The theory of relativity in the real world: GPS

People often ask me: "What is the theory of relativity so good for?" Usually they think of it as a kind of abstract, mystical mathematical theory that has nothing to do with everyday life. In fact, this is not at all the case.


Imagine for a moment that you are flying in a commercial airliner whose pilot and crew are directing the aircraft to their destination using the Global Positioning System (GPS). Moreover, many expensive cars now come with built-in navigation systems, including GPS receivers with digital maps, and you can buy a handheld GPS navigator that will show your location (latitude, longitude, altitude) with an accuracy of 5 to 10 meters, weigh just a few ounces and cost about a hundred dollars.

GPS was developed by the United States Department of Defense to provide satellite navigation to the U.S. Army. Later it was transferred under the joint control of the Ministry of Defense and the Ministry of Transport for use in both military and civilian purposes.

The current GPS configuration consists of 24 satellites orbiting the earth in high orbits. Each satellite in the “constellation” of GPS flies at an altitude of about 20 thousand kilometers above the earth, its orbital speed is about 14 thousand kilometers per hour (orbital period of about 12 hours - contrary to popular belief, GPS satellites are not in geostationary or geosynchronous orbits). The satellite orbits are distributed so that at any time at least four satellites are visible from any position on Earth (up to 12 satellites can be seen at a time). Each satellite carries an atomic clock on its board that ticks with an accuracy of one nanosecond (one billionth of a second). A GPS receiver in an airplane determines its current position and direction by comparing time signals received from different GPS satellites (usually from 6 to 12) andtrilateration at the current position of each of the satellites. This achieves remarkable accuracy: even a simple pocket-sized GPS receiver can determine your absolute position relative to the Earth’s surface with an accuracy of 5 to 10 meters in just a few seconds (using difference techniques comparing two nearby receivers, accuracy is often on the order of centimeters or millimeters in relative position reached within an hour or so). The GPS receiver in the car can receive exact values ​​of location, speed and direction in real time!

To achieve this accuracy, time signals from GPS satellites must be known with an accuracy of 20-30 nanoseconds. However, due to the constant motion of satellites relative to the observer on Earth, in order to achieve the desired 20-30 nanosecond error, it is necessary to take into account the effects predicted by the general and special theory of relativity.

Since the observer on earth sees satellites in motion, the special theory of relativity (STR) states that we should see that their clock counts time more slowly (see the lecture on STR ). STO says that the onboard atomic clocks on satellites should be delayed by about 7 microseconds a day compared with the Earth’s clock due to the lower speed due to the relativistic time dilation.

In addition, the satellites are in orbits at a great distance from the Earth, where the curvature of space-time due to the mass of the Earth is less than on the earth's surface. The forecast of the general theory of relativity (GR) is that the course of the clocks located closer to the massive object will seem slower than those located further from it (see the lecture on black holes ). In fact, being observed from the earth's surface, the clocks on satellites seem faster than similar clocks on earth. Computations based on general relativity indicate that the clock on each GPS satellite should be in a hurry relative to the earth at 45 microseconds per day.

The combination of these two relativistic effects means that the clock on board each satellite should go faster than a similar clock on the ground by about 38 (45 - 7 = 38) microseconds per day! It sounds like a small quantity, but the high accuracy required in the GPS system requires nanosecond errors, while 38 microseconds are equal to 38 thousand nanoseconds. If these effects were not taken into account, then the coordinates calculated on the basis of the cloud of GPS satellites would be incorrect after two minutes, and errors in global locations would continue to accumulate at a speed of about 10 kilometers per day!The whole system would be completely unsuitable for navigation after a very small period of time. This kind of cumulative error is akin to finding my location when I stand on the porch of my house in Columbus, Ohio one day, and when I do the same search a week later, my GPS receiver would tell me that I was standing on my porch and I'm about 5 thousand meters high somewhere above Detroit.

GPS engineers included these relativistic effects in their calculations during system design and deployment. For example, to neutralize the effect described by GR, they slowed down the progress of the atomic clock before it was launched, so that, being in their orbits, they would go at the same speed as the reference atomic clock at GPS ground stations. In addition, a microcomputer is integrated in each GPS receiver, which (among other things) performs the necessary relativistic calculations at the time of determining the user's location.

Relativity is not just some kind of abstract mathematical theory: understanding it is a prerequisite for the correct operation of GPS!