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Like GPS? Thank Relativity

The Crux
By C. Renée James
Aug 29, 2014 10:42 PMNov 20, 2019 12:56 AM


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In 1971—16 years after Einstein's death—the definitive experiment to test Einstein's relativity was finally carried out. It required not a rocket launch but eight round-the-world plane tickets that cost the United States Naval Observatory, funded by taxpayers, a total of $7,600. The brainchild of Joseph Hafele (Washington University in St. Louis) and Richard Keating (United States Naval Observatory) were "Mr. Clocks," passengers on four round-the-world flights. (Since the Mr. Clocks were quite large, they were required to purchase two tickets per flight. The accompanying humans, however, took up only one seat each as they sat next to their attention-getting companions.) The Mr. Clocks had all been synchronized with the atomic clock standards at the Naval Observatory before flight. They were, in effect, the "twins" (or quadruplets, in this case) from Einstein's famous twin paradox, wherein one twin leaves Earth and travels nearly at the speed of light. Upon returning home, the traveling twin finds that she is much younger than her earthbound counterpart. In fact, a twin traveling at 80 percent the speed of light on a round-trip journey to the Sun's nearest stellar neighbor, Proxima Centauri, would arrive home fully four years younger than her sister. Although it was impossible to make the Mr. Clocks travel at any decent percentage of the speed of light for such a long time, physicists could get them going at jet speeds—about 300 meters (0.2 mile) per second, or a millionth the speed of light—for a couple of days. In addition, they could get the Mr. Clocks out of Earth's gravitational pit by about ten kilometers (six miles) relative to sea level. And with the accuracy that the Mr. Clocks were known to be capable of, the time differences should be easy to measure.

Hafele and Keating with two Mr. Clocks on their initial flight.

Clock Complications

This particular experiment had a gauntlet of computational complications, though, and Hafele attempted to account for all of them by imagining the experiment unfold from a great, stationary distance. For one, an altitude of ten kilometers is not much. Gravitationally speaking, Earth's surface is actually 6,371 kilometers (3,958 miles) from the center of mass, and different airports are at slightly different altitudes. Another ten kilometers turns out to make very little difference. On top of that, Earth is not simply a stationary body but is spinning on its axis (fortunately, Earth and its associated objects are all free-falling around the Sun, so its orbital motion becomes irrelevant in this experiment). This rotation forces everything on the planet to be in an accelerating frame of reference. Even the stationary ground-based clocks are not really stationary unless they're at the poles, so the clock used to synchronize the others experiences its own relativistic effects. Also, any airplane traveling eastward relative to the ground is actually traveling much faster absolutely than an airplane traveling westward at the same speed relative to the ground. Other problems arise from the fact that airplanes cannot simply fly completely around the world at a cruising altitude of ten kilometers. They have to land, refuel, and drop off and pick up other ticket-holding passengers. Quite often, in fact: on their trip around the world, they landed and took off over ten times each. These were, after all, ordinary commercial airliners, not dedicated physics labs. Even time spent at the gate at a latitude different from that of the Naval Observatory clock in Annapolis, Maryland, would make a difference. Hafele carefully computed the tiny effects from motion (special relativity) and altitude (general relativity), effects that added up and sometimes even canceled out.

One of the actual atomic clock units used in the Hafele–Keating experiment. Image by Binarysequence

Results From Mr. Clock

The Mr. Clocks were up to the task of measuring predicted time differences in the tens of nanoseconds, though, even when other factors (temperature differences or external vibrations, for instance) were accounted for. And so off they went—two around the world to the east and two around the world to the west. The westward clocks were expected, after all was said and done, to gain 275 nanoseconds (billionths of a second) compared with the standard in Annapolis. Their special relativistic contribution from speed should have been about a third of this total, the greatest effect on the westward flights being the quicker aging due to altitude. The eastward ones, on the other hand, should have lost 40 nanoseconds, as their speed would have stretched time out more than their altitude would have compressed it. That was the theoretical computation, anyway. Each trip took about three days, during which the attending physicist would continually check on the Mr. Clocks' health (and consequently sleep very little). There was some uncertainty, as each clock suffered its own "drift" in timekeeping. The initial result showed that the westward clocks gained an average of 160 nanoseconds, while the eastward ones lost 50 nanoseconds. Certainly these numbers were in the correct sense, but the westward figure was distressingly off. Some computational gymnastics attempting to account for each clock's drift put the numbers at 273 nanoseconds and 59 nanoseconds, more in line with Einstein's prediction. Again.

A Better Timekeeper

The general uncertainties in the famed Hafele-Keating experiment were troubling, though, so physicists devised more and more tests to verify relativity. Atomic clocks, which had experienced quite a growth spurt in their early years, ultimately became more and more portable. They also became more and more precise. These days, with state-of-the-art laser-driven aluminum ion clocks, even a 33-centimeter (one-foot) difference in altitude will produce a measurable difference in the rate of time. That's tabletop physics at its most bizarre, something Einstein would have been delighted to see. But he already knew that it would play out in this fashion. Once, when asked what he would have done if the Eddington expedition had failed to observe the bending of light, he stated, "Then I would have felt sorry for the dear Lord—the theory is correct." Adding layers of bizarre to the bending of light and the stretching of time is the current Gravity Probe B experiment, which shows that Earth's rotation drags space-time with it like a giant honey dipper twisting the thick, sticky contents of Winnie-the-Pooh's pot.

Mapping via Relativity

Other than stroking the egos of physicists, though, what can relativity do for anyone? It's one thing to joke that children simply age slower than grownups because they're closer to Earth, but another to make any practical use of the nanosecond consequences of our flexible space-time. In a single billionth of a second, though, light can go an entire foot. In the 275 billionths of a second accumulated by the three-day Hafele-Keating experiment, light could travel the length of a football field. But what about things moving much faster than jets? At much higher altitudes? Over time, their clocks would be less and less synchronized with clocks on Earth.

Artist's rendering of a Block IIF GPS satellite over the Earth. For the most part, this lack of agreement doesn't make a bit of difference to your average citizen, but it does if your average citizen is on the surface of Earth expecting a satellite more than 20,000 kilometers (12,400 miles) up and moving four kilometers (2.5 miles) per second to treat time the same way we ground-based creatures do. This is exactly why GPS satellites need to account for this difference. Initially conceived during the space race, our global positioning system would be pointless were it not for the fact that each space vehicle contains its own portable atomic clock. This time it's not to verify Einstein's relativity but to incorporate it. At the speeds and altitudes of these satellites, time would tick along nearly 39 millionths of a second faster each day. In a single microsecond, light can travel 300 meters (0.2 mile), so the 39-microsecond error from a mere day in orbit would mean confusing the position of an earthbound target by more than ten kilometers. To keep this from happening, the clocks on board the space vehicles are adjusted so that their frequencies are ever so slightly smaller (about one part in two billion) than their counterparts on Earth. Now every member of the small armada of GPS satellites sends out its signal 24/7 telling anyone who will listen what the exact time is. Your receiver picks up several of these signals, does a bit of calculating, and figures that if satellite A says it's this time, and satellite B says that it's this other time, and satellite C says it's some third time, this must mean that you are at the intersection of Gray Street and 17th Avenue.

Grateful for GPS

If you have any sort of GPS system, you owe a debt of gratitude to Einstein and to the decades of physicists who worked, very often on the public dime, to create portable, highly precise clocks, whose oscillators were not springs or pendulums but atoms themselves. The century-old thought experiments of a curious character led to ever more rigorous tests of bizarre predictions, and those led to an industry that is expected to be worth over $26 billion by 2016. It has saved untold hours of time that would have been spent driving around aimlessly. It has led people out of danger, saved injured hikers, and helped motorists find the nearest gas station when they were running on fumes. Parents use it to track their children, receiving a text message if the babysitter has taken the child beyond a certain distance. In 2013, German researchers showed that GPS could better map actual shifts in Earth's crust and provide life-saving minutes of warning for those in the path of a deadly tsunami. It has become so commonplace, in fact, that the idea of getting a new cell phone without GPS capabilities is virtually unheard of. Why would you? It costs almost nothing these days, and it is practically guaranteed to be useful, if for no other reason than finding the nearest restaurant on your next road trip.

Top image by Dolinsk / Shutterstock

From Science Unshackled: How Obscure, Abstract, Seemingly Useless Scientific Research Turned Out to Be the Basis of Modern Life, by C. Renée James. Published by Johns Hopkins University Press. Reprinted by permission of the publisher.

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