Satellite Orbit synchronization

[name123]

Consider a large non-rotating sphere, and one series of satellites, series A, in free fall orbit at the same velocity and altitude slightly above its "equator" going from east to west. Forming a ring in which all the satellites are equal distance apart, and another series of satellites, series B, forming a similar ring but in the opposite direction. And that in space to both the North and South of the sphere were two space ships at rest with respect to the sphere, at a distance far enough away from the sphere that all the satellites could observe one or the other at all times. These two spaceships have synchronised clocks which flash a bright light every second.

My question is whether it is possible that all the clocks on the series A satellites could be synchronised such that they could agree upon when the flashes from the two space ships happened?

The reason I am asking is that if they could then there seems to me to be a contradiction when certain assumptions are made. The clocks in series A satellites could be in synch with each other, as could the clocks in the series B satellites. Yet each time a satellite from series A passes a satellite from series B they could observe each other firing torches off a mirrored ceiling and conclude that the clock on the other satellite from the other series was running slower than its own, as light appeared to travel a longer distance during for the same clock time (on the other clock). However since the clocks in each series would be in synch, each time a satellite from series A passed one from series B it could be seen the no more time had expired for the clocks on either series. So how could the conclusion that the clocks on the other series be considered correct if the interval between each time a satellite from series A passed on from series B was constant and equal according to the clocks in each series? The conclusion based on the assumption would appear to be contradicted by the experimental results of comparing the amount of time that had expired.

Admittedly this is quite similar to a thread that was closed, but the question is different. <forum feedback removed>

 

[Dale]

My question is whether it is possible that all the clocks on the series A satellites could be synchronised such that they could agree upon when the flashes from the two space ships happened?

Be aware, this is a general relativity question, and so it can be very messy.

The answer is yes. You can simply have them all agree to use any given reference frame. Then they will all be synchronized and they will all agree on the timing of any flash. There is no need for them to use a reference frame where they are at rest or where their clocks keep coordinate time. This approach is taken in the GPS system.

The clocks in series A satellites could be in synch with each other, as could the clocks in the series B satellites.

If they all agree to use the same reference frame then they can all be in synch with each other at all times, both A and B. Again, they do not need to use a reference frame where they are at rest and their clocks keep coordinate time.

 

[name123]

If they all agree to use the same reference frame then they can all be in synch with each other at all times, both A and B. Again, they do not need to use a reference frame where they are at rest and their clocks keep coordinate time.

Thanks Dale. Just to be clear, I am not assuming the clocks involve any adjustments based on some calculation. Just standard wind-up clocks, or atomic clocks maybe. Is that still ok?

 

[Dale]

Thanks Dale. Just to be clear, I am not assuming the clocks involve any adjustments based on some calculation. Just standard wind-up clocks, or atomic clocks maybe. Is that still ok?

That is fine, but proper time alone does not define a reference frame.

I think that you need to explicitly write down the reference frame you have in mind. You should write it either in terms of the metric in that frame or in terms of a transformation from one of the standard reference frames on the Schwarzschild spacetime. (Or both)

 

[name123]

That is fine, but proper time alone does not define a reference frame.

I think that you need to explicitly write down the reference frame you have in mind. You should write it either in terms of the metric in that frame or in terms of a transformation from one of the standard reference frames on the Schwarzschild spacetime.

I am not really too concerned about reference frames. My concern is that (as I understand it) with certain assumptions it may be concluded by a satellite in series A when it watches a passing satellite in series B bounce torch light off its mirrored ceiling that because the light seemed to travel further for the same time interval that a light bounce takes in the A satellite, that proper time is passing slower on the B series satellites (it's clocks are running slower). As that conclusion would be proved to be incorrect each time an A series satellite passed a B series satellite and the clocks could be seen to still be in synch.

 

[Dale]

I am not really too concerned about reference frames.

You cannot discuss synchronization at all without a reference frame. (more precisely without a coordinate chart)

 

[name123]

You cannot discuss synchronization at all without a reference frame. (more precisely without a coordinate chart)

Well I was considering the clocks to be synchronised if the observers in the satellites agree with what time their clocks were showing when the spaceships flashed their lights. Perhaps they could have some digital clocks, and sensors to detect the light and the results be written out to a log. Even if that is not in line with an official definition it seems sufficient for the issue I am considering. The satellites could also display their digital clocks on the sides so that their readings could be filmed as the satellites passed each other.

The issue was simply whether it would be wrong for a satellite in the A series to conclude that the clock on a satellite in the B series was running slower simply because the light (flashed from a torch) on the B series seemed to travel further per second of the B series clock.

 

[Dale]

Well I was considering the clocks to be synchronised if the observers in the satellites agree with what time their clocks were showing when the spaceships flashed their lights.

Yes, I understood that from your original post. That requires a reference frame.

The issue was simply whether it would be wrong for a satellite in the A series to conclude that the clock on a satellite in the B series was running slower simply because the light (flashed from a torch) on the B series seemed to travel further per second of the B series clock.

This comparison also requires a reference frame. (again, more precisely a coordinate chart).

A coordinate chart defines both what is meant by "when the spaceships flashed their lights" and "running slower". You simply cannot avoid it for this question. I am sorry, you clearly do not like that response, but I cannot give you another response for this question.

 

[name123]

Yes, I understood that from your original post. That requires a reference frame.

Why at the end can the logs not just be compared to see if they agree?

This comparison also requires a reference frame. (again, more precisely a coordinate chart).

A coordinate chart defines both what is meant by "when the spaceships flashed their lights" and "running slower". You simply cannot avoid it for this question. I am sorry, you clearly do not like that response, but I cannot give you another response for this question.

What I meant by when the spaceships flashed their lights is when those light signals were detected by each satellite, and that will be indicated by the clock reading written out to the logs.

What I meant by one clock running slower than another is that if the clocks (or clocks in synch with them) were to be compared at a later time the clock which was running slower would indicate that less time had passed. Like with gravity time dilation for example, where a clock under a stronger gravitational field will run slower than one at rest with respect to it, but under a weaker gravitational field.

What concern have you with those answers?

 

[jbriggs444]

Yet each time a satellite from series A passes a satellite from series B they could observe each other firing torches off a mirrored ceiling and conclude that the clock on the other satellite from the other series was running slower than its own, as light appeared to travel a longer distance during for the same clock time (on the other clock).

Given the unusual synchronization scheme, it will no longer be true that light travels at c according to the coordinates in use. It will have an obvious anisotropy, traveling more slowly "upstream" and more rapidly "downstream".

This means that one has to reason more carefully than a simplistic "light travels farther, therefore takes more time" to obtain a contradiction.

 

[jartsa]

@name123 If the satellites moved around using rockets, then we could safely say that acceleration of a satellite has some effect on that satellite's idea about ticking rates of distant clock.(Making the satellites move by their own power would not change the logic of the scenario)

 

[Dale]

Why at the end can the logs not just be compared to see if they agree?

Synchronization requires specification of a coordinate chart. Logs are irrelevant without it. You can have the same logs written indicating synchronization or not depending on your coordinate chart. Evidence, such as logs, does not specify simultaneity, the reference frame does. This isn't a question of technology or method, it is a question of definition.

This is getting exhausting. You can either specify a reference frame and get a specific answer to this question, or you can ask a different question.

 

[name123]

@name123 If the satellites moved around using rockets, then we could safely say that acceleration of a satellite has some effect on that satellite's idea about ticking rates of distant clock.(Making the satellites move by their own power would not change the logic of the scenario)

Yes, if you had a series C and series D satellites using rockets, then their clocks would be ticking slower than those of the A and B series and that could be observed, when comparing their spaceship flash detection logs, and when they pass each other.

 

[name123]

Synchronization requires specification of a coordinate chart. Logs are irrelevant without it. You can have the same logs written indicating synchronization or not depending on your coordinate chart. Evidence, such as logs, does not specify simultaneity, the reference frame does.

This is getting exhausting. You can either specify a reference frame and get a specific answer to this question, or you can ask a different question.

Perhaps you could suggest how you would approach the problem

 

[jbriggs444]

Yes, if you had a series C and series D satellites using rockets, then their clocks would be ticking slower than those of the A and B series and that could be observed, when comparing their spaceship flash detection logs, and when they pass each other.

Since we have not reached a meeting of the minds on a scenario with satellites in regular circular orbits, it seems premature to try to reason about powered rocket ships traveling willy nilly.

 

[Dale]

Perhaps you could suggest how you would approach the problem

I would specify the coordinate chart as a transformation from one of the standard charts on the Schwarzschild spacetime (as I said in post 4). Probably just the standard Schwarzschild coordinates. I would also specify the metric, but I would start with the transformation.

I would also explicitly express the comparisons I was interested in either in terms of the coordinates or in terms of invariants.

 

[name123]

I would specify the coordinate chart as a transformation from one of the standard charts on the Schwarzschild spacetime (as I said in post 4). Probably just the standard Schwarzschild coordinates. I would also specify the metric, but I would start with the transformation.

And when you did would the series A and series B clocks be considered to be synchronised? Or perhaps to put it a different way, for each event of an A series satellite passing a B series satellite would their clocks be predicted to be showing the same time?

 

[Dale]

And when you did would the series A and series B clocks be considered to be synchronised?

You could choose coordinates either way. If you want them to be synchronized then you can choose such coordinates, if you don't want them to be synchronized then you can choose other coordinates. The choice is up to you.

Btw, I will not work this explicitly for you. It is a messy problem. If you are that interested in it then you will need to go through the effort yourself.

 

[name123]

You could choose coordinates either way. If you want them to be synchronized then you can choose such coordinates, if you don't want them to be synchronized then you can choose other coordinates. The choice is up to you.

Btw, I will not work this explicitly for you. It is a messy problem. If you are that interested in it then you will need to go through the effort yourself.

So are you saying that there is a coordinate system that you could choose in which the event of an A series satellite passing a B series satellite would not be predicted as being an event in which their clocks showed the same time? Perhaps imagine that the A satellites have a hole in them which the B series satellites pass through, so with a sausage roll analogy the A satellite would be the pastry and the B satellite the sausage.

 

[jartsa]

Since we have not reached a meeting of the minds on a scenario with satellites in regular circular orbits, it seems premature to try to reason about powered rocket ships traveling willy nilly.

Well, the traveling twin in the twin paradox uses a rocket. If he used deflection by the gravity field of a neutron star instead, can you explain what happens in that case?

The explanation would probably be useful in this current scenario too.

 

[Dale]

So are you saying that there is a coordinate system that you could choose in which the event of an A series satellite passing a B series satellite would not be predicted as being an event in which their clocks showed the same time?

That is a different question. The answer to that is an invariant and does not depend on the coordinate system.

 

[Nugatory]

So are you saying that there is a coordinate system that you could choose in which the event of an A series satellite passing a B series satellite would not be predicted as being an event in which their clocks showed the same time? Perhaps imagine that the A satellites have a hole in them which the B series satellites pass through, so with a sausage roll analogy the A satellite would be the pastry and the B satellite the sausage.

No, we are not saying that.

Everyone always has to agree about what two clocks read when they are side by side at the same point in space. Depending on how they were initially set and how they got to that same point they may or may not read the same - but everyone will agree about the numbers that will appear on their face when they are side by side.

However, when the two clocks are not colocated, any comparison between them is necessarily frame-dependent and coordinate-dependent. I can choose a coordinate system in which the two clocks always read the same, or I can choose a coordinate system in which one clock is slower than the other, or faster, or whatever. This is because any comparison between two clocks that are not colocated is basically asking "what do the two clocks read at the same time?"; "at the same time" means "at the points on their worldlines that have the same $t$ coordinate"; and that obviously depends on how you choose your coordinate system.

 

[name123]

No, we are not saying that.

Everyone always has to agree about what two clocks read when they are side by side at the same point in space. Depending on how they were initially set and how they got to that same point they may or may not read the same - but everyone will agree about the numbers that will appear on their face when they are side by side.

However, when the two clocks are not colocated, any comparison between them is necessarily frame-dependent and coordinate-dependent. I can choose a coordinate system in which the two clocks always read the same, or I can choose a coordinate system in which one clock is slower than the other, or faster, or whatever. This is because any comparison between two clocks that are not colocated is basically asking "what do the two clocks read at the same time?"; "at the same time" means "at the points on their worldlines that have the same $t$ coordinate"; and that obviously depends on how you choose your coordinate system.

So can you choose a coordinate system where each time an A series satellite passes a B series satellite their clocks display the same time and all the A and B series clocks log the same time for the spaceship flashes, but in which the clocks in the A and B series are not synchronous (some are running faster or slower than the others)?

 

[Dale]

So can you choose a coordinate system where each time an A series satellite passes a B series satellite their clocks display the same time and all the A and B series clocks log the same time for the spaceship flashes, but in which the clocks in the A and B series are not synchronous (some are running faster or slower than the others)?

I don't know. You would have to actually work out the messy math.

 

[name123]

I don't know. You would have to actually work out the messy math.

When you say you do not know, do you mean that you do not know whether there is such a coordinate system, or do you mean you know that there is, but you do not know what it is?

 

[Nugatory]

So can you choose a coordinate system where each time an A series satellite passes a B series satellite their clocks display the same time and all the A and B series clocks log the same time for the spaceship flashes, but in which the clocks in the A and B series are not synchronous (some are running faster or slower than the others)?

Before I answer that question I'm going to have to ask you to be precise about what you mean by "some are running faster or slower than the others". That, is exactly what comparison do you make between the two clocks? The answer to your question is going to be "yes" no matter how you answer my question, but depending on how you answer that question, it may be easy or wildly insanely difficult to actually show you the coordinate system that does what you want.

In my next post I will show you one of the easy answers.

 

[Dale]

When you say you do not know, do you mean that you do not know whether there is such a coordinate system, or you only mean that you know there is, but you do not know what the coordinate system is?

I don't know if there is such a system.

 

[name123]

Before I answer that question I'm going to have to ask you to be precise about what you mean by "some are running faster or slower than the others". That, is exactly what comparison do you make between the two clocks? The answer to your question is going to be "yes" no matter how you answer my question, but depending on how you answer that question, it may be easy or wildly insanely difficult to actually show you the coordinate system that does what you want.

In my next post I will show you one of the easy answers.

I mean that if there were holes in the sphere around the equator and the satellites were after a certain amount of time to fly into them and go to the centre of the sphere and stop, and then their clocks were compared, that some clocks would not be showing the same time.

Actually ignore that, I assume that no coordinate system is going to have that happening. So an example where according to the coordinate system the clocks on the satellites are not synchronous but give the results described. I assume that would be the case for any ship passing the sphere at a reasonably fast velocity at the level of the equator should provide that, but I would be interested in the example. As I do not know how it handles the logging of the flashes.

 

[Dale]

after a certain amount of time to fly into them and go to the centre of the sphere and stop

This is exceptionally vague. I am curious to see what @Nugatory has in mind, but I don't know how he will proceed with this.

 

[Nugatory]

In my next post I will show you one of the easy answers.

As promised...
Let us equip each satellite with identical flashing lights. Their flash rate is very high, let's say a flash every nanosecond when at rest...
Satellite X will compare its clock against the clock on satellite Y by considering the arrival time (X's clock) at which two consecutive flashes arrive. For each flash, we will take the distance it traveled (this may be different for the two flashes because the two satellites are in motion relative to one another so their relative positions may have changed between flashes), divide the distance by $c$ to get the time in flight, subtract that from the arrival time to get the times the two flashes left satellite Y. If those times differ by less than one nanosecond on X's clock we conclude that Y's clock is faster than X's, if they differ by one nanosecond we know both clocks are ticking at the same rate, and if they differ by more than one nanosecond we know that Y's clock is slower than X's.

If this is how we're going to compare the clocks, you will find that there is an obvious drop-dead easy coodnate system does what you ask. Choose the inertial frame in which the central mass is at rest and all the satellites are rotating/counterrotating at the same speed. The clocks of the rotating and counter rotating satellites will always be ticking at different rates so are not synchronized, yet they change by the exact same amount on each orbit and receive the flashes from the spaceships at the same time. Note that all the satellite clocks will run uniformly slow compared with a clock at rest on the planet or the spaceships; only the planet and spaceship clocks will accurately track the value of the $t$ coordnate.

I mean that if there were holes in the sphere around the equator and the satellites were after a certain amount of time to fly into them and go to the centre of the sphere and stop, and that their clocks were compared, that some clocks would not be showing the same time.

That's not a comparison of the clocks while they're in orbit and not colocated, it's a comparison after you've removed them from orbit and brought them to the center of the planet. It doesn't tell us anything about whether they were or were not synchronized while in orbit. (For what is it worth, I do not believe that there is a more natural easy way of doing the comparisn than what I describe above)

 

[name123]

As promised...
Let us equip each satellite with identical flashing lights. Their flash rate is very high, let's say a flash every nanosecond when at rest...
Satellite X will compare its clock against the clock on satellite Y by considering the arrival time (X's clock) at which two consecutive flashes arrive. For each flash, we will take the distance it traveled (this may be different for the two flashes because the two satellites are in motion relative to one another so their relative positions may have changed between flashes), divide the distance by $c$ to get the time in flight, subtract that from the arrival time to get the times the two flashes left satellite Y. If those times differ by less than one nanosecond on X's clock we conclude that Y's clock is faster than X's, if they differ by one nanosecond we know both clocks are ticking at the same rate, and if they differ by more than one nanosecond we know that Y's clock is slower than X's.

If this is how we're going to compare the clocks, you will find that there is an obvious drop-dead easy coodnate system does what you ask. Choose the inertial frame in which the central mass is at rest and all the satellites are rotating/counterrotating at the same speed. The clocks of the rotating and counter rotating satellites will always be ticking at different rates so are not synchronized, yet they change by the exact same amount on each orbit and receive the flashes from the spaceships at the same time. Note that all the satellite clocks will run uniformly slow compared with a clock at rest on the planet or the spaceships; only the planet and spaceship clocks will accurately track the value of the $t$ coordnate.

So let's imagine that you use the centre of the sphere as the reference frame. How do you explain the satellites logging the flashes of the spaceships at the same time? Why would the light from the spaceship take longer to reach one satellite than another?

Just to be clear what I mean is that if in your chosen frame of reference each satellite receives each light flash from the spaceships at the same time, and at that time (according to your calculations for that frame of reference) some of their clocks were going at different rates to other ones, then how comes when they come to log the time they detected the flash of light they all report the same time on their clocks. It seems like you would have predicted them to have reported different times (some clocks would have been running faster or slower than others), and be shown to be wrong by experimental results (the log readings). I realize I may have misunderstood you, but I just wanted to make clear my concern.

 

[Dale]

So let's imagine that you use the centre of the sphere as the reference frame.

That is a location, not a reference frame. A reference frame specifies three spatial and one time coordinate at each event in the spacetime.

 

[russ_watters]

I don't want to derail the great job others are doing, but want to pop in and point something out real quick:

So can you choose a coordinate system where each time an A series satellite passes a B series satellite their clocks display the same time and all the A and B series clocks log the same time for the spaceship flashes...

As stated before, this sounds a lot like a somewhat more complicated than necessary description of how GPS timekeeping works. GPS satellite clocks are set up with the transformation built-in so that every time they fly over the ground station, they find themselves to be roughly in sync with its clock.

They could have been set up without the transformation, but anyone with the proper information would be able to figure it out after-the-fact (38ns a day gained, times the number of days in orbit). This choice of how to do the timekeeping is somewhat arbitrary, and as long as it is done correctly, it will produce a usable system.

...but in which the clocks in the A and B series are not synchronous (some are running faster or slower than the others)?

...but this still seems like the same contradiction you've repeated perhaps a dozen times already, saying clocks that read the same time every time they pass are somehow not synchronized. I'll let someone else sort that out...

 

[Ibix]

So let's imagine that you use the centre of the sphere as the reference frame.

You can use the inertial frame in which the centre of the sphere is at rest as a reference frame. The centre of the sphere is not, in itself, a reference frame.

How do you explain the satellites logging the flashes of the spaceships at the same time?

Because, in this frame, all the satellites are always the same distance from the ships. And their clocks were started at the same time in this reference frame and the satellites are all moving at the same speed in this reference frame. There's literally nothing to choose between the satellites so they must behave the same.

Why would the light from the spaceship take longer to reach one satellite than another?

In this frame, they wouldn't.

It is trivial to construct a coordinate system in which the satellites do have different tick rates - just pick any system (or almost any other system? Can't think of a counter example) other than this highly symmetric one.

It seems like you would have predicted them to have reported different times (some clocks would have been running faster or slower than others), and be shown to be wrong by experimental results (the log readings).

Barring calculation error, every frame will agree that any pair of satellite clocks read the same when they pass (assuming you set the experiment up to do this, which I think was the plan). What that pair of satellites think happens to the other member of the pair between times they meet up is frame dependent. You can construct a coordinate system in which the other clock ticks at a steady rate. You can construct a coordinate system in which they tick at different rates. It's up to you which you prefer. You can't directly measure "what time is that clock over there measuring right now".

 

[name123]

How do you explain the satellites logging the flashes of the spaceships at the same time?

Because, in this frame, all the satellites are always the same distance from the ships. And their clocks were started at the same time in this reference frame and the satellites are all moving at the same speed in this reference frame. There's literally nothing to choose between the satellites so they must behave the same.
.

As I had written:

Just to be clear what I mean is that if in your chosen frame of reference each satellite receives each light flash from the spaceships at the same time, and at that time (according to your calculations for that frame of reference) some of their clocks were going at different rates to other ones, then how comes when they come to log the time they detected the flash of light they all report the same time on their clocks. It seems like you would have predicted them to have reported different times (some clocks would have been running faster or slower than others), and be shown to be wrong by experimental results (the log readings). I realize I may have misunderstood you, but I just wanted to make clear my concern.

Is it that you are suggesting that from the selected frame of reference the clocks on the satellites would be time dilated but uniformly so, and so synchronous?