Satellite Orbit synchronization

[Dale]

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?

They needn’t even be time dilated if you choose your time coordinate appropriately.

 

[name123]

They needn’t even be time dilated if you choose your time coordinate appropriately.

I'm curious...

 

[Ibix]

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?

In the inertial frame of reference of the centre of the sphere, yes. But you can construct coordinate in which the flash that all satellite clocks regard as time zero was simultaneous but later ones weren't. You can construct systems in which that flash was not simultaneous but others were (I can't think why you'd want to do it, but you can do it).

Literally all you are doing by choosing a coordinate system is choosing which 3d surfaces in 4d spacetime you want to call "now". And hence how many ticks of a given clock there are between "at the same time as my watch reads time t" and "at the same time as my watch reads a second later".

 

[Ibix]

I'm curious...

Simply scale the time coordinate to match the tick rate of this specific set of moving clocks instead of stationary ones. Again, it's not obvious why you'd want to do that (edit: except to hammer home the freedom to do so), but it's perfectly fine to do so.

Edit 2: Einstein said "time is what clocks measure". Traditionally, we read that as "proper time is what a clock measures; coordinate time is what a systematic layout of clocks, synchronised in some sense, measures". The bit before the semicolon is non-negotiable. The part after has more or less infinite freedom to define "systematic layout" and "synchronised in some sense". That's in total opposition to our Newtonian intuition - but tough on our intuition.

 

[name123]

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?

In the inertial frame of reference of the centre of the sphere, yes. But you can construct coordinate in which the flash that all satellite clocks regard as time zero was simultaneous but later ones weren't. You can construct systems in which that flash was not simultaneous but others were (I can't think why you'd want to do it, but you can do it).

Ok, so the advice is contradictory to that given by Nugartory.

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.

But that is ok. I suspect you are correct. I would just like it clear if you are contradicting previous advice, otherwise it is confusing.

So could you provide a coordinate system in which the flashes are not received simultaneously by the satellites, and in which they agree on their clocks when they pass each other, and agree on the time they received their flashes from the spaceships?

 

[PeterDonis]

it's not obvious why you'd want to do that

It is if you're on a rotating planet. This is exactly what the GPS frame of reference (and more generally most Earth Centered Earth Fixed frames) does: the time coordinate is set to match the tick rate of clocks at rest on the geoid of the rotating Earth (which according to an inertial frame centered on the Earth are moving at whatever their local rotation velocity is).

 

[Ibix]

Ok, so the advice is contradictory to that given by Nugartory.

See what @Nugatory says. I've been known to make mistakes.

So could you provide a coordinate system in which the flashes are not received simultaneously by the satellites, and in which they agree on their clocks when they pass each other, and agree on the time they received their flashes from the spaceships?

Any other coordinate system. One in which the sphere is moving, for example.

 

[name123]

.
Any other coordinate system. One in which the sphere is moving, for example.

Could you do the maths perhaps, because I am not sure what is at rest in that scenario. I could add in a spaceship passing in the x direction, and then take it from that spaceship's perspective (with that spaceship being at rest), where the satellites are not time dilated uniformly and not synchronous, and their path is not be circular but elliptical and the sphere no longer spherical because of length contraction in the x direction. From that perspective the distance from the spaceships flashing their lights would be less for the ones on the sides the semi-minor axes touch than the ones on the sides the semi-major axes touch. I realize there that it is just a visualisation and doesn't involve the maths.

The problem I have with the visualisation it is that it would seem reasonable to from a satellite's perspective to reduce it to a series of such perspectives. Rather like a cartoon strip, to get a "movie" as it were of what it would be like from a satellite's perspective. As I imagine it (from a satellites perspective) the shape of the sphere would keep changing, with length contraction in the direction of the tangent to its orbit. So a type of oblate spheroid where the largest circumference is North to South, and follows the satellite around. If I were then to imagine beams of light coming out from the sphere indicating angles; from the frame of reference of an A series satellite for example, it is always at the point of the ellipse where the semi major axis touches, throughout its orbit. Thus it would be making an orbit of a higher altitude than any other satellite (other than the one opposite it). The series A satellites at either + or - 90 degrees from its angle (considered from the centre of the sphere) are always at the point of the ellipse where the semi-minor axis touches. And because they log the same time for the light reaching them from the spaceships, even though the light would have had less distance to travel, then given certain assumptions their clocks must be going slower. 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.

 

[Dale]

Ok, so the advice is contradictory to that given by Nugartory.

It is only contradictory in the sense that different reference frames are different conventions. In England they drive on the left, in the USA we drive on the right. It is that sort of contradiction.

 

[name123]

It is only contradictory in the sense that different reference frames are different conventions. In England they drive on the left, in the USA we drive on the right. It is that sort of contradiction.

I was thinking more in the lines of from a certain frame of reference it being claimed that the satellite clocks would not be synchronous (Nugartory) and another claim that from the same frame of reference that they would be synchronous (Ibix). But maybe I am confused.

 

[Dale]

And because they log the same time for the light reaching them from the spaceships, even though the light would have had less distance to travel,

These reference frames are non inertial so there is no requirement that the coordinate speed of light equals c.

 

[Dale]

I was thinking more in the lines of from a certain frame of reference it being claimed that the satellite clocks would not be synchronous (Nugartory) and another claim that from the same frame of reference that they would be synchronous (Ibix). But maybe I am confused.

They were talking about different frames.

 

[name123]

These reference frames are non inertial so there is no requirement that the coordinate speed of light equals c.

What frame is non-interial?

 

[Ibix]

Could you do the maths perhaps

For time-like and null geodesics in the spacetime of a moving gravitational source? What's your budget and timescales for this work?

their path not be circular but elliptical

Not elliptical - remember they're moving around a moving object, so it's some kind of flattened cycloid. Otherwise, you seem to have got what I was aiming at.

The problem I have with the visualisation it is that it would seem reasonable to from a satellite's reduce it to a series of such perspectives.

You are mixing frames. The satellites' perspective is the same whether there's a ship passing by or not. You can view from the ship. You can view from the sphere. You can view from a satellite. You can't view from the satellite and the ship at the same time.

 

[name123]

They were talking about different frames.

What was the difference?

 

[Ibix]

They were talking about different frames.

I'm not sure. If I understood correctly the frame that Nugatory was talking about then I think he made a typo. Note that there are at least two conditionals in that sentence - which is why I want to see what Nugatory says.

 

[name123]

You are mixing frames. The satellites' perspective is the same whether there's a ship passing by or not. You can view from the ship. You can view from the sphere. You can view from a satellite. You can't view from the satellite and the ship at the same time.

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).

 

[name123]

I'm not sure. If I understood correctly the frame that Nugatory was talking about then I think he made a typo. Note that there are at least two conditionals in that sentence - which is why I want to see what Nugatory says.

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, ...

I am not sure he made a mistake, but who of us has never made one?

 

[PeterDonis]

instead just point out the conceptual error

Your basic conceptual error is to think of frames as fundamental and trying to understand everything else in terms of them. That is confusing you because frames are abstractions: they are conventions we adopt. But you are thinking of them as actual physical things. They're not.

 

[Janus]

Could you do the maths perhaps, because I am not sure what is at rest in that scenario. I could add in a spaceship passing in the x direction, and then take it from that spaceship's perspective (with that spaceship being at rest), where the satellites are not time dilated uniformly and not synchronous, and their path is not be circular but elliptical and the sphere no longer spherical because of length contraction in the x direction. From that perspective the distance from the spaceships flashing their lights would be less for the ones on the sides the semi-minor axes touch than the ones on the sides the semi-major axes touch. I realize there that it is just a visualisation and doesn't involve the maths.

Its a bit more complicated than that.

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, it from its perspective, the ring of satellites would be like this, with the ellipse showing the shape of the orbit and the dots the relative positions of the satellites.

If he were watching the satellites as they orbit, he would see them speed up and spread out as they moved to the bottom part the orbit shown here and slow down and bunch up as moved to the top part of the orbit.

 

[name123]

Its a bit more complicated than that.

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, it from its perspective, the ring of satellites would be like this, with the ellipse showing the shape of the orbit and the dots the relative positions of the satellites.
View attachment 228680

If he were watching the satellites as they orbit, he would see them speed up and spread out as they moved to the bottom part the orbit shown here and slow down and bunch up as moved to the top part of the orbit.

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.

 

[Dale]

What frame is non-interial?

All frames in curved spacetime are non inertial.

 

[name123]

All frames in curved spacetime are non inertial.

I did not realize that all frames in theoretical curved spacetime were non inertial. I thought that it would at least be possible that at least one imaginary perspective was considered to be stationary given the equations. An absolute frame of reference so to speak.

 

[PeterDonis]

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

"Stationary" is not the same as "inertial". Given an object on a timelike worldline, you can always find a coordinate chart in which that object is stationary, i.e., at rest. But in curved spacetime, or even in flat spacetime if the object has nonzero proper acceleration, the coordinate chart in which the object is stationary will not be inertial.

An absolute frame of reference so to speak.

There is no such thing, because there is no such thing as "stationary" in any absolute sense.

 

[name123]

"Stationary" is not the same as "inertial". Given an object on a timelike worldline, you can always find a coordinate chart in which that object is stationary, i.e., at rest. But in curved spacetime, or even in flat spacetime if the object has nonzero proper acceleration, the coordinate chart in which the object is stationary will not be inertial..

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

 

[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?