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I've changed my answer substantially after further thought.

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.

In this frame the length $L$ the light travels is given by

$L = \frac{z}{\gamma} - vt $

$L = \frac{z}{\gamma} - vL/c$

$L = \frac{z}{\gamma(1+\frac{v}{c})}$

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.

I've changed my answer substantially after further thought.

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 = \frac{z}{\gamma} - vt $

$L = \frac{z}{\gamma} - vL/c$

$L = \frac{z}{\gamma(1+\frac{v}{c})}$

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.

I've changed my answer substantially after further thought.

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.

In this frame the length $L$ the light travels is given by

$$ L = \frac{z}{\gamma} - vt

L = \frac{z}{\gamma} - vL/c

L = \frac{z}{\gamma(1+\beta)} $$

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.

I've changed my answer substantially after further thought.

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.

In this frame the length $L$ the light travels is given by

$L = \frac{z}{\gamma} - vt $

$L = \frac{z}{\gamma} - vL/c$

$L = \frac{z}{\gamma(1+\frac{v}{c})}$

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.

I've changed my answer substantially after further thought.

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 length $L$ the light travels is given by

$L = \frac{z}{\gamma} - vt $ $L = \frac{z}{\gamma} - vL/c $ $L = \frac{z}{\gamma(1+\beta)}$

I've changed my answer substantially after further thought.

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.

In this frame the length $L$ the light travels is given by

$$ L = \frac{z}{\gamma} - vt

L = \frac{z}{\gamma} - vL/c

L = \frac{z}{\gamma(1+\beta)} $$

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.