Undergrad Satellite Orbit synchronization

[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?

 

[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?