The distance between interference lines in the double-slit experiment is: w=z labmda /d Where: w: Distance between fringes, z: Distance from slits to screen, labmda: Wave length of light, d: Distance between slits

Sanai Ball 2022-10-09 Answered
The distance between interference lines in the double-slit experiment is:
  w = z λ / d
w: Distance between fringes
z: Distance from slits to screen.
λ: Wave length of light
d: Distance between slits.
Now, looking at the setup from a reference frame traveling in the direction of the distance 𝑧 between slit and screen and moving away from the slit in direction of the screen, the light gets redshifted and the distance 𝑧 gets Lorentz contracted:
  w = z γ γ ( 1 + v c ) λ / d = z ( 1 + v c ) λ / d
This only holds for the first fringe or so, that aren't too far from the center, because the red-shift formula changes farther out. However, it's obvious that the distance between fringes has increased!
Where is my mistake?
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Answers (1)

Johnathon Mcmillan
Answered 2022-10-10 Author has 7 answers
In the frame at rest with respect to the slits, interference is determined by how many wavelengths fit between one slit and the screen compared to how many wavelengths fit between the other slit and the screen. In the frame in motion with respect to the screen, three things are different:
1. The wavelength is altered by the Doppler effect
2. The screen-slit distance is altered by a Lorentz contraction, and
3. The observer claims the light hitting the screen doesn't come from where the slits are at that instant but from where they were located a short time ago. The path length for the light is different than the instantaneous separation between the slit and screen.
In this frame the length 𝐿 that the light travels is given by
L = z γ v t
L = z γ v t
L = z γ ( 1 + v c )
This is the same factor by which the Doppler effect alters the wavelength, so observers in both reference frames agree on the number of wavelengths that lie along the path that the light travelled.
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