# Global Positioning System (GPS) and Relativity

The Global Positioning System (GPS) is nowadays considered as the prime example for the everyday importance of Relativity. It is claimed that without the relativistic corrections (which amount to 38 microseconds/day) the error in the determination of the position would accumulate quickly to values much larger then the observed accuracy (Ref.1 , Ref.2).
However, in reality the positions are actually not obtained by comparing the time signal received from the satellite with the receiver time, but by observing the difference between the time signals obtained from a number of different satellites (see the Wikipedia GPS article for details (note that I have linked this now to an older version of the article, as the latest version is not as clear in this respect)).

Consider for simplicity a one dimensional problem where the receiver is located somewhere on the line connecting the two transmitters. In this case the signal from transmitter 1 reaches the receiver at time

(1)     t1 = t0+ x1/c

and the signal from transmitter 2 reaches the receiver at time

(2)     t2 = t0+ x2/c ,

where t0 is the time the signal is being sent out (assuming both transmitter clocks are synchronized), x1 is the distance of the receiver from transmitter 1, x2 the distance of the receiver from transmitter2, and c the speed of light.
Now if one subtracts Eqs.(1) and (2) one gets

(3)     x1-x2 = c. [t1-t2].

One knows therefore the position of the receiver just by comparing the time signals from the two transmitters (the receiver clock is completely irrelevant).
If one assumes now that the transmitter clocks are running fast or slow by a relative factor (1+ε), one has instead:

(4)     x1-x2 = c.[(1+ε).t1 -(1+ε).t2] = c.(1+ε).(t1-t2)

which means that the position will simply be wrong by a relative factor ε, but there is obviously no accumulation as the transmitter clocks run at the same rate relatively to each other (assuming that all satellites have identical heights and speeds, i.e. identical relativistic time dilations).

Now the quoted relativistic correction of 38 microseconds/day corresponds to ε=4.4.10-10. As the satellites are at a distance of around 20000 km (=2.109 cm), the positional error due to relativity should actually only be 4.4.10-10 . 2.109 cm = 0.8 cm! This is even much less than the presently claimed accuracy of the GPS of a few meters, so the Relativity effect should actually not be relevant at all!

For a further discussion of this issue please see my discussion pages.

Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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