The first derivation is similar to *here*.

Lorentz transformations for space with time

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 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 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*,

which is the “Lorentz factor”.

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* (*x – vt*),

giving

*ct′ = γ* (*ct – x*(*v*/*c*)),

so that

*t′ = γ* (*t – x*(*v*/*c*²)),

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

*C = γ*,

*D = −γv*/*c*².

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

Lorentz transformations for time with space

Let unprimed *x* and *t* be from the timeframe K and primed *x′* and *t′* be from the timeframe K′. Since time 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 lenticity *w* in the K frame. Hence the transformation must yield *x*′ = 0 if *x* = *t*/*w*. Therefore, *B* = −*A*/*w* and the first equation becomes

*x′ = A* (*x – t*/*w*).

*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 lenticity in the opposite direction, i.e., replacing *w* with *−w*:

*x = A* (*x′* − (−*t′*/*w*))* = A* (*x′ + t′*/*w*).

*Determining the constants of the first equation*

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

*x′* = *A* (1 − *k/w*) *x*,

*x = A* (1 + *k*/*w*) *x′*.

Multiplying these two equations together gives,

*xx′ = A*² (1 + *k*²/*w*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 − *k*²/*w*²) = *γ*,

which is the “Lorentz factor”.

When the transformation equations are required to satisfy the light signal equations in the form *x* = *t*/*k* and *x*′ = *ct*′/*k*, 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* = *t*/*k* and *x*′ = *t*′/*k*, by substituting term by term into the earlier obtained equation for the spatial coordinate

*x′ = γ* (*x – t*/*w*),

giving

*t′*/*k = γ* (*t/*k* – x*(*k*/*w*)),

so that

*t′ = γ* (*t – x*(*k²*/*w*)),

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

*C = γ*,

*D = −γ**k*²/*w*.

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