# Relativity in GNSS (GPS)

I was curious about the use of relativity in GNSS (GPS) systems (since I only know very basic relativity but I wouldn't mind having a good excuse to learn more... ), and I've found some conflicting results. Some say that the relativistic variations in satellite clock speed have a HUGE effect on location accuracy, since microseconds of clock error lead to kilometres of distance error when you're talking about signals traveling at light speed.

However, there's an opposing argument which points out that at least 4 satellites are always used anyway (rather than 3) because of the cheap, inaccurate clocks used for the receiver. The use of 4 satellites allows us to cancel out this receiver clock error and find position. (This fact seems to be agreed upon by both camps -- see the IEEE paper I reference later) Thus, the argument goes, if all the satellites have the same clock error, the same process used to eliminate the receiver clock error should also eliminate (most of) the clock error due to relativistic effects. If you look at the CPS navigation equations on Wikipedia (http://en.wikipedia.org/wiki/Global_Positioning_System#Navigation_equations), it does seem clear that if all the satellite clocks have the same error, it's mathematically equivalent to the receiver clock having error, and so it should cancel out in the calculation.

The result of these conflicting arguments is that claims for the position error due to ignoring relativistic effects range from fractions of centimetres to over ten kilometres. That's a pretty significant difference, and I'd really like to know what the truth is.

By the way, here are some of the places I've been reading from:

http://www.physicsmyths.org.uk/gps.htm
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
"GPS, atomic clocks and relativity" by Allan W. Love (Published in IEEE Potentials, April 1994)

Unfortunately, the two sites which oppose the idea of needing relativity in GNSS calculations seem a little less than reputable (they also appear to reject the idea of special relativity altogether in other articles). However, their arguments on this particular front make sense to me. That is, the time error should cancel out in the calculation process in exactly the same way it cancels for the receiver clock error.

That said, my gut also tells me that that's probably an over-simplification, and it may be that clock error due to relativistic effects is still important. Error canceling out looks nice on paper, but how well it works in practice may be a completely different story. I suspect the position error might come out to be significant, even if it's not 11 km per day.

Summary/(tl;dr):
Does anyone know for certain how important relativistic effects actually are in GNSS systems? I would appreciate references/calculations if you happen to have them.

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russ_watters
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"...if all the satellites have the same clock error."

You see, you don't need to know the nuts and bolts of GPS position calculations to know that Relativity is required for it to be accurate: you can't keep the clocks synchronized without Relativity.

And even if you do want to get into the nuts and bolts, you'll have to be speculating about how a different system might work. Such arguments tend to be unclear and ad hoc; an endless string of "well what if it did this instead?" questions.

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"...if all the satellites have the same clock error."

You see, you don't need to know the nuts and bolts of GPS position calculations to know that Relativity is required for it to be accurate: you can't keep the clocks synchronized without Relativity.

Do you mind elaborating? The impression I got was that if the satellites are moving with the same speed and distance from the earth (a quick search tells me they are?), then the relativistic effect on their clock speed should be the same. If that's true, then the satellite clocks should (ideally) all stay synced up with each other, regardless of whether relativistic effects are present or not, shouldn't they? And even if the speeds/radii are slightly different from satellite to satellite, the error should still cancel out to some extent, shouldn't it?

Am I misunderstanding something about the relativistic changes in clock speed?

russ_watters
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No two clocks are identical, so you have to actively and continuously correct their synchronizations. It might only be a few nanoseconds a day, but it would add up.

So the real GPS system keeps the clocks synchronized with a ground station, so the 38ms a day must be (and is) corrected for. So while it may be possible to do it a different way and have it work, for the REAL GPS system, the correction is programmed-in to its operation and the system likely wouldn't even work at all if the predicted clock error had been off by that much (a program probably would have discarded such a faulty reading).

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russ_watters
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To say it another way: the question of whether the accuracy of GPS depends on Relativity is a non sequitur to the question of if GPS proves Relativity and conversely if Relativity is needed in the real GPS system. It is irrelevant because GPS proves Relativity much more directly, based on its clock synchronization method.

Sorry for dragging up this thread, but I had a chance to talk to some engineers who actually work on GPS systems, and I thought the results of that discussion were worth posting. Doing Google searches for things like "relativity and GPS" seems to lead to a lot of over-simplified and incorrect information, so I hope this will lead people in the right direction...

First of all, while I wouldn't endorse the physicsmyths website because there's a fair bit of crackpot material on there, we came to the conclusion that <<Crackpot Link Deleted>> is almost certainly correct... in a sense. That is, in an idealized GPS system, if all the satellite clocks are wrong by the same amount (i.e. 38 μs/day due to relativity), it wouldn't cause very much positioning error.

That kind of error where the satellite clocks are all off by the same amount is almost identical to error in the clock within the receiver, and receiver clock error is something that real GPS systems routinely eliminate (by using at least 4 satellites rather than 3) because the receiver clocks are cheap and not very accurate. Thus, the claim that ignoring the 38 μs/day of relativistic error in the satellite clocks will lead to a positioning error of 11 km/day is almost certainly wrong. That 38 μs won't stop you from getting an accurate position.

However, that's not the whole picture. While the crackpot sites trying to disprove relativity are likely correct that the whole 11-km-of-position-error-per-day-if-we-ignored-relativity argument is wrong, GPS is still a great example of a real world system which wouldn't work without relativity. There are a few reasons for this.

First of all, the GPS is system is not ideal. The statement that all satellite clocks have the same error of 38 μs/day because of relativity is actually incorrect. The earth is not a perfect sphere, and satellite orbits are not perfectly circular. Both of these things lead to fluctuations in the relativistic clock error that GPS satellites experience; on average they'll run faster by about 38 μs/day, but because of small variations in speed and gravity, the clock error will fluctuate a bit around that 38 μs/day mark. These fluctuations are not the same from satellite to satellite, and so they don't cancel out like the average 38 μs/day error does.

The result is that GPS systems constantly have to make position corrections of up to ~10 m to account for these fluctuations. One of the engineers actually checked this on an real receiver, so this isn't just the result of some simple calculations; real GPS receivers in use today are actually making position corrections of anywhere from 0 to 10 m to account for the fluctuations in satellite clock speed due to fluctuations in gravity and orbit geometry. Sure, it's not 11 km of error, but it's certainly not negligible. If they were to ignore these errors, the GPS system would not work as well as it does. Whether or not you buy the 11-km-of-error-per-day argument (which I don't), GPS system operation does indeed depend on relativistic corrections for accuracy.

So position measurements do depend on relativistic corrections, just not to the extent that's usually claimed. However, there's another problem with the argument that the 38 μs of error per day just cancels out in the position calculation: position calculations aren't the only thing that GPS satellite clocks are used for. GPS satellites are actually used in the calculation of absolute time: UTC (Coordinated Universal Time). For this application, 38 μs/day of error would certainly have a large impact. For this reason, GPS satellites are sent up with a slightly reduced clock frequency so that once they're in orbit, that 38 μs/day is eliminated.

Additionally, while we didn't come to a solid conclusion on this, it should be noted that relativistic error probably also has an effect on other aspects of the system as well. The satellite has to beam down information to us, and clock accuracy is pretty important when it comes to sending this information (e.g. analog-to-digital and digital-to-analog conversion). Relativistic clock error may cause distortions in the information that gets sent down to earth, which would of course have a negative impact on how well the GPS system can work. Again, I'm not sure about this part, but it does seem likely that there would be other aspects of the real-world system which wouldn't function correctly if we ignored the relativistic clock errors.

Summary/(tl;dr):
Real world GPS satellites are sent up with a slightly reduced clock frequency to account for the average 38 μs/day error due to relativity. The crackpot sites trying to disprove relativity are likely right about one thing: this 38 μs/day probably doesn't have much impact on the position calculation because it's the same for all satellites. The statement that ignoring relativity would lead to 11 km of position error per day is probably quite wrong.

However, real-world GPS systems are pretty complicated, and it's not as simple as all the satellite clocks running slow by 38 μs/day. The fact that earth is not a perfect sphere and satellites do not orbit in perfect circles means that the relativistic clock error fluctuates around that 38 μs/day mark. These fluctuations do lead to significant errors because they're not the same for each satellite, and real GPS systems have to constantly adjust the position by anywhere from 0 to 10 m to account for those fluctuations.

GPS clocks are also used in the calculation of absolute time (UTC) for which 38 μs of error per day would be a big deal. It's also pretty likely that the 38 μs/day of error would have an impact on other aspect of the system, such as communication between the satellite an earth. Even if the 38 μs/day cancels out for position calculations (which it probably does) it will still have an impact on other aspects of the system.

In short, the common claim that GPS would gain 11 km of position error per day if relativity were ignored is probably wrong. However, relativity is an important part of GPS systems, and they would not work correctly if relativity were ignored. GPS systems are still a great example of relativity used in day-to-day life, but probably not for the reasons typically claimed.

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Dale
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That kind of error where the satellite clocks are all off by the same amount is almost identical to error in the clock within the receiver, and receiver clock error is something that real GPS systems routinely eliminate (by using at least 4 satellites rather than 3) because the receiver clocks are cheap and not very accurate. Thus, the claim that ignoring the 38 μs/day of relativistic error in the satellite clocks will lead to a positioning error of 11 km/day is almost certainly wrong. That 38 μs won't stop you from getting an accurate position.
While the primary civilian usage of GPS requires 4 satellites and utilizes a cheap reciever clock, many military applications use an accurate reciever clock to ensure navigation availability with only 3 satellites. So the statement that ignoring the relativistic error in the satellite clocks is correct, even if it doesn't apply to all GPS use scenarios.

While the primary civilian usage of GPS requires 4 satellites and utilizes a cheap reciever clock, many military applications use an accurate reciever clock to ensure navigation availability with only 3 satellites. So the statement that ignoring the relativistic error in the satellite clocks is correct, even if it doesn't apply to all GPS use scenarios.

I wasn't aware of that, but I guess it makes sense.

Still, unless I'm mistaken and that 38 μs/day does lead to very large errors even in the four-satellite case, I think it's somewhat misleading to claim that kilometres of position error would result from ignoring relativity, because that's not true for the GPS systems most people are familiar with where four satellites are used rather than three.

By the way, it's still under the assumption that 4 satellites are used, but the wikipedia article on Error analysis for the Global Positioning System seems to agree with my post above (emphasis added by me):

When combining the time dilation and gravitational frequency shift, the discrepancy is about 38 microseconds per day, a difference of 4.465 parts in 1010.[11] Without correction, errors in the initial pseudorange of roughly 10 km/day would accumulate. This initial pseudorange error is corrected in the process of solving the navigation equations. In addition the elliptical, rather than perfectly circular, satellite orbits cause the time dilation and gravitational frequency shift effects to vary with time. This eccentricity effect causes the clock rate difference between a GPS satellite and a receiver to increase or decrease depending on the altitude of the satellite.

PeterDonis
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Does anyone know for certain how important relativistic effects actually are in GNSS systems? I would appreciate references/calculations if you happen to have them.

The definitive reference for relativity and the GPS system is this article by Neil Ashby on the Living Reviews in Relativity site:

http://relativity.livingreviews.org/Articles/lrr-2003-1/fulltext.html [Broken]

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