# GPS and relativity

sophiecentaur
Gold Member
I'm not going over the same stuff unless I have to. Error is the wrong word but who's nitpicking? You need to do the project because time dilation effects are not apparent. IE relative time and distance intervals are real natural phenomena. Since you intend to introduce that 'nonsense' catch you later.

There are plenty of real errors in a system like this but the main effect of orbital speed and the presence of the Earth's mass is surely not an error but a main feature of the whole system - and would have been included in the very first, idealised 'back of a fag packet' system concept.

I used the word 'nonsense' because I can see no reason how a measured position could possibly be regarded as drifting off at the rate of c. Can you?

There is an atomic clock in each GPS sat. Because of their speed and the different gravity at their alititude, the clocks run 38 microseconds fast compared to Earth. Putting the numbers in the formula gives an error of about 300 meters per microsecond. That would be 11 kilometers per day, give or take. Sounds right to me.

Someone said if the clocks were ALL up in the satellites (they are, most GPS receivers do not contain atomic clocks) it wouldn't matter as they would all be fast by the same amount, and no earth clock was used. That is not true.

The problem is distance = speed X time. The speed is of light is constant. So you need to know the time in order to measure how long the signal takes to reach you. You're relying on the GPS clock then to give you a 'start' and 'end' time for the signal, and if that clock is fast, how can your position be correct?

edit: forgot speed, writing too fast

edit: start/end might not be how it works but the key thing is the receiver uses the GPS clock to do all the timings, and you need at least two values to time something.

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Someone said if the clocks were ALL up in the satellites (they are, most GPS receivers do not contain atomic clocks) it wouldn't matter as they would all be fast by the same amount, and no earth clock was used. That is not true.

The problem is distance = speed X time. The speed is of light is constant. So you need to know the time in order to measure how long the signal takes to reach you. You're relying on the GPS clock then to give you a 'start' and 'end' time for the signal, and if that clock is fast, how can your position be correct?

The 'distance = speed X time issue' has been considered before in this thread: https://www.physicsforums.com/showthread.php?t=543848#13

The systematic error would be in a region of 1 cm. So GPS would still work as well.

Do you agree that, given that Earth-time is not used in position calculations, there would be no 11 km/day GPS position error build up? That is what the whole argument is about.

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sophiecentaur
Gold Member
You do not need to know the absolute time of the signal arrival. Just knowing the difference in arrival time puts you somewhere on a possible hyperboloid with relation to a pair of satellites. The original land based 'hyperbolic' navigation systems (Decca Navigator) used simple LC oscillators in the receivers to get a phase difference between received signals and even they gave an accuracy of a few tens of m. (enough to find lobster pots in the fog, it was said).
That system was simpler because the ship's location was static relative to the (static) transmitters and it was essentially a two dimensional model. Three transmitters would give you your position at the intersection of three hyperbolae. The oldest systems involved reading numbers off a receiver and referring to charts with sets of hyperbolae actually drawn on them. The GPS is more complicated because of the fact that everything is on the move but all that is taken into account. The Earth is rotating at a constant rate relative to the frame of the satellite network and the receiver is taking all this into account whilst it is 'chasing' the relative phases of the signals it's looking at.

The GPS system could, in fact, be looked upon as telling you either your rapidly changing position in 3D or your fixed position on a spinning globe.

Very cleverly implemented but not too hard to grasp the fundamentals what's going on if you relate the problem to the older land-based system ideas. More available satellite signals gives you better accuracy because they each represent a time reference for the others.

atyy
Do you agree that, given that Earth-time is not used in position calculations, there would be no 11 km/day GPS position error build up? That is what the whole argument is about.

This guy gets something similar to what you are thinking?

http://osg.informatik.tu-chemnitz.de/lehre/old/ws0809/sem/online/GPS_presentation.pdf [Broken]
http://osg.informatik.tu-chemnitz.de/lehre/old/ws0809/sem/online/GPS.pdf [Broken]

"When using three satellites for position determination (equation (4)) this corresponds to an error up to 12km per day. For measurements based on four satellites, only the term Δ of equation (5) is affected by clock drifts. To correct this error, the time base of the GPS satellites is modified ... 39 μs.d-1.

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This guy gets something similar to what you are thinking?

http://osg.informatik.tu-chemnitz.de/lehre/old/ws0809/sem/online/GPS_presentation.pdf [Broken]
http://osg.informatik.tu-chemnitz.de/lehre/old/ws0809/sem/online/GPS.pdf [Broken]

"When using three satellites for position determination (equation (4)) this corresponds to an error up to 12km per day. For measurements based on four satellites, only the term Δ of equation (5) is affected by clock drifts. To correct this error, the time base of the GPS satellites is modified ... 39 μs.d-1.

Thanks. This is proof that a normal GPS receiver would determine the position accurately even if clock frequency on satellites wasn't corrected, because a normal receiver uses at least 4 satellite signals so equation 5 in the paper applies (the term Δ in equation 5 is calculated independently of coordinates r, which remain unaffected as the author states).

So the fact that without relativistic time drift compensation, GPS positioning would result in position error of 38000 feet per day and GPS would not be usable is a widespread misconseption, because no device without an atomic clock can afford to use just 3 satellites.

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atyy
Thanks. This is proof that a normal GPS receiver would determine the position accurately even if clock frequency on satellites wasn't corrected, because a normal receiver uses at least 4 satellite signals so equation 5 in the paper applies (the term Δ in equation 5 is calculated independently of coordinates r, which remain unaffected as the author states).

So the fact that without relativistic time drift compensation, GPS positioning would result in position error of 38000 feet per day and GPS would not be usable is a widespread misconseption, because no device without an atomic clock can afford to use just 3 satellites.

That seems correct, very interesting indeed. I certainly had the 38000 ft.d-1 number in my head, but didn't know it had to be qualified by "if 3 satellites are used, whereas in reality at least 4 satellites and a lousy clock are used"

Do knowledgeable folks like sophiecentaur agree?

I have looked into how a gps receiver works and it seems as you say to get a time signal from the satellite and then compare the satellite's signal to one that it generates itself, comparing the two to determine the distance.

If the satellite clock is fast, so time is running faster for the satellite? Would not it therefore produce a compressed signal? And the compression of this signal would continue to increase as time difference continued to build up? So the receiver would get increasingly less accurate? So the GPS error would indeed accumulate? In the same way as the guy who flies close to light speed ages much more slowly than the guys he left behind on his planet?

How does the 4th satellite prevent this? What exactly is the local clock in the Mario Haustein paper?

edit: Oh i get it now, It seems that it doesnt accumulate with >3 satellites.

edit again: No, I don't get it. I'll read it some more. Unless anyone can explain?

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I'd read it some more and knocked up a quick bit of Lego that can determine its position from observing two beacons.

I don't want to comment on my own post but I can't edit it. Basically before I accept the assertion that error would not accumulate I need that formula (4) and (5) from the Haustein paper explained to me - because from where I sit it seems to violating basic engineering principles, so I'm really not sure the guy is correct.

phinds
Gold Member
2021 Award
I think that the error accumulates because position determination involves time differences between clocks on different satellites. I'll try to elaborate on this possibly cryptic statement later today or tomorrow, because, right now, my five-year-old daughter is not letting me concentrate sufficiently to think or type.

George, I look forward to your feedback on this concept.

Suxxor has made some points that I'm not competent to refute and he is convinced that the correction is NOT necessary. This "feels" wrong to me, but I well know how little the universe cares about how I "feel" about things.

Here's the thing: Since engineers hate to complicate things for no reason, why would the designers of the system have added in an unneeded correction?

Even atomic clocks are not perfect. If they were then I think the OP would have been right. For everything to stay synchronized the clocks on each satellite must re-synchronize from time to time with a master clock. The master clock is ground based so if relativistic effects were not compensated for then a satellite that just had its clock synchronized would be out of sync with a satellite that had been synchronized less recently. having satellites out of sync would certainly cause error, though I don't know how much.

http://tycho.usno.navy.mil/gpsinfo.html

Also, the GPS system is designed to provide 4 co-ordinates, not 3. The 4'th being time. If the master clock were space based and time dilation were not accounted for the time co-ordinate would drift. A cheap receiver with a fallible clock would not know how much drift to account for.

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