What follows are four derivations of the Lorentz transformation from the complete Galilei (Galilean) transformations for the time domain and the distance domain. These derivations focus on one dimension. Other dimensions may be reached by rotations of length or duration.

I. The Time Domain

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 length *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* = *κ*^{−1}, is the same in both frames, so for the point *P*:

*x = ct*, and *x′ = ct′* (also *κ**x = t*, and *κ**x′ = t′*).

A. Velocity (cf. inverse lenticity)

Define the *velocity* *v* as the time rate of displacement *v *= *ds*/*dt*. Consider the standard Galilean transformation with *x* = *ct* and 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 primes are interchanged and 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).

Therefore, the Lorentz transformation for the time domain is

*x′* = *γ*(*x* − *v**t*), and

*ct′* = *γ*(*ct* − *βx*).

B. Inverse velocity (cf. lenticity)

Define the inverse velocity as *w* = *v*^{−1} = (*dx*/*dt*)^{−1}. Consider the transposed dual standard Galilean transformation with *κ* = *c*^{−1}, *t* = *κ**x* and a factor *λ*, which is to be determined and may depend on *α* = *w*/*κ*:

*t′* = *λ*(*t* − *wx*) = *λ*(*t* − *α**t*) = *λ**t*(1 − *α*).

The inverse transformation is the same except that the primes are interchanged and the sign of *α* is reversed:

*t′* = *λ*(*t* + *wx*) = *λ*(*t* + *α**t*) = *λ**t*(1 + *α*).

Multiply these two equations to get

*tt′* = *λ*²*tt′*(1 − *α*²).

Divide out *tt′* to get

*λ*² = 1/(1 − *α*²), or

*λ* = 1/√(1 − *α*²).

Therefore, the dual Lorentz transformation for the distance domain with inverse velocity is

*κx′* = *λκ**x*(1 − *α*), and

*t′*/*κ* = *λ*(*t*/*κ* − *α**x*).

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

II. The Distance Domain

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 length *z* and *z′* from the origins of *O* and *O′* respectively. According to the second postulate of the special theory of relativity the pace of light, *κ*, is the same in both frames, so for the time-point *Q*:

*z* = *κ**s*, and *z′* = *κ**s′*.

A. Lenticity (cf. inverse velocity)

Define the *lenticity* *w* as the distance rate of dischronment *w *= *dz*/*ds*. Consider the dual standard Galilean transformation with *z* = *κ**s* and a factor *λ*, which is to be determined, which may depend on *α* = *w*/*κ*:

*z′* = *λ*(*z* − *ws*) = *λ*(*z* − *α**z*) = *λz*(1 − *α*).

The inverse transformation is the same except that the primes are interchanged and the sign of α is reversed:

*z* = *λ*(*z′* + *ws′*) = *λ*(*z′* + *α**z′*) = *λz′*(1 + *α*).

Multiply these two equations to get

*zz′* = *λ*²*zz′*(1 − *α*²).

Divide out *zz′* to get

*λ*² = 1/(1 − *α*²), or

*λ* = 1/√(1 − *α*²), cf. *γ* above.

Therefore, the dual Lorentz transformation for the distance domain is

*z′*/*κ* = *λ*(*z*/*κ* − *α**s*) and

*κs′* = *λ*(*κ**s* − *α**z*).

B. Inverse lenticity (cf. velocity)

Define the *inverse lenticity* as *u* = *w*^{−1} = (*dz*/*ds*)^{−1}. Consider the transposed dual standard Galilean transformation with *c* = *κ*^{−1}, *z* = *κ**s* and a factor *γ*, which is to be determined and may depend on *β* = *κ*/*w* = *u*/*c*:

*s′* = *γ*(*s* − *uz*) = *γ*(*s* − *βs*) = *γ**s*(1 − *β*).

The inverse transformation is the same except that the primes are interchanged and the sign of *β* is reversed:

*s′* = *γ*(*s* + *uz*) = *γ*(*s* + *βs*) = *γ**s*(1 + *β*).

Multiply these two equations to get

*ss′* = *γ*²*ss′*(1 − *β*²).

Divide out *ss′* to get

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

Therefore, the Lorentz transformation for the distance domain with inverse lenticity is

*s′* = *γ*(*s* − *uz*) and

*cz′* = *γ*(*cz* − *βs*).

*Revised 2021-06-24, 2024-04-23, 2024-07-20, and 2024-07-23.*