Doppler shift of a signal reflected in a mirror moving away from the observer.

In summary, the original question was about whether the relativistic Doppler shift of a reflection from a mirror moving away from the observer was the same as the Newtonian equation, specifically when the mirror is orthogonal to the direction of motion. After some calculations and simplification, it was determined that the equations are indeed the same. This result was inspired by a discussion on the role of time dilation and red shift in a scenario where a mirror is moving parallel to the observer. The conclusion is that they are not factors in this specific scenario, and length contraction and the Lorentz transform may play a role in explaining the result.
  • #1
yuiop
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I have re-written this as as I accidently deleted my original post. I was wondering if the relativistic Doppler shift of a reflection from a mirror moving away from the observer was the same as the Newtonian equation in the special case that the mirror is orthogonal to the direction of motion.

I referred to equation (13) in this paper http://arxiv.org/ftp/physics/papers/0409/0409014.pdf and set theta to zero for this special case.

I now think I have figured out the answer to my own question.

The equation I gave in my my first post:

[tex] f = f_0 \frac{1-2v/c+v^2/c^2}{1-v^2/c^2} [/tex]

Can be re-arranged:

[tex] f = f_0 \frac{(1-v/c)(1-v/c)}{(1-v/c)(1+v/c)} [/tex]

and simplified:

[tex] f = f_0 \frac{(1-v/c)}{(1+v/c)} [/tex]

and this is exactly the same as the none relativistic equation for Doppler radar.

Length contraction and time dilation is not involved in this special case of reflection in a moving mirror.
 
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  • #2
Yes, this is kind of a trivial result though. You can just move to the coordinates of the mirror and argue that because of symmetries (light beam coming in 90 degrees wrt the mirror), outgoing angle must be same as incoming angle.
 
  • #3
clamtrox said:
Yes, this is kind of a trivial result though. You can just move to the coordinates of the mirror and argue that because of symmetries (light beam coming in 90 degrees wrt the mirror), outgoing angle must be same as incoming angle.

The question was not about angles but about red shift of a reflected signal. I chose the 90 degree angle wrt the mirror to eliminate complications due to relativistic aberration. I was inspired to ask the question because in another thread there was originally some doubt about whether time dilation or red shift are factors in the signal reflected from a mirror moving parallel to the observer. I am now fairly sure that they are not factors in that scenario.
 
  • #4
yuiop said:
The question was not about angles but about red shift of a reflected signal. I chose the 90 degree angle wrt the mirror to eliminate complications due to relativistic aberration. I was inspired to ask the question because in another thread there was originally some doubt about whether time dilation or red shift are factors in the signal reflected from a mirror moving parallel to the observer. I am now fairly sure that they are not factors in that scenario.

So more explicitly (if you want to think in terms of length contractions), you get a relativistic correction because the Lorentz transform from mirror coordinates to observer coordinates changes the angle of the mirror. If the mirror is perpendicular to the beam, you can just trivially transform to mirror coordinates and you have a moving observer observing a beam (bouncing off a stationary mirror, but you know what happens there already).
 
  • #5
clamtrox said:
So more explicitly (if you want to think in terms of length contractions), you get a relativistic correction because the Lorentz transform from mirror coordinates to observer coordinates changes the angle of the mirror. If the mirror is perpendicular to the beam, you can just trivially transform to mirror coordinates and you have a moving observer observing a beam (bouncing off a stationary mirror, but you know what happens there already).

Thanks, yes it helpful to think of it in that context. :smile:
 

FAQ: Doppler shift of a signal reflected in a mirror moving away from the observer.

1. How does the Doppler shift occur in a signal reflected in a mirror?

The Doppler shift occurs when there is relative motion between the source of the signal and the observer. In this case, the mirror is moving away from the observer, causing the reflected signal to have a lower frequency or longer wavelength.

2. Why does the Doppler shift occur in a signal reflected in a mirror?

The Doppler shift occurs due to the change in distance between the source of the signal (in this case, the mirror) and the observer. As the mirror moves away from the observer, the distance between them increases, causing the wavelength of the signal to appear longer.

3. Is the Doppler shift always noticeable in a reflected signal from a moving mirror?

No, the Doppler shift may not always be noticeable depending on the speed of the moving mirror and the frequency of the signal. If the mirror is moving at a very slow speed or the frequency of the signal is very low, the Doppler shift may not be detectable.

4. How does the speed of the mirror affect the amount of Doppler shift in the reflected signal?

The speed of the mirror directly affects the amount of Doppler shift in the reflected signal. The faster the mirror moves away from the observer, the greater the change in frequency and wavelength of the signal will be.

5. Can the Doppler shift in a reflected signal from a moving mirror be used for any practical applications?

Yes, the Doppler shift in a reflected signal from a moving mirror can be used for various practical applications such as determining the speed and direction of a moving object, measuring the flow of fluids, and studying the movement of celestial bodies.

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