# Modification of Law of Reflection from a (moving) mirror in Special Relativity

• maverick280857
In summary, the conversation discussed the relations between the angle of reflection and angle of incidence in case of reflection from a moving plane mirror. The procedure used to solve the problem was outlined, and it was mentioned that Snell's Law may still hold in the laboratory frame for low velocities. However, a short article suggested that there may have been a mistake in the procedure. The conversation also touched on the possibility of different behaviors for different polarizations of light and the equivalence of reflected and emitted light. There was also a suggestion to include a case with a proper 30 degree angle for the mirror.
maverick280857
Hi everyone

A few weeks ago, I had worked out the relations between the angle of reflection and angle of incidence in case of reflection from a plane mirror,

(1) moving toward the incident ray
(2) moving normal to the incident ray

(PS -- This is not homework.)

The way I did it was (for the first case, as an example)

1. Assume the angle of incidence is $\theta$ in the lab frame.
2. The velocity of the mirror is

$$\mathbf{v}_{m} = -v_{m}\sin\theta\hat{x} + v_{m}\cos\theta\hat{y}$$

3. The velocity of the light ray in the lab frame is,

$$\mathbf{v}_{l} = c\sin\theta\hat{x} - c\cos\theta\hat{y}$$

4. Find the velocity of the incident light ray in the mirror frame,

$$\mathbf{v}_{i;m} = \frac{\mathbf{v}_l + (\gamma-1)\frac{\mathbf{v_l}\cdot\mathbf{V_m}}{V_m^2}\mathbf{V}_m -\gamma\mathbf{V}_m}{\gamma\left(1-\frac{\mathbf{v_l}\cdot\mathbf{V_m}}{c^2}\right)} = c\cos i \hat{x} - c\sin i \hat{y}$$

where $i$ is the angle of incidence in the frame of the mirror.

5. Use the law of reflection in the frame of the mirror, to write the velocity of the reflected ray as

$$\mathbf{v}_{r;m} = c\cos i \hat{x} + c\sin i \hat{y}$$

6. Transform this velocity back to the lab frame, using the inverse of formula 4 (with i --> r, for the reflected ray of course).

7. From this transformed velocity, find the angle of reflection in the lab frame.

The original question was to find the modification to Snell's Laws for a moving mirror. I thought the above procedure is a correct way of doing it. I was able to show that for low velocities, Snell's law does also hold in the laboratory frame (we have assumed in step 5 that it always holds in the frame of the mirror -- which seems reasonable to me).

But I came across this short article: http://home.c2i.net/pb_andersen/pdf/aberration.pdf , and it seems to suggest that I made some mistake.

What is going wrong? Is the procedure I outlined above correct?

Does Snell's Law hold even in the laboratory frame for a moving mirror?

Would appreciate inputs/insights.

Vivek

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You can check your answer by working out the problem normally in the rest frame of the mirror, then Lorentz transforming to the lab frame.

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maverick280857 said:
(1) moving toward the incident ray

maverick280857 said:
(2) moving normal to the incident ray

I think it is generally accepted that in this special case, the reflected light ray will return along the same path as the incident light ray for any relative velocity of the mirror.

Also, in the frame of the light source, a mirror with motion components that are parallel and normal to the incident light ray, can be treated as if it only had the parallel motion component as far as working out the angle of reflection is concerned.

If the plane mirror (dielectric of index of refraction n) is moving perpendicular to the incident light, and the reflection is observed at a velocity such that the light is incident at Brewster's angle, will light of one polarization be reflected and the other refracted? Suppose the mirror is stationary, and the observer is moving at a velocity such that the angle of incidence is Brewster's angle?
Bob S

yuiop said:

Also, in the frame of the light source, a mirror with motion components that are parallel and normal to the incident light ray, can be treated as if it only had the parallel motion component as far as working out the angle of reflection is concerned.

If refraction was considered, could it be that incident,reflected and refracted were not in plane, depending on the velocity of the mirror ?

yuiop said:

I think it is generally accepted that in this special case, the reflected light ray will return along the same path as the incident light ray for any relative velocity of the mirror.

Also, in the frame of the light source, a mirror with motion components that are parallel and normal to the incident light ray, can be treated as if it only had the parallel motion component as far as working out the angle of reflection is concerned.
It appears from the cases you considered that there is no essential difference between reflected light and light emitted at the same angle. Do you think this would be a general equivalence?
Obviously beaming and aberration are interchangable being essentially the same phenomenon, is it possible that reflection is also basically a consequence of the conservation of momentum wrt light and comp[letely equivalent??
This also seems to suggest that Doppler shift would apply equally to reflected and emitted light what do you think?

It would have been nice if you had included a case where the mirrow was actually inclined at a proper 30 deg angle not just a Thomas angle.
Thanks

## 1. How does the law of reflection change in special relativity when the mirror is moving?

In special relativity, the law of reflection states that the angle of incidence equals the angle of reflection. However, when the mirror is moving, the law of reflection still holds true but the angles are measured differently due to the effects of time dilation and length contraction.

## 2. What factors affect the modification of the law of reflection in special relativity?

The main factors that affect the modification of the law of reflection in special relativity are the speed of the mirror, the speed of light, and the relative velocity between the observer and the mirror. These factors impact the measurement of angles and the perceived position of the mirror.

## 3. How does the speed of the mirror affect the modification of the law of reflection in special relativity?

The speed of the mirror has a significant impact on the modification of the law of reflection in special relativity. As the mirror moves faster, the angle of reflection appears to be smaller due to length contraction, and the perceived position of the mirror is also shifted due to the effects of time dilation.

## 4. Can the modification of the law of reflection in special relativity be observed in real-life scenarios?

Yes, the modification of the law of reflection in special relativity has been observed in real-life scenarios, such as in experiments using high-speed mirrors and in observations of objects moving at high speeds. These observations have confirmed the predictions made by the theory of special relativity.

## 5. How does the modification of the law of reflection in special relativity impact our understanding of light and motion?

The modification of the law of reflection in special relativity highlights the interplay between light and motion, as well as the relativity of space and time. It also demonstrates the importance of considering the observer's frame of reference when analyzing the behavior of light in different scenarios. This concept is crucial in understanding the fundamental principles of special relativity and its applications in modern physics.

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