Light Clock Thought Experiment

[ynot]

In the classic Light Clock Thought Experiment the clock is shown standing perpendicular (at a right angle) to the plane of the clock’s travel. What however if the same clock was tilted forward at an angle (say 45 degrees) as shown in the image below?

The light “blip” (red line) reflected from the top mirror of position 1 strikes the bottom mirror of position 2 at a particular angle of incidence (AOI). The Law Of Reflection dictates that the blip is reflected from the mirror at an equal and opposite angle of relection (AOR) as indicated by the dotted line. For the blip to reach and be refected from the top mirror at position 3 however, it has to be refelected at a completely different angle (AO?).

Also the distance the blip travels from the top mirror of position 1 to the bottom mirror of position 2 is considerably shorter than the distance it travels from the bottom mirror of position 2 to the top mirror of position 3 (in the same time).

Seems this light clock would have to break both the Law of Reflection and the constant speed of c. How can this be so?

 

[BowlOfRed]

Also the distance the blip travels from the top mirror of position 1 to the bottom mirror of position 2 is considerably shorter than the distance it travels from the bottom mirror of position 2 to the top mirror of position 3 (in the same time).

In your frame, it doesn't take the same time. The shorter path takes less time in your frame. c holds, but the timing of the events is not the same in both frames.

AOI = AOR is actually only true when the mirror is at rest in your frame. Mirrors don't often move at a significant fraction of c, so this simplification is fine most of the time.

 

[ynot]

In your frame, it doesn't take the same time. The shorter path takes less time in your frame. c holds, but the timing of the events is not the same in both frames.

Blip from top mirror p1 to bottom mirror p2, and then from bottom mirror p2 to top mirror p3 are both in the same “moving” frame being observed from a “stationary” frame, but they represent two different degrees of time dilation.

I’ve added a second (green) blip that travells in the opposite direction and both blips reflect from opposite mirrors at the same time when the clock is being observed as being stationary. Are you saying that when the clock is observed as moving, the red blip would reach bottom mirror p2 before the green blip reaches top miror p2? In other words, although the whole clock is travelling at the same speed it’s simultaneously measuring two different degrees of time dilation?

AOI = AOR is actually only true when the mirror is at rest in your frame. Mirrors don't often move at a significant fraction of c, so this simplification is fine most of the time.

As I understand the properties of light the blip wouldn’t gain any momentum from a moving mirror so I don’t see how a mirror moving at any speed would change the AOR from the AOI. Light always strikes and reflects from a mirror at c regardless of mirror movement. Please explain how mirror movement would cause AOI to not = AOR (not using math).

ETA – I’ve just noticed that the distance from top mirror p1 to bottom mirror p1 (stationary clock) is greater than the distance from top mirror p1 to bottom mirror p2 (moving clock). Does that mean the red blip part of the clock has time increased (sped up) up while the green blip part has time dilated?

 

[BowlOfRed]

I’ve added a second (green) blip that travells in the opposite direction and both blips reflect from opposite mirrors at the same time when the clock is being observed as being stationary. Are you saying that when the clock is observed as moving, the red blip would reach bottom mirror p2 before the green blip reaches top miror p2?

Absolutely. Events that are simultaneous in one frame (like the light pulses reaching the end of the tube in the moving frame) are not necessarily simultaneous in another frame (the light pulses do not reach the end simultaneously in your frame).

As I understand the properties of light the blip wouldn’t gain any momentum from a moving mirror

Light has momentum and it can exchange momentum with other objects. This exchange is how a solar sail can work. The light can either lose or gain momentum from this exchange.

 

[Delvo]

In the classic Light Clock Thought Experiment the clock is shown standing perpendicular (at a right angle) to the plane of the clock’s travel... The Law Of Reflection dictates that the blip is reflected from the mirror at an equal and opposite angle of reflection (AOR)...

Even in the conventional diagram with the mirrors aligned with the direction of travel, the zigzag path already doesn't match the path that the light actually takes relative to the device with the mirrors. That stays perpendicular at all times no matter how the whole device is oriented. The zigzag path describes the light's movement from an outside perspective.

To accurately predict the zigzag shape, you would need to apply a different rule of reflection, which converts from one reference frame to the other by making all paths lean "forward" in some way from where they would be from the mirrors' reference frame, which means it would no longer make any sense from within the mirrors' reference frame. (And that new modified rule would need to allow for the possibility that either the zigs or the zags can actually point at an angle back toward the back of the vehicle, not just always forward; to see why, just picture the clock laying down so the only possible light paths are exactly entirely forward & backward. It still bounces, perpendicular to the mirrors as seen from the inside, but from the outside, it follows a single straight path, sometimes retracing parts of it by alternating between longer forward segments and shorter backward segments.)

I have not seen any proposed version of this concept with mirrors angled in such a way as to cause light reflections at any angle other than perpendicular, as perceived from within the vehicle/clock. It could be worked out, but it would be a needless, even counterproductive, complication for an idea that's meant to make things as simple & clear as they can be.

Also the distance the blip travels from the top mirror of position 1 to the bottom mirror of position 2 is considerably shorter than the distance it travels from the bottom mirror of position 2 to the top mirror of position 3 (in the same time).

Yes, that's a standard result of this kind of setup when the directions of the light's path and the device's movement are anything but perpendicular to each other. It's usually described in a more pure form in which the two vectors are exactly the same direction, like putting the mirrors on the floor and ceiling of a hypothetical super-elevator or a rocket while it's launching, or just a spaceship where the mirrors are at the front & back walls of a room instead of the floor & ceiling... but any other angle between the parallel case and the perpendicular case can be adjusted for by just treating the parallel and perpendicular aspects separately and combining the resulting vectors. The resulting diagram would feature a ziazag in which, if the zigs are long & forward, the zags are short & backward.

 

[alexi_drago]

I might be wrong but one thing that strikes me about your diagram is that the distance the blip travels is different in each direction so the time taken to travel in each direction is also different but the mirrors position changes consistently.
On the blips downward travel the mirrors won't have travelled as far as they do on the blips upward travel, the mirrors at positions 2 and 4 should be shifted to the left.

 

[ynot]

Thanks for the responses. I'm quite ill at present and aren't in a condition to reply.

 

[Ziggurat]

Even in the conventional diagram with the mirrors aligned with the direction of travel, the zigzag path already doesn't match the path that the light actually takes relative to the device with the mirrors. That stays perpendicular at all times no matter how the whole device is oriented. The zigzag path describes the light's movement from an outside perspective.

Actually, it doesn't.

There are two errors at play here in regards to the depiction from the outside perspective. The first is that length contraction of the device means that from the outside perspective, the corners of the device will no longer be right angles. It will be skewed. The second is that because the path lengths for the light are different for the "up" and "down" paths, the time for each segment will also be different. Which means that each successive snapshot of the position won't be equally spaced either.

Now, none of those changes will actually resolve the problem ynot has discovered: namely, that a moving mirror need not reflect like at the same angle a stationary mirror will reflect it at. But the problem can be resolved.

To accurately predict the zigzag shape, you would need to apply a different rule of reflection, which converts from one reference frame to the other by making all paths lean "forward" in some way from where they would be from the mirrors' reference frame

So this is one way to do it: start with normal reflection in the object's frame, and then just transform everything, including the light's path, into the frame where the object is moving.

But it's not the only way. You can do it starting in the moving frame too, by using Huygen's Principle plus Doppler shift. For a stationary mirror, Huygen's principle leads to the conclusion that an incident wave at some angle will be reflected at the same angle. But if the mirror is moving, we need to make two adjustments. First, the mirror will not be at the same place when different parts of the wave hit it. Second, the reflected wave will be at a different wavelength than the incident wave. So we can still use Huygen's principle to calculate the angle, but the details get messier, and the reflected angle may be much different than one might naively expect.

 

[RussDill]

BTW, if you are confused by the fact that the zig and zag take different times, turn the experiment all the way on it's side:

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In this case, light traveling against the relative direction of motion will reach the other mirror very quickly, but light traveling with the relative direction of motion may take an arbitrarily long amount of time depending on how close to the speed of light the relative motion is.