UPDATE: correction {here will have to multiply the Gravitational shift by a 19.758 (4.4451*4.4451), the 4.44 went to the denominator instead of the numerator. A funny mistake since quasi-consciously I left teh 10^-10 factor where it belonged, in the numerator since at some level I realized I am doing some mistake otherwise. You will see it’s not easy to guess but you need a pen and paper. Luckily that qusi-conscious thing didn’t change the order of time shift and we are quite close to correct ans. SO all-in-all 0.5468 fs >> 10.804 fs. The fractional shift in time is not an inverse of fractional shift in frequency (as I stated or intended to calculate), it is same as the fractional shift in frequency, exact to the order of smallness of the fractional shift of frequency itself. Since that factor is very small in our case, the fractional shift in time can be set equal to a fractional shift in frequency, a very important point.}

The above update is summarized in all detail in the following 4 scanned pages:

refer this: http://infyinfo.files.wordpress.com/2011/10/006.jpg

I had made a quick calculation of how relative velocity of GPS satelite of ~4 km/s wrt the baseline (it has to be transverse, hence a transverse Doppler effect was actually calculated) introduces a time dilation of 0.216 ps (~.25 ps). But this was wrt the time of flight of 2.43 milli-secs (730 kms at speed of light) In that case this becomes 88.89 ps for 1 second of time of flight. (that is if the satelite were at the level of moon it would introduce that kind of fractional time dilation to processes measured on the baseline)

Weinberg calculates the gravitational red-shift of the atomic clocks on the GPS satelite by neglecting earth spin (that is a stationary earth field for the satelite) He also assumes the satelite to be under the perigee so that first order Doppler shifts are absent. I quickly put the values into this equation, frac. freq. shift = -3.47 x 10^-10 {3*R/(R+H) -2} with R=6371 kms and H=20200 kms and obtain the time dilation to be (freq.= inv. of time) 0.225 ps for 1 second of time of flight. This is actually a time contraction or violet shift since H > R/2. (it would be a red shift for H < R/2) Also we have to scale this to our time of flight of 2.43 milli-secs which makes the total time contraction to be 0.5468 fs. Compared to the Doppler effect of o.216 ps this is negligible (also compare for 1 sec tof: 0.225 ps of gravitational shift Vs 88.89 ps Doppler transverse shift, that’s what we would get if these satelites were at height of moon neglecting moon’s gravity and earth spin and for the time being not worrying that the orbital dynamics would also change, just a casual analogy for now)

this just proves there is nothing to worry from relativity reg. time dilation as long as a synchronization is made between GPS satelite and earth. The synch between both ends of the detector is an automatic benefit of GPS system. Plus if you are worried about earth spin, remember the principle of equivalence: it says the rotational acceleration of the earth being about 1% the shift towards the satelite time-relay would be again similarly effected. {recalling my centripetal and coriolis effects in classical case one has to see how they transform under a general relativistic approach, one is 0.3% of acc. due to gravity the other is 0.6% of the acc. due to gravity, One can just compute the potential energy functions and use principle of equivalence}

http://www.nature.com/nature/journal/v228/n5274/abs/228849a0.html

Read later to this article: http://wp.me/p1wdOw-4u and http://wp.me/p1wdOw-4T (there are a few others that are related, will try to link later, basically all the articles that describe time dilations and classical rotattion case)