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# Article of the Month – March 2023

### Implications of Relativistic Effects on the Global Positioning System (GPS)

Raghav Sharma, BSc Physical Science with Electronics, University of Delhi

Abstract

Relativity has no obvious consequences in daily life, but one close look at the working of a GPS device is sufficient to highlight the enormous implications of relativistic effects on situations where velocity, gravity, and accuracy are involved. Clocks on a moving satellite do not appear to tick at the same intervals as clocks on Earth. This is especially problematic in high-accuracy systems like the GPS. Understanding the mathematical principles behind these effects allows for a derivation of precise offset values- adjustments that need to be made to satellite clocks to correct any time difference caused by relativity.

Introduction

The Global Positioning System (GPS) is a highly accurate, satellite-based positioning and navigation system. To maintain that accuracy, time-dilating relativistic effects arising from both the general and special theory of relativity need to be taken into account. This is achieved by adjusting the rates of onboard satellite clocks and incorporating mathematical corrections. This article explores time dilation effects on GPS and describes some calculations and adjustments that are made to account for them. The correction for special relativistic time dilation is derived in detail.

1. An Overview of Time-Dilation Effects

A handheld GPS receiver can determine the absolute position on the surface of the Earth to within 5 to 10 metres. Achieving a navigational accuracy of 5 metres requires knowing the onboard GPS satellite time to an accuracy of about 17 nanoseconds, which is the time taken by light to travel 5 metres. Because satellites are constantly moving with respect to the Earth-centred (approximately inertial) frame and are further away from the Earth’s gravitational well, one must consider time dilation caused by both special and general relativistic effects. If these effects were left uncompensated, navigational errors would accumulate at a rate in excess of 10 kilometres per day, rendering the system unusable within about 2 minutes.

1. Special Relativity

GPS Satellites are not geosynchronous because that would limit coverage. They have a time period of about 12 hours (so that any satellite passes over the same location each day) and a corresponding orbital velocity of about 3874 m/s relative to the centre of the Earth.

According to the Special Theory of Relativity, moving clocks run slower. The time dilation amount is determined by Lorentz transformations. The time measured on-board the satellite is reduced by the Lorentz factor γ:

where τ_{Ground} and τ_{GPS} are time intervals measured on the Earth’s surface and by the satellite clock, respectively.

The derivation of the time by which satellite clocks lag behind surface clocks, Δτ, is given below:

Using binomial expansion for small values of (v/c):

Taking v=3874 m/s and c=2.998×108m/s:

For a time-interval of 1 day (86,400s) on the Earth’s surface:

Therefore, GPS clocks lose about 7μs a day due to special relativistic time dilation.

1. General Relativity

GPS satellites have an orbital altitude of 20,184 km measured from the surface. According to the General Theory of Relativity, a clock in a gravitational field runs slower. This effect is given by:

Where τ_0 is the time interval measured near a mass (i.e., in a gravitational well), and is the time interval measured far away from the mass.

For small values of (M/r):

The clocks on the Earth’s surface are a distance of R_Earth=6378.1 km from the gravitational centre, so the time dilation with respect to GPS satellites is twofold. It is stated without proof that due to general relativistic time dilation effects, clocks onboard the satellites gain about 45μs per day, with respect to ground-based clocks.

1. Error Correction

The combination of general and special relativistic time dilation means that GPS clocks gain about 38μs a day. As stated before, the desired accuracy can be as high as 17 nanoseconds. Thus, it is crucial to correct any time difference.

A time offset of 38μs corresponds with a fractional change of +4.465×10^-10, i.e. the satellite clocks need to be slowed down by this fraction. The fundamental L-band frequency produced by the atomic clocks on-board is 10.23 MHz. This needs to be offset by the aforementioned fraction. Therefore, the actual frequency of the GPS clocks is set to 10.22999999543 MHz before launch.[3-4]

The variation in these changes due to the eccentricity (deviation from circularity) of the satellite orbit also needs to be taken care of. Built-in microcomputers used in GPS receivers help in any additional timing calculations required using satellite-provided data.

Conclusion

Relativity dictates that clocks aboard GPS satellites do not tick at the same rate as those on the Earth. Both general and special relativistic time dilation effects are at play. Neglecting to adjust for these would render GPS useless in a few minutes. Correcting them involves giving the onboard atomic clocks a slight offset in frequency, so that they may appear to run at the same rate as ground-based clocks. This correction is one of many needed to maintain a navigational accuracy of up to a few metres.

References:

1. Pogge, Richard W. (2017): Real-World Relativity: The GPS Navigation System
2. Will, Clifford M.: Einstein’s Relativity and Everyday Life
3. Nelson, Robert A. (1999): The Global Positioning System- A National Resource
4. Oxley, Alan (2017): Uncertainties in GPS Positioning- A Mathematical Discourse, Pages 71-80