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*).

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* and *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) = γct(1 − β) *and *x′* = γ(*x − βct*) = *γx*(1 − *β*) 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 ℓ 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 an instant *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 instant *Q*:

*t* = *kx*, and *t′* = *kx′*.

A. Space lenticity

Define lenticity ℓ as the space lenticity ℓ_{s} = *dt*/*ds*. Consider the dual standard Galilean transformation of *t* and *kx* with a factor *λ*, which is to be determined and may depend on *β*, where 1/*β* = ℓ/*k*:

*kx′ = t′* = *λ*(*t* − ℓ*x*) = *λ*(*t* − *kx*/*β*) = *λt*(1 − 1/*β*).

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

*t* = *λ*(*t* + ℓ*x*) = *λ*(*t′* + *kx′*/*β*) = *λt′*(1 + 1/*β*).

Multiply these two equations to get

*tt′* = *λ*²*tt′*(1 − 1/*β*²).

Divide out *tt′* to get

*λ*² = 1/(1 − 1/*β*²) = *β*²/(*β*² − 1) = −*β*²*γ*², or

*λ* = 1/√(1 − 1/*β*²) = *β*/√(*β*² − 1) = −*β**γ*.

B. Time lenticity

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

*t′ = kx′* = *λ*(*kx* − ℓ*t*/*k*) = *λ*(*kx* − *t*/*β*) = *λkx*(1 − 1/*β*).

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

*kx* = *λ*(*kx* + ℓ*t*/*k*) = *λ*(*kx′* + *t′*/*β*) = *λkx′*(1 + 1/*β*).

Multiply these two equations to get

*k*²*xx′* = *λ*²*k*²*xx′*(1 − 1/*β*²).

Divide out *k*²*xx′* to get

*λ*² = 1/(1 − 1/*β*²) = *β*²/(*β*² − 1) = −*β*²*γ*², or

*λ* = 1/√(1 − 1/*β*²) = *β*/√(*β*² − 1) = −*β**γ*.

Therefore, the dual Lorentz transformation for time with space is

*t′ = kx′ = −βγ*(*t* − ℓ*x*) = *−βγ*(*t* − *kx*/*β*) = −*βγt*(1 − 1/*β*) = −*γ*(*βt* − *kx*) = *γ*(*kx* − *βt*) *= **γt*(1 − *β*)

*kx′* = *t′* = −*β**γ*(*kx* − ℓ*t*/*k*) = −*β**γ*(*kx* − *t*/*β*) = −*βγkx*(1 − 1/*β*) = *γ*(*t* − *βkx*) = *γkx*(1 − *β*).

*cf*. *ct′* = *γ*(*ct* − *βx*) and x′ = *γ*(*x* − *βct*). In short, it is equivalent to interchanging *c* ↔ *k* and *x* ↔ *t*.