Undergrad Satellite Orbit synchronization

[PeterDonis]

In the example do the satellites' have a nonzero proper acceleration?

No, they are in free fall orbits. But spacetime is curved because of the presence of the gravitating mass.

 

[Dale]

I thought that it would at least be possible that at least one imaginary perspective was considered to be stationary given the equations.

A stationary perspective is easy. That is completely different from an inertial frame. (I am not sure why you use the word imaginary here)

I really recommend that you stop this line of inquiry. You are so far away from understanding the issues involved that this haphazard approach is getting you confused. You need a more systematic approach.

To understand the issues you are raising you need to be able to understand at least the first two chapters of Sean Carroll’s Lecture Notes On General Relativity.

I was just examining it from the satellites perspective.

In relativity “perspective” usually means reference frame. You haven’t examined anything from any perspective because you have not defined any reference frame. This is why you need to learn some background information first.

so perhaps instead just point out the conceptual error (no maths required).

The conceptual error is precisely avoiding the maths that are required to answer the question. If you had the mathematical background then you would understand

1) why I am going on about reference frames
2) why @PeterDonis states that frames are not physical
3) why @Ibix is talking about finding and timelines

Would you like to start learning the background necessary to understand this material?

 

[name123]

The conceptual error is precisely avoiding the maths that are required to answer the question. If you had the mathematical background then you would understand

1) why I am going on about reference frames
2) why @PeterDonis states that frames are not physical
3) why @Ibix is talking about finding and timelines

Would you like to start learning the background necessary to understand this material?

Please enlighten the forum in relation to the scenario provided...

 

[Dale]

Please enlighten the forum in relation to the scenario provided...

Certainly, here are the lecture notes I mentioned above:
https://arxiv.org/abs/gr-qc/9712019

Here is a briefer introduction also by Sean Carroll:
https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf

And here is the first lecture in a video course by Leonard Susskind:

 

[name123]

Certainly, here are the lecture notes I mentioned above:
https://arxiv.org/abs/gr-qc/9712019

Here is a briefer introduction also by Sean Carroll:
https://preposterousuniverse.com/wp-content/uploads/2015/08/grtinypdf.pdf

And here is the first lecture in a video course by Leonard Susskind:

Could you possibly be more specific as it seems that even your mentors have disagreed and you yourself have stated regarding whether there is a coordinate system in which the satellites from the A and B series would match clockwise when they passed, and for their logging of the spaceship flashing events:

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

Presumably you knew the information in the links beforehand, and if it wasn't obvious to you but you have since worked it out, then perhaps share.

off to bed, be back later

 

[Dale]

Could you possibly be more specific

Specifically, you should be comfortable with the material in the first two chapters in the Lecture notes. The third chapter would be optional. If that is too difficult then start with the brief introduction and the video lectures.

it seems that even your mentors have disagreed

Most of the disagreements are disagreements about how to rigorously interpret what you are saying, not about the underlying theory.

Presumably you knew the information in the links beforehand, and if it wasn't obvious to you but you have since worked it out, then perhaps share.

It is still not obvious to me (you still have not specified the reference frame), and as I told you this is a difficult problem that I am not willing to work through. It is difficult, but I see minimal value to it. Presumably you feel it is a valuable question, so then you are the one that has the motivation to dig through the math to get the answer.

 

[PeterDonis]

even your mentors have disagreed

On what?

whether there is a coordinate system in which the satellites from the A and B series would match clockwise when they passed

Whether or not the satellite clocks match each other every time they pass is an invariant fact, independent of any choice of coordinates: they do. The fact that you keep asking about whether there is a coordinate system in which this is true, even though it is a fact independent of any choice of coordinates, is an indication that, as I said earlier, you are just confusing yourself by focusing on coordinates and frames instead of on physical invariants.

Please enlighten the forum in relation to the scenario provided...

Here is a description of the scenario without once mentioning any coordinates or frames:

We have a gravitating sphere with two satellites in free-fall circular orbits around it, in opposite directions. The satellites meet every half orbit. The sphere is transparent (this is an idealized thought experiment so we can stipulate this even though it obviously wouldn't work for a real planet), and there is a clock at the exact center of the sphere that sends out a spherical light pulse in all directions every time it ticks. Each light pulse is time stamped with the time of its emission according to the clock at the center of the sphere.

The satellites will observe the following: if they synchronize their clocks on one meeting, their clocks will be synchronized at every meeting. Also, if their clocks are synchronized as just described, and they record the time by their clocks when they receive each time stamped light pulse from the center of the sphere, both satellites will observe the exact same relationship between times received and timestamps of the received pulses.

 

[Janus]

So how does the visualisation you provided explain the same time logged for the spaceship flashes, between the satellite to the left and right of the satellite passing the passing ship (of whose perspective we are imagining) and the satellites to the left and right of the opposite satellite (from the one passing the ship)? Presumably those distances from north/south flashing spaceships are not the same. Nor their next positions.

well as with any scenario where you have relative motion, you do have to take the relativity of simultaneity in account.
To demonstrate we will just consider linear motion.
Below is a rod from which light flashes initiate at the ends To the left is how events occur according to the rod itself. The flashes start at the same time and the expanding light meets at the center of the rod.
On the right are the same events according to someone for which the rod is moving at 0.5c to the right.
The flash from the left end of the rod starts first, and the light expands outward at c while the rod continues to move to the right.
After the rod has moved some distance, the second flash leaves the right end, and expands out at c. Both flashes continue to expand as the rod continues to move to the right until they meet, again at the middle of the rod.

The fact that the light meets at the center of the rod is an invariant fact for both frame. Whether or not each flash started at the same time is not.
So in your scenario, the fact that flashes from the spacecraft arrive at all the satellites simultaneously in one frame does not mean that they arrive simultaneously in all frames. In the same way, the fact that two satellites are always in sync when they pass each other is an invariant fact doesn't mean that they remain in sync at all points of their orbits in all frames.

 

[Ibix]

I was just examining it from the satellites perspective. But if I have made a mistake then please point it out. I would be happy to see the perspective mathematically from the satellite's point of view, but given your request for money presumably you aren't going to, so perhaps instead just point out the conceptual error (no maths required).

The basic conceptual problem is that "the satellite's point of view" is not a precise definition of anything. You can get away with such imprecision in flat spacetime for non-accelerating objects because pretty much any reasonable construal of "point of view" turns out to mean standard issue Einstein frames. Not so in the general case.

It's possible to write down what each satellite literally sees through a telescope if it watches one of the counter-orbiting ones. To do this you just need to determine what the watched satellite's clock reads when it emits light that passes through the sphere and arrives at the watching satellite at a specified time. Since this involves finding time-like geodesics of Schwarzschild's interior solution it is non-trivial (hence why I was joking about your budget) and probably contains implausible assumptions about the behaviour of glass under pressure. But it's unambiguous.

What you cannot then do is correct for the travel time of light to work out what the watched satellite's clock is "really doing". At least, not in a unique way. You can do it in flat spacetime because there's an unique "natural" answer to the question of how to split up spacetime into space and time, one which makes sense for any inertial observer. That gives us a clear candidate for what you mean by "now" when you ask what rate some satellite's clock is ticking at now. That is not the case in a curved spacetime case - there are multiple different approaches to defining "now", many of which won't let you answer the question anyway.

For example, you can use SR locally. So let's assume adjacent satellites in one ring are close enough together to use it and try to synchronise clocks. We have set the satellite clocks running synchronised in the sphere's rest frame, so the Lorentz transforms tell us that the clock in the satellite ahead will be slightly ahead as seen by the satellite behind it. So we jet over to the satellite infront and wind the clock back a bit. But we can make the same argument here about the next satellite forwards in the chain, and go and set its clock back too - double the correction of the first one, in fact. And we can keep making the same argument until eventually we get back to our starting satellite and come to the conclusion that this satellite's clock is not synchronised to itself since it, too, needs to be set back.

This is not a problem with GR, rather it's that the chain-small-Einstein-frames-together methodology doesn't produce a consistent notion of simultaneity if you push it too far (it turns out you've created a corkscrew-like slice of spacetime, not a plane). You can come up with methods that do work, but they're either mathematically complex (e.g. radar coordinates) or don't really match the notion of "perspective of the satellite" (e.g. the sphere-centred coordinate system I used earlier).

 

[Nugatory]

The basic conceptual problem is that "the satellite's point of view" is not a precise definition of anything...

I agree with everything that Ibix says in this post. However, I would like to suggest a simplification: suppose that we treat the satellite orbits as a classical central force problem instead of using gravity to keep them in their orbits... Now we can avoid the complexities of curved spacetime and work the problem in a single global inertial frame (the one in which Earth and spaceships are at rest is of course a sensible one) using the simpler methods of SR with a natural way of correcting for light travel time.

This simpler setup still captures what I think is the essence of @name123's problem: How can it be that all the satellites log the same arrival time for the flashes, and their clocks match when they meet every half-orbit, even though the clocks on the satellites are mutually time-dilated so that the satellite observers will always find all the other satellite clocks to running slow for the entire orbit?

 

[Ibix]

This simpler setup still captures what I think is the essence of @name123's problem: How can it be that all the satellites log the same arrival time for the flashes, and their clocks match when they meet every half-orbit, even though the clocks on the satellites are mutually time-dilated so that the satellite observers will always find all the other satellite clocks to running slow for the entire orbit?

Indeed. So we have two co-axial counter-rotating merry go rounds with clocks on their rims. I think the only real difference this makes to my answer is that the maths of working out what each clock actually sees from a counter-rotating clock is merely annoying, not actively difficult.

In particular, chaining together local Einstein frames still fails, and for the same reason. Global inertial frames are available, but any chosen clock is only instantaneously at rest, or never at rest, in any chosen one.

 

[Dale]

This simpler setup still captures what I think is the essence of @name123's problem

I agree, that is a much better problem. Of course, even in flat spacetime there is not one unique definition of the reference frame for a non-inertial observer. So the OP will still need to specify what reference frame he wants to use to indicate the satellite's reference frame as a transformation from some inertial frame.

 

[name123]

well as with any scenario where you have relative motion, you do have to take the relativity of simultaneity in account.
To demonstrate we will just consider linear motion.
Below is a rod from which light flashes initiate at the ends To the left is how events occur according to the rod itself. The flashes start at the same time and the expanding light meets at the center of the rod.
On the right are the same events according to someone for which the rod is moving at 0.5c to the right.
The flash from the left end of the rod starts first, and the light expands outward at c while the rod continues to move to the right.
After the rod has moved some distance, the second flash leaves the right end, and expands out at c. Both flashes continue to expand as the rod continues to move to the right until they meet, again at the middle of the rod.
View attachment 228688
The fact that the light meets at the center of the rod is an invariant fact for both frame. Whether or not each flash started at the same time is not.
So in your scenario, the fact that flashes from the spacecraft arrive at all the satellites simultaneously in one frame does not mean that they arrive simultaneously in all frames. In the same way, the fact that two satellites are always in sync when they pass each other is an invariant fact doesn't mean that they remain in sync at all points of their orbits in all frames.

I had already explained in post #43 how I was visualising it. And there it was clear that I was not thinking that from a satellite's perspective all the satellites were receiving the signal at the same time (because their distance to the shapeship flashing would be different). My surprise with your visualisation in post #55 was that you were envisaging length contraction not just in the x direction, but also in the y direction perpendicular to it. I did not realize there would be length contraction in the y direction also. I was imagining it without length contraction in the y direction, and wondered how if there was no length contraction in the y, such that the satellites either side of the satellite opposite the one whose perspective it is where the same distance from the spaceship as the ones either side of the satellite whose perspective it is, how it could be explained them receiving the light at the same time if they were thought to be in the position you depicted. So a confusion on my part. The answer presumably being that there would be considered to be length contraction in the y. Anyway, as I mentioned I did not realize there would be length contraction in the y direction as it isn't mentioned in the special relativity equations.

Anyway, my surprise there does not effect the part that I was confused about and mentioned in #43:

The problem I would have with it is that if their clocks are going slower how do they show the same time per orbit. Could you perhaps explain it in terms of mathematical example, or explain conceptually the error I made? I imagine it as rotating a cardboard cutout around an axis, (different spoke lengths for different satellites) but having a problem of the clocks on some satellites (rather than their velocities) going slower than others but measuring the same time per orbit.

I don't need a mathematical example, or the exact figures, I would be happy with someone just pointing out what was wrong with the visualisation of the problem (the contraction in the y-direction doesn't alter the problem I have with the conception).

 

[Nugatory]

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.

We are using coordinates in which the Earth and the two spaceships are at rest. This is an inertial frame in which all the satellites are moving as they orbit; because it is an inertial frame the speed of light is always $c$. Using these coordinates, the distance from either spaceship to any of the satellites is the same, so the light travel time from spaceship to satellite is the same for all the satellites. All of the satellite clocks are running slow, meaning that at the same time that the spaceship/earth clocks read $T$ the satellite clocks read $T-\Delta$; thus if a flash reaches a satellite at the same time that the earth/spaceship clocks read $T$ the timestamp logged by the satellite will be $T-\Delta$. However, it turns out that the value of $\Delta$ is the same for all the satellites (which is not at all surprising when you consider that they're all moving at the same speed relative to Earth so you'd expect the same amount of time dilation). Because the value of $\Delta$ is the same for all the clocks (equivalently, the clocks are all slow by the same amount) they all end up with the same timestamps.

So far we're fine, there's not even a hint of an apparent paradox in any of this. Furthermore, we're happy with the idea that the clocks on the orbiting and counter-orbiting satellites will agree when they pass each other - just consider that a flash of light arriving at two satellites at the moment that they're passing one another will get the same timestamp.

The apparent contradiction you're seeing appears when we use the procedure I described in post #30 to compare the rate at which the clock on one satellite ticks with the rate of a clock on another satellite. How can the timestamps all match, and the clocks match when the rotating and counter-rotating satellites meet at every orbit, when the clocks are always running at different rates? That's where the relativity of simultaneity comes in.

A few paragraphs back I said that "at the same time that the spaceship/earth clocks read $T$ the satellite clocks read $T-\Delta$" - and relativity of simultaneity means that things that happen at the same time in one frame may not happen at the same time in another frame. We're using the frame in which the Earth and the two spaceships are at rest and we find that all the satellite clocks read $T-\Delta$ at the same time; but when we use the procedure of post #30 we're using a different frame (the inertial frame in which satellite X is momentarily at rest) and in this one the events "satellite X clock reads $T-\Delta$" and "satellite Y clock reads $T-\Delta$ do not happen at the same time. However, in this frame the distances between the spaceships and the two satellites are also not the same so the light travel time is also different; the two effects cancel each other out so that the timestamps end up the same. So that's how the timestamps come out the same even though the clocks aren't synchronized.

Relativity of simultaneity will also explain how the clocks on satellite X and satellite Y end up agreeing when they pass one another but the explanation is a bit harder to visualize; you are probably better off starting with the "Doppler explanation" section of this FAQ. The situation is different because we're dealing with a complicated circular path instead of straight-line out and back paths, but the result has to be the same: the total number of flashes that reach X from Y in a half-orbit has to be equal to the total number of flashes X emits during that half orbit; the two clocks tick once with for every flash, so they both have to tick off the same amount of time during their half-orbits between one meeting and the next.
Now consider the situation on satellite X $t$ seconds before it passes satellite Y with both of their clocks reading $T$; clearly X's clock reads $T-x$. However, at the same time (using the frame in which X is momentarily at rest) that X's clock reads $T-x$ the clock on satellite Y will not read $T-x$; it reads some value that added to the number of time Y's clock will tick as it finishes its half-orbit will be equal to $T$. (If you are not familiar with the "Andromeda paradox", google for it - the small changes in the relative speed of X and Y as they orbit make for large and counterintuitive changes in what they are doing "at the same time").

 

[Dale]

I had already explained in post #43 how I was visualising it. And there it was clear that I was not thinking that from a satellite's perspective all the satellites were receiving the signal at the same time (because their distance to the shapeship flashing would be different).

The problem with your description of post 43 is that you were describing a lot of features and details that you assumed that "the satellite's frame" would have without actually writing down what you mean by "the satellite's frame" and showing that it actually has those features. There is simply no way to avoid it. The satellites are non-inertial, so there is no default frame to use that we consider to be their frame. You actually have to specify it explicitly, e.g. as a transform between the ship's frame which is inertial.

Basically, you are asking a question with a crucial undefined term in it. You are struggling to understand the answers, but you need to work on understanding your own question first. Work on the question first, that is, work on defining what you mean exactly by "the satellite's frame".

 

[name123]

I thought it was obvious in post #43 I was considering that the satellite's perspective could be approximated by imagining a series of short lines where it is traveling at the velocity it is in the direction of it's tangent. Each consideration could be of a duration of 0.0000000000000000000000000000000000000000001 nanoseconds while the orbit could be a billion years. I had described how such a configuration would lead to the imagining of the situation, and it would not be too different from that considered by Janus in post #55

Assuming you had a spaceship( the red arrow in the following image) skimming the orbit of a ring of satellites such that it momentarily matches the velocity of the satellite it is passing, then,...

Except I had not imagined any length contraction in the y direction. If you think both my and Janus's consideration that the orbits of the other satellites would appear elliptical is wrong, then please say so. If not then the point I am making stands. As it goes around, a satellite (in the same series) appearing to be closer to the centre of the sphere/spheroid would seem to remain at that distance the whole way around. Which would mean the light from the spaceship flashes would have taken less time to reach it. It could then be concluded that it's clocks were going slower because they register the same time reading for the light reaching them. What I am having trouble with is the reconciliation of the clocks being considered as going slower on the other satellite the whole orbit, and the orbit being measured as taking the same time according to their clock, and the satellite never overtaking. The reason is that if their clock was going slower, then the observing satellite would presumably be concluding that the orbit took less time than the slower clock showed, but if it were orbiting in less time than the observing satellite was orbiting then why does in not do more orbits over a long period?

 

[jbriggs444]

series of short lines

The short lines are not a problem. The bends between the short lines are. They have to be accounted for. Each bend changes the simultaneity convention used between the one tangent inertial frame and the next.

 

[name123]

The short lines are not a problem. The bends between the short lines are. They have to be accounted for. Each bend changes the simultaneity convention used between the one tangent inertial frame and the next.

So is it that you think both my and Janus's consideration that the orbits of the other satellites would appear elliptical from the satellite's perspective is wrong? If then please say so. If not then I am not sure what difference you are thinking it makes to the problem I mentioned. Perhaps you could explain.

 

[Dale]

If you think both my and Janus's consideration that the orbits of the other satellites would appear elliptical is wrong, then please say so.

Yes, I think it is wrong. You cannot make a valid non-inertial frame by stitching together a bunch of inertial frames. When we say "the satellite's perspective" in relativity then we are talking about a reference frame where the satellites are at rest. Since they are non-inertial then there is no standard reference frame and it must be explicitly defined. A sequence of inertial frames does not represent the satellite's frame.

 

[name123]

Yes, I think it is wrong. You cannot make a valid non-inertial frame by stitching together a bunch of inertial frames. When we say "the satellite's perspective" in relativity then we are talking about a reference frame where the satellites are at rest. Since they are non-inertial then there is no standard reference frame and it must be explicitly defined. A sequence of inertial frames does not represent the satellite's frame.

Ok, well however it would look for satellites of its own series, do you accept that it would look that way the whole way around given the symmetry?

 

[Dale]

Ok, well however it would look for satellites of its own series, do you accept that it would look that way the whole way around given the symmetry?

Only if you define the transformation that way! You can define it largely arbitrarily, so it is up to you to include that feature if you see it as an important feature to include. This is something that is up to you to decide, and you do that by specifying the satellite's reference frame mathematically.

 

[name123]

Only if you define the transformation that way! You can define it largely arbitrarily, so it is up to you to include that feature if you see it as an important feature to include. This is something that is up to you to decide, and you do that by specifying the satellite's reference frame mathematically.

Just to be clear what I meant, is that at no matter what part of the orbit the satellite is, its observational relationship to the other satellites in its series does not change. So that if there were no defining features on the "sphere" and no other things in the universe other than those being considered, you could not tell from photographs which part of the orbit the satellite was in. If you were already clear on that, then I am not sure what is arbitrary about it. Perhaps you could give an example of how it is arbitrary?

 

[Ibix]

Ok, well however it would look for satellites of its own series, do you accept that it would look that way the whole way around given the symmetry?

You want to know what a satellite's clock shows at the same time as another satellite's clock reads 00.01, 00.02, 00.03 etc.

1. In relativity there is no general meaning to "at the same time".
2. For inertial bodies in flat spacetime there is a sensible default guess as to what "at the same time" means (Einstein frames).
   
3. This scenario is either not in flat spacetime (your formulation) or the satellites are not inertial (Nugatory's formulation).
4. Thus there is no default sensible meaning to "at the same time".
5. The only attempt you've made to define "at the same time" (chaining together small inertial frames) does not produce a consistent meaning (see post #69).
6. You are free to define "at the same time" in many different ways. Pretty much the only constraints are that the other satellite's clock must always be advancing and we must not assign two times to the same event. The rate depends on your definition.
7. Since you don't have a definition of "at the same time" forced on you or pre-selected by some mechanism then the question is pointless. The answer is "whatever you define it to be".

Can you tell us what step in that chain of reasoning is escaping you?

 

[PeterDonis]

I thought it was obvious in post #43 I was considering that the satellite's perspective could be approximated by imagining a series of short lines where it is traveling at the velocity it is in the direction of it's tangent.

If this was what you were thinking, unfortunately, it's not correct. You cannot construct a valid global frame this way.

 

[Dale]

Perhaps you could give an example of how it is arbitrary?

Yes, you have the freedom to define the transformation as something like:
$t=\gamma T$
$r=R$
$\phi=\Phi+\omega T$
$z=Z$
or
$t=\gamma T$
$x= X \cos(\Phi+\omega T) + \gamma Y \sin(\Phi+\omega T)$
$y= Y\cos(\Phi+\omega T) - \gamma X \sin(\Phi+\omega T)$
$z=Z$
or
...

It is up to you.

 

[russ_watters]

I What I am having trouble with is the reconciliation of the clocks being considered as going slower on the other satellite the whole orbit...

Again, this is not what happens or is observed. If a satellite is watching another satellite's clock (or via radio), when they are moving apart each will observe the other's clock to be running slower and when they are approaching each will observe the other's clock to be running faster.

@Nugatory provided the details of this in Post #74. I feel like your not wanting to accept this has been your entire issue all along. From the link he provided:

Let us focus on what Stella and Terence actually see with their own eyes. (Just to emphasize that we're talking about direct observation here, I'll put the verb "see" and its brothers in the HTML strong font throughout this section.) To make things interesting, we'll equip them with unbelievably powerful telescopes, so each twin can watch the other's clock throughout the trip. If each twin saw the other clock run slow throughout the trip, then we would have a contradiction. But this is not what they see.

Just in case it's too hard to read the clock hands through the telescope, we'll add a flash unit to each clock, set to flash once a second. You might guess at first that Terence sees Stella's clock flashing once every 7 seconds (with the time dilation factor we've chosen) and vice versa. Not so! On the Outbound Leg, Terence sees a flash rate of approximately one flash per 14 seconds; on the Inbound Leg, he seesher clock going at about 14 flashes per second. That is, he sees it running fast! Stella sees the same behavior in Terence's clock.

 

[Dale]

Again, this is not what happens or is observed. If a satellite is watching another satellite's clock (or via radio), when they are moving apart each will observe the other's clock to be running slower and when they are approaching each will observe the other's clock to be running faster.

Where "observe" refers to the frame invariant Doppler shift rather than some coordinate-dependent quantity.

 

[Ibix]

Where "observe" refers to the frame invariant Doppler shift rather than some coordinate-dependent quantity.

Indeed. If you want to know what a video camera mounted on one satellite and tracking another will actually record, @name123, then that's possible in principle. Even with Nugatory's simplified scenario it would involve numerical approximation, but it's possible. It's just that you can't work backwards to subtract out the Doppler effect without first picking a definition of "at the same time".

 

[name123]

You want to know what a satellite's clock shows at the same time as another satellite's clock reads 00.01, 00.02, 00.03 etc.

1. In relativity there is no general meaning to "at the same time".
2. For inertial bodies in flat spacetime there is a sensible default guess as to what "at the same time" means (Einstein frames).
   
3. This scenario is either not in flat spacetime (your formulation) or the satellites are not inertial (Nugatory's formulation).
4. Thus there is no default sensible meaning to "at the same time".
5. The only attempt you've made to define "at the same time" (chaining together small inertial frames) does not produce a consistent meaning (see post #69).
6. You are free to define "at the same time" in many different ways. Pretty much the only constraints are that the other satellite's clock must always be advancing and we must not assign two times to the same event. The rate depends on your definition.
7. Since you don't have a definition of "at the same time" forced on you or pre-selected by some mechanism then the question is pointless. The answer is "whatever you define it to be".

Can you tell us what step in that chain of reasoning is escaping you?

No you I was not asking what a satellites clock shows at the same time as another satellites clock reads 00.01, 00.02 etc. I had tried to explain what I meant in post #82 when I wrote:

Just to be clear what I meant, is that at no matter what part of the orbit the satellite is, its observational relationship to the other satellites in its series does not change. So that if there were no defining features on the "sphere" and no other things in the universe other than those being considered, you could not tell from photographs which part of the orbit the satellite was in. If you were already clear on that, then I am not sure what is arbitrary about it. Perhaps you could give an example of how it is arbitrary?

You can imagine that the pictures do not give you sight of whether the other satellites (in the same series) have clocks or not.

 

[name123]

Yes, you have the freedom to define the transformation as something like:
$t=\gamma T$
$r=R$
$\phi=\Phi+\omega T$
$z=Z$
or
$t=\gamma T$
$x= X \cos(\Phi+\omega T) + \gamma Y \sin(\Phi+\omega T)$
$y= Y\cos(\Phi+\omega T) - \gamma X \sin(\Phi+\omega T)$
$z=Z$
or
...

It is up to you.

I am not quite clear on what the different symbols mean, but are you suggesting that if one of those transformations were chosen, that you could tell from pictures taken from one of the satellites (imagine none contain clocks, and so no clock can be pictured), that you could tell where it was in its orbit (perhaps from how far apart its neighbouring satellites appeared from it or from the centre of the "sphere")?