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# GPS and Relativity

Posted Aug26-11 at 03:42 AM by stevebd1

Time dilation due to gravity (GR)-

$$d\tau=dt\sqrt{1-\frac{2M}{r}}$$

where $M=Gm/c^2$

Time dilation due to velocity (SR)-

$$d\tau=dt\sqrt{1-\frac{v^2}{c^2}}$$

Quantities-

Earth's mass- 5.9736e+24 kg

Earth's (mean) radius- 6.371e+6 m

Satellite's altitude- 2e+7 m

Satellite's velocity- 3.889e+3 m/s

86400 seconds in a day

Difference due to gravity-

dT sat (r=6.371e+6 + 2e+7)- 0.999999999833
dT Earth (r=6.371e+6)- 0.999999999304

difference- 0.000000000529

Time dilation on Earth relative to satellite is 4.562265e-05 seconds per day or 45.62265 microseconds

Difference due to velocity-

dT sat (v=3.889e+3)- 0.999999999916
dT Earth- 1 (0.999999999999 at the equator, i.e. ~1)

difference- 0.000000000084

Time dilation on satellite relative to Earth is 7.26931e-06 seconds per day or 7.26931 microseconds

Total time dilation- 38.35334 microseconds per day on Earth, atomic clocks on satellites need to be slowed down to match atomic clocks on Earth.

The ISS-

altitude- 2.78e+5 to 4.6e+5 m

velocity- 7701.11 m/s

The time dilation on the ISS relative to Earth is between 24.35197 and 25.8877 microseconds a day (depending on the altitude) meaning the atomic clocks on the ISS have to be speeded up to match those on Earth in contrast to the GPS satellites (which means that since its inception in 1998, the ISS has travelled forward in time by approx. 0.1 of a second).
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