Although there are many experimental methods available to measure the speed of light, the underlying principle behind all methods [is] the simple kinematic relationship between constant velocity, distance and time given below:
c = D / t (1)
In all forms of the experiment, the objective is to measure the time required for the light to travel a given distance. (Ref.)
From the perspective of the experimenter, light is an object whose speed is to be determined. Even though the distance traversed is fixed, it is placed in the numerator because this speed is to be compared with the speeds of other objects. For the same reason the quantity to be measured, time, is placed in the denominator.
But if we take the perspective of the experiment, of what is measured, then the fixed distance is the independent variable, which is placed in the denominator. The dependent variable is the time, which is placed in the numerator, so the pace of light is measured:
c´ = t / D (2)
Obviously, c and c´ are related:
c´ = 1 / c (3)
But (1) is written as if the quantity measured is the distance, D, whereas (2) is written as it occurs, which is that the time, t, is what is measured. This simple cognitive switch from distance to time has larger implications, as we shall see.
It was discovered that the speed of light in a vacuum is a constant for all observers. Because of this the speed of light can be an independent quantity with which to compare any motion. Let us then form the ratio of a velocity, v, with the speed of light:
β = v / c (4)
We may also form the ratio of a lenticity, u, with the pace of light:
β´ = u / c´ = c / v (5)
Next consider the transformation of variables from one observer to another. For the distance variable, which is treated as the dependent variable, we have:
x´ = x – vt (6)
From the experimental perspective, this corresponds to the measured dependent variable, the time:
t´ = t – ux (7)
Given that these transformations must work for light as well, we incorporate the following:
x = ct (8)
t = c´x = x/c (9)
Combining (8) with (6) and then (7) yields the following:
x´ = x – vt = x – vx/c = (1 – β) x (10)
and ct´ = ct – vt = (c – v) t
and so t´ = (1 – β) t (11)
also x´ = ct´ = ct – vx/c
and so t´ = t – (v/c²) x (12)
t´ = t – ux = t – uct = (1 – β´) t (13)
c´x´ = c´x – ux = (c´ – u) x
and so x´ = (1 – β´) x (14)
also t´ = c´x´ = c´x – uct
and so x´ = x – (c/v²) t (15)
Compare the earlier post Symmetric laws of physics. The Lorentz transformation includes the gamma factor with equations (6) and (12):
x´ = γ (x – vt) (16)
t´ = γ (t – (v/c²) x) (17)
Similarly, we may include a factor gamma prime with equations (7) and (15):
t´ = γ´ (t – ux) (18)
x´ = γ´ (x – (c/v²) t) (19)
If we replace these with β-equations (10) and (11), we have
x´ = γ (1 – β) x (20)
t´ = γ (1 – β) t (21)
As well as the β´-equations (13) and (14):
x´ = γ´ (1 – β´) x (22)
t´ = γ´ (1 – β´) t (23)
The derivation of gamma shows γ = (1 – β²)-1/2. Similarly, gamma prime is γ´ = (1 – β´²)-1/2 (see here).
To derive transformations that include both these factors, multiply equations (20) and (22) together, and equations (21) and (23) together:
x´² = γγ´ (1 – β) (1 – β´) x² (24)
t´² = γγ´ (1 – β) (1 – β´) t² (25)
Simplify and take the square root:
x´ = ((1 – β) (1 – β´) / ((1 + β) (1 + β´)))1/4 x (26)
t´ = ((1 – β) (1 – β´) / ((1 + β) (1 + β´)))1/4 t (27)
This is the fully symmetric relativity transformation.