GPS and Relativity

How GPS Works

The GPS constellation consists of at least 24 satellites orbiting Earth at an altitude of approximately 20,200 kilometers, with orbital periods of about 12 hours. Each satellite carries multiple atomic clocks (cesium and rubidium) that broadcast precise timing signals. A GPS receiver on the ground picks up signals from at least four satellites simultaneously and calculates its position by measuring the time each signal took to arrive. Because radio signals travel at the speed of light, a timing error of even a few nanoseconds translates into a position error of about one meter.

The system requires extraordinary clock synchronization. All satellite clocks must be synchronized with ground-based reference clocks (maintained by the U.S. Naval Observatory and other institutions) to within about 20 to 30 nanoseconds for the system to achieve its specified accuracy of a few meters. This level of precision means that relativistic effects, which produce timing differences on the order of microseconds per day, cannot be ignored.

Special Relativistic Effect: Kinematic Time Dilation

GPS satellites orbit at approximately 3.87 kilometers per second. According to special relativity, clocks moving relative to an observer tick more slowly. The time dilation factor for the satellite orbital speed is gamma = 1/sqrt(1 - v²/c²), which for v = 3.87 km/s gives a fractional slowing of about 8.3 x 10^-11. This means satellite clocks tick slower than ground clocks by about 7.2 microseconds per day due to their orbital motion.

This effect alone would cause the satellites to fall behind ground clocks by 7.2 microseconds every day, which would translate to a position error of about 2.2 kilometers per day (since light travels about 300 meters per microsecond).

General Relativistic Effect: Gravitational Time Dilation

General relativity predicts that clocks in weaker gravitational fields (higher altitude) tick faster than clocks in stronger fields (lower altitude). GPS satellites orbit at 20,200 km altitude where Earth gravitational potential is significantly weaker than at the surface. The fractional speedup due to the gravitational effect is approximately 5.3 x 10^-10, causing satellite clocks to run faster than ground clocks by about 45.8 microseconds per day.

This gravitational effect is about six times larger than the kinematic effect and works in the opposite direction. It is by far the dominant relativistic correction for GPS.

The Net Relativistic Correction

The kinematic effect (satellite clocks run slow by 7.2 microseconds/day) and gravitational effect (satellite clocks run fast by 45.8 microseconds/day) partially cancel, leaving a net effect where satellite clocks run faster than ground clocks by approximately 38.6 microseconds per day.

If this correction were ignored, the accumulated timing error after one day would be about 38.6 microseconds. Since light travels approximately 300 meters per microsecond, this corresponds to a position error of about 11.6 kilometers per day. After a week without correction, the error would exceed 80 kilometers, and the system would be completely useless for any practical navigation purpose.

The GPS system handles this correction in two ways. First, before launch, the satellite clock frequencies are adjusted. The onboard clocks are set to tick at 10.22999999543 MHz instead of the nominal 10.23 MHz, so that once in orbit, the combined relativistic effects bring the effective frequency to exactly 10.23 MHz as observed from the ground. Second, the ground control segment monitors satellite clocks continuously and uploads small corrections to compensate for residual drift and other effects.

Additional Relativistic Effects in GPS

Beyond the primary time dilation corrections, several other relativistic effects must be accounted for. The Sagnac effect, a consequence of the rotation of the Earth, must be corrected when comparing times between satellites or between satellites and rotating ground stations. For receivers in aircraft or other fast-moving platforms, the receiver own velocity introduces additional kinematic time dilation that must be modeled.

The GPS orbital paths are slightly elliptical rather than perfectly circular, which means the satellite altitude and speed vary throughout each orbit. This creates a periodic variation in the relativistic correction of up to about 46 nanoseconds, which is corrected by the receiver software using a formula specified in the GPS interface control document. This periodic correction is known as the eccentricity correction and accounts for the varying combination of gravitational and kinematic dilation along the elliptical orbit.

The GPS system also accounts for the Shapiro delay, a general relativistic effect where signals passing near a massive object (in this case, Earth) are delayed by the curvature of spacetime along the signal path. For GPS signals, this delay is small (on the order of a few centimeters of equivalent range) but must be modeled for the highest precision applications.

GPS as a Test of Relativity

GPS constitutes a continuously running test of both special and general relativity. The system simply would not work at its designed accuracy if the relativistic corrections were wrong. Every time a GPS receiver calculates a position fix accurate to a few meters, it is implicitly confirming that Einstein theories predict the behavior of clocks in motion and in gravitational fields correctly.

In fact, GPS data has been used to make precision tests of general relativity. The gravitational redshift predicted by Einstein has been confirmed using GPS signals to an accuracy of about 0.01%. Future satellite navigation systems with even more precise clocks may provide tests of relativity at levels that approach the predictions of alternative theories of gravity, potentially distinguishing between general relativity and competing theories.