# 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.)

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– 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
– João Portela Nov 13 '12 at 11:29
– João Portela Nov 13 '12 at 11:38

## 4 Answers

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

So $50ns$ error in timekeeping corresponds to $15m$ error in distance prediction.
Hence, for $38\mu s$ error in timekeeping corresponds to $11km$ error in distance prediction.

If we do not apply corrections using GR to GPS then $38\mu s$ 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! :-)

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You simply discuss special relativity where, where as general relativity is in fact a very important effect for GPS systems. – Noldorin Dec 10 '10 at 20:58
@Noldorin: the main GR correction is included, see $T_2$ – Retarded Potential Mar 28 at 18:04

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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.

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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.

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GPS signals do not return to the satellite, they only go to the receiver AFAIK... – Thomas O Nov 18 '10 at 13:53

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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

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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

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– endolith Nov 18 '10 at 15:16

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@endolith : ... if you bring an atomic clock with you ! – Frédéric Grosshans Nov 18 '10 at 18:14
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After dealing with GPS algorithms for significant part of my lifetime, I'd say in one word: GPS is a precise device. It is really state of the art to determine location from timing (aka DTOA, differential time of arrival), catching electromagnetic waves which are so damn fast.

That has to account for many subtle effects which might be insignificant in simple life - such as athmospheric disturbances, relativistic effects, ridiculously hard electronics design issues, etc.

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