# Why does GPS depend on relativity?

I am reading A Brief History of Time by Stephen Hawking, and in it he mentions that without compensating for relativity, GPS devices would be out by miles. Why is this? (I am not sure which relativity he means as I am several chapters ahead now and the question just came to me.)

• astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html – kennytm Nov 18 '10 at 13:52
• I'm trying to locate my sources on this, but I have read that even if you don't account for general relativity (by slowing down the clocks prior to launch) your GPS would work just fine because the error is the same for all satelites. The only issue would be that the clocks would not be synchronized with the ground, but that is not necessary for calculating your current position. Can anyone confirm this? – João Portela Nov 13 '12 at 11:27
• Found something: physicsmyths.org.uk/gps.htm can anyone comment on this? – João Portela Nov 13 '12 at 11:29
• found something else in this same site: physics.stackexchange.com/q/17814/3177 (some answers mention this) – João Portela Nov 13 '12 at 11:38
• I looked at that uk site hurriedly and there seem to be some crank "disproofs" of special relativity, so I doubt that that site is trustworthy. There are cranks on stack exchange, too, of course....and on Wikipedia, and in academia, and .....yours truly, – joseph f. johnson Dec 5 '15 at 23:11

Error margin for position predicted by GPS is $15\text{m}$. So GPS system must keep time with accuracy of at least $15\text{m}/c$ which is roughly $50\text{ns}$.

So $50\text{ns}$ error in timekeeping corresponds to $15\text{m}$ error in distance prediction.
Hence, for $38\text{μs}$ error in timekeeping corresponds to $11\text{km}$ error in distance prediction.

If we do not apply corrections using GR to GPS then $38\text{μs}$ error in timekeeping is introduced per day.

You can check it yourself by using following formulas

$T_1 = \frac{T_0}{\sqrt{1-\frac{v^2}{c^2}}}$ ...clock runs relatively slower if it is moving at high velocity.

$T_2 = \frac{T_0}{\sqrt{1-\frac{2GM}{c^2 R}}}$ ...clock runs relatively faster because of weak gravity.

$T_1$ = 7 microseconds/day

$T_2$ = 45 microseconds/day

$T_2 - T_1$ = 38 microseconds/day

use values given in this very good article.

And for equations refer to HyperPhysics.

So Stephen Hawking is right! :-)

• Is $R$ the radius of the earth, or the orbit radius? – ja72 May 11 '14 at 16:32
• But what's relevant for GPS is the difference between timestamps from different satellites, right? And since they are on the same altitude they should be time shifted by the same amount, so the differences should be basically the same as without relativity. I mean it doesn't matter how much the error in the clocks is after a day, since the localization error is not cumulative, because the satellites' clocks don't drift away from each other. – isarandi Jun 14 '15 at 23:31
• As noted in this answer, it is important to note that the values given correspond to the difference between the factors on earth and in orbit - meaning that the expressions for $T_1$ and $T_2$ as given don't evaluate to the values given, although the values given are correct. Tip of the hat to Michael Seifert who pointed this out. – Floris Jul 28 '15 at 19:41
• @Dims 15/300000000 != 100*10^(-6), it equals 5*10^(-8). I got my answer by just typing it into google, but it should be easy to see that 15 divided by 3 is going to be a leading 5, not a leading 1. – Shufflepants Apr 27 '18 at 14:21
• Lots of misinformation here. As per the US Naval Observatory (the creators of GPS to replace LORAN): GPS does NOT use relativity calculations at all (repeat, it does NOT use relativity calculations). – MC9000 Jan 16 at 0:26

There's the article from Ohio State University http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html which explains quite well why the clocks on a GPS satellite are faster by about 38 microseconds every day. The article then claims that not compensation for these 38 microseconds per day would cause a GPS to be off by about 11 km per day, plainly unusable, and claims that this (the fact that we need to compensate for the 38 microseconds to get GPS working) is proof for General Relativity.

The problem is that while the clocks are indeed off by 38 microseconds per day and General Relativity is all fine, we wouldn't actually have to compensate for it. The GPS in your car or your phone doesn't have an atomic clock. It doesn't have any clock precise enough to help with GPS. It doesn't measure how long the signal took to get from satellite A to GPS. It measures the difference between the signal from satellite A and the signal from satellite B (and two more satellites). This works if the clocks are fast: As long as they are all fast by the exact same amounts, we still get the right results.

That is, almost. Satellites don't stand still. So if we rely on a clock that is 38 microseconds fast per day, we do the calculations based on the position of a satellite that is off by 38 microseconds per day. So the error is not (speed of light times 38 microseconds times days), it is (speed of satellite times 38 microseconds times day). This is about 15 cm per day. Well, satellite positions get corrected once a week. I hope nobody thinks we could predict the position of a satellite for long time without any error.

Back to the original assumption, that without compensation the error would be 11km per day: The satellite clocks are multiplied by a factor just shy of 1 so that they go at the correct speed. But that wouldn't work. The effect that produces 38 microseconds per day isn't constant. When the satellite flies over an ocean, gravity is lower. The satellite speed changes all the time because the satellite doesn't fly on a perfect circle around a perfectly round earth made of perfectly homogenous material. If GR created an error of 11km per day uncompensated, then it is quite unconceivable that a simple multiplication of the clock speed would be good enough to reduce this to make GPS usable.

• Nice. But I have to say that from a the philosophical position of an experimenter, a machine that makes it operators tear their hair out (which GPS would in the absence of of GR) isn't working until those behaviors are understood (which would happen when someone invented GR to explain the anomaly). But that's a philosophical point. – dmckee Dec 6 '15 at 1:07
• This is the one correct answer on this page. GPS was significant evidence for GR because we can compare the speed of clocks in orbit to those on earth. However, the accuracy of the GPS system doesn't depend on the satellites keeping exact time. As long as they keep the same time, the system works. – Robert Stiffler Dec 6 '15 at 3:39
• Actually, GPS is a poor "proof" of GR for the reason you state. gnasher has the correct answer - Einstein field equations are not used in GPS at all (imagine the number crunching involved and the computer power necessary wasting all that energy - not to mention added weight to satellites - especially a few decades ago) – MC9000 Jan 16 at 0:29

You can find out about this in great detail in the excellent summary over here: What the Global Positioning System Tells Us about Relativity?

In a nutshell:

1. General Relativity predicts that clocks go slower in a higher gravitational field. That is the clock aboard the GPS satellites "clicks" faster than the clock down on Earth.
2. Also, Special Relativity predicts that a moving clock is slower than the stationary one. So this effect will slow the clock compared to the one down on Earth.

As you see, in this case the two effects are acting in opposite direction but their magnitude is not equal, thus don't cancel each other out.

Now, you find out your position by comparing the time signal from a number of satellites. They are at different distance from you and it then takes different time for the signal to reach you. Thus the signal of "Satellite A says right now it is 22:31:12" will be different from what you'll hear Satellite B at the same moment). From the time difference of the signal and knowing the satellites positions (your GPS knows that) you can triangulate your position on the ground.

If one does not compensate for the different clock speeds, the distance measurement would be wrong and the position estimation could be hundreds or thousands of meters or more off, making the GPS system essentially useless.

The effect of gravitational time dilation can even be measured if you go from the surface of the earth to an orbit around the earth. Therefore, as GPS satellites measure the time it's messages take to reach you and come back, it is important to account for the real time that the signal takes to reach the target.

• GPS signals do not return to the satellite, they only go to the receiver AFAIK... – Thomas O Nov 18 '10 at 13:53
• But the main point still holds, and it is that more time passes on Satellite's clock than your clock back on earth, with respect to either one of you. – Cem Nov 18 '10 at 13:59
• Interestingly general relativity is not use per se in calculations for GPS systems. Rather, a nice little trick involving special relativity (applying a series of Lorentz transformations in infinitesimal steps) is what it does. This turns out to be sufficiently accurate and a lot easier computationally. – Noldorin Nov 18 '10 at 14:22
• You can detect time dilation just by spending a few days in the mountains. leapsecond.com/great2005/index.htm – endolith Nov 18 '10 at 15:16
• @endolith : ... if you bring an atomic clock with you ! – Frédéric Grosshans Nov 18 '10 at 18:14

GPS satellite navigation system doesn't use, doesn't need and doesn't prove Einstein's General Relativity.

The GPS satellites use classical (Newtonian) relativistic principles to work. These are the same relativistic principles that make sense in the everyday world, that most people equate with 'common sense'. GPS calculates positions based on geometric principles. The atomic clocks on the satellites have their rates preset in order to match experimentally observed effects. No General Relativity is used or needed.

The website dedicated by the USA government to the GPS system, www.gps.gov, has a lot of information for those who want to know how GPS works, but does not put forward anything about General Relativity (GR).

The equations used by the GPS system are based on geometry and classical relativity. They dont include compensations for GR effects. The satellite simply sends out a signal at regular intervals with some information. The receiving device does all the required calculations with the signals it receives. You can check out the basics at http://en.wikipedia.org/wiki/Global_Positioning_System

When the clocks of each satellite were set, the rate of each clock was adjusted according to previously observed effects. Experimental observations over decades, showed how speed and altitude affected the rate of ticking of atomic clocks. It had been experimentally observed that atomic clocks speed up at higher altitudes. It was also observed that the speed of the clock also affects the rate of ticking. Based on these experimental observations, the GPS engineers had to adjust the rate of the atomic clocks based on what had been observed - regardless of what any theory said. The engineers knew how the clocks would be affected in orbit because of experimental observations. Not the theory of general relativity. Supporters of Einsteins GR had to make sure their theory fits these observations. The GPS engineers didnt use GR to work out how much to change the clocks.

More detail:

www. freerepublic.com /focus/chat/1025790/posts

"In the 1990's, he [Van Flandern] worked as a special consultant to the Global Positioning System (GPS), a set of satellites whose atomic clocks allow ground observers to determine their position to within about a foot. Van Flandern reports that an intriguing controversy arose before GPS was even launched. Special Relativity gave Einsteinians reason to doubt whether it would work at all. In fact, it works fine."

"At high altitude, where the GPS clocks orbit the Earth, it is known that the clocks run roughly 46,000 nanoseconds (one-billionth of a second) a day faster than at ground level, because the gravitational field is thinner 20,000 kilometers above the Earth. The orbiting clocks also pass through that field at a rate of three kilometers per second -- their orbital speed. For that reason, they tick 7,000 nanoseconds a day slower than stationary clocks. To offset these two effects, the GPS engineers reset the clock rates, slowing them down before launch by 39,000 nanoseconds a day. They then proceed to tick in orbit at the same rate as ground clocks, and the system "works." Ground observers can indeed pin-point their position to a high degree of precision."

"in Einstein's theory the relevant speed is always speed relative to the observer, it was expected that continuously varying relativistic corrections would have to be made to clock rates. This in turn would have introduced an unworkable complexity into the GPS. But these corrections were not made. Yet "the system manages to work, even though they use no relativistic corrections after launch," Van Flandern said."

www.alternativephysics.org/book/GPSmythology.htm

"The presence of Special and General Relativity effects has no bearing on the accuracy of GPS operation. In summary, it wouldn’t matter whether clocks aboard GPS satellites ran faster or slower than Earth’s clocks or even changed their speed each day. Just so long as the satellites’ clocks remained synchronised with each other and the time-difference relative Earth’s clocks didn’t become too large, GPS receivers would continue to calculate their correct position."

The following articles make the claim that GPS requires General Relativity, but in error. They refer to the initial clock rate presets as being due to GR, when the reality is that these presets had to be done, theory or not, because of experimental observations.

www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

This article is cited in many articles and publications that are supportive of the theory of general relativity, inclucing wikipedia references. Yet, it is just opinion. In it, Richard W. Pogge makes these 2 claims without supporting them in the text. The references cited at the end of this article dont refer to relativity at all - they link to the GPS FAQ at the FAA.

1 "However, because the satellites are constantly moving relative to observers on the Earth, effects predicted by the Special and General theories of Relativity must be taken into account to achieve the desired 20-30 nanosecond accuracy." 2 "For example, to counteract the General Relativistic effect once on orbit, they slowed down the ticking frequency of the atomic clocks before they were launched so that once they were in their proper orbit stations their clocks would appear to tick at the correct rate as compared to the reference atomic clocks at the GPS ground stations."

Wandera asks: In an article that claims GPS as a real world example for relativity, why doesnt the author back up such important claims? The firt claim isnt even explained, and isnt backed up at all. The second claim labels the effect G Relativistic, but this is another unsupported opinion.

www.aticourses. com/global_positioning_system.htm

"The corrections that must be applied include signal propagation delays caused by the ionosphere and the troposphere, the space vehicle clock error, and the user’s receiver clock error." "Other error sources and modeling errors continue to be investigated. For example, a recent modification of the Kalman filter has led to improved performance. Studies have also shown that solar radiation pressure models may need revision and there is some new evidence that the earth’s magnetic field may contribute to a small orbit period variation in the satellite clock frequencies." "RELATIVITY The precision of GPS measurements is so great that it requires the application of Albert Einstein’s special and general theories of relativity for the reduction of its measurements."

Wandera says: The claim about general relativity isnt true and the article makes no attempt to back up this claim. Its nothing but opinion.

• Note: Van Flandern defended non-mainstream crap: like the face of Mars, exploding planets, the non-existance of the Big Bang, FTL gravity... – jinawee Apr 19 '14 at 11:49
• jinawee, are you saying that Van Flandern is lying about the GPS setup? Otherwise, how are his other comments relevant? The big bang theory says that the current laws of physics didnt apply at the start. isnt that worse than any face on mars? – 497362 Apr 19 '14 at 11:54
• "The engineers knew how the clocks would be affected in orbit because of experimental observations. Not the theory of general relativity." Whether they knew it because of experimentation, theory or reading about it in the tabloids, the bottom-line is that there is a correction for GR effects - hence GPS depends on relativity. – David van Driessche Apr 19 '14 at 17:22
• This answer is unhelpful, appears to be flat-out wrong, and cites a senile deranged man as its only source of evidence. – DumpsterDoofus Apr 21 '14 at 22:52
• @wandera: That is unnecessary, as it has already been done. Read the above posts, including the one by Pavel Radzivilovsky, who works on airborne navigation and mission-critical mapping hardware. His answer has been up for 4 years already. – DumpsterDoofus Apr 22 '14 at 19:23

Lorenz created the theory of slowing clocks prior to Einstein, not Einstein in a GR theory. My understanding of GPS is that it is calculated by the one-way time of flight of radiation from satellite-to-receiver. The synchronization of the clocks of the two is essential. But, just because a clock slows down or speeds up does not mean that time is changing, only that the shape of the atom is changing (as Lorenz postulated). The Cesium atoms are being compressed as they move through the ether. So, the equations are pure time-of-flight equations, and the clocks are adjusted and the pre-adjustments (i.e. intentional slowing of the satellite clocks) matches the equations of Lorenz. In response to David van Driessche, GPS does depend on adjustment of clocks and so LET relativity, but it does not depend on GR. GR is an as yet unproven theory. It's most often cited success is that it was used the predict the precession of the planet Mercury. Unfortunately, more recent calculations of Mercury's orbit, using only Newtonian mechanics, shows that there is close to zero precession. Therefore, the precession of Mercury disproves GR, itis not predicted by GR.[i.e. in science, a theory is put aside when it does not predict correctly].

• General relativity is an accepted theory, it is not an 'unproven theory.' Of course, it fails at or below the Planck scale, just as quantum field theories are not reliable beyond some UV scale, which is why a theory of quantum gravity is in development. – JamalS May 11 '14 at 12:58
• There is a lot wrong with this post. As a particular example, the precession of the perihelion of Mercury is an observed fact, therefore cannot be disproven by a calculation in Newtonian mechanics. – Martin May 11 '14 at 13:27

## protected by Qmechanic♦May 11 '14 at 13:08

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).