The first derivation is similar to *here*.

Lorentz transformation for the *time domain*:

Let unprimed *x* and *t* be from inertial frame K and primed *x′* and *t′* be from inertial frame K′. Since space is assumed to be homogeneous, the transformation 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*

A body at rest in the K′ frame at position *x*′ = 0 moves with constant velocity *v* in the K frame. Hence the transformation must yield *x*′ = 0 if *x* = *vt*. Therefore, *B* = −*Av* and the first equation becomes

*x′ = A* (*x – vt*).

*Using the principle of relativity*

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame K′ to frame K should have the same form as the original but with the velocity in the opposite direction, i.e., replacing *v* with *−v*:

*x = A* (*x′* − (−*vt′*))* = A* (*x′ + vt′*).

*Determining the constants of the first equation*

Since the mean speed of light is the same in all frames of reference, for the case of a light signal, the transformation must guarantee that *t* = *x*/*c* when *t*′ = *x*′/*c*, with mean speed of light *c*. Substituting for *t* and *t*′ in the preceding equations gives:

*x′* = *A* (1 − *v/c*) *x*,

*x = A* (1 + *v*/*c*) *x′*.

Multiplying these two equations together gives,

*xx′ = A*² (1 + *v*²/*c*²) *xx′*.

At any time after *t* = *t*′ = 0, *xx*′ is not zero, so dividing both sides of the equation by *xx*′ results in

*A* = 1/√(1 − *v*²/*c*²) = *γ _{t}*,

which is the “Lorentz factor of the time domain”.

When the transformation equations are required to satisfy the light signal equations in the form *x* = *ct* and *x*′ = *ct*′, by substituting the *x* and *x′*-values, the same technique produces the same expression for the Lorentz factor.

*Determining the constants of the second equation*

The transformation equation for time can be easily obtained by considering the special case of a light signal, again satisfying *x* = *ct* and *x*′ = *ct*′, by substituting term by term into the earlier obtained equation for the spatial coordinate

*x′ = γ _{t}* (

*x – vt*),

giving

*ct′ = γ _{t}* (

*ct – x*(

*v*/

*c*)),

so that

*t′ = γ _{t}* (

*t – x*(

*v*/

*c*²)),

which determines the transformation coefficients *C* and *D* as

*C = γ _{t}*,

*D = −γ*/

_{t}v*c*².

So *C* and *D* are the unique constant coefficients necessary to preserve the constancy of the mean speed of light in the primed system of coordinates.

Lorentz transformation for the *distance domain*:

Let unprimed *z* and *s* be from inertial frame K and primed *z′* and *s′* be from inertial frame K′. Since space is assumed to be homogeneous, the transformation must be linear. The most general linear relationship is obtained with four constant coefficients, *A*, *B*, *C*, and *D*:

*z′ = Az + Bs*

*s′ = Cs + Dz*

A body at rest in the K′ frame at position *z*′ = 0 moves with constant lenticity *w* in the K frame. Hence the transformation must yield *z*′ = 0 if *z* = *ws*. Therefore, *B* = −*Aw* and the first equation becomes

*z′ = A* (*z – ws*).

*Using the principle of relativity*

According to the principle of relativity, there is no privileged Galilean frame of reference: therefore the inverse transformation for the position from frame K′ to frame K should have the same form as the original but with the velocity in the opposite direction, i.e., replacing *w* with *−w*:

*z = A* (*z′* − (−*ws′*))* = A* (*z′ + ws′*).

*Determining the constants of the first equation*

Since the mean pace of light is the same in all frames of reference, for the case of a light signal, the transformation must guarantee that *s* = *z*/*κ* when *s*′ = *z*′/*κ*, with mean pace of light *κ*. Substituting for *s* and *s*′ in the preceding equations gives:

*z′* = *A* (1 − *w/κ*) *z*,

*z = A* (1 + *w*/*κ*) *z′*.

Multiplying these two equations together gives,

*zz′ = A*² (1 + *w*²/*κ*²) *zz′*.

At any time after *s* = *s*′ = 0, *zz*′ is not zero, so dividing both sides of the equation by *zz*′ results in

*A* = 1/√(1 − *w*²/*κ*²) = *γ*_{s},

which is the “Lorentz factor of the distance domain”.

When the transformation equations are required to satisfy the light signal equations in the form *z* = *κ**s* and *z*′ = *κ**s*′, by substituting the *z* and *z′*-values, the same technique produces the same expression for the Lorentz factor.

*Determining the constants of the second equation*

The transformation equation for time can be easily obtained by considering the special case of a light signal, again satisfying *z* = *κ**s* and *z*′ = *κ**s*′, by substituting term by term into the earlier obtained equation for the spatial coordinate

*z′ = γ _{s}* (

*z – ws*),

giving

*κs′ = γ _{s}* (

*κ*

*s – z*(

*w*/

*κ*)),

so that

*s′ = γ _{s}* (

*s – z*(

*w*/

*κ*²)),

which determines the transformation coefficients *C* and *D* as

*C = γ _{s}*,

*D = −γ*/

_{s}w*κ*².

So *C* and *D* are the unique constant coefficients necessary to preserve the constancy of the mean pace of light in the primed system of coordinates.

*Revised 2024-05-31*