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Old Mar 18th 2016, 04:47 PM   #1
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special relativity and a reflection paradox

The law of reflection as stated on farside is:
The law of reflection states that the incident ray, the reflected ray, and the normal to the surface of the mirror all lie in the same plane. Furthermore, the angle of reflection is equal to the angle of incidence .
Now, before getting to my question, suppose I use a non relativistic example first. If I have a spherical mirror and have a light source "dead centre" of the mirror then any light rays hit the mirror at perpendicular angle to the surface of the mirror. This is true irrespective of which the angle to the horizontal the light ray leaves its source. It is true whether it is horizontal, vertical or at a diagonal. (I'm hoping this is understandable without visuals. I"m using a faulty laptop so its difficult to draw them out.) Anyway, my point here is that every ray reflects off the spherical mirror and comes back on its previous path and passes back through the centre again.

Now for the relativistic case: If the spherical mirror is travelling past our "stationary" observer at speed close to that of light (perhaps 0.9c if that will work). Then, to our observer, it seems no longer to be a sphere but an ellipsoid. Now if a light source at the mirror's centre emits a ray of light at any angle other than horizontal or vertical then it will encounter a surface that is no longer perpendicular to itself but at a non right angle and therefore that ray will not reflect and return to the centre.

Oh No! Doesn't special relativity assume all the laws of the universe must be the same for every observer. However, an additional observer stationed inside our mirror would see a very different reflected path. How can this be?
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Old Mar 23rd 2016, 05:32 PM   #2
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I don't see any issue at all. To the stationary observer (let's call him 'A') the mirror is an ellipsoid, so the light he shines at the mirror is not all reflected back to him, as you noted. But if an observer is traveling with the mirror (person 'B') then to him it's spherical, and any light he shines is all reflected back to him.

Let's consider this from the point of view of the observer B who travels with the sphere. To him the sphere is indeed a sphere, he gets the reflections you suggest for a sphere. Then he notices person A passing by at 0.9c, and that when A is at the center of the sphere he shines his light, which reflects off the inside of the sphere and bounces back to the center - but by then A has moved past the center, so most of the reflected light misses him.

No issues at all.
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Old Apr 5th 2016, 01:42 PM   #3
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Aha! I found another wrinkle.

Ok, I can see you are right about the scenario I posed. There is no paradox because the sphere has moved when the light returns.

However I thought of a modification. This is actually a kind of reverse to what I started with. Let's say our traveller is inside an ellipsoidal mirror but the mirror is orientated so that its longest ellipsoidal axis is along the direction of travel. Now let's say our "stationary" observer sees it travel past at just the right speed so that the length contraction makes it appear, to him, to be a sphere. Then for the traveller shining a light ray, from the centre he would expect that light ray to not return to the centre (unless the light ray is perfectly horizontal or vertical as it is an ellipsoid) but to miss the centre by some margin. However, our stationary observer, who sees the ellipsoid now as a sphere, does not have that expectation!! How can this be??
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Old Apr 6th 2016, 07:02 AM   #4
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The solution to this "paradox" is the same as previously. To the stationary observer the mirror appears as a sphere, but because light source is moving, by the time the light reflects off the inside of the sphere the source has moved, and hence the reflected light does not hit the source.
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Old Apr 6th 2016, 01:41 PM   #5
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But doesn't that violate the law of reflection; the angle of incidence equals the angle of reflection, for the stationary observer? Or is that law null and void for moving mirrors?
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Old Apr 6th 2016, 02:04 PM   #6
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No, I don't think it violates that at all. The stationary observer sees the light reflect off the spherical mirror and go precisely back to where it started (angle of incidence = angle of reflection for the stationary observer). By then the light source has moved, and so most (not all) of the returning light misses the source. The exception is light that was projected straight ahead or behind, which does come back to the center, as does light that was projected at right angles to the direction of travel. For the moving observer the light reflects off the elipsoid mirror, and for him angle of incidence = angle of reflection and most of the light does not come back to him, except for the light that was originally projected either directly to the front, rear, or to the side.
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Old Apr 6th 2016, 02:23 PM   #7
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Ok, well let me explain my problem I am having with that.

If the angle of incidence = angle of reflection for a stationary observer seeing a spherical mirror in motion. No matter which direction the ray travels in it is making a trajectory towards what appears to be a sphere. It hits the mirror and then must appear, at that instant to retrace its path on a bearing to where the centre of the sphere is at that instant. You argue that this reflected ray will miss the centre because the centre is moving. My problem is that according to my understanding light always travels in straight lines for any inertial frame. The only way it can now miss the centre is to veer off in a curve, yet that would imply some acceleration. Either the reflected ray is not a straight line or its angle of reflection does not retrace itself at the same perpindular angle. How can this be otherwise?
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Old Apr 6th 2016, 03:41 PM   #8
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Ok, I get it. The ray retraces its path but the path is not retracing itself for the traveller, only the stationary observer which implies interesting things about the uniqueness of a set of co-ordinates under a lorentz transformation. in one frame they are distinct and unique and in another they overlap. I kind of didn't see that coming.
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