Einstein's Relativity and Everyday Life
Global Positioning System (GPS)
Clifford M. Will
What good is fundamental physics to the person on the street?
This is the perennial question posed to physicists by their non-science friends, by students in the humanities and social sciences, and by politicians looking to justify spending tax dollars on basic science. One of the problems is that it is hard to predict definitely what the payback of basic physics will be, though few dispute that physics is somehow "good."
Physicists have become adept at finding good examples of the long-term benefit of basic physics: the quantum theory of solids leading to semiconductors and computer chips, nuclear magnetic resonance leading to MRI imaging, particle accelerators leading to beams for cancer treatment. But what about Einstein's theories of special and general relativity? One could hardly imagine a branch of fundamental physics less likely to have practical consequences. But strangely enough, relativity plays a key role in a multi-billion dollar growth industry centered around the Global Positioning System (GPS).
When Einstein finalized his theory of gravity and curved spacetime in November 1915, ending a quest which he began with his 1905 special relativity, he had little concern for practical or observable consequences. He was unimpressed when measurements of the bending of starlight in 1919 confirmed his theory. Even today, general relativity plays its main role in the astronomical domain, with its black holes, gravity waves and cosmic big bangs, or in the domain of the ultra-small, where theorists look to unify general relativity with the other interactions, using exotic concepts such as strings and branes.
But GPS is an exception. Built at a cost of over $10 billion mainly for military navigation, GPS has rapidly transformed itself into a thriving commercial industry. The system is based on an array of 24 satellites orbiting the earth, each carrying a precise atomic clock. Using a hand-held GPS receiver which detects radio emissions from any of the satellites which happen to be overhead, users of even moderately priced devices can determine latitude, longitude and altitude to an accuracy which can currently reach 15 meters, and local time to 50 billionths of a second. Apart from the obvious military uses, GPS is finding applications in airplane navigation, oil exploration, wilderness recreation, bridge construction, sailing, and interstate trucking, to name just a few. Even Hollywood has met GPS, recently pitting James Bond in "Tomorrow Never Dies" against an evil genius who was inserting deliberate errors into the GPS system and sending British ships into harm's way.
But in a relativistic world, things are not simple. The satellite clocks are moving at 14,000 km/hr in orbits that circle the Earth twice per day, much faster than clocks on the surface of the Earth, and Einstein's theory of special relativity says that rapidly moving clocks tick more slowly, by about seven microseconds (millionths of a second) per day.
Also, the orbiting clocks are 20,000 km above the Earth, and experience gravity that is four times weaker than that on the ground. Einstein's general relativity theory says that gravity curves space and time, resulting in a tendency for the orbiting clocks to tick slightly faster, by about 45 microseconds per day. The net result is that time on a GPS satellite clock advances faster than a clock on the ground by about 38 microseconds per day.
To determine its location, the GPS receiver uses the time at which each signal from a satellite was emitted, as determined by the on-board atomic clock and encoded into the signal, together the with speed of light, to calculate the distance between itself and the satellites it communicated with. The orbit of each satellite is known accurately. Given enough satellites, it is a simple problem in Euclidean geometry to compute the receiver's precise location, both in space and time. To achieve a navigation accuracy of 15 meters, time throughout the GPS system must be known to an accuracy of 50 nanoseconds, which simply corresponds to the time required for light to travel 15 meters.
But at 38 microseconds per day, the relativistic offset in the rates of the satellite clocks is so large that, if left uncompensated, it would cause navigational errors that accumulate faster than 10 km per day! GPS accounts for relativity by electronically adjusting the rates of the satellite clocks, and by building mathematical corrections into the computer chips which solve for the user's location. Without the proper application of relativity, GPS would fail in its navigational functions within about 2 minutes.
So the next time your plane approaches an airport in bad weather, and you just happen to be wondering "what good is basic physics?", think about Einstein and the GPS tracker in the cockpit, helping the pilots guide you to a safe landing.
Clifford M. Will is Professor and Chair of Physics at Washington University in St. Louis, and is the author of Was Einstein Right? In 1986 he chaired a study for the Air Force to find out if they were handling relativity properly in GPS. They were.