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Posted by Bruce on May 18, 2001 04:48:04 UTC

The reason the satellite clocks run faster than the ground based clocks is because the gravitational time dilation effect is greater than the time dilation effect due to relative motion (for the GPS system case). Starting with the Schwarzchild metric (the first solution to Einstein's equation). In geometric units:
dt2=(1-2M/r)dt2 - dr2/(1-2M/r) - r2df2

Set dr=0, divide the metric by the square of bookkeeper time dt2 and set -r2(df/dt)2= -v2 to obtain for either satellite or Earth clocks

(dt/dt)2=(1-2M/r)-v2

We can write the above equation for both satellite and Earth clocks and find the ratio of time lapses recorded on the clocks.

dtsat/dtEarth=(1-2M/rsat-v2sat)1/2(1-2M/rEarth-v2Earth)-1/2

We can apply (1+d)n ~ 1+nd provided |d|

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