t’* + *βx*/*k*) = *λt’*(1 + *β*).

Multiply these two equations to get

*tt′* = *λ*²*tt′*(1 − *β*²).

Divide out *tt′* to get

*λ*² = 1/(1 − *β*²) = *γ*², or

*λ* = 1/√(1 − *β*²) = *γ*.

B. Time lenticity

Define lenticity *w* as the time lenticity *w*_{t} = (*ds*/*dt*)^{−1}. Consider the transposed dual standard Galilean transformation of *t* = *kx* with a factor *λ*, which is to be determined and may depend on *β* = *k*/*w*:

*t’ = kx′* = *λ*(*kx* − *tk*/*w*) = *λ*(*kx* − *βt*) = *λkx*(1 − *β*).

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

*kx = λ(kx’ + t’k/w) = λ(kx’ + βt’) = λkx’(1 + β).*

Multiply these two equations to get

*k*²*xx′* = *λ*²*k*²*xx′*(1 − *β*²).

Divide out *k*²*xx′* to get

*λ*² = 1/(1 − *β*²) = *γ*², or

*λ* = 1/√(1 − *β*²) = *γ*.

Therefore, the dual Lorentz transformation for time with space is

*x′* = *γ*(*x* − *βt*/*k*) = *γ*(*x* − *t/*w),

*t′* = *γ*(*kx* − *βt*) = *γ*(*kx* − *tk*/*w*).

Compare *ct′* = *γ*(*ct* − *βx*) and x′ = *γ*(*x* − *βct*). Duality is equivalent to interchanging *c* ↔ *k* and *x* ↔ *t*.

*Revised 2021-06-24.*