Is Einstein Still Right?

by Clifford M. Will

  The actual experiment, coordinated by J. C. Hafele, then at Washington University in St. Louis, and Richard Keating of the US Naval Observatory, used cesium-beam atomic clocks. Because of their limited budget they could not simply charter planes to circumnavigate the globe non-stop, but instead they had to fly the clocks on commercial aircraft during regularly scheduled flights. (Because of government regulations, they couldn’t even fly first class!) No, the clocks were not strapped into their seats like the other passengers. Actually on most of the flights they were positioned against the front wall of the coach-class cabin to protect them against sudden motions on landing and to connect them more easily to the airplane’s power supply. The flights included numerous stopovers during the course of the experiment, and the air speeds, altitudes, latitudes and flight directions all varied. But by keeping careful logs of the flight data, they could calculate the expected time differences for each flight. The eastward trip took place between 4 and 7 October and included 41 hours in flight, while the westward trip took place between 13 and 17 October, and included 49 hours in flight. For the westward flight the predicted gain in the flying clock was 275 nanoseconds (billionths of a second), of which two thirds was due to the gravitational blueshift; the observed gain was 273 nanoseconds. For the eastward flight, the time dilation was predicted to give a loss larger than the gain due to the gravitational blueshift, the net being a loss of 40 nanoseconds; the observed loss was 59 nanoseconds. Within the experimental errors of plus or minus 20 nanoseconds, attributed to inaccuracies in the flight data and intrinsic variations in the rates of the cesium clocks, the observations agreed with the predictions!

  What is the main ingredient missing from the two experiments we have just described? Height. Up to a limiting value, the size of the gravitational redshift increases with the difference in height between the emitter and receiver or between the two clocks. The limit arises because the higher you go, the weaker gravity gets, so that eventually added height makes no difference. The gamma-ray experiment of Pound and Rebka, unfortunately, could not go to larger differences in height, because the gamma rays are emitted equally in all directions from the sample of Fe57 crystals. A consequence of this is that as the height is increased, the number of gamma rays received by the absorber becomes so small as to be unusable. The jet-lagged clocks experiment was financially limited to typical commercial aircraft altitudes. But what about putting an atomic clock on a satellite or rocket? By the time of the Hafele–Keating experiment, plans for such an experiment were already under way.

  The idea of a satellite test of the redshift had been suggested as early as 1956, just before the first Earth satellites were launched, and had been tried in 1966, with modest success, at the 10 percent level. But the experiment that was being worked on in 1971 was truly ambitious. The idea was to get two of the best atomic clocks in existence, called hydrogen maser clocks, put one on the top of a rocket, blast it up to a couple of times the radius of the Earth, and compare its rate to the clock left on the ground as the rocket ascends and later descends back to Earth. In principle, the accuracy achievable in a measurement of the shift was one part in ten thousand, or one hundredth of a percent. Fortunately, the experiment brought together the two sets of experts required to pull it off.

  The first set of experts consisted of Robert Vessot and Martin Levine of the Smithsonian Astrophysical Observatory at Harvard University. Their laboratory was at the forefront of development of this new kind of atomic clock. Soon after the invention of the hydrogen maser clock in 1959 by Harvard physicists Norman Ramsey, Daniel Kleppner and H. Mark Goldenberg, Vessot, who then worked for Varian Associates, pioneered the development of a commercial, portable version of the new timepiece. By 1969 Vessot had left industry for Harvard, and now wanted to make use of these devices in fundamental physics experiments. The other set of experts was the National Aeronautics and Space Administration (NASA), National Aeronautics and Space Administration (NASA) which would provide the launch vehicle, tracking and other facilities required to get the clock aloft and measure its frequency shift.

  The hydrogen maser clock was ideal for this experiment. It is based on a transition between two atomic energy levels in hydrogen that emits light in the radio portion of the spectrum, with a frequency of 1,420 million cycles per second (1,420 megahertz), or a wavelength of about 21 centimeters. The spread in frequencies is so narrow that the actual frequency is known to twelve significant digits, or to an accuracy of one part in a hundred billion.

  The original plan was to put one of these clocks in orbit, but by 1970 it was evident that the cost of the Titan 3C rocket and the 2,000-pound payload required to achieve this was more than the NASA budget would permit, and a more modest plan was developed. The new plan was to send the clock on a suborbital flight to an altitude of about 10,000 kilometers, a thousand times higher than typical commercial jet altitudes, using a cheaper Scout D rocket and a smaller payload of a few hundred pounds. NASA denoted the project Gravity Probe-A. But there were two problems that had to be overcome to make the experiment work.

  The first was to build a lightweight maser clock that would withstand the 20 g acceleration it would experience during launch. The second was how to detect the gravitational redshift. Consider what happens during the ascent of the rocket, say, when the rocket clock emits its signal, and the signal is received at the ground and compared with the frequency of the ground clock. The received frequency differs from the ground clock frequency because of two effects: the gravitational blueshift, caused by the height difference, and special relativity’s time dilation, caused by the rocket’s rapid motion. However, the received frequency is also shifted toward the red because of the usual Doppler shift produced by the rocket’s motion away from the ground clock (during descent of the rocket, the Doppler effect would produce a blueshift), and this Doppler shift is 100,000 times larger than the gravitational redshift for a typical Scout D velocity of several kilometers per second.

  One clearly must find a way to eliminate this huge effect somehow, in order to see the much smaller effects of interest, and this was done in a very elegant way as follows. Suppose a signal is emitted from the ground clock toward the ascending rocket clock (this is called the “uplink,” while a downward signal is called the “downlink”). Incorporated into the rocket payload is a “transponder,” a device that takes a received signal and sends it right back with the same frequency (and with a little more power, to make up for any losses during transmission up). Right when the transponder sends the transponded downlink, a second “one-way” downlink is sent to the receiver on Earth (see Figure 2.4).

  Figure 2.4 Gravity Probe-A. As a hydrogen maser clock rises over the Atlantic in the nose cone of a Scout rocket, a signal from an identical ground maser clock is sent toward it. When the signal is received by the rocket, it is sent back, and a signal directly from the rocket clock is sent along with it. Because the rocket clock is at a different height and is moving at a different velocity than the ground clock, the frequency of the one-way signal received by the ground clock is changed by the Doppler effect, the gravitational redshift, and special relativity’s time dilation. For the two-way signal, though, the ground clock is both emitter and receiver, so there is no gravitational redshift or time dilation (the signal is emitted and received at the same height and velocity), and the Doppler shift contributes twice. The transponder that turns the signal sees a signal redshifted by the Doppler effect because it is moving away from the ground, and the ground receiver sees the signal Doppler shifted again because the transponder that turned it around is moving away from the ground. Thus, if half of the frequency change of the round-trip signal is subtracted from the frequency change of the one-way signal, the Doppler effect will cancel out completely.

  What happens to the frequencies of the transponded signal and that of the one-way downlink signal when received back on Earth? Initially, when the uplink signal arrives at the ascending rocket clock, the received frequency differs from that of t
he rocket clock by the Doppler shift, and by the gravitational redshift and time dilation. Then, when the transponded signal is received back on Earth, its frequency is further redshifted by the Doppler effect, because the transponder (which you recall is attached to the ascending rocket) is receding from Earth, but it is now gravitationally blueshifted by an amount that exactly cancels the gravitational redshift experienced by the signal on the uplink. The signal is also changed by time dilation, but also in a way that exactly cancels the change experienced on the uplink. Therefore, when received back at the ground, this two-way signal has had its frequency changed by exactly twice the Doppler shift, and that’s all. On the other hand, the one-way downlink signal sent by the ascending rocket clock and received at the ground has been changed by only one factor of the Doppler shift and by the gravitational blueshift and the time dilation. All one has to do then is take the frequency change on the two-way signal, divide by two, and subtract it from the one-way frequency change, and presto: no Doppler effect. This Doppler cancelation scheme was in fact incorporated directly into the electronics that gathered the data from the two radio links, and so it disappeared from the experiment altogether.

  After the years of development of the clocks, of making one of them spaceworthy, of testing and retesting them to simulate launch conditions, the time had come to actually do the experiment. It was a perfect day for a launch, a pleasant June morning in 1976, with just some high, thin clouds in the sky. Vessot was in charge of the rocket clock at NASA’s launch facility on Wallops Island, one of the many small islands that hug the eastern coast of the narrow Virginia peninsula separating Chesapeake Bay from the Atlantic Ocean. Levine took care of the ground clock at the NASA tracking station at Merritt Island, right next to Cape Canaveral in Florida. As is often the case, the period leading up to launch was not without its crises. One countdown had already been aborted because of a problem with some ammonia refrigerant. A misbehaving monitor designed to keep track of conditions in the rocket clock was brought into line by Vessot through the elegant technique of dropping it on the floor, a sort-of old-fashioned “hard-reboot.”

  Finally, the countdown reached the end, and at 6:41 a.m. Eastern Standard Time the Scout D roared into the Virginia skies. At 6:46, the payload containing the clock separated from the fourth stage of the rocket, and was in free fall thereafter. At this point data could be taken, because the rocket clock was no longer affected by the high accelerations and vibrations of launch. For about three minutes the one-way downlink frequency from the rocket clock (with the Doppler piece canceled automatically, remember) was lower than that of the ground clock, because the high velocity of the rocket caused a time dilation redshift to lower frequencies, while the altitude was not yet large enough to produce a gravitational blueshift. At 6:49 the frequencies of rocket and ground clock were exactly the same, because the gravitational blueshift canceled exactly the time dilation redshift. After that, as the altitude increased and the speed of the rocket decreased, the gravitational blueshift dominated more and more. The peak of the orbit occurred at 7:40. Here, the shift was predominantly the gravitational blueshift, amounting to almost 1 hertz out of 1,420 megahertz, or four parts in ten billion. Because both the rocket clock (after separation from the fourth stage of the Scout D) and the ground clock maintained their intrinsic frequencies stably to one part in a million billion, they could measure these changes in frequency to very high accuracy. Data taking continued during descent, with the cancelation between gravitational blueshift and time dilation occurring again at 8:31. At 8:36, the payload was too low in the sky to be tracked reliably, and shortly thereafter, some 900 miles east of Bermuda, the rocket and its onboard atomic clock crashed as planned into the Atlantic Ocean. This two-hour flight produced more than two years of data analysis for Vessot and his colleagues, but when all was said and done, the predicted frequency shifts agreed with the observed shifts to a precision of seventy parts per million, or to 7/1000 of a percent.

  Modern atomic clocks maintain time so precisely that the gravitational redshift now touches our daily lives. This remarkable convergence between fundamental physics and everyday life is due to GPS, the Global Positioning System. Deployed originally for military navigation, GPS has rapidly transformed itself into a thriving commercial entity with countless applications. It is based on an array of as many as thirty-two satellites orbiting the Earth, each carrying a precise atomic clock based either on cesium or rubidium atoms. Using a GPS-enabled device, which detects radio emissions from any of the satellites which happen to be overhead, users can determine their absolute latitude, longitude and altitude to an accuracy of 15 meters, and local time to fifty billionths of a second. In addition to GPS there are the Russian GLONASS, the Chinese BeiDou and the European Galileo systems, all in various stages of operation and development.

  Apart from the obvious military uses, GPS has found applications in airplane navigation, oil exploration, wilderness recreation, bridge construction, sailing and interstate trucking, to name just a few. Another crucial application is finding lost smartphones and lost pets! If you purchased this book from a vendor such as Amazon, its route to your door was tracked every step of the way using barcode scanners with GPS. Even Hollywood has met GPS, pitting James Bond in the 1997 film Tomorrow Never Dies against an evil genius who was inserting deliberate errors into the GPS system and sending British ships into harm’s way. For better or for worse, GPS is everywhere and it is here to stay.

  But how does GPS actually work? On a piece of paper in two dimensions, the problem is one faced by every high school mathematics student: given two fixed points representing two satellites, find the point, representing your cell phone, that is a given distance from one and a given distance from the other (Figure 2.5). The problem is solved by drawing arcs of the given radii using a compass and finding where they intersect. Your cell phone does something similar, albeit a bit more complicated. First, the GPS receiver in your cell phone calculates the distance between itself and each satellite with which it communicated, using the time at which the signal was emitted, as determined by the on-board atomic clock and encoded into the signal, the time at which the signal was received by your phone, and using the speed of light. Then, given readings from four GPS satellites, it is a simple matter to use the same principle as our high school math student to compute the receiver’s precise location, both in space and in time (the GPS receiver does it by solving equations embedded in the computer chip, not with compasses). 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.

  Figure 2.5 GPS and relativity. As many as thirty-two satellites of the US GPS system orbit the Earth in an array of orbital planes designed so that a user has a strong likelihood of connecting to three or four satellites at any given time. Knowing the distances from any two satellites on a plane, a user can determine his location on that plane by the analogue of finding where two circles of the given radii intersect on the plane, as shown in the right panel. With three satellites in view, the user can determine his location in three dimensions. With four satellites, the user can also determine local time, i.e. the fourth dimension.

  But, the orbiting clocks are 20,000 kilometers above the Earth in accurately known orbits that circle twice per day, and experience gravity that is four times weaker than that on the ground. Because of the gravitational redshift effect, the orbiting clocks tick slightly faster, by about 45 microseconds (millionths of a second) per day, than ground clocks. The satellite clocks are also moving at 14,000 kilometers per hour, much faster than clocks on the surface of the Earth, and special relativity says that such rapidly moving clocks tick more slowly, by about 7 microseconds per day, because of time dilation. 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. Compare that with the 50 nanoseconds precision required! 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 about 7 meters per minute!

  When the first GPS satellite was launched in 1977, it was already recognized that incorporating relativity would be necessary. Initially this was done by electronically adjusting the rates of the satellite clocks so that they artifically ticked at the same rate as ground clocks.

  But in 1983, after some relativity experts argued that the relativistic effects were being implemented incorrectly, the US Air Force, which ran the GPS program at the time, began to worry. Six satellites were already in orbit with more scheduled for launch, and it became important to determine if these critics were correct. Accordingly, they asked the Air Force Studies Board (AFSB), a part of the National Research Council, to conduct an independent analysis of their methods. That board then asked Cliff (Nico was only three years old at the time!) to put together a committee of experts and to chair the study. After examining the methods used by the Air Force to account for relativistic effects, and after studying the analysis of the critics, Cliff’s committee concluded that the criticisms were not correct and that the Air Force was properly implementing methods that had become standard in the atomic clock community. The committee then turned to a list of tasks of a more operational nature for which the Air Force wanted advice.

  An awkward moment occurred midway through the year-long study when the AFSB staff person asked Cliff when he had become a naturalized US citizen. Cliff replied that he was still a Canadian citizen on a permanent resident’s green card (a fact evidently overlooked on the curriculum vitae that he had submitted to the AFSB). This was a problem, because by AFSB rules all studies are initially classified, even if they do not actually deal with classified material, as was true of Cliff’s study. After some urgent phone calls the staffer managed to get a general to declassify the study retroactively, so that the work could continue.