Since the Earth is moving around the Sun, which is moving around Milky Way, etc... What reference frame is used for the complete motion of the begin/end points (which are non-inertial right?)?
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1$\begingroup$ Considering that this is a fishy question, as Ben Crowell noted, I voted to close it, but I think that this would be unfair to those that answered it in good faith. $\endgroup$– Ron MaimonCommented Nov 12, 2011 at 22:10
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3 Answers
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They trust the method used by GPS for geodesy, where the claim is they can go down to picosecond and cm accuracy if necessary (for military use).
GPS error analysis takes even general relativity into account.
In GPS signal propagation (PDF) the systematic errors of a simple GPS setup are given, but the OPERA experiment has more sophisticated use of four satellites.
There has been no analysis of the GPS systematics further in the paper published in the archive, that is why I say they trust it.
Among other corrections, GPS corrects back to the velocity of light in vacuum. The meter is defined as a fraction of the velocity of light in vacuum (the second is defined by the caesium clock at normal temperature and pressure).
In my opinion, it is possible that some of the very sophisticated corrections of GPS values might be systematically off, with an end result of effectively redefining the meter. This would not show up in navigation or geodesy because the lengths probed by the OPERA experiment are very large (732 km) and the errors very small, 20 cm. A tiny systematic offset of what a meter is for GPS would not show up in the normal world use of it, but it would show up in measuring the neutrino speed with this method.
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$\begingroup$ It seems odd to me that the anonymous poster accepted this answer. Everything said here is correct, but none if it answers the question. $\endgroup$– user4552Commented Nov 12, 2011 at 21:49
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$\begingroup$ @BenCrowell The answer is in the link in the first words. GPS takes into account all the relative motions and they used the GPS definitions. $\endgroup$– anna vCommented Nov 13, 2011 at 12:34
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$\begingroup$ "GPS takes into account all the relative motions[...]" I think it would be more accurate to say that GPS takes into account the relative motions that are relevant, and doesn't need to take into account the relevant motions that are not relevant (such as the earth's orbital motion around the sun). The distinction between the two is important. This has to do with the notion of an inertial frame in GR, which is different from that in Newtonian mechanics. $\endgroup$– user4552Commented Nov 13, 2011 at 19:49
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$\begingroup$ @BenCrowell if you read a bit about GPS, they also have general relativity corrections so they would not ignore the obvious. $\endgroup$– anna vCommented Nov 13, 2011 at 20:56
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$\begingroup$ "(for military use)" This is a common misconception. Several manufacturers were able in the past to reverse-engineering the signal on the second frequency of GPS (enabling cm level precision). Furthermore, the new GPS band (L5) does not have encryption, and new open signal are being deployed also on the L2 band. $\endgroup$– pinponCommented Oct 15, 2019 at 8:45
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GPS uses a nonrotating coordinate system in which the time coordinate is essentially a time that would be defined by radio signals broadcast from a hypothetical clock at the center of the earth: http://relativity.livingreviews.org/Articles/lrr-2003-1/ GPS uses general relativity heavily. In GR, an inertial frame is defined as a free-falling one, and since the earth is free-falling, a nonrotating frame fixed to the earth's center is inertial. The earth's orbital motion around the sun, etc., are therefore irrelevant.
There is a special-relativistic correction that needs to be applied in order to convert from a nonrotating frame to one moving with the rotation of the earth. This correction comes out to be a few nanoseconds. I assume the OPERA team took it into account, but even if they hadn't, it would have been far to small to explain the claimed 60 ns shift.
The motion of the GPS satellites does not need to be corrected for, contrary to claims by van Elburg, who is an idiot. Presumably the reason the popular press has paid so much attention to van Elburg is that his arguments use only freshman physics, so science journalists can understand them. As described above, and as discussed in the Living Reviews article, GPS results are stated in a nonrotating frame tied to the earth, not in a frame moving with the satellites, as van Elburg seems to have imagined.
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1$\begingroup$ I think this is unfair to van Elberg. He gave an order of magnitude estimate of rotational corrections to the GPS distance/time estimates, and showed these explain OPERA. It is obvious that any effect is an artifact, and these corrections are correct order of magnitude. The details are probably impossible to get right given that OPERA is not releasing the details of their frame corrections. $\endgroup$ Commented Nov 12, 2011 at 23:10
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May be the case that this problem has to do with the «one-way» light speed and the referential that is used.
Afaik the only known measures of the c are done in a «two-way» version (mean value in a closed path).
When a photon is released in space it starts its journey at c speed independently of the source and of the receiver. The CMB referential clearly is the only referential to «observe» the light as isotropic. As the Earth moves we observe a dipole, and in different directions we measure different wavelengths for the same physical object (photon).
This paper (Cosmological Principle and Relativity - Part I) analyses the anisotropy of light speed for a moving observer.
Fig.3 and eq. 18, pag 14
The one-way light speed is : $c_{A}^{r}=\frac{c_{0}}{1+V/c_{0}\cdot\cos\phi_{A}}$
Using $c_0=299792.458$ Km/s is two-way light speed, $V\;$ is the speed of the lab in relation to the CMB: $V=V_{SS}+V_E$=369$\pm$30 km/s (data from here)
gives the max value of $\frac{\left|c_{V\pm\delta V}-c_{V}\right|}{c_{V}}\cdot10^{5}$=10.2.
All experimental measures of |v-c|/c are within this limit.
Anyway Einstein is correct.
I suspect that the syncronization used in the GPS is in the same as in the above paper and not as Einstein did.
In the pic 
Sat A must be synchronized with C at the same time thru the shortest red path and thru the longest blue path. At the same time B is in sync with C thru other paths with different lengths. IMO this is only possible if they are synchronised as in the above paper (instant observer) and not in the Einstein way that only considers one path between the observer and any other point ("Synchronisation around the circumference of a rotating disk gives a non vanishing time difference that depends on the direction used").
