GPS relativity error - really?

[Prometeus]

There are many popscience articles about relativity and how relativity is essential for GPS and in many is mentioned something like this: If GPS wasnt corrected for time dilation predicted by GRT, there would be an error of 12 kilometers in GPS positioning per day.

But when I checked how GPS really works, it seems to me, that this statement is wrong. It is wrong because the actual GPS is using at least 3 satellites for positioning and 4th satellite for time. In reality the time on satellites is corrected by ground signal at least every 30 seconds, but even if the time wasnt corrected for 24 hours, there would be no significant GPS positioning error, because all four satellites are on the same orbit height and are experiencing practically the same gravitational time dilation.

This 12 km positioning error would happen only in case if for GPS calculation would be used the time from the handheld GPS on Earth, because then there would be a time difference between time on surface of Earth and time on the GPS satellite, but this is not the case in real GPS, because it would be ridiculous to sell user GPS with atomic clocks. It would be very heavy and very expensive compared to cheap solution which is used in real GPS with cheap handheld GPS.

This seems quite simple and common sense to me, but there are no real sources about it, probably because it does not fit the popular media coverage how GPS would not work without GRT. The only serious source I have found is german version of Wikipedia about GPS:
https://de.wikipedia.org/wiki/Global_Positioning_System#Relativistische_Effekte

 

[FactChecker]

The numbers I have seen are only calculations of the time distortion due to the altitude and velocity of a general satelite in the GPS orbit. I think that they do not get into the exact workings of the GPS system. But it wouldn't surprise me if they have not gone to great lengths to examine the accuracy of a hypothetically flawed GPS system.
That being said, I think that your top-level analysis of the situation is not complete. Updating the clock time using a signal that takes time to arrive from Earth and adjusting the calculations accordingly would not be simple. It would be much simpler if the on-board clock is already correctly adjusted to account for relativity. That would make any further adjustments from a ground signal much smaller. I think that the relativity adjustment is not so difficult. It's essentially a constant adjustment. So my guess is that they would have adjusted the clocks to account for relativity. It is the simpler solution.

 

[Prometeus]

The numbers I have seen are only calculations of the time distortion due to the altitude and velocity of a general satelite in the GPS orbit. I think that they do not get into the exact workings of the GPS system. But it wouldn't surprise me if they have not gone to great lengths to examine the accuracy of a hypothetically flawed GPS system.
That being said, I think that your top-level analysis of the situation is not complete. Updating the clock time using a signal that takes time to arrive from Earth and adjusting the calculations accordingly would not be simple. It would be much simpler if the on-board clock is already correctly adjusted to account for relativity. That would make any further adjustments from a ground signal much smaller. I think that the relativity adjustment is not so difficult. It's essentially a constant adjustment. So my guess is that they would have adjusted the clocks to account for relativity.

Updating the clock time using a signal from Earth bases is quite simple, you just account for the time it takes radio signal to get to satellite using simple Newton level math. And in fact this is how it works in real GPS, time on satellites is updated from ground bases at least every 30 seconds in older satellites and less than every 15 seconds in newer types of satellites.

But this is different topic, the main topic is that GPS would work without significant positioning errors for 24 hours without any relativity time correction.

 

[FactChecker]

I would definitely adjust the GPS satellites clocks for the effects of relativity, since it is a constant adjustment based on known altitude and velocity.

Once that is done, any correction from a varying distance to an Earth source would be much smaller, if any. It seems important to keep that adjustment small so that ongoing calculations can be corrected more easily and with greater accuracy.

 

[weirdoguy]

but there are no real sources about it, probably because it does not fit the popular media coverage how GPS would not work without GRT.

Or maybe because your common sense is not correct? It's not only popular media coverege, you'll find it in most textbooks on relativity. Do scientist lie to their readers?

 

[Prometeus]

Or maybe because your common sense is not correct? It's not only popular media coverege, you'll find it in most textbooks on relativity. Do scientist lie to their readers?

Have you actually read what I have been writing or have you just read the title of the thread? Could you please point out specifically what is wrong with my common sense approach?

Inbetween I have found some source in English, which confirms my claim.
<Link to unacceptable reference removed>


It is quite off topic to discuss if scientists do generally lie to their readers, but everybody who is claiming this GPS position error is lying. Could you post some link with serious scientist claiming this GPS error?

 

[Prometeus]

I would definitely adjust the GPS satellites clocks for the effects of relativity, since it is a constant adjustment based on known altitude and velocity.

Once that is done, any correction from a varying distance to an Earth source would be much smaller, if any. It seems important to keep that adjustment small so that ongoing calculations can be corrected more easily and with greater accuracy.

Im not getting how this is relevant to the main topic of the thread - there is no significant GPS position error without relativity correction?

 

[FactChecker]

Corrections, if any, from the ground signal are much smaller if the clocks on the satellites are essentially correct in the first place. That requires an easy adjustment for the effects of relativity. By all accounts, those relativity adjustments are done -- why not? They are easy. It is a matter of speculation how well, without the relativity adjustments, the necessarily larger adjustments from a ground source would work. You would have to work through all the GPS calculations and algorithms to figure that out.

Each GPS satellite has multiple atomic clocks that are accurate to 100 billionths of a second. I do not know if there are cross-checks done to select a correct time from among the clocks, but I assume that is done. (see https://www.gps.gov/applications/timing/)

 

[Prometeus]

Corrections, if any, from the ground signal are much smaller if the clocks on the satelites are essentially correct in the first place. That requires an easy adjustment for the effects of relativity. By all accounts, those relativity adjustments are done -- why not? They are easy. It is a matter of speculation how well, without the relativity adjustments, the necessarily larger adjustments from a ground source would work. You would have to work through all the GPS calculations and algorithms to figure that out.

In reality those relativity clock adjustments are made automatically on atomic clocks on satellites by frequency modulaton, but main topic of this thread is not the time dilation difference between satellites and Earth, but time dilation difference between several satellites clocks on orbit which should be due to relativity theory close to zero even in case of uncorrected time. Therefore claim of GPS positioning error in case of uncorrected time dilation on satellites is not true. It is not matter of speculation, it is just about physics which applies in this case and can be calculated.

Or do you think that there is time dilation difference between satellites with practically same orbital speed and same orbital height?

 

[Ibix]

I think the reason for the ground correction as well as highly accurate clocks is military belt-and-braces approach. If the clock accuracy degrades for some reason (enemy action or unexpected space conditions) then the ground stations still work. If the ground stations fail the space-based clocks should be right anyway. Both need to break to stop the system working.

If I understand correctly, the satellite clocks need to be ultra-precise because the difference in distances to the satellites needs to be precise. The ground based clocks don't need to be too precise because they are used to estimate orbital velocity of the satellites at the present time, and hence the gamma factor. This doesn't require high precision because it doesn't change very fast.

 

[FactChecker]

Or do you think that there is time dilation difference between satellites with practically same orbital speed and same orbital height?

No. That is the same always. On the other hand, the signal delays from Earth are affected by location and unpredictable atmosphere. I would trust the times of redundant, multiple, frequency adjusted atomic clocks more than the time on a signal from ground.

 

[Ibix]

No. That is the same always.

I'm not sure I agree. My understanding is that the satellites aren't all in the same orbit, so their relative velocities change. For example, two could be right next to each other and almost stationary at one time, and later on opposite sides of the world with very high relative speeds.

 

[Ibix]

Inbetween I have found some source in English, which confirms my claim.

Shan't bother quoting the link since it'll get deleted. You can safely ignore anything on the physicsmyths site, as far as I have seen.

 

[CWatters]

But when I checked how GPS really works, it seems to me, that this statement is wrong. It is wrong because the actual GPS is using at least 3 satellites for positioning and 4th satellite for time. In reality the time on satellites is corrected by ground signal at least every 30 seconds, but even if the time wasnt corrected for 24 hours, there would be no significant GPS positioning error, because all four satellites are on the same orbit height and are experiencing practically the same gravitational time dilation.
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The position calculation in the receiver doesn't only use time difference data. It also uses data sent from the satellite about its own position and that depends on the satellite knowing the time accurately. If relativity wasn't taken into account the satellite's knowledge about its own position would drift. So they would all send incorrect data about their position.

Yes they can and did tweak the clocks before launch and yes they do update the satellite's position regularly from the ground but remember this is a military system and there may be reasons why they don't want satellite's to work independently from signals from the ground.

I believe receivers also have to make some additional real time relativity corrections but I don't know why.

Edit: I've no idea why some of the text in this post is small font and italic. There aren't any tags I can see.

 

[FactChecker]

I'm not sure I agree. My understanding is that the satellites aren't all in the same orbit, so their relative velocities change. For example, two could be right next to each other and almost stationary at one time, and later on opposite sides of the world with very high relative speeds.

Good point. Is it the relative speed between the GPS satellites that counts in the system, the relative speed versus a position on Earth that counts, or something else? I am not expert on this and had not thought enough before making the assumption that the correction was constant.

 

[CWatters]

I think most of the correction 47uS? is due to them being in a weaker gravitational field due to their altitude.

 

[jbriggs444]

I think most of the correction 47uS? is due to them being in a weaker gravitational field due to their altitude.

Careful. It is not gravitational field strength that correlates with time dilation. It is the gravitational potential.

 

[Prometeus]

I'm not sure I agree. My understanding is that the satellites aren't all in the same orbit, so their relative velocities change. For example, two could be right next to each other and almost stationary at one time, and later on opposite sides of the world with very high relative speeds.

The satellites are on the same orbit height, but yes, they are not on the same orbit heading in the same direction and yes, therefore there is some time dilation difference between satellites due to special relativity. But there are two reasons why this time dilation difference does not cause error in GPS position calculation:
1. In short term - in the moment of the GPS positioning calculation which takes only few secons to calculate by using the time from one satellite and position of at least three other satellites is the time dilation caused by special relativity irrelevant, because the time needed for GPS calculation is only few seconds. It would be significant, if the calculation of GPS position would take hours to calculate which is in reality obviously not the case.
2. In the long term the satellites are orbiting Earth every 11 hours, so the average time dilation difference between satellites in the time scale of days is zero.

Here is a good picture how the satellites move and how they are used for GPS positioning:
https://en.wikipedia.org/wiki/Global_Positioning_System#/media/File:GPS24goldenSML.gif

 

[PeroK]

In summary:

Premise: It is stated in many science books that GPS requires times adjustments for relativistic effects on clocks on board satellites.

Initial Assumption: this must be to maintain synchronisation of those clocks with each other.

Observation: satellite clocks could (perhaps more simply) be kept synchronised from a master clock on Earth.

Conclusion (1): perhaps the adjustment for relativity is needed for something more than simple synchronisation. In fact, how does GPS really work? What are all the calculations that must be made by satellites and receivers?

Conclusion (2): relativity is bogus and there has been a global scientific conspiracy for 100 years.

I vote for conclusion (1).

 

[Prometeus]

In summary:

Premise: It is stated in many science books that GPS requires times adjustments for relativistic effects on clocks on board satellites.

Initial Assumption: this must be to maintain synchronisation of those clocks with each other.

Observation: satellite clocks could (perhaps more simply) be kept synchronised from a master clock on Earth.

Conclusion (1): perhaps the adjustment for relativity is needed for something more than simple synchronisation. In fact, how does GPS really work? What are all the calculations that must be made by satellites and receivers?

Conclusion (2): relativity is bogus and there has been a global scientific conspiracy for 100 years.

I vote for conclusion (1).

Completely off topic. I have never claimed any of this. I think that what you are doing is called straw man and I am surprised that official science advisor and insights author is producing such low quality post.

 

[russ_watters]

In reality those relativity clock adjustments are made automatically on atomic clocks on satellites by frequency modulaton, but main topic of this thread is not the time dilation difference between satellites and Earth, but time dilation difference between several satellites clocks on orbit which should be due to relativity theory close to zero even in case of uncorrected time. Therefore claim of GPS positioning error in case of uncorrected time dilation on satellites is not true. It is not matter of speculation, it is just about physics which applies in this case and can be calculated.

Or do you think that there is time dilation difference between satellites with practically same orbital speed and same orbital height?

There isn't significant time dilation between the satellites, it's true. But in order to locate itself, the ground receiver has to know where the satellites are. And the ground receiver knows where the satellites are because the satellites tell them. But if the satellite doesn't know what time it is, it also doesn't know where it is, and sends the ground receiver it's own location incorrectly.

The commonly cited 10 or 12km a day is a convenient guess based on the speed of light and clock error (I get 11.4, but why quibble?). But maybe the real deviation is on the base of the triangle and therefore much smaller. Or maybe the lack of synchronization would trigger a program error and no position reading at all. It's impossible to know for sure and it definitely is not important.

I don't know why people bother arguing this (it comes up occasionally). Yes engineers probably could have made a GPS system that worked almost as well without the correction. But so what? Why would anyone care?

 

[PeroK]

Completely off topic. I have never claimed any of this. I think that what you are doing is called straw man and I am surprised that official science advisor and insights author is producing such low quality post.

I would like to see the mathematics:

What data is maintained by the satellites and sent to the receiver?
How do the sateliites maintain this data?
What calculations are carried out by the receiver?

The other thing you are missing is that time errors get multiplied up by the speed of light. The receiver calculates the distance to the sateliite by calculating how long the signal has traveled and multiplying this by $c$.

Suppose a satellite is $1 \mu s$ out of sync, then that gets multiplied up to $300m$. I.e. the receiver thinks the satellite is $300m$ from where it really is/was. So, time errors that you are assuming are irrelevant make a big difference to these distance calculations. That's where the 12km per day comes from: the receivers calculation of the satelittes position if the satelitte's clock is relativistically 24 hours out of sync with an Earth clock.

I think you are, in general, oversimplifying the basic mathematical calculations needed to triangulate with moving satellites. It's not at all obvious how you triangulate with things that are orbitting the Earth.

Until you analyse all the mathematics and calculations required, you cannot conclude that the GPS system would work with a simplified set of data and without relativistic calculations.

 

[Prometeus]

There isn't significant time dilation between the satellites, it's true. But in order to locate itself, the ground receiver has to know where the satellites are. And the ground receiver knows where the satellites are because the satellites tell them. But if the satellite doesn't know what time it is, it also doesn't know where it is, and sends the ground receiver it's own location incorrectly.

I don't know why people bother arguing this (it comes up occasionally). Yes engineers probably could have made a GPS system that worked almost as well without the correction. But so what? Why would anyone care?

The ground receiver gets the time from one satellite and position from at least three other satellites. If there would be time difference between the time received from one satellite and time on other three satellites, then the calculation would be wrong. But there is no time difference between satellites, so the calculation is accurate. This is one quite simple part, which unfortunately only a part of discussion participants understands.

The second, much more complicated topic would be: If the navigation data of the satellites would not include the relativity corrections done by frequency modificantion on atomic clocks, would then a significant difference between alleged and real position of satellites arise over course of let's say 24 hours?

Im not sure, but I would say not, because the time dilation for all satellites is in the time scale of one day the same. Generally speaking due to several other small imperfections the navigation data of the satellites have to be corrected despite the fact, that there is relativity correction done by frequency modification on atomic clocks. But what I am sure is that the navigation data drift caused by missing relativity correction certainly would not produce GPS position calculation error of 12 km per day.

 

[FactChecker]

The fact is that the GPS satellite clocks DO include a frequency adjustment to account for the effects of relativity:
(From https://www.e-education.psu.edu/geog862/node/1714
"Therefore, on balance, the clocks in the GPS satellites in space appear to run faster by about 38 microseconds a day than the clocks in GPS receivers on earth. So, to ensure the clocks in the satellites will actually produce the correct fundamental frequency of 10.23 MHz in space, their frequencies are set to 10.22999999543 MHz before they are launched into space.")

Furthermore, any signal from the Earth is subject to unpredictable delays due to atmospheric effects (see https://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System#Atmospheric_effects). That would complicate using a signal from ground to synchronize GPS clocks.

How the time signals from the ground are currently used in the presence of already corrected clocks is not a reliable indicator of how well they would work without the relativity corrections. Anyone who proposes that the relativity corrections are not necessary would have to do a lot of detailed, expert work to back that up. I don't see any convincing evidence of that sort.

 

[Prometeus]

I would like to see the mathematics:

What data is maintained by the satellites and sent to the receiver?
How do the sateliites maintain this data?
What calculations are carried out by the receiver?

The other thing you are missing is that time errors get multiplied up by the speed of light. The receiver calculates the distance to the sateliite by calculating how long the signal has traveled and multiplying this by $c$.

Suppose a satellite is $1 \mu s$ out of sync, then that gets multiplied up to $300m$. I.e. the receiver thinks the satellite is $300m$ from where it really is/was. So, time errors that you are assuming are irrelevant make a big difference to these distance calculations. That's where the 12km per day comes from: the receivers calculation of the satelittes position if the satelitte's clock is relativistically 24 hours out of sync with an Earth clock.

I think you are, in general, oversimplifying the basic mathematical calculations needed to triangulate with moving satellites. It's not at all obvious how you triangulate with things that are orbitting the Earth.

Until you analyse all the mathematics and calculations required, you cannot conclude that the GPS system would work with a simplified set of data and without relativistic calculations.

OK here are basics and math:
Time difference between Earth and satellites caused by relativistic time dilation is growing by 38 microseconds per day. When we calculate actual speed of satellite of 4 km/s this gets us 4000 m/s *0,000038 s = 0,15 m position drift per day. Which makes average position drift of 4,5 m per month for all satellites. This monthly position drift is well within 8 m acceptable position error stated in official GPS guideline and in reality the position of GPS satellites must be corrected at least once per week, mostly once per day, because they are drifting away also from other reasons than due to relativity.

I don't see any realistic way how to get to GPS position error of 12 km per day due to uncorrected relativity effects. If you study about this topic and make the calculations, you should get the same result.

 

[Prometeus]

The fact is that the GPS satellite clocks DO include a frequency adjustment to account for the effects of relativity:
(From https://www.e-education.psu.edu/geog862/node/1714
"Therefore, on balance, the clocks in the GPS satellites in space appear to run faster by about 38 microseconds a day than the clocks in GPS receivers on earth. So, to ensure the clocks in the satellites will actually produce the correct fundamental frequency of 10.23 MHz in space, their frequencies are set to 10.22999999543 MHz before they are launched into space.")

Furthermore, any signal from the Earth is subject to unpredictable delays due to atmospheric effects (see https://en.wikipedia.org/wiki/Error_analysis_for_the_Global_Positioning_System#Atmospheric_effects). That would complicate using a signal from ground to synchronize GPS clocks.

How the time signals from the ground are currently used in the presence of already corrected clocks is not a reliable indicator of how well they would work without the relativity corrections. Anyone who proposes that the relativity corrections are not necessary would have to do a lot of detailed, expert work to back that up. I don't see any convincing evidence of that sort.

So what? Surely there is 38 microseconds time dilation shift per day and it is corrected by frequency modulation. I have written the same too.

 

[FactChecker]

The issue is not only the position of the satellites. The errors of the GPS ground locations are a function of how the satellite signals triangulate on the sphere of the Earth. That is a very different problem.

 

[Prometeus]

The issue is not only the position of the satellites. The errors of the GPS ground locations are a function of how the satellite signals triangulate on the sphere of the Earth. That is a very different problem.

Yes, this is different but related topic and I have explained and calculated it in post number 25.

 

[FactChecker]

Yes, this is different but related topic and I have explained and calculated it in post number 25.

No. Post #25 does not address that issue.

 

[PeroK]

OK here are basics and math:
Time difference between Earth and satellites caused by relativistic time dilation is growing by 38 microseconds per day. When we calculate actual speed of satellite of 4 km/s this gets us 4000 m *0,000038 s = 0,15 m position drift per day. Which makes average position drift of 4,5 m per month for all satellites. This monthly position drift is well within 8 m acceptable position error stated in official GPS guideline and in reality the position of GPS satellites must be corrected at least once per week, mostly once per day, because they are drifting away also from other reasons than due to relativity.

I don't see any realistic way how to get to GPS position error of 12 km per day due to uncorrected relativity effects. If you study about this topic and make the calculations, you should get the same result.

$38 \mu s \times c \approx 12km$

That's the error that the receiver will make if the time is uncorrected.

 

[Nugatory]

We're two pages in and no closer to a resolution, so we're closing it.

There are a number of good technical papers on the detailed operation of the GPS system; if someone wants to recommend a few we can post links into this thread.

 

[PeterDonis]

This classic article by Neil Ashby is relevant:

https://www.aapt.org/doorway/TGRU/articles/Ashbyarticle.pdf