# Special Relativity and GPS

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.

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

Last edited by a moderator:

Doc Al
Mentor
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!)

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
Homework Helper
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.

Last edited:
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

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.

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.

George Jones
Staff Emeritus
Gold Member
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.

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.

Last edited by a moderator:
mgb_phys
Homework Helper
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

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 :)

mgb_phys said:
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 alot like the formula for Lorentz factor?

Last edited by a moderator:
mgb_phys
Homework Helper
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.

Cool thanks a bundle!

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

And thanks again :) Really appreciate it!

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.

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

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.

Last edited:
George Jones
Staff Emeritus
Gold Member
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

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

Last edited by a moderator:
Doc Al
Mentor
It's a notorious anti-relativity crank site. That's why I deleted the link! :grumpy:

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

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
Staff Emeritus
Gold Member
It's a notorious anti-relativity crank site. That's why I deleted the link! :grumpy:

cesiumfrog said:

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.

cesiumfrog said:
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 [Broken]

Last edited by a moderator:
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.

Last edited by a moderator:
mgb_phys
Homework Helper
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!

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
Staff Emeritus
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.

Last edited:
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
Staff Emeritus
Gold Member
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