Calculating GPS Locations: The Impact of Special Relativity

[p4h]

So I'm doing a report on special relativity and GPS and kinda at a loss what to do:

1. I've actually found several sites denoting the need to calibrate for relativistic effects when calculating locations, and I'm writing a report on just the opposite, that how is the system effected / how much difference does it make. So which should I believe? My Physics / Math teacher or for example, this guy: < link to crackpot site deleted by Doc Al >
2. If I am to prove that it has an effect, what equation should I use to see the diffirence? Should I use:
t^1 =γ ((t - v*x)/c^2 )
x^1 =γ (x - v*t)
where γ is Lorentz factor.

Thanks in advance.

PS: Sorry for any Typos, English aint my first language.

 

[Doc Al]

That particular site you linked is a crank site--don't waste your time. (And it's against our policy to allow linking to such sites, so I will delete it. Nothing personal!)

 

[p4h]

That particular site you linked is a crank site--don't waste your time. (And it's against our policy to allow linking to such sites, so I will delete it. Nothing personal!)

No need, edited my post, as I got the answer to the question too.. Just saw his title and figured he might be right so.

 

[mgb_phys]

There are two effects.
Time runs quicker for moving satelites (due to special realtivity) by 7us / day.
Time runs slower for the satelite in lower gravity ( general relativity) by 45us/day.

So overall the effect is the opposite to what you would expect from just the motion.

There was a posting in the forums which compared the two effects , I can't find it but I think it was by astronuc.
edit. found this page giving the details http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

 

[p4h]

I've read that too, but I need the actual math involved, as I can't just write 7 ms / day without any explanation as to why it's 7 ms / day and not 200

 

[p4h]

Apt title; Physicsmyths is crack pottery. If you believed every claim of that site's author, you would have to think that the entirety of modern physics is wrong. The particular page you linked looks like a "strawman" argument (go find a reputable mainstream source, and you can tell us yourself why his reasoning is irrelevant on every point).

Well I did a search on google and more than one site popped up, so I figured there might be an actual debate on the subject, in which case I'd be screwed. Which is why I also posted it here, to see whether it was just some crack site or not.

 

[cesiumfrog]

You must learn to critically read between the lines for yourself.

Well I did a search on google and more than one site popped up, so I figured there might be an actual debate on the subject, in which case I'd be screwed. Which is why I also posted it here, to see whether it was just some crack site or not.

Laziness. Google will always pop up more than one page (don't panic), there's no point asking questions before you've actually read through a few of them (since until then your answers are all just going to be "google finds sites that answer your question").

 

[George Jones]

There are two effects.
Time runs quicker for moving satelites (due to special realtivity) by 7ms / day.
Time runs slower for the satelite in lower gravity ( general relativity) by 45ms/day.

So overall the effect is the opposite to what you would expect from just the motion.

There was a posting in the forums which compared the two effects , I can't find it but I think it was by astronuc.
edit. found this page giving the details http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

In https://www.physicsforums.com/showpost.php?p=731738&postcount=5", I give a simple-minded (without proper justification) analysis of GPS that is based on Project A from the book Exploring Black Holes by Taylor and Wheeler. It calculates the effect all in one go, i.e., no split into special relativistic and general relativistic effects.

Even though the post is simple-minded, it still might be a bit too technical for the intended project. To get some idea for the size of the accumulated error, convert to a distance by multiplying by the speed of light.

 

[mgb_phys]

Back of the envelope calcs.
GPS orbit is around 26,000km period is around 12hours.
So speed V = 2 pi 26.6E6 m / 12 * 3600 s = 3900 m/s

contraction = sqrt( 1 - v^2/c^2 ) = sqrt ( 1 - 1.6e-10 ) = 0.99999999991699995
times 24hours = 7 microsecs

 

[p4h]

In https://www.physicsforums.com/showpost.php?p=731738&postcount=5", I give a simple-minded (without proper justification) analysis of GPS that is based on Project A from the book Exploring Black Holes by Taylor and Wheeler. It calculates the effect all in one go, i.e., no split into special relativistic and general relativistic effects.

Even though the post is simple-minded, it still might be a bit too technical for the intended project. To get some idea for the size of the accumulated error, convert to a distance by multiplying by the speed of light.

Yeah it's a bit out of my league also. But thanks :)

Back of the envelope calcs.
GPS orbit is around 26,000km period is around 12hours.
So speed V = 2 pi 26.6E6 m / 12 * 3600 s = 3900 m/s

contraction = sqrt( 1 - v^2/c^2 ) = sqrt ( 1 - 1.6e-10 ) = 0.99999999991699995
times 24hours = 7 microsecs

So two questions, as I want to understand it also, rather than just writing it down.

1. V = 2 pi 26.6E6 m / 12 * 3600 s: Where do you get the 2 pi from? (Might be a newbie question)

2. And from where do you derive the formula for contraction, as it seems to look a lot like the formula for Lorentz factor?

 

[mgb_phys]

26,000 km is the orbital radius, so the distance in one orbit is the circumference ie "2 pi r"
Length / time / mass all have the same lorentz contraction formula, it's one of the clever things about special relativity.

 

[p4h]

Cool thanks a bundle!

Might revive this thread later this week, but we'll see..

And thanks again :) Really appreciate it!

 

[RandallB]

There are two effects.
Time runs quicker for moving satelites (due to SR) by 7us / day.
Time runs slower for the satelite in lower gravity (GR) by 45us/day.

So overall the effect is the opposite to what you would expect from just the motion.

You typed too fast and have this backwards; relative to the Earth bound GPS user;
1)Time runs slower for moving satellites (due to SR) by 7 us/day.
2)Time runs fast for the satelite in lower gravity at altitude (GR) by 45us/day.

Net of 1) & 2) is 38 micro seconds fast per day for the satellites over surface with GR lower density gravity due to distance, dominating SR speed effect.

The link you provided:
http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html
Has this correct.

 

[cesiumfrog]

Distance is determined by timing signals that pass between the ground and the satellites. A rough estimate on the error in position that this causes can be obtained by calculating the distance that a signal moving at the speed of light travels in 39 millionths of a second. This gives 11.6 kilometres.

To get some idea for the size of the accumulated error, convert to a distance by multiplying by the speed of light.

The argument in physicsmyths is that GPS works by simply comparing the lag between signals from different satellites, hence drift between surface clocks and geostationary satellite clocks will not translate to positioning error. So the fact that GPS is accurate to better than 10km is not in itself evidence for GR.

However, I remember (or imagine) a detailed article describing the positioning error that would result if the GPS switched GR corrections off. Presumably this would result from the roughly-geosynchronous satellites not being exactly geostationary, and hence very slowly losing record of their own positions (from which to triangulate from), but while the measured clock drift aboard geostationary satellites may indeed have verified GR (as the crackpot failed to note) it nonetheless is not obvious that an accurately working GPS requires distinguishing between Newtonian gravity and GR.

 

[George Jones]

The argument in physicsmyths is that GPS works by simply comparing the lag between signals from different satellites, hence drift between surface clocks and geostationary satellite clocks will not translate to positioning error. So the fact that GPS is accurate to better than 10km is not in itself evidence for GR.

I had before never heard of
<crank site link deleted>

but after poking around there for a few minutes, it seems to me that this an anti-relativity crank site.

 

[Doc Al]

It's a notorious anti-relativity crank site. That's why I deleted the link!

 

[cesiumfrog]

it seems to me that this an anti-relativity crank site.

How much of this thread did you read??

I think your unfounded suggestion that GPS would fail by 10km/day (if GR weren't true) lends the crank more credibility than any hyper-link has.

 

[George Jones]

It's a notorious anti-relativity crank site. That's why I deleted the link!

If I gave the link that you deleted in the original thread, I apologize. I started reading the thread after the link had been deleted.

How much of this thread did you read??

I scanned the whole thread. Reading more slowing, it appears that one of your posts has been deleted, so I didn't read the post that p4h quoted in post#6. Since I scanned every post, I wasn't paying attention to quotes. Without seeing either your deleted post or the original link, I, erroneously, took your post #14 seems to cite a certain crank site as an authority.

I think your unfounded suggestion that GPS would fail by 10km/day (if GR weren't true) lends the crank more credibility than any hyper-link has.

I have been under the impression that this is a way to get a rough handle on errors. For example, Neil Ashby, noted authority on GPS (the author of the livingreview on GPS) did the same thing in his presentation to the American Association of Physics Teachers. See slide 15.

http://www.aapt-doorway.org/TGRUTalks/Ashby/ashbyGRworkshop.pdf

 

[cesiumfrog]

Sure, that calculation gives a rough handle on something, but how relevant is it? You're implying it equals the magnitude of error which GPS would accumulate if GR corrections were switched off.

Do you really think that implication is true (if so, please explain where the *crank website* argument fails), or has the crackpot correctly identified a mistake in the way we present evidence of GR to our students?

--
Kind of like how those that do not understand subtleties of the photoelectric effect may end up teaching non-truths about which phenomena demonstrates that light is composed of discreet photons. In denouncing crackpots we must be careful not to become them.

 

[mgb_phys]

You typed too fast and have this backwards; relative to the Earth bound GPS user;

Doh - I originally wrote the effects the other way round and forgot to swap the values!

 

[p4h]

26,000 km is the orbital radius, so the distance in one orbit is the circumference ie "2 pi r"
Length / time / mass all have the same lorentz contraction formula, it's one of the clever things about special relativity.

So if I am to draw a link between the accumilated time error for a day and the accumilated error in length for a day, how do I do it?

Reason I want to find the connection is that everywhere I've read about GPS, atomclocks and whatnot, they do a big deal to explain what is done to revert the effects of the special theory of relativity and general theory of relativity (Althought I am to ignore gravitational effects, and look at the effect of special relativity alone)

 

[pervect]

Sure, that calculation gives a rough handle on something, but how relevant is it? You're implying it equals the magnitude of error which GPS would accumulate if GR corrections were switched off.

Do you really think that implication is true (if so, please explain where the physicsmyths argument fails), or has the crackpot correctly identified a mistake in the way we present evidence of GR to our students?

First, some history.

From http://www.leapsecond.com/history/Ashby-Relativity.htm

At the time of launch of the first NTS-2 satellite (June 1977), which contained the first Cesium clock to be placed in orbit, there were some who doubted that relativistic effects were real. A frequency synthesizer was built into the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was that predicted by GR, then the synthesizer could be turned on bringing the clock to the coordinate rate necessary for operation. The atomic clock was first operated for about 20 days to measure its clock rate before turning on the synthesizer. The frequency measured during that interval was +442.5 parts in 1012 faster than clocks on the ground; if left uncorrected this would have resulted in timing errors of about 38,000 nanoseconds per day. The difference between predicted and measured values of the frequency shift was only 3.97 parts in 1012, well within the accuracy capabilities of the orbiting clock. This then gave about a 1% validation of the combined motional and gravitational shifts for a clock at 4.2 Earth radii.

At present one cannot easily perform tests of relativity with the system because the SV clocks are actively steered to be within 1 microsecond of Universal Coordinated Time (USNO).

It seems obvious to me that what George (and Ashby before him) calculated is in fact a prediction of GR - it's a prediction of how much one has to warp the clock frequency of the spaceborne clock so that the space-born clock can be kept synchronized with the Earthbound clock.

This was not designed as a formal test of GR, you can think of it as an informal test of GR which GR nonetheless passed easily.

The crank website is not relevant to the topic. If you think you've got some argument about how we teach relativity (and I don't see your point at this time), please describe it *without* bringing any crank websites into the discussion.

I'm getting tired of deleting links to crank websites - they don't need any advertising, and I'm not going to give them any on PF.

 

[p4h]

It's not to be rude or anything, but if this is such a heated discussion, maybe it'd be better to just make a new thread, get the arguments from both sides and start from there?

And I'll state my question again:

So if I am to draw a link between the accumilated time error for a day and the accumilated error in length for a day, how do I do it?

Reason I want to find the connection is that everywhere I've read about GPS, atomclocks and whatnot, they do a big deal to explain what is done to revert the effects of the special theory of relativity and general theory of relativity (Althought I am to ignore gravitational effects, and look at the effect of special relativity alone)

 

[George Jones]

Sure, that calculation gives a rough handle on something, but how relevant is it? You're implying it equals the magnitude of error which GPS would accumulate if GR corrections were switched off.

Do you really think that implication is true (if so, please explain where the *crank website* argument fails), or has the crackpot correctly identified a mistake in the way we present evidence of GR to our students?

Maybe this should be split off into a different thread.

Differential GPS involves comparing times of clocks on the ground and in satellites. Essentially, GPS devices at known locations on the Earth's surface are treated as satellites with orbital radii given by the their distances from an Earth-centered frame, and with orbital periods equal to a day. A differential receiver uses both real satellites and Earth-bound "satellites" to determine its position, so there are GR timing differences between two "satellites" due to their much different r coordinates.

 

[LURCH]

I had before never heard of
<crank site link deleted>

but after poking around there for a few minutes, it seems to me that this an anti-relativity crank site.

Well heck, I could tell that just from the title!

 

[p4h]

Maybe this should be split off into a different thread.

Differential GPS involves comparing times of clocks on the ground and in satellites. Essentially, GPS devices at known locations on the Earth's surface are treated as satellites with orbital radii given by the their distances from an Earth-centered frame, and with orbital periods equal to a day. A differential receiver uses both real satellites and Earth-bound "satellites" to determine its position, so there are GR timing differences between two "satellites" due to their much different r coordinates.

But how does it determine positions from just time? I may seem stupid but I can't make the connection.

 

[mgb_phys]

So if I am to draw a link between the accumilated time error for a day and the accumilated error in length for a day, how do I do it?

It's not that simple - the position solution comes from the time and empheris of mutliple satelites. It's not just a question of time error * speed of light = distance error.
Since the initial fix is just based on correlating the time difference to the satelite it doesn't actaully matter what the absolute time on the satelite's clock is, and any change in frequency of the transmitted signal due to time dialtion will be much smaller than the change due to doppler shift anyway.

The main errors are propogation delays due to atmospheric effects.

 

[p4h]

So I can't draw a direct line between it, and I'd be in way over my head if I were to try and calculate this? Man my teachers didn't do their homework :(

 

[mgb_phys]

This might be a clearer explanation
http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/GPS/Neil_Ashby_Relativity_GPS.pdf

 

[RandallB]

From the OP

So I'm doing a report on special relativity and GPS and kinda at a loss what to do:

Later

So I can't draw a direct line between it, and I'd be in way over my head if I were to try and calculate this? Man my teachers didn't do their homework :(

p4h
From what I've seen your making this homework problem much harder than needed and likely off track from where your teacher is pointing you.
You don't really think they expect you to do a multi point GPS calculation do you?
Have you figured out that GPS requires synchronization between all measuring/signaling points?
Any problems with keeping thinks synchronized considering SR simultaneity issues?

If you need your teacher or us to tell you to bring up SR simultaneity issues in your report then it is you not your teacher that has some homework to do. Even more homework if you do not already know and understand “simultaneity”; you need that before working GPS issues.
It is a lot to cover in just a report, and so working only with a SR discussion seems reasonable; with maybe mentioning the GR effects are even more significant worthy of discussing in a separate report.

The “heated discussion” you referred to is typical in THIS FORUM when members have differing ideas, and is the intent of this forum to air them.
Your right this thread is running two subjects.
This site does have a separate HOMEWORK FORUM intended for homework problems, you can find it at the top of the main home page.

 

[russ_watters]

But how does it determine positions from just time? I may seem stupid but I can't make the connection.

What the code time signals tell you is that one satellite is some distance further from you than another. Since the receiver knows how far apart the satellites are, that is enough information to triangulate your distance from them and your position on Earth (giving a two-position solution).

 

[cesiumfrog]

A differential receiver uses both real satellites and Earth-bound "satellites" to determine its position, so there are GR timing differences between two "satellites" due to their much different r coordinates.

It seems to me like the maximum positional error caused by relativistic effects will be (38\mus/day)*(velocity of satellite, not of light). What do you think?

 

[mgb_phys]

Roughly correct.
The position fix relies on knowing where the satelite was when it broadcast the data block, this is based on it's onboard clock - so you would be out by the distance the satelite travels in 38us ( actually half that since it makes two orbits/day)

In practice I don't think it changes your actual position very much, if you have good geometry with satelites both east and west of you then the errors due to clock difference will apply in both directions, you will get a poorer fit but will be in roughly the same place.

( Except that the designers of the satelites knew about relativity and set the clocks to run fast ;-)

 

[cesiumfrog]

And that's for differential GPS. For normal GPS I'd expect the relativistic error to be smaller again, since the satellites have less time dilation relative to each other than to ground stations (presuming that the satellite positions are determined by triangulation from ground stations, as would seem obvious and consistent, rather than by some kind of comparison between satellite and ground station clocks).

 

[mgb_phys]

The satelite position is calcualted in the receiver based on the tranmsitted empheris.
The onboard empheris (e the satelites knowledge of it's orbit) is recalculated (and uploaded) every few days when the satelite passes over one of the ground stations - not sure if they fix the satelite position optically with a transit telescope (which would be most accurate) or a radio fix.
The satelites store and transmit an almanac giving each others approximate positions - which helps single channel receivers find other sats - but they don't send signals between each other (actaully they do but this is nothing to do with the GPS position)

Differential GPs just receives the GPS signal at a fixed ground station and retransmits it to the reciever (either via satelite WAAS/EGNOS or by radio starFire) it takes out large scale ionospheric effects.

Ironically the simplest GPS to understand is the highest accuracy RTK mode - where you have two receivers close together looking at the same satelites - this just uses the pure time difference of the high frequency carrier wave to get a distance between the two receivers directly to an accuracy of cm. It then uses the low accuracy GPS position to work out where it is to a much poorer accuracy.

 

[p4h]

Kinda afraid to post after the verbal spanking Randall gave me, but got another question:

contraction = sqrt( 1 - v^2/c^2 ) = sqrt ( 1 - 1.6e-10 ) = 0.99999999991699995
times 24hours = 7 microsecs

I'm sorry to seem stupid, but I've tried two hours now to get 0.99999999991699995
times 24hours = 7 microsecs.. I just don't see how?

 

[mgb_phys]

I'm sorry to seem stupid, but I've tried two hours now to get 0.99999999991699995
times 24hours = 7 microsecs.. I just don't see how?

Remember it's the fraction of a day so,
1 - 0.99999999991699995 = 8.3 E-11 of a day

And 24 * 60 * 60 * 8.3E-11 = 7.2 E-6 seconds

 

[p4h]

Ah! Thanks again..

If I do:

\sqrt{1-\frac{3900}{299792458}} = .9999934955

So we don't get the same. >.<

Where do you get the 1 - .9999934955 from?

 

[mgb_phys]

The result of the sqrt is the contraction - this is the fraction of a day that the satelite expereinces, so a satelite's view of a day is 0.9999934955 of an Earth's view of a day.
The difference between these is simply 1 - 0.9999934955
or if you prefer, satelite day = 0.9999934955 * 24 * 3600 seconds, Earth day = 1 * 24 * 3600 seconds.
The difference is 7.2 us.

edit sorry just realized you are asking about the actual numerical value
You forgot to square the v and c, it should be sqrt( 1 - (3900^2)/(3E8^2))

 

[p4h]

The result of the sqrt is the contraction - this is the fraction of a day that the satelite expereinces, so a satelite's view of a day is 0.9999934955 of an Earth's view of a day.
The difference between these is simply 1 - 0.9999934955
or if you prefer, satelite day = 0.9999934955 * 24 * 3600 seconds, Earth day = 1 * 24 * 3600 seconds.
The difference is 7.2 us.

edit sorry just realized you are asking about the actual numerical value
You forgot to square the v and c, it should be sqrt( 1 - (3900^2)/(3E8^2))

Yeah thanks it was the top part I wanted..

And ye forgot to square them, but that only got my answer closer to 1

But thanks yet again!

 

[RandallB]

Kinda afraid to post after the verbal spanking Randall gave me, but got another question:

No reason to fear posting – often newbes accidentally work on homework in the discussion forums instead of the homework forum. If someone doesn’t tell you, you won’t learn how to best use PF.

By the way if a Mentor has not already (I am not one) welcome to the PF forums.

PS: Also, no need to abuse your teacher behind their back (post #28), at least be sure what they want you to learn from doing a report. Remember your writing a report not a textbook.

 

[p4h]

Well I just got alittle frustrated because I couldn't see my way out of the project, since I also have this graph I can't figure out.

 

[p4h]

Okay I got two more things left before I stop reviving this thread, as I figure I might as well use this one, instead of starting a new one.

1. When thinking of what impact the theory of relativity (both SR and GR) have on things around us, I couldn't think of anything except Redshift / Doppler effect. I'm sure there are a lot more areas but I can't think of any?

2. The image attached is a graph I'm supposed to be able to figure out. I'm supposed to be able to figure out the distance between a Satelite and a GPS reciever.
X axis: Time / seconds
Y axis: Signal strength / no unit
There is no legend to the graph, so I don't know which curve represents the satellite and which represents the reciever.

I'd appreciate it if only hints is dropped, so I can fiddle with it myself, but I may ask again if I'm lost :)

PS: I realize this should be in the homework section, but as I already have a thread with this subject, I might as well use it.

 

[RandallB]

Cute problem.
I assume you can translate the fairly intuitive answer of about “0.065 tid/s” into the appropriate units of distance.
The correct value answer of course is not the important part.
I’m sure they expect you to report on what detailed information has to be known at the receiver to use in observing this one satellite.
And why using SR & GR in so important to maintaining that information.

A good problem in that it is easy to answer, once you understand the issues of running a GPS.
Hint: think simple

Mentors can move the thread if they feel it is needed, interesting approch at least.

 

[p4h]

I assume I have to make a connection between length and signal strength, but to me, when I think of signal strength, I just think that it has to vary a lot due to different factors, being it the ionozation in the Ionosphere, or being it the humidity in the Troposphere? I may be overthinking it.

 

[RandallB]

Signal strength has nothing to do with it.
You are making it harder than it is.
Radio photons are no different than light photons.
Think simple & how do you measure distance with them.