Effects of relativity on GPS

Relativity produces several measurable effects on the GPS System that uncorrected would render the system useless. The most common effect is on GPS time measurements. Each satellite in the GPS constellation carries one or more atomic clocks in order for a GPS receiver to compare the signals from different satellites and triangulate its position. Three relativistic effects are the time dilation, gravitational frequency shift, and eccentricity effects.

The time dilation effect is predicted by the theory of Special Relativity. The theory states that relative motion between two clocks will cause the moving clock to appear slower than the stationary one. Thus, the clock on board a GPS satellite will lag behind an identical GPS receiver clock. Relativity predicts that the clock frequency will be slowed by a factor of v^2/(2c^2 )≈ 10 ^(-10) , or result in a delay of about 7 μs/day, where the orbital velocity is v = 4 km/s, and c = the speed of light. The time dilation effect have been measured and verified using the GPS system.

The effect of gravitational frequency shift on the GPS system is due to General Relativity, which states that a clock closer to a massive object will be slower than a clock farther away. Applied to the GPS system, the receivers are much closer to Earth than the satellites, causing the GPS clocks to appear faster by a factor of 5×10^(-10), or about 45 μs/day. Gravitational frequency shift is also a noticeable effect.

Combining the time dilation and gravitational frequency shift, clocks on the GPS satellites tick approximately 38 μs/day faster than clocks on the ground or in GPS receivers. Without correcting for these effects, errors in position determination of roughly 10 km/day would accumulate, resulting in a worthless system. In addition, because GPS satellite orbits are not perfectly circular, their elliptical orbits cause the time dilation and gravitational frequency shift effects to vary with time. This eccentricity effect causes the clock rate difference between a GPS satellite and a receiver to increase or decrease depending on the velocity orbital altitude of the satellite.

Several steps are taken to correct for relativistic effects and maintain a usable GPS system. First, each GPS satellite clock is programmed with a “factory offset” to match the 38 μs/day time shift. Second, GPS receivers are equipped with a microcomputer to perform relativistic calculations which help to correct for the aforementioned effects. Lastly, GPS satellite clock speeds are updated daily to be within 1 μs of Universal Coordinated Time as regulated by the US Naval Observatory.

References

(1) Pogge, Richard W., “Real-World Relativity: The GPS Navigation System” [http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html Accessed 25 January 2008] .

(2) Ashby, Neil, “General relativity in the global positioning system” [http://www.phys.lsu.edu/mog/mog9/node9.html Accessed 25 January 2008] .

(3) Van Flandern, Tom, “What the Global Positioning System Tells Us about Relativity.” [http://metaresearch.org/cosmology/gps-relativity.asp Accessed 25 January 2008] .