I was curious about the use of relativity in GNSS (GPS) systems (since I only know very basic relativity but I wouldn't mind having a good excuse to learn more... ), and I've found some conflicting results. Some say that the relativistic variations in satellite clock speed have a HUGE effect on location accuracy, since microseconds of clock error lead to kilometres of distance error when you're talking about signals traveling at light speed.(adsbygoogle = window.adsbygoogle || []).push({});

However, there's an opposing argument which points out that at least 4 satellites are always used anyway (rather than 3) because of the cheap, inaccurate clocks used for the receiver. The use of 4 satellites allows us to cancel out this receiver clock errorandfind position. (This fact seems to be agreed upon by both camps -- see the IEEE paper I reference later) Thus, the argument goes, if all the satellites have the same clock error, the same process used to eliminate the receiver clock error should also eliminate (most of) the clock error due to relativistic effects. If you look at the CPS navigation equations on Wikipedia (http://en.wikipedia.org/wiki/Global_Positioning_System#Navigation_equations), it does seem clear that if all the satellite clocks have the same error, it's mathematically equivalent to the receiver clock having error, and so it should cancel out in the calculation.

The result of these conflicting arguments is that claims for the position error due to ignoring relativistic effects range from fractions of centimetres to over ten kilometres. That's a pretty significant difference, and I'd really like to know what the truth is.

By the way, here are some of the places I've been reading from:

http://www.physicsmyths.org.uk/gps.htm

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

crackpot link deleted

"GPS, atomic clocks and relativity" by Allan W. Love (Published in IEEE Potentials, April 1994)

Unfortunately, the two sites which oppose the idea of needing relativity in GNSS calculations seem a little less than reputable (they also appear to reject the idea of special relativity altogether in other articles). However, their arguments on this particular front make sense to me. That is, the time error should cancel out in the calculation process in exactly the same way it cancels for the receiver clock error.

That said, my gut also tells me that that's probably an over-simplification, and it may be that clock error due to relativistic effects is still important. Error canceling out looks nice on paper, but how well it works in practice may be a completely different story. I suspect the position error might come out to be significant, even if it's not 11 km per day.

Summary/(tl;dr):

Does anyone know for certain how important relativistic effects actually are in GNSS systems? I would appreciate references/calculations if you happen to have them.

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# Relativity in GNSS (GPS)

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