# Lorentz transformation derivations

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′ = A² (1 + v²/c²) 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 − v²/c²) = γt,

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′ = γt (x – vt),

giving

ct′ = γt (ct – x(v/c)),

so that

t′ = γt (t – x(v/c²)),

which determines the transformation coefficients C and D as

C = γt,
D = −γtv/c².

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′ = A² (1 + w²/κ²) 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 − w²/κ²) = γs,

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 = κs and z′ = κs′, 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 = κs and z′ = κs′, by substituting term by term into the earlier obtained equation for the spatial coordinate

z′ = γs (z – ws),

giving

κs′ = γs (κs – z(w/κ)),

so that

s′ = γs (s – z(w/κ²)),

which determines the transformation coefficients C and D as

C = γs,
D = −γsw/κ².

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