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Let's assume there's a high speed spaceship moving right relative to us, and it's carrying two light clocks. One of them is set up so that the bouncing light pulse is oriented horizontally, and in the other one the pulse moves vertically, relative to us. Also, connected to both clocks there's a multicolored indicator lightbulb that flashes red every time the pulse in the horizontal light touches a wall, and flashes blue every time the pulse in the vertical light touches a wall.

Now let's consider the horizontal clock as observed from our non-ship reference frame. Since the ship is moving right, when the light pulse moves right we will see the clock's right wall running away from it. The contrary will be observed when the pulse goes left: the clocks' left wall will run to meet the light pulse. This will cause one of the two trips of the light beam to be shorter than the other, so the indicator lightbulb will flash red, followed by a long wait, then flash twice in quick succession, then a long wait again, and so on.

Now, the vertical clock won't be affected by the ship's sideways motion, so we will perceive the blue pulses in the indicator to occur evenly, thus not coinciding with the red ones.

Furthermore, the horizontal clock being length-contracted in our reference frame, the discrepancies become even greater.

Now, someone on the ship won't observe any of these effects that result from the ship's motion, and should therefore see both of the light pulse's trips as taking the same time, which would apparently cause the red and blue light flashes to happen in prefect sync.

How is this discrepancy resolved?

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Nice question.

The key is that 'at the same time' is dangerous language. It can be justified if two events occur at the same point in space, but otherwise requires careful handling.

Let's suppose the clocks and the light bulbs are in the bottom left corner of a square room on the spaceship. When the light pulses strike the left wall and the bottom wall, the red and blue lamps flash, and both observers (I.e. Us outside, and anyone moving with the ship) will agree.

Alternate blue flashes for the top wall will happen halfway between the bottom wall flashes, and by symmetry both observers will agree.

But alternate red flashes for the right wall will happen halfway between the left wall flashes, and for us outside the ship there is no symmetry as the time from left to right is not the same as the time from right to left.

So we will see the red and blue flashes as coinciding, but for us the red flashes will not be simultaneous with the pulses hitting the right wall. Even though they are for an observer in the ship. It's a nice illustration of the Relativity of Simultaneity.

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  • $\begingroup$ Thanks for the reply, but it seems like I didn't make my setup as clear as I'd hoped. The red and blue lights were supposed to be the very same multicolored lightbulb, to avoid discrepancies due to relativity of simultaneity. Fortunately, I managed a way to get rid of that bulb altogether. Let's set up the two light clocks so that they cross in the middle. For an observer on the ship then, the clocks being in perfect sync, their pulses will meet in the middle every time. The "stationary" observer, however, will not see the photons coincide at the center of the cross. $\endgroup$ – uKER Feb 13 at 18:26

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