# GPS & Relativity: Position Error Lower Than 38000 Feet?

• Suxxor
In summary: I think it's called a 'fix' or 'alignment'. So as long as the receiver always has the satellites in view, it should never lose track of time and therefore any relativistic effects would be negligible.

#### Suxxor

I was watching QI http://www.youtube.com/watch?v=gU7McFm_cKQ&t=7m09s and prof Brian Cox said that time runs roughly 38000 ns per day faster on GPS satellites than on the ground.

From that he concluded that since light travels roughly 1 foot per nanosecond, GPS would generate a positional error 38000 feet per day, if relativity effects weren't compensated.

The same conclusion is brought here: http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

***

This kind of reasoning seems incorrect, because GPS satellites are all located in a very similar gravitational field and the ground clock on receiver is constantly reset to follow the more accurate time signals from the satellites ( http://electronics.howstuffworks.com/gadgets/travel/gps3.htm ), so the absolute time on ground would not matter.

As a result, the position error of GPS would be much lower than 38000 feet per day. What do you think?

I think I don't understand your point. As described in the articles, the relativistic effects are compensated, so the error is indeed much less. Are you trying to claim that the compensation is not necessary??

Bill_K said:
I think I don't understand your point. As described in the articles, the relativistic effects are compensated, so the error is indeed much less. Are you trying to claim that the compensation is not necessary??

Yes, pretty much.

Actually I was trying to say that the 38000 feet per day position error drift estimate, if the relativistic effects weren't compensated, is wrong. The error would never accumulate like that over time.

Bill_K said:
I think I don't understand your point. As described in the articles, the relativistic effects are compensated, so the error is indeed much less. Are you trying to claim that the compensation is not necessary??

Yes, pretty much.

Actually I was trying to say that the 38000 feet per day position error drift estimate, if the relativistic effects weren't compensated, is wrong. The error would never accumulate like that over time. But is this reasoning justified?

Suxxor said:
Yes, pretty much.

Actually I was trying to say that the 38000 feet per day position error drift estimate, if the relativistic effects weren't compensated, is wrong. The error would never accumulate like that over time.

Why not? Show us your calculation, rather than just stating it is wrong.

Zz.

Let's try to be more clear.

There's an oscillator of some sort (cesium, rubidium, it varies) on the satellite clocks. N cycles of this frequency are considered to be a second. The value of N on the satellite clocks has deliberately been set to an incorrect value, or rather a value that would not be correct on a ground clock, to keep the space- clocks synhchronized with accurate ground clocks.

If you did not compensate the oscillators, the primary effect would be that the space-clocks would not stay synchronized with their ground counterparts.

While the usual GPS receivers do not use accurate ground clocks, such accurate atomic clocks (of the same sort that are on the spacecraft ) do exist.

It would be possible to find a software work-around for the fact that the space-clocks were not keeping synch with the ground clocks, but the difference in synchronization would be obvious and significant, and of the magnitude quoted in the literature.

I think what the OP means is that a GPS receiver constantly syncs it's clock with the clock on the satellite anyway, so that the clock on the receiver would be accurate enough for any kind of useful positioning. As I understand, he is asking that considering the way the standard GPS receiver works, shouldn't there not be any build up of a positioning error even if the relativistic effects aren't accounted for? Because if the error isn't accounted for, it has very little time to rack up and since the average clock in a GPS receiver isn't very accurate anyway, would the error caused by relativity be of any significance between synchronizations?

chingel said:
I think what the OP means is that a GPS receiver constantly syncs it's clock with the clock on the satellite anyway, so that the clock on the receiver would be accurate enough for any kind of useful positioning. As I understand, he is asking that considering the way the standard GPS receiver works, shouldn't there not be any build up of a positioning error even if the relativistic effects aren't accounted for? Because if the error isn't accounted for, it has very little time to rack up and since the average clock in a GPS receiver isn't very accurate anyway, would the error caused by relativity be of any significance between synchronizations?

As I understand it, the clock in the receiver doesn't need to be that good as it is basically only comparing the relative times / phases of the signals received from all the satellites it can see. It's the relative times that gives the position.

chingel said:
I think what the OP means is that a GPS receiver constantly syncs it's clock with the clock on the satellite anyway, so that the clock on the receiver would be accurate enough for any kind of useful positioning. As I understand, he is asking that considering the way the standard GPS receiver works, shouldn't there not be any build up of a positioning error even if the relativistic effects aren't accounted for? Because if the error isn't accounted for, it has very little time to rack up and since the average clock in a GPS receiver isn't very accurate anyway, would the error caused by relativity be of any significance between synchronizations?

Thanks. That is precisely what I meant. It seems that the huge position error accumulation estimate of 38000 feet per day is nonsense, because it is based on the assumption that receiver uses absolute local time, thus error accumulates. Whereas in reality receiver resets it very frequently using signal from one of the satellites.

sophiecentaur said:
As I understand it, the clock in the receiver doesn't need to be that good as it is basically only comparing the relative times / phases of the signals received from all the satellites it can see. It's the relative times that gives the position.

Yeah, so the errors would be caused by desynchronization between satellite clocks rather than desynchronization between the satellite and ground times.

Because the satellites are all located at similar distance from the center of Earth, thus similar gravitational field and moving with similar velocity, relativistic effects would be also similar on all satellites. So relativistic effects wouldn't cause satellites to desync with each other and that is all that counts on receiver position calculation.

ZapperZ said:
And in case anyone is interested in the gory detail of this, one can read this paper:

Zz.

BTW, the paper describes the relative time shift between orbital and ground based reference frames (pages 15, 16), but does not state how the GPS position error would have behaved, if clock frequency wasn't compensated according to EQ 36.

It could be that the compensation was put to place to simplify syncing with the ground station and because GPS is also used for time transfer, so it wouldn't be nice if time would be running slightly faster on satellites than on the ground.

But positioning could work nearly as well without the compensation. The paper doesn't state opposite.

Suxxor said:
BTW, the paper describes the relative time shift between orbital and ground based reference frames (pages 15, 16), but does not state how the GPS position error would have behaved, if clock frequency wasn't compensated according to EQ 36.

It could be that the compensation was put to place to simplify syncing with the ground station and because GPS is also used for time transfer, so it wouldn't be nice if time would be running slightly faster on satellites than on the ground.

But positioning could work nearly as well without the compensation. The paper doesn't state opposite.

This is puzzling. Are you saying that even if there is timing error, there would be NO positioning error?

Try it. It takes something with velocity v, a time t to go travel a distance x. If all you have are v and t, and you wish to use those to find x, are you saying that an error in t will NOT produce an error in x?

Again, as I've asked earlier, show your own calculation to prove your point. All you have done so far is make some vague, hand-waving argument. This will be the last time I will ask that before this thread becomes a speculative, unverified topic that is in violation of the PF Rules that you had agreed to.

Zz.

ZapperZ said:
This is puzzling. Are you saying that even if there is timing error, there would be NO positioning error?

Try it. It takes something with velocity v, a time t to go travel a distance x. If all you have are v and t, and you wish to use those to find x, are you saying that an error in t will NOT produce an error in x?

$x = 20200 km = 20 200 000 m$ (approximate height of the orbit)
$c = 3 \cdot 10^{8} m/s$
$\Delta \tau= 38 \mu s / day = \frac{38 \cdot 10^{-6} s}{24 \cdot 60 \cdot 60 s} = 4.4 \cdot 10^{-10} s/s$ (relativistic time drift on satellite compared to ground; taken from literature)

$t_{without \:relativistic\: effects} = x / c$
$t_{measured} = t_{without\: relativistic\: effects} + t_{without\: relativistic\: effects} \cdot \Delta \tau = t_{without\: relativistic\: effects} (1 + \Delta \tau)$
$x_{measured} = c \cdot t_{measured} = c \cdot t_{without\: relativistic\: effects} (1 + \Delta \tau) = c \cdot x / c \cdot (1 + \Delta \tau) = x \cdot (1 + \Delta \tau)$

$\Delta x = x_{measured}-x = x \cdot \Delta \tau = {20,2 \cdot 10 ^6 m} \cdot 4.4 \cdot 10^{-10} = 8.9 \cdot 10^{-3} m = 8,9 mm$

This error of distance caused by relativistic time drift is negligible - under one cm. Even if the error would be 10 times as large, GPS would still be as usable.

ZapperZ said:
Again, as I've asked earlier, show your own calculation to prove your point. All you have done so far is make some vague, hand-waving argument. This will be the last time I will ask that before this thread becomes a speculative, unverified topic that is in violation of the PF Rules that you had agreed to.

Zz.

The nature of my argument was that there is no systematic error accumulation.

Brian Cox claimed that the error accumulation is $\Delta\epsilon = c \cdot \Delta \tau = 38000 feet / day$, where $\Delta \tau= 38 \mu s / day$.

But since absolute ground time is being synced constantly (link to prove that is in first post),
actually $\Delta \tau= 0 \mu s / day$ and $\Delta\epsilon = c \cdot 0 = 0 feet / day$.

BTW ZapperZ, I haven't waved my hand once, only slapped my forehead a couple of times after reading your posts in this thread.

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A simple page I found talking about GPS:

http://www.kowoma.de/en/gps/positioning.htm [Broken]

The clock on the receiver isn't accurate enough to measure the time it took to receive the signal with high accuracy anyway, it just uses relative times, hoping the clock is accurate enough that it doesn't change it's speed too much and the relative time proportions are relatively accurate. Then it calculates the position and also corrects the clock by making the spheres of the distances from the satellites intersect.

So it seems that the receiver isn't just synchronizing the clock from time to time, it is actually figuring out the correct time based on the relative times of the signals for each position calculation, otherwise it would have much less accuracy. This should mean that there would be no error build-up.

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pervect said:
Why would that be?
It seems to me the poster has a point. If the clocks are frequently synchronized then, even if we were to remove the adjustment for relativistic effects, systemic clock errors would not accumulate.

Do you disagree with that?

Furthermore if this is actually true then it would be interesting to know if we leave out the relativistic corrections how much it would actually matter.

I'm REALLY glad you guys didn't design the GPS system since I prefer driving on the road to driving through corn fields and buildings which is what would happen if the relativistic effects were not accounted for.

phinds said:
I'm REALLY glad you guys didn't design the GPS system since I prefer driving on the road to driving through corn fields and buildings which is what would happen if the relativistic effects were not accounted for.

Did you use smaller font for all the reasoning to back up your argument? Because I cannot see any.

To prove your point, you'd have to give at least some - in particular show that the two simple calculations in post #13 are wrong, incomplete or based on the wrong assumptions. Even Einstein had to back his theories up with reason for others to accept them. (And I believe in you. You are just like Einstein.)

What they're saying is: If satellite and ground based clocks are initially synched, separated, and put into operation without correcting for relativistic effects the accuracy of the system would fail by 1 foot/ns. That's a fact. Read Ashby's paper or better yet do this project on the GPS.

Student project on the Global Postioning System [Taylor and Wheeler Exploring Black Holes]

Here's a video by the Perimeter Institute regarding GPS & GR/SR

Source- http://www.perimeterinstitute.ca/Perimeter_Inspirations/General/Perimeter_Inspirations/ [Broken]

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Brucep said:
What they're saying is: If satellite and ground based clocks are initially synched, separated, and put into operation without correcting for relativistic effects the accuracy of the system would fail by 1 foot/ns. That's a fact.

Who is saying that? No-one is saying the system would fail 1 foot/ns. Some sources and Brian Cox is claiming the system would fail 38 000 feet per day which is not the same as 1 foot per ns.

1 foot/ns is the speed of light and means that 1 ns desync between GPS signals means approximately 1 foot of position error calculated in GPS receiver. This applies to all kind on GPS signal desynchronizations, not only those caused by not correcting relativistic effects.

This thread argues that since relativistic effects do not cause desynchronization build-up between satellite signals, there would not be GPS position error build-up and GPS would still work.

So far there is not a single argument in this thread to rebute that claim.

Brucep said:

The paper has been recommended before in this thread, but does not contain information how position error would behave if correction would not be used.

Brucep said:
or better yet do this project on the GPS. Student project on the Global Postioning System [Taylor and Wheeler Exploring Black Holes]

The project has a statement
"Of course there is a wrinkle: The clock in your hand-held
receiver is not nearly so accurate as the atomic clocks carried
in the satellites. For this reason, the signal from a fourth
overhead satellite is employed to check the accuracy of the
the hand-held receiver to process GPS signals as though it
contained an atomic clock.
"

So, the paper admits that ground clock is in sync with one of the satellites.

But then suddenly on page A-4, it starts calculating the timing differences on satellite and ground and translating this to position error. How can it do that, when just before it concluded that ground time does not matter for position error, because it is synced by the satellite? This is a contradiction.

"To one signiﬁcant ﬁgure, the satellite clocks and Earth clock go
out of synchronism by about 50 000 nanoseconds per day due to their difference in altitude alone.
"

"In 1 nanosecond a light signal (or a radio wave)
propagates approximately 30 centimeters, or about one foot. So a difference of, say, hundreds of nanoseconds will create difﬁculties.
"

***

IMO, the argument is now even stronger than before, because we know from multiple sources, that only satellite signals are used on position calculation and ground time is not at all used.

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Suxxor:
"Who is saying that? No-one is saying the system would fail 1 foot/ns. Some sources and Brian Cox is claiming the system would fail 38 000 feet per day which is not the same as 1 foot per ns."

Relativistic physics is saying the GPS system would fail at 1 foot for every ns the GPS satellite and ground based clocks fell out of synch. That would be 38,403 ns/day.

Everything else about your argument is obfuscating nonsense. You argue that the correction isn't needed for the GPS to work. Complete nonsense.

Suxxor said:
Did you use smaller font for all the reasoning to back up your argument? Because I cannot see any.

To prove your point, you'd have to give at least some - in particular show that the two simple calculations in post #13 are wrong, incomplete or based on the wrong assumptions. Even Einstein had to back his theories up with reason for others to accept them. (And I believe in you. You are just like Einstein.)

I don't think Einstein would have felt compelled to waste his time refuting nonsense. I certainly don't.

It is puzzling when someone drops by to say how the whole discussion is too ridiculous for him and how he couldn't care less to answer, yet still finds the time to say just that. Then please don't waste your time anymore and don't say anything anymroe.

Brucep said:
Relativistic physics is saying the GPS system would fail at 1 foot for every ns the GPS satellite and ground based clocks fell out of synch. That would be 38,403 ns/day.

Everything else about your argument is obfuscating nonsense. You argue that the correction isn't needed for the GPS to work. Complete nonsense.

Perhaps you mean that the speed of light says that when the clock is wrong by 1 ns the inaccuracy would be 1 foot, not relativistic physics? Maybe you meant to say that according to relativity the time on the satellites advances 38 microseconds faster each day?

How exactly is the correction needed for GPS to work?

Searching online, I found that good and accurate quartz clocks are wrong every day by +/- 0,02 seconds. That is 20 milliseconds or 20 000 microseconds each day. A 38 microsecond deviation would get lost somewhere in there. GPS relies on only the relative times and uses them to calculate an intersecting point.

chingel said:
It is puzzling when someone drops by to say how the whole discussion is too ridiculous for him and how he couldn't care less to answer, yet still finds the time to say just that. Then please don't waste your time anymore and don't say anything anymore.

I think he has a point though.
I haven't determined exactly what you are arguing about. Is it the fact that clocks on satellites are not in sync with the ground? Is it that the difference in sync would produce laughably wrong results, if not corrected for? Or is it that, by compensating for this, it is easy (i.e. very cheap) to produce a receiver that gives very good answers? You just seem to be getting ratty with each other by not understanding where you're each coming from. In your own terms you could both / all not be too wrong at all.

Feedback is a wonderful thing and it is one of the secrets of the high accuracy of the system. The transmitter clocks are constantly being corrected from the ground by looking at the positioning errors at reference sites on the ground. The clocks in receivers just don't need anything special in the way of absolute timing. They just need to be able to look at differences. Not trivial, of course, but very doable using quartz oscillators.

It might be more fruitful to be discussing how such incredibly weak signals, received from the satellite network, can be processed by a handheld receiver with an antenna that is built into the case of an affordable mobile phone. The effective noise bandwidth must be way below 1Hz to get the system to work. Thats why it can take so long for a receiver to lock on, if it's been out of touch for any length of time (that's where the receiver clock accuracy comes in.

btw, I took exception to the statement about "triangulation" early on in that Utube movie. There is no directional information used in GPS - as we all know - and there was no real description about how GPS works at all. The SR and GR bits were useful, though.

Brucep said:
Relativistic physics is saying the GPS system would fail at 1 foot for every ns the GPS satellite and ground based clocks fell out of synch. That would be 38,403 ns/day.

But ground based clock is not used at all! How can it cause an error then? You have missed the point.

Brucep said:
Everything else about your argument is obfuscating nonsense. You argue that the correction isn't needed for the GPS to work. Complete nonsense.

It is not. Several posters in this thread have understood the issue: #8, #9, #15, #16 .

The problem is, you have not understood the argument, but rant at me claiming I talk nonsense. If several other people understand the issue and you don't, don't you think the problem could be your own limitations.

I feel dissapointed, because I took time to respond to your post and explain the same thing again. None of it reached the target.

sophiecentaur said:
I think he has a point though.
I haven't determined exactly what you are arguing about.

We are arguing whether GPS positioning would work if relativistic time drift in satellites wasn't compensated.

One camp is saying that the position error build-up would be 38 000 feet per day (without compensation). The argument is based on the fact that ground clock would go out of sync with satellite clocks.

The other camp (me and few other people) are saying that there would be no error build-up, since position calculation involves only satellite signals and ground clock is not at all used.

***

The problem is that the 38000-feet camp does not understand the basis of our argument.

sophiecentaur said:
I think he has a point though.
I haven't determined exactly what you are arguing about. Is it the fact that clocks on satellites are not in sync with the ground? Is it that the difference in sync would produce laughably wrong results, if not corrected for? Or is it that, by compensating for this, it is easy (i.e. very cheap) to produce a receiver that gives very good answers? You just seem to be getting ratty with each other by not understanding where you're each coming from. In your own terms you could both / all not be too wrong at all.

It was very clear to me from the first post what the question was. He saw a video, where someone said that if relativity were not taken into account, the GPS would accumulate an 38000 feet error per day. The OP said that he thinks absolute time isn't important and such an accumulation wouldn't happen, then he asked what we think of it.

I said I think the error wouldn't accumulate either. Then someone said that it would and we would be driving in corn fields if we don't take relativity into account, which I am arguing against, but unfortunately the guy who said it doesn't want to discuss it because it is nonsense to him.

Then there is also someone else who says it is nonsense that correction for relativity isn't needed to make GPS work, but it seems he doesn't understand the question either.

I don't understand why is such a hard time given to someone who just asks a simple question. I think the question was very clear.

chingel said:
I don't understand why is such a hard time given to someone who just asks a simple question. I think the question was very clear.
I am not surprised, often on this forum when someone has a question about relativity the first reaction of many here is to be defensive and assume relativity as a theory is questioned. It is rather annoying and a hostile learning environment.

I do not think anybody here questions that clocks in GPS satellites run faster due to the gravitational field of the Earth even if we subtract the time delay of relative motion.

The question here is if we would not compensate for relativistic and gravitational effects would the error accumulate over time. That to me seems a legitimate question.

The argument is simple for me. GR & SR predict the GPS failure rate would be ~ 1 foot per nanosecond with no relativistic correction and perform, as designed, with the correction.

Suxxor says that's not true. The GPS would function just fine without the relativistic correction. Fortunately the GPS operation proofs he's wrong while being considered a best test for GR in the weak field.

I did provide a detailed explanation with the link to the GPS project in Edwin Taylor's and John Wheeler text Exploring Black Holes.

In geometric units, used in the project, this is the weak field approximation derived from the Schwarzschild metric.

dt_satellite/dt_earth = 1 - M_earth/r_sat - v^2_sat/2 + M_earth/r_earth + v^2_earth/2

The GPS correction

dt_sat - 4.4453EE-10 = dt_earth

86,400 seconds/day * 4.4453EE-10 = 38,407 nanoseconds/day

Suxxor is claiming that doesn't matter. The reason you have to account for this correction is that light travels 1 foot per nanosecond and GR says it's required. The GPS is unique as a weak field experiment where the miniscule effects of gravity can't be ignored.

I called his comments nonsense because they are. He intimated that the GR correction is so small that it would hide inside 'other corrections'. Ignoring the fact that all the signals travel at 1 foot per nanosecond. So what you have is somebody who knows very little about the literature trying to challenge it.

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I don't think anyone could question the straightforward predictions of SR & GR in this context. Any serious challenge would not be worth this length of thread.
There must have been a misunderstanding coupled with a bit of testosterone, I think.

Please keep the discussion dispassionate and free from insult, either explicit or implicit.

Brucep said:
I called his comments nonsense because they are. He intimated that the GR correction is so small that it would hide inside 'other corrections'. Ignoring the fact that all the signals travel at 1 foot per nanosecond. So what you have is somebody who knows very little about the literature trying to challenge it.

I don't think his argument is that an error exists due to SR/GR, I think he is implying that the error is static, not compounding. This appears to be more of an engineering/computing than a physics question.

Others correct me if I am mistaken, but the reason the error compounds is because in order to determine your location, the GPS receiver uses distances to the satellites. Since the satellites constantly change position relative to the receiver (they are orbiting the earth), you must determine distance through timing, which will necessarily involve your planetside clock and the relativity corrections that go with it.

I actually think everyone here is wrong - if uncorrected the error is neither static nor strictly compounding; it is semi-periodic (there is a maximum error for satellite position, even in a system of dozens of satellites).

Brucep said:
I called his comments nonsense because they are. He intimated that the GR correction is so small that it would hide inside 'other corrections'. Ignoring the fact that all the signals travel at 1 foot per nanosecond. So what you have is somebody who knows very little about the literature trying to challenge it.

You are still not getting the point. I don't think it can be explained more clearly than chingel and Passionflower have done in posts #28 and #29.

I would not mind getting adequate counter-arguments, but this is getting frustrating.

Nevertheless, I'll try explaining one more time.

***
I accept that time on satellites would go 38 us per day faster on the satellites than on ground without the correction - no question about that.

If ground time was used in GPS position calculation, the position error would indeed build up 38 000 ns/day.

However, ground time is not used for position calculation. Even your own Taylor and Wheeler paper confirms that. That's why they use 4 satellites instead of 3.

Do you see now what I mean? Ground time is not used. So for position calculation, time difference between ground and satellite does not matter. Because ground time is not used. Get it?

Suxxor said:
This kind of reasoning seems incorrect, because GPS satellites are all located in a very similar gravitational field and the ground clock on receiver is constantly reset to follow the more accurate time signals from the satellites ( http://electronics.howstuffworks.com/gadgets/travel/gps3.htm ), so the absolute time on ground would not matter.

I wonder whether the spheres used in the resetting take relativity into account?