6. Derivation of the constant speed of light from the Lorentz transforms
Velocity of light in the x direction in the moving frame:
We consider a light pulse traveling in the x direction, crossing the point x1 then the point x2. The distance traveled is related to the time it takes by:
x2 − x1 = c (t2 − t1) .
We use this and the Lorentz transforms (4.8) to calculate the velocity of light in the primed or moving reference frame.
(6.1)
Equation (6.1) shows that in the x direction, the velocity of light is the same value in the moving frame as in the stationary frame, i.e. c' = c .
Velocity of light in the y or z direction in the moving frame:
We next derive a similar result in a direction perpendicular to the relative motion. We will use the y direction for convenience, but we could equally well use any perpendicular direction.
We start by noting that to get light to appear to go vertically in the moving frame, it must be angled in the stationary frame as illustrated in the sketch at the right. This is similar to rain in a windless day coming straight down, but appearing to be coming down at an angle directed towards you when viewed from a moving automobile or bicycle.
As noted on the sketch, we immediately have two relationships:
Δx = VΔt and diagonal = cΔt .
Using Pythagoras's theorem, we solve for the remaining side:
This can be solved for Δt: Δt = γb/c .
Now to calculate the perpendicular velocity of light in the moving (primed frame):
.
Note that we also used the fact that the y coordinate is unchanged during the Lorentz transformation, making Δy' = Δy = b .
Thus, c' = c . As it was in the x direction, the velocity of light in the y direction is not changed by relative motion.
Summary
The velocity of light is unchanged by Lorentz transformations in both the parallel and perpendicular directions. We will address this question again for the arbitrary direction case after we have derived transformations for velocity (in Chapter 9).