# Special relativity mirror clock experiment inconsistency

Say I set up a relativistic mirror clock experiment in which a spaceship contains a set of mirrors with a photon bouncing between them. Say the photon's motion is parallel to the direction of motion of the space ship, and the spaceship is moving at near to the speed of light, say 0.99c.

From the perspective of someone inside the ship, the photon bounces back and forth as normal, at a rate of one bounce per second or whatever it is. No problems there.

From the perspective of someone at some "stationary" point, the speed of the photon is c and the speed of the plates is 0.99c, so the photon must "race" to approach the front mirror and then when it bounces of the front mirror it should almost immediately hit the back mirror and start racing forward again.

So the observer on the ship sees regular periods but the observer outside the ship sees the first "half" of the second take much longer than the second "half". This seems to me like a paradox or a contradiction or something.

Where did my logic go wrong?

That's a very long ship.

Your logic didn't go wrong. The concept of simultaneity is broken with special relativity. Although the clock doesn't appear symmetric to the stationary observer, the ticks (a complete trip) remain consistent with the speed of light when you account for length contraction. And yes, to the stationary observer, it would seem like the trip going in the direction of the ship's velocity would take longer. That's because the light is essentially covering more distance. The trip back would shorter for the same reason.

The answer has to do with the relativity of simultaneity. Each observer can be imagined to measure the speed of anything, including a light ray, using clocks attached to different positions along their own ruler which have been synchronized in their own frame. So for example, if the ruler is 10 meters long, and an object passes the clock at the back end when that clock reads 6 seconds, and passes the clock at the front end when that clock reads 8 seconds, then the moving object was measured to travel 10 meters in 2 seconds, or 5 meter/second relative to that ruler/clock system. So, we can imagine the observer on board the ship has a ruler next to him which is at rest relative to the ship, and which stretches from one mirror to another, with clocks attached to each end next to the mirrors. We can likewise imagine an observer outside the ship has a ruler which is at rest relative to himself, and which is aligned right alongside the ship so that this observer can see which marking on his ruler is lined up with a mirror each time the light ray hits it, and what the reading on the clock sitting on that marking was at that moment. In this way, each observer can use their own ruler and synchronized clocks to measure the speed of the same light ray bouncing back and forth between the mirrors.

Each observer synchronizes their own clocks using the Einstein synchronization convention, which is based on the assumption that light should move at the same speed in both directions relative to themselves; for example, for any given pair of clocks in an observer's system, that observer could set off a flash of light at the midpoint between the two clocks, and set both clocks to read the same time at the moment the light from the flash reached them. This will necessarily have the consequence that each observer defines the other observer's clocks to be out-of-sync. For example, if the observer on the ship sets off the flash at the midpoint between two clocks on the ship, the other observer outside will see the clock closer to the front of the ship moving away from the marking on his ruler that was next to the flash when it happened, while the clock closer to the back is moving towards that marking, so he must say that in his own frame the light reached the back clock before the front one. But the observer on the ship set both clocks to read the same time when the light hit them, so from the perspective of the observer outside, the clock at the back will have a time that's ahead of the clock at the front.

To better understand the details of how both observers measure the speed of light to be c in both directions, it's helpful to consider not only the relativity of simultaneity but also time dilation (which causes each observer to measure the other observer's clocks running slow) and length contraction (which causes each observer to measure the other observer's ruler to be shortened). Again, suppose the observer on the ship has a ruler that extends from one mirror to the other, with clocks attached to each end. But now consider how the rulers and clocks of this observer will look in my frame, if I see the observer to be moving at some velocity v along my x-axis (with my ruler parallel to the x-axis). From my perspective, the ruler which the moving observer used to measure the distance is shrunk by a factor of $\sqrt{1 - v^2/c^2}$ due to length contraction, the time between ticks on the clocks of the moving observer expands by $1 / \sqrt{1 - v^2/c^2}$ due to time dilation (or equivalently, in $T$ seconds of time in my frame I only see the moving observer's clock tick forward by $T \sqrt{1 - v^2/c^2}$), and the rear clock's time-reading is ahead of the front clock's reading by $vL/c^2$ due to the relativity of simultaneity, where $L$ is the distance between the clocks in the observer's own frame, as measured by their own ruler.

To flesh this out, let's look at a numerical example. Say that the ruler is 50 light-seconds long in its own rest frame, moving at 0.6c in my frame. In this case the relativistic gamma-factor $1 / \sqrt{1 - v^2/c^2}$ (which determines the amount of length contraction and time dilation) is 1.25, so in my frame the ruler's length is 50/1.25 = 40 light seconds long. At the front and back of the ruler are clocks which are synchronized in the ruler's rest frame; because of the relativity of simultaneity, this means that in my frame they are out-of-sync, with the front clock's time being behind the back clock's time by $vL/c^2$ = (0.6c)(50 light-seconds)/$c^2$ = 30 seconds.

Now, when the back end of the moving ruler (which is next to the back mirror on the ship) is lined up with the 0-light-seconds mark of my own ruler, say there's a light flash at that position. Let's say at this moment the clock at the back of the moving ruler reads a time of 0 seconds, and since the clock at the front is always behind it by 30 seconds in my frame, then in my frame the clock at the front must read -30 seconds at that moment. 100 seconds later in my frame, the back end will have moved (100 seconds)*(0.6c) = 60 light-seconds along my ruler, and since the ruler is 40 light-seconds long in my frame, this means the front end will be lined up with the 100-light-seconds mark on my ruler. Since 100 seconds have passed, if the light beam is moving at c in my frame it must have moved 100 light-seconds in that time, so it will also be at the 100-light-seconds mark on my ruler, just having caught up with the front end of the moving ruler, where the front mirror is located.

Since 100 seconds passed in my frame, this means 100/1.25 = 80 seconds have passed on the clocks at the front and back of the moving ruler. Since the clock at the back read 0 seconds when the flash was set off, it now reads 80 seconds; and since the clock at the front read -30 seconds, it now reads 50 seconds. And remember, the ruler was 50 light-seconds long in its own rest frame! So in its frame, where the clock at the front is synchronized with the clock at the back, the light flash was set off at the back when the clock there read 0 seconds, and the light beam passed the clock at the front when its time read 50 seconds, so since the ruler is 50-light-seconds long, the beam must have been moving at 50 light-seconds/50 seconds = c as well! So you can see that everything works out--if I measure distances and times with rulers and clocks at rest in my frame, I conclude the light beam moved at 1 c, and if a moving observer measures distance and times with rulers and clocks at rest in his frame, he also concludes the same light beam moved at 1 c.

After reaching the front end of the moving ruler at 100 seconds in my frame, the light then bounces off the front mirror, then immediately begins traveling in the opposite direction towards the back end. Then at 125 seconds in my frame the light will be at a position of 75 light-seconds on my ruler, and the back end of the moving ruler, which is next to the back mirror, will be at that position as well. Since 125 seconds have passed in my frame, 125/1.25 = 100 seconds will have passed on the clock at the back of the moving ruler. Now remember that on the clock at the front read 50 seconds when the light reached it, and the ruler is 50 light-seconds long in its own rest frame, so an observer on the moving ruler will have measured the light to take an additional 50 seconds to travel the 50 light-seconds from front end to back end. So again, both me and the observer on the ship have measured the light to travel from front mirror to back mirror at a speed of 1 light-second per second.