Simultaneity & Digital Clockfaces: Understanding Relativistic Time Syncing

[dface]

I am having a hard time understanding a scenario that I have not found discussed elsewhere. It has to do with simultaneity and information content.

There are two digital clocks on either end of a spaceship. Clock L on the left, Clock R on the right. They have digital displays, showing time numerically when activated.

There is a man in the middle of the spaceship, call him S.
There is a man, at rest at position x=0, call him O.
The spaceship is moving at uniform relativistic speed to the right.

Previously, S has synchronized clocks L and R using light pulses sent from his position in the middle of the spaceship, while moving at this same relativistic speed. I understand that O and S will not agree about the clocks being synchronized under these conditions, but let us imagine that S programmed clock L to indicate a later time than clock R when it is first synchronized (for example 0:00 on L compared to 0:01 on R). Let this time offset be appropriate so that to O, the clocks appear to be in sync. From that point onwards, S knows the clocks will be out of sync by this offset, but that doesn't bother him because he is aware of it.

At t=0, O and S are both at x=0. The spaceship is still moving to the right, at the same relativistic speed used at synchronization. S sends out two photons, one towards the left clock and one towards the right clock. When these photons hit the digital clocks, the clocks radiate the numbers from their displays at that moment. These numbers radiate to the eyes of O and S.

Will O and S see different numbers from each clock? It appears to me that O and S will observe different pairs of numbers, but how can this be if the radiation was created at a single event? Clock L radiates numbers once, and clock R radiates once. How can that radiation carry different "information content" to the different observers?

 

[dauto]

You're mistaken. the clock's will radiate different numbers because from the point of view of S one of the clocks is running late (on purpose) and they radiate simultaneously two different numbers, and from the point of view of O the clocks are synchronized (and running slow due to time dilation) and radiate two different numbers at different times because for O the two events are not simultaneous.

 

[dface]

Are you suggesting (completely arbitrary numbers, just for example):
S sees 0:04 from Clock L, and 0:03 from Clock R (since they were deliberately offset)
O sees 0:02 from Clock L, and 0:05 from Clock R (since the two radiation events occurred at a different time from his perspective)

If so, I am very confused as to how different numbers for each individual clock were radiated. The radiation "contains" a number, so how can L (or R) radiate two different numbers to two different observers?

 

[dauto]

Are you suggesting (completely arbitrary numbers, just for example):
S sees 0:04 from Clock L, and 0:03 from Clock R (since they were deliberately offset)
O sees 0:02 from Clock L, and 0:05 from Clock R (since the two radiation events occurred at a different time from his perspective)

If so, I am very confused as to how different numbers for each individual clock were radiated. The radiation "contains" a number, so how can L (or R) radiate two different numbers to two different observers?

It cannot

 

[dface]

Can you walk me through what happens to each photon and the subsequent number radiation? I'd really appreciate it, as I am having a hard time visualizing what numbers reach O and S.

 

[ghwellsjr]

I am having a hard time understanding a scenario that I have not found discussed elsewhere. It has to do with simultaneity and information content.

There are two digital clocks on either end of a spaceship. Clock L on the left, Clock R on the right. They have digital displays, showing time numerically when activated.

There is a man in the middle of the spaceship, call him S.
There is a man, at rest at position x=0, call him O.
The spaceship is moving at uniform relativistic speed to the right.

Previously, S has synchronized clocks L and R using light pulses sent from his position in the middle of the spaceship, while moving at this same relativistic speed. I understand that O and S will not agree about the clocks being synchronized under these conditions, but let us imagine that S programmed clock L to indicate a later time than clock R when it is first synchronized (for example 0:01 on L compared to 0:00 on R). Let this time offset be appropriate so that to O, the clocks appear to be in sync.

What does "appear to be in sync" mean? It's possible that at one instant in time, O will see both clocks reading the same value but later and before he won't see them at the same time. Since you cannot sync moving clocks, I don't see how this has any bearing on the rest of your scenario.

From that point onwards, S knows the clocks will be out of sync by this offset, but that doesn't bother him because he is aware of it.

At t=0, O and S are both at x=0. The spaceship is still moving to the right, at the same relativistic speed used at synchronization. S sends out two photons, one towards the left clock and one towards the right clock. When these photons hit the digital clocks, the clocks radiate the numbers from their displays at that moment. These numbers radiate to the eyes of O and R.

Will O and R see different numbers from each clock? It appears to me that O and R will observe different pairs of numbers, but how can this be if the radiation was created at a single event?

Did you mean S everywhere that I put an R in bold?

Clock L radiates numbers once, and clock R radiates once. How can that radiation carry different "information content" to the different observers?

Of course any observer will see the same number on L as any other observer and any observer will see the same number on R as any other observer.

 

[dface]

Did you mean S everywhere that I put an R in bold?

Yes, I corrected it above. Sorry for the confusion.

What does "appear to be in sync" mean?

By this I mean that S arranges things so that at t=0, if O could check the times on clocks L and R, they would agree for him. I understand that they will tick at a slower rate than any clock in Os frame. This is probably an unnecessary complication.

I think I understand this now. To summarize:

From the perspective of S, the radiation brings a number from clock L (L) and a number from clock R (R). L and R will differ (L<R), because he deliberately arranged it such that they are offset. No surprises for S.

From the perspective of O, if he could have glimpsed the clocks at t=0, he would have seen them both showing the same number (L=R) thanks to the arrangement made above. From his perspective, clock L is hit by a photon first. It radiates out a number (L). From his perspective, clock R is hit afterwards, it radiates out a number (R). He expects L<R. This is what he sees. In addition, the numbers L and R are the same for both O and S.

 

[ghwellsjr]

What does "appear to be in sync" mean?

By this I mean that S arranges things so that at t=0, if O could check the times on clocks L and R, they would agree for him. I understand that they will tick at a slower rate than any clock in Os frame. This is probably an unnecessary complication.

This really doesn't answer the question. How does O "check the times on clocks L and R"? Does he just look at them and see that they have the same time on them? Or does he do something more complicated? If S had previously synchronized L and R to his own clock, then because they are equidistant from him at all times, any time he looks at them, he will see the same time displayed on both of them and that time will always be offset by some constant time earlier than his own clock because they are always equidistant from him. But, assuming that S has set the clocks out of sync, O cannot do the same thing that S could do because he is only equidistant from them at only one instant in time. More about this later.

I think I understand this now. To summarize:

From the perspective of S, the radiation brings a number from clock L (L) and a number from clock R (R). L and R will differ (L<R), because he deliberately arranged it such that they are offset. No surprises for S.

Maybe you do understand but for the sake of others who might still be confused, I have made some spacetime diagrams to illustrate an example that generally follows your scenario, except that I have made the time difference be more than 1 second (just to make it easier to draw simple diagrams). First I show the rest frame of the spaceship with the clocks having been set out of sync. The dots show one-second intervals of time for all the observers and clocks and I start them all at their own times of 0 seconds. The spaceship is moving at 0.6c to the right with respect to the man at rest who is shown in green and labeled O (for Observer, I presume) but since this is the spaceship's rest frame, O is moving at 0.6c to the left:

The man in the middle of the spaceship is shown in blue and labeled S (for Spaceship, I presume). He previously set the Left clock, L, shown in black, to 3 seconds earlier than his own clock and he previously set the Right clock, R, shown in red to 3 seconds later than his own clock for a delta of 6 seconds. I have drawn in how the images of those two clocks left them at the Coordinate Time of t=0 (in the Spaceship's rest frame) and propagated to S at his time of 8 seconds at which point he sees the left clock displaying 0 and the right clock displaying 6 seconds for a delta of 6 seconds. At any other point in time, S will see the Left clock, L, displaying 8 seconds earlier than his own clock and the Right clock, R, displaying 2 seconds earlier than his own clock, always with a delta of 6 seconds.

From the perspective of O, if he could have glimpsed the clocks at t=0, he would have seen them both showing the same number (L=R) thanks to the arrangement made above.

We use the Lorentz Transformation process to get to the rest frame of O but I have drawn in different light paths corresponding to the Coordinate Time of t=0 when both the L and R clocks were reading 0 seconds. When you say that O glimpsed the clocks at t=0, I have to presume you mean that he sees the readings of 0 on both those clocks but he has to wait 4 seconds for those two images to reach him so he sees them at his time of 4 seconds:

But any other time that O looks at those two clocks, he will see them displaying two different times. For example, I think it is easy to see that at O's time of 6 seconds, he will see the Left clock at 4 seconds and the Right clock at 1 second. They don't track like they do for S.

From his perspective, clock L is hit by a photon first. It radiates out a number (L). From his perspective, clock R is hit afterwards, it radiates out a number (R). He expects L<R. This is what he sees.

Here is a more complicated diagram showing O's rest frame and two photons sent out by S at t=0. One hits the Left clock at its time of 2 seconds and the other one hits the Right clock at its time of 8 seconds. The images of these two displayed times propagate toward O and arrive at his time of 5 seconds from the Left clock (displaying 2) and at his time of 20 seconds for the Right clock (displaying 8):

In addition, the numbers L and R are the same for both O and S.

Yes, they are but S sees them arrive at the same time, his time of 10 seconds whereas O sees them arrive at significantly different times. If it weren't for the fact that the displays lock in the times that the clocks were at when the photons arrived, it would hardly be anything to comment about, don't you think?

Here is a diagram of the previous one transformed to the rest frame of the Spaceship:

As you can see, it doesn't matter which frame we use to depict a scenario, they all show exactly the same information about what is happening--the only difference being in the values of the coordinates.

Now I want to go back and address the issue that I think is important to you in this thread--the issue of syncing moving clocks. This is really bad terminology. It would be better to talk about the simultaneity of events on the moving clocks because that has a well-defined meaning and a well-defined process to establish such simultaneity. It doesn't happen by mere appearances. It requires the observer to send out radar signals and wait for their return echoes. He must also carefully keep a log of all transmissions and then correlate them after the fact to determine which previously observed events were simultaneous.

Here is a diagram depicting the process for the green Observer O to establish that the three events of the Spaceship's three clocks reaching 8 seconds were simultaneous according to O's rest frame:

It's important to realize that O is continually sending out radar signals because he has no way of knowing ahead of time which ones will be significant but I'm only showing three significant ones that he sent out at his time of 0, 4 and 8 seconds. They are significant because they happen to be the ones for which their averages with the times of the return echoes are the same, they all average to 10 seconds. This is an application of Einstein's second postulate that the light takes the same amount of time to get from an observer to a target as the return echo takes to get back to the observer. Of course, in this diagram, that's exactly what it looks like but this is no coincidence--that's how we created the coordinate system for the frame in the first place.

But, like I said before, it doesn't matter what frame we depict a scenario in--they all depict exactly the same information. Just to emphasize this point, I will transform from the rest frame of Observer O to the rest frame of the Spaceship:

Even though in this frame the outgoing radar signals do not take the same amount of time as their return echoes, Observer O can still establish the same events as being simultaneous according to his application of Einstein's second postulate or according to his own rest frame.

 

[Dale]

It has to do with simultaneity and information content.
...
How can that radiation carry different "information content" to the different observers?

It does not. Whatever information content is transmitted is frame invariant. The only frame variant thing is whether the two signals were transmitted simultaneously or not.

I think that there is a misunderstanding regarding what you are asking in some of the responses above. The information is invariant, the timing and location is frame variant.