Question about GPS clock synchronization

[Arkalius]

I'm familiar with the combined effects of special and general relativity making clocks on GPS satellites tick faster than those on Earth by about 38 microseconds per day. The clocks on the satellites are adjusted to stay in sync with Earth clocks, and all is fine an dandy in that regard.

However, there is an element of this I cannot reconcile on my own. It's clear the clocks will stay in sync with Earth clocks by this means, but I'm wondering how GPS clocks stay in sync with each other. They're all at the same altitude so there is no GR effects between them, but they are all moving relative to each other. These motions are, as I understand it, inertial from the perspective of general relativity (they are following curved inertial paths in spacetime, experiencing no actual acceleration). It would seem then to me that they should all see each other's clocks ticking more slowly and end up way out of sync with each other.

A more simple example I can think of is two satellites in identical orbits, except one is inclined 90 degrees from the other. They both pass the intersection points at nearly the same time (so as to just miss colliding). When they pass one of those points, their clocks are synced. Now, like in the twin paradox, they move away from each other, and each should notice the other's clock ticking slower than its own. But because of curved spacetime they end up moving back to meet up again without any acceleration. Since there was no acceleration, and thus no change of reference frame occurred, both should see the other's clock having passed less time when they meet again, an obvious contradiction.

Now, I realize I must be wrong about this. It seems to obvious a flaw to go unnoticed if it were real. I just don't quite see why its wrong. I'm sure it has to do with the fact that I don't have as deep an understanding of general relativity as I do special relativity. So, I'd love to know what I'm missing in this scenario.

It's especially frustrating because I ended up in an argument with this individual who has published a paper claiming to disprove special relativity, and one of his main talking points is how GPS satellites cannot possibly stay in sync with each other under the model of SR, and since they obviously do, SR must be wrong. Based on his other arguments its clear he doesn't fully understand SR, but I don't have any compelling rebuttal to that particular argument and I wish I did.

 

[Dale]

but I'm wondering how GPS clocks stay in sync with each other

Synchronization isn't something physical that happens or does not happen. It is a convention that you adopt.

The convention adopted for GPS is that the GPS satellites are synchronized in the ECI frame. They are therefore not synchronized in other frames. Specifically, they are not synchronized in the rest frames of the individual satellites.

 

[Arkalius]

Synchronization isn't something physical that happens or does not happen. It is a convention that you adopt.

The convention adopted for GPS is that the GPS satellites are synchronized in the ECI frame. They are therefore not synchronized in other frames. Specifically, they are not synchronized in the rest frames of the individual satellites.

I realize they aren't likely in constant synchronization with each other. However, it seems to me that the difference in timing between any two would have to vary in a cyclical fashion. The idea that they can just drift apart endlessly doesn't sit well with me. More specifically, I'd like to see the contrived example I specified be addressed, because it creates a kind of twins paradox that doesn't involve acceleration, and I don't understand the solution.

 

[Dale]

I ended up in an argument with this individual who has published a paper claiming to disprove special relativity, and one of his main talking points is how GPS satellites cannot possibly stay in sync with each other under the model of SR, and since they obviously do, SR must be wrong.

Unfortunately, even a crackpot can occasionally have a kernel of truth in their argument. Technically, any measurement where tidal gravity is important can be used to invalidate SR. This is well known and is in fact the primary motivation for GR.

However, a version of his argument could be formulated using big centrifuges in space far from any significant gravitation. In such a case you could develop a flat spacetime GPS that would still work similarly. In such a system you would simply use the SR correction and not the gravity correction. Then the clocks would be synchronized in the "centrifuge centered inertial frame"

 

[Dale]

However, it seems to me that the difference in timing between any two would have to vary in a cyclical fashion.

By design the difference in timing in the ECI frame is 0 at all times. All clocks are and remain synchronized in the ECI frame so there is never any difference in timing in the ECI frame between any pair at any time in the ECI frame. They don't drift apart in the ECI frame, ever.

Can I emphasize that this is all in the ECI frame again?

 

[Arkalius]

By design the difference in timing in the ECI frame is 0 at all times. All clocks are and remain synchronized in the ECI frame so there is never any difference in timing in the ECI frame between any pair at any time in the ECI frame. They don't drift apart in the ECI frame, ever.

Can I emphasize that this is all in the ECI frame again?

I understand that...

Let's move away from GPS specifically for a second and look at my fictional example. It seems you have (like in the twins paradox) two satellites whose clocks are synced at one meeting point, which will then move relative to each other (thus each should see the other's clock ticking slower) and then meet up again later in the orbit, all without either of them ever accelerating. In the twins paradox, the shift in reference frame from one twin's acceleration is what resolves the apparent paradox, but we don't have that here. When the satellites meet again each one should expect the other's clock to read earlier than his own. I'm sure this must be wrong, but I don't know why.

 

[jbriggs444]

I understand that...

Let's move away from GPS specifically for a second and look at my fictional example. It seems you have (like in the twins paradox) two satellites whose clocks are synced at one meeting point, which will then move relative to each other (thus each should see the other's clock ticking slower) and then meet up again later in the orbit, all without either of them ever accelerating. In the twins paradox, the shift in reference frame from one twin's acceleration is what resolves the apparent paradox, but we don't have that here. When the satellites meet again each one should expect the other's clock to read earlier than his own. I'm sure this must be wrong, but I don't know why.

Because zero proper acceleration for an object that is continuously at the origin of a coordinate system is not sufficient to make that coordinate system inertial.

An inertial coordinate system is one in which the laws of mechanics hold good. For instance, any object subject to no net force must continue in uniform motion. Note that from any satellite's point of view, other satellites are subject to no net force but do NOT continue in uniform motion. Accordingly, no satellite's rest frame is inertial.

In general relativity and in the presence of tidal gravity there is no such thing as a globally inertial frame. There are only sufficiently small local regions so that one can define frames that are approximately inertial.

 

[Arkalius]

Because zero proper acceleration for an object that is continuously at the origin of a coordinate system is not sufficient to make that coordinate system inertial.

An inertial coordinate system is one in which the laws of mechanics hold good. For instance, any object subject to no net force must continue in uniform motion. Note that from any satellite's point of view, other satellites are subject to no net force but do NOT continue in uniform motion. Accordingly, no satellite's rest frame is inertial.

OK now we're getting somewhere. I suspected the answer would be something along those lines. Can we take this a step further then? In many thought experiments in SR, we often note what an observer in one frame would see on a clock in a moving frame in a given instant for the observer in order to show time dilation. Can you give me an idea of what one satellite would observe on the clock of the other in each unit of proper time of the first one, from the point of initial synchronization to the point when they again meet? Any background as to how to understand this scenario better would be helpful, though I realize it may involve some complex concepts. Could it be as simple (relatively speaking) as treating each satellite as being in an accelerating reference frame in SR? That would, it seem, resolve the problem.

 

[Dale]

I understand that...

Let's move away from GPS specifically for a second and look at my fictional example. It seems you have (like in the twins paradox) two satellites whose clocks are synced at one meeting point, which will then move relative to each other (thus each should see the other's clock ticking slower) and then meet up again later in the orbit, all without either of them ever accelerating. In the twins paradox, the shift in reference frame from one twin's acceleration is what resolves the apparent paradox, but we don't have that here. When the satellites meet again each one should expect the other's clock to read earlier than his own. I'm sure this must be wrong, but I don't know why.

Tidal gravity is significant in this scenario so the SR equations do not apply. You have to use GR. In GR what you do is integrate the spacetime interval over the worldline. That will give the same result for both.

 

[Dale]

Can you give me an idea of what one satellite would observe on the clock of the other in each unit of proper time of the first one,

To answer this would require you to define in a very rigorous way what you mean by "would observe on the clock of the other". Unlike in SR, there is no standard reference frame applied by convention for a given observer. If you really want to pursue this then I would start with Carroll's lecture notes on GR.

 

[pervect]

Can you give me an idea of what one satellite would observe on the clock of the other in each unit of proper time of the first one, from the point of initial synchronization to the point when they again meet? Any background as to how to understand this scenario better would be helpful, though I realize it may involve some complex concepts.

The underlying problem is that space-time is curved.

One can gain some insight in the problem of understanding curved space-time by considering what it's like to live on the surface of the Earth, and asking how to represent it on a flat sheet of paper as a flat-lander from Abbot's book "Flatland" would see it.

One can devise many ways of representing the curved surface of the Earth on a flat sheet of paper. None of them will be to scale, a unit distance on the flat map will not be a unit distance on the actual territory, which is curved.

One approach they might take is to measure the angle and distance of all points from themselves, then mark the positions of every object on a flat piece of paper with the same angles and distances. This will be good for some purposes, but not others. The distance from the central point to any other point will be given accurately on the map, but the distance between two points that are both not on center will not be to scale.

For very nearby points, the innacuracies will be minor. This is similar to the way that even though the surface of the Earth is curved, one can navigate within a city quite well using a flat map, because the inaccuracies in the flat map caused by curvature aren't significant.

If one tries to naviagate a long distance, though, one needs to learn spherical trignometry, the actual geometry of the Earth. The situation for the observer in space-time is similar. In a nearby region, they can ignore the curavature, and get a fairly accurate map of space-time. But over large distances, this approach breaks down, and to get an accurate representation of the universe, one needs to understand the geometry of general relativity.

Using Fermi-normal coordinates is rather like the approach of taking bearings and distances from other objects, and using that to draw a map of a curved Earth on a flat piece of paper near one particular point. The calculations are rather involved, unfortunately - and the results aren't really any better than just saying that if you are in a small space-station, orbiting the Earth, you can navigate without a lot of difficulty just as you can walk around a small town without taking into account the fact that the Earth is curved.

If you're really curious, though, you can read about Fermi-Normal coordinates, if you think that it will help. But it'll probalby be a bit too advanced, and not really all that helpful, though it may be good to know that it exists.

I often thought it would be interesting to have a graphic of a map of a large section of the Earth drawn using a Fermi-normal like approach, but I've never seen one or attempted to create one. The distortions would serve as a good example of why this isn't done except in a small area, though.

Being three dimensional entities, we can have some intuitive understanding of a curved 2d surface. But this won't help us understand how geometry works when it's curved, it's simply doing flat geometry in a space of higher dimension. To understand that takes a rather abstract approach, abandoning Euclidean geometry in favor of geometry that is not flat, geometry that does not obey Euclid's postulates.

 

[Nugatory]

I understand that...
In the twins paradox, the shift in reference frame from one twin's acceleration is what resolves the apparent paradox, but we don't have that here.

Acceleration has nothing to do with the resolution of the twin paradox. What's going on is that the two twins are following different paths through spacetime between the separation event and the reunion event; these paths have different lengths so different amounts of proper time elapse on them. Acceleration only comes into the picture because we can't put the twins on different paths through spacetime without accelerating one of them (unless, as in your example, we use gravity for that purpose).

To properly analyze your example we have to calculate the proper time along each satellite's path through spacetime between one meeting event and the next. If it is the same, the clocks will agree at every meeting; velocity-dependent and gravitational time dilation are irrelevant. In your example, we can use the symmetry of the situation to conclude that the proper time is indeed the same on both paths.

This would also be a really good time to work through the Twin Paradox FAQ. Pay particular attention to the section on the Doppler effect; it's the best way of understanding what each observer actually "sees" as they move apart and back together on their opposing orbits.

 

[Arkalius]

The underlying problem is that space-time is curved.

I appreciate your reply. I suspected a rigorous solution to this quandary would likely involve much of the complex math that permeates general relativity. I didn't expect I'd get a fully detailed solution to it (or if I did, that I would fully understand it). I just wanted to get an idea of the concepts I'm misunderstanding that led to my confusion, and other posters have managed to do that for me.

Unfortunately I don't feel I have a deep enough understanding of the resolution to be able to confidently rebut the objection from that author I mentioned in my original post, and I fear there isn't likely a sufficiently rigorous way of rebutting it without a significantly more detailed understanding of the math involved. Maybe one day I'll try to tackle that challenge. I can live without rebutting a crackpot... but he was just so arrogant and smug about it and I hated not being able to address his objection in a compelling way. Oh well :)

 

[Arkalius]

Acceleration has nothing to do with the resolution of the twin paradox. What's going on is that the two twins are following different paths through spacetime between the separation event and the reunion event; these paths have different lengths so different amounts of proper time elapse on them. Acceleration only comes into the picture because we can't put the twins on different paths through spacetime without accelerating one of them (unless, as in your example, we use gravity for that purpose).

I disagree. In a flat spacetime, the only way two worldlines with a non-zero relative velocity can intersect more than once is if at least one of them includes acceleration. The naïve expression of the twins paradox involves invoking the principle of relativity that suggests both twins should see the other's clock tick slower than his own, so that when they meet again, each should expect the other to be younger, hence paradox. The resolution is to point out that one twin had to accelerate to turn around and this shifted his frame of reference, which ultimately leads to the situation you describe, of him having a shorter path through spacetime than the other. I often find myself having to explain how acceleration is not relative to people who inevitably suggest that from the traveling twin's perspective it looks like Earth accelerated. So it seems to me that the acceleration (and understanding its effects on the situation) are important to the scenario.

To properly analyze your example we have to calculate the proper time along each satellite's path through spacetime between one meeting event and the next. If it is the same, the clocks will agree at every meeting; velocity-dependent and gravitational time dilation are irrelevant. In your example, we can use the symmetry of the situation to conclude that the proper time is indeed the same on both paths.

Hmm, that is a good way of looking at it. I like that resolution. It doesn't help as much with the GPS example since GPS satellites don't ever really "meet up" to compare clocks, but each orbit of a GPS satellite will involve the same amount of proper time for each, so I suppose that is an effective way of recognizing that they won't get "out of sync" as it were with each other.

 

[Dale]

In a flat spacetime, the only way two worldlines with a non-zero relative velocity can intersect more than once is if at least one of them includes acceleration.

That is correct. The symmetry is broken which rebuts the argument of symmetry in the first place.

However, if you want to know the proper time elapsed for each twin then one more piece is needed: the spacetime interval. Geometrically, it is the equivalent of length in spacetime. To obtain the amount of elapsed time on a twins journey between two events you simply calculate the spacetime length of that path.

The same thing holds in curved spacetime. To get elapsed proper time you calculate the spacetime interval along the worldline. The difference is that in flat spacetime there are some simplifying rules that don't hold in curved spacetime. (e.g. straight worldlines can only intersect once and the longest interval is a straight worldline)

 

[PAllen]

Your example of two identical orbits except orientation, such that they intersect twice, they would both show the same time on meeting successive meetings. No Lorentz frame could possibly be applicable over an orbit because you have diverging geodesics that ultimately re-converge. If you stick to observations, they would both visually observe the other's clock slow, then fast.

Now, you could construct non-identical orbits with recurring intersections. These would also be pure inertial motion. Their clocks would drift further and further apart on each intersection.

The general rule in GR is that connecting two events for which time like paths between them are possible, is a path of maximal proper time. This path is a geodesic - i.e. inertial. However, there may be any number of different geodesics connecting those events. In general, these will have less proper time, and there will be non geodesic paths with more proper time than these alternative geodesics.

 

[Mister T]

So it seems to me that the acceleration (and understanding its effects on the situation) are important to the scenario.

The effect can be made negligible. That tells you how important it is.

 

[pervect]

Acceleration has nothing to do with the resolution of the twin paradox.

I disagree. In a flat spacetime, the only way two worldlines with a non-zero relative velocity can intersect more than once is if at least one of them includes acceleration. The naïve expression of the twins paradox involves invoking the principle of relativity that suggests both twins should see the other's clock tick slower than his own, so that when they meet again, each should expect the other to be younger, hence paradox. The resolution is to point out that one twin had to accelerate to turn around and this shifted his frame of reference, which ultimately leads to the situation you describe, of him having a shorter path through spacetime than the other. I often find myself having to explain how acceleration is not relative to people who inevitably suggest that from the traveling twin's perspective it looks like Earth accelerated. So it seems to me that the acceleration (and understanding its effects on the situation) are important to the scenario.

Acceleration has little to do with the twin paradox.

Let me give a longish quote on the power of the "principle of maximal aging" in special relativity. From E.F. Taylor's "A Call to Action". <link>

The twin paradox helps us to predict how a high speed free particle moves in flat space–time. The traveling twin streaks away to a distant star, returning to greet her stay-at-home sister, who has relaxed in an inertial frame between the events of departure and return. The two alternative journeys are most clearly described using a world line, a position versus time plot. When the twins meet at the end of the trip, the reading on the wristwatch of the stay-at-home twin (who follows the natural straight world line) is greater than the reading on the wristwatch of the round-trip twin. The aging between the events of departure and arrival is greater along the straight world line than along every nearby alternative world line between these two events. This example illustrates nature's command to the stone: Follow the world line of maximal aging!

As Taylor explains, in flat space-time, the principle of maximal aging is very powerful. It has close ties with a formulation of physics known as the Lagrangian formulation. (It will be much easier to follow Taylor's paper if one is familiar with Lagrangian mechanics). The principle of maximal aging (or it's equivalent, the principle of least action) is powerful enough to replace Newton's laws as a way to determine a body's natural "force-free" motion.

So far from being "paradoxical", in flat space-time, given the ability to calculate proper time (aging) from the path a particle takes (it's worldline), one can find the path a particle takes from the principle that it ages the most.

The principle is rather similar to the idea that the triangle inequality in space, where the shortest distance between two points is a straight line, and a path that's not straight must be longer. One of the differences between the spatial analogy and the space-time analogy is a matter of sign, in space, the shortest distance between two points is a straight line, in space-time the equivalent worldline has the maximal aging, rather than the shortest aging.

Taylor doesn't do a great job of talking about what happens in curved space-time, alas. He does correctly observes that - in a small enough region - one can still use the principle of maximal aging to do physics, in either curved or flat space-time. But he doesn't stress the important point enough - the point about the requirement that the region be small enough - though he does mention it briefley.

What happens in curved space-time? It's similar to what happens in curved space, to the principle that a straight line, renamed as a "geodesic", is the shortest distance between two points.

On a curved surface, two paths of shortest distance - geodesics, the shortest distance between two points that lies on the surface - can intersect more than once, even though both lines are "straight". It's only in flat space that a pair of straight lines can only intersect in at most one point.

For instance, on the surface of a sphere, curves of shortest distance, geodesics, are great circles. And every pair of great circles intersects twice. So in a curved, non-euclidean geometry, "straight lines" can and do intersect. It's only in flat space-time that two straight lines

If one confines oneself to a small enough region where geodesics don't intersect, nothing important changes between curved space and flat space (or between curved space-time, and flat space-time.

But, as the example of the sphere shows us, in curved space, two geodesics ("straigh lines") CAN intersect. We ask ourselves - if we have two straight lines that intersect in a curve space, do they have to be the same length? If the answer is negative, then the description of a straight line as the shortest distance between two points is only valid locally, not globally. Two unequal paths cannot both be the shortest.

It's fairly easy to come up with an example in which there are two "straight" paths in space that intersect on a curved surface, which are of unequal length. Consider two towns on the opposite side of a mountain. There is a path from one to the other that goes over the tip of the mountain that is "straight". But it's not the shortest distance between the two towns, it's better to go around the mountain, than over the top. In fact, the curve over the top of the mountain is longer than some non-straight paths that go around it, though the straight path that avoids the peak will be shorter than non-straight paths that avoid the peak.

The case of the orbiting clock is the space-time analogy of our two cities on the opposite sides of the mountain. There are multiple geodesic worldines ("straight lines") that connect two events in space-time. Like the case of the mountain, they are not all of equal length, though they are all "straight". For this reason, we replace the principle of "maximal aging" with the principle of "extremal aging".

Extremal aging can be formulated mathematically as Lagrange's equation, and it says that a straight path is shorter than any nearby path, without claiming that it's shorter than all possible paths.

Curved spatial geometries may not be intuitively familiar. But there is nothing paradoxical about them. But certainly, the curved case is more complicated, and some cherished notions have to be revised (slightly) to deal with curved, non-Euclidean geometries. It's not good to pile up the unfamiliarities due to curvature with other unfamiliarityies, such as trying to learn special relativity. Thus, pedagogically, it's better to learn special relativity first - which means avoiding gravity until one has learned how things work without curvature, first.

Let me go back to the point about acceleration for a bit. Acceleration makes a line "not straight". But it doesn't directly cause aging, any more than angles cause distance to increase.

Typically, people who think about "acceleration" as solving the twin paradox are making an important error. They are attempting to hold onto the familiar notions of "absolute time", modifying these deeply held ideas by thinking of some sort of "absolute time" in the background, that's modified by some sort of mechanism.

Unfortunately, this sort of half-measure simply doesn't work. Abosolute time can't be modified or tweaked to work with special relativity. It needs to be abandoned totally, not "patched up".

 

[Arkalius]

Let me give a longish quote on the power of the "principle of maximal aging" in special relativity. From E.F. Taylor's "A Call to Action".

I appreciate the detailed response, and I have no real disagreements with it. I guess the reason I like to point out the acceleration is because understanding what it does is actually what unlocked my ability to understand special relativity (and it seems to be so for many others I encounter). The acceleration may not be of formal importance to the scenario, but it certainly helps for visualizing the resolution to the paradox.

 

[Nugatory]

The acceleration may not be of formal importance to the scenario, but it certainly helps for visualizing the resolution to the paradox.

It may feel helpful to you... but do consider that thinking in those terms is the source of the confusion that led to you to ask the question in post #1 and #6 of this thread.

 

[A.T.]

The acceleration may not be of formal importance to the scenario, but it certainly helps for visualizing the resolution to the paradox.

Acceleration is like changing direction in space-time, so it obviously affects the path in space-time, and hence the accumulated proper time between two events. Just like changing direction in a car affects the distance traveled between two points.

There is just no general relation between the acceleration and the rate of aging at some time-point. Just like there is no general relation between the rate of turning and the speed of a car.

 

[Arkalius]

It may feel helpful to you... but do consider that thinking in those terms is the source of the confusion that led to you to ask the question in post #1 and #6 of this thread.

Not really... curved spacetime follows different rules, which makes applying principles of special relativity more complicated (or impossible). I have a better sense of it now. I think my point about acceleration is that it is the thing that is different between both twins. I think many people see the scenario and while they realize there's acceleration I think many people assume acceleration is relative like velocity and thus is not a "difference" between the two twins.

 

[PeterDonis]

my point about acceleration is that it is the thing that is different between both twins

In the standard twin paradox, which is set in flat spacetime (no gravity), yes, this is true. In fact it's easy to show that for any two objects to separate and then meet up again in flat spacetime, at least one must experience nonzero proper acceleration.

In curved spacetime (gravity present), however, this is no longer the case (as you noted in an earlier post); you can have pairs of twins that separate and then meet up again, with both being in free fall at all times, yet they still show different elapsed times when they meet up again. In these cases you obviously can't explain the different elapsed times by acceleration, since neither twin felt any. This is a key reason why the more general approach described by Nugatory in post #12 is worth learning; it handles all cases in a uniform manner.