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′ = 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) = 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/c²)),
which determines the transformation coefficients C and D as
C = γ,
D = −γv/c².
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′ = A² (1 + k²/w²) 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 − k²/w²) = 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 = −γk²/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.