# Hafele–Keating experiment

Hafele and Keating aboard a commercial airliner, with two of the atomic clocks and a stewardess.
One of the actual HP 5061A Cesium Beam atomic clock units used in the Hafele–Keating experiment

The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity.

## Overview

### Kinematic time dilation

According to special relativity, the rate of a clock is greatest according to an observer who is at rest with respect to the clock. In a frame of reference in which the clock is not at rest, the clock runs more slowly, as expressed by the Lorentz factor. This effect, called time dilation, has been confirmed in many tests of special relativity, such as the Ives–Stilwell experiment and time dilation of moving particles. Considering the Hafele–Keating experiment in a frame of reference at rest with respect to the center of the earth, a clock aboard the plane moving eastward, in the direction of the Earth's rotation, had a greater velocity (resulting in a relative time loss) than one that remained on the ground, while a clock aboard the plane moving westward, against the Earth's rotation, had a lower velocity than one on the ground.

### Gravitational time dilation

General relativity predicts an additional effect, in which an increase in gravitational potential due to altitude speeds the clocks up. That is, clocks at higher altitude tick faster than clocks on Earth's surface. This effect has been confirmed in many tests of general relativity, such as the Pound–Rebka experiment and Gravity Probe A. In the Hafele–Keating experiment, there was a slight increase in gravitational potential due to altitude that tended to speed the clocks back up. Since the aircraft flew at roughly the same altitude in both directions, this effect was approximately the same for the two planes, but nevertheless it caused a difference in comparison to the clocks on the ground.

### Results

The results were published in Science in 1972:[1][2]

nanoseconds gained predicted measured gravitational 144±14 −184 ± 18 −40 ± 23 −59 ± 10 179±18 96±10 275±21 273±7

The published outcome of the experiment was consistent with special and general relativity. The observed time gains and losses were different from zero to a high degree of confidence, and were in agreement with relativistic predictions to within the ~10% precision of the experiment.

## Repetitions

A more complex and precise experiment of this kind was performed by research group of the University of Maryland between September 1975 and January 1976. Three atomic clocks were brought to an altitude of 10 km above Chesapeake Bay in Maryland, and three other atomic clocks were at the ground. A turboprop plane was used, flying at only 500 km/h, in order to minimize the velocity effect. The plane was steadily observed using radar, and its position and velocity were measured every second. Five flights were carried out, each of 15 hours duration. Special containers protected the clocks from external influences such as vibrations, magnetic fields, or temperature variations. The time difference was measured by direct clock comparison at the ground before and after the flight, as well as during the flight by laser pulses of 0.1 ns duration. Those signals were sent to the plane, reflected, and again received at the ground station. The time difference was observable during the flight, before later analysis. An overall difference of 47.1 ns was measured, which consisted of the velocity effect of -5.7 ns and a gravitational effect of 52.8 ns. This agrees with the relativistic predictions to a precision of about 1.6%.[3][4]

A reenactment of the original experiment by the NPL took place in 1996 on the 25th anniversary of the original experiment, using more precise atomic clocks during a flight from London to Washington, D.C. and back again. The results were verified to a higher degree of accuracy. A time gain of 39 ± 2 ns was observed, compared to a relativistic prediction of 39.8 ns.[5] In June 2010, NPL again repeated the experiment, this time around the globe (London - Los Angeles - Auckland - Hongkong - London). The predicted value was 246 ± 3 ns, the measured value 230 ± 20 ns.[6]

Because the Hafele–Keating experiment was reproduced by increasingly accurate methods, there has been a consensus among physicists since at least the 1970s that the relativistic predictions of gravitational and kinematic effects on time have been conclusively verified.[7] Criticisms of the experiment did not address the subsequent verification of the result by more accurate methods, and have been shown to be in error.[8]

## Similar experiments with atomic clocks

Measurements in which the only effect was gravitational have been conducted by Iijima et al. between 1975 and 1977. They carried a commercial cesium clock back and forth from the National Astronomical Observatory of Japan in Mitaka, at 58 m above sea level, to Norikura corona station, at 2876 m above sea level, corresponding to an altitude difference of 2818 m. During the times when the clock stayed at Mitaka, it was compared with another cesium clock. The measured change in rate was (29±1.5)×10−14, consistent with the result of 30.7×10−14 predicted by general relativity.[9]

In 1976, Briatore and Leschiutta compared the rates of two cesium clocks, one in Turin 250 m above sea level, the other at Plateau Rosa 3500 m above sea level. The comparison was conducted by evaluating the arrival times of VHF television synchronization pulses and of a LORAN-C chain. The predicted difference was 30.6 ns/d. Using two different operating criteria, they found differences of 33.8±6.8 ns/d and 36.5±5.8 ns/d, respectively, in agreement with general relativity.[10] Environmental factors were controlled far more precisely than in the Iijima experiment, in which many complicated corrections had to be applied.

In 2010, Chou et al. performed tests in which both gravitational and velocity effects were measured at velocities and gravitational potentials much smaller than those used in the mountain-valley experiments of the 1970s. It was possible to confirm velocity time dilation at the 10−16 level at speeds below 36 km/h. Also, gravitational time dilation was measured from a difference in elevation between two clocks of only 33 cm.[11][12]

Nowadays both gravitational and velocity effects are, for example, routinely incorporated into the calculations used for the Global Positioning System.[13]

## Equations

The equations and effects involved in the experiment are:

Total time dilation

$\Tau = \Delta\tau_v + \Delta\tau_g + \Delta\tau_s$

Velocity

$\Delta\tau_v = - \frac{1}{2c^2} \sum_{i=1}^{k}v_i^2 \Delta t_i$

Gravitation

$\Delta\tau_g = \frac{g}{c^2} \sum_{i=1}^{k} (h_i - h_0) \Delta t_i$

Sagnac effect

$\Delta\tau_s = - \frac{\omega^2}{2c^2} \sum_{i=1}^{k} R_i^2 \cos^2 \phi_i \Delta t_i$

Where c = speed of light, h = height, g=acceleration of gravity, v = velocity, $\omega$ = angular velocity of Earth's rotation and $\Delta \tau$ represents the proper duration/distance of a section of the flight while $\Delta t$ represents the coordinate duration/distance of a section of the flight as measured from an inertial frame centered in the center of the Earth (ECIF). The effects are summed over the entire flight, since the parameters will change with time.

## Historical and scientific background

In his original 1905 paper on special relativity,[14] Einstein suggested a possible test of the theory: "Thence we conclude that a spring-clock at the equator must go more slowly, by a very small amount, than a precisely similar clock situated at one of the poles under otherwise identical conditions." Because he had not yet developed the general theory, he did not realize that the results of such a test would in fact be null, since the surface of the earth is a gravitational equipotential, and therefore the effects of kinematic and gravitational time dilation would precisely cancel. The kinematic effect was verified in the 1938 Ives–Stilwell experiment and in the 1940 Rossi-Hall experiment. General relativity's prediction of the gravitational effect was confirmed in 1959 by Pound and Rebka. These experiments, however, used subatomic particles, and were therefore less direct than the type of measurement with actual clocks as originally envisioned by Einstein.

Hafele, an assistant professor of physics at Washington University in St. Louis, was preparing notes for a physics lecture when he did a back-of-the-envelope calculation showing that an atomic clock aboard a commercial airliner should have sufficient precision to detect the predicted relativistic effects.[15] He spent a year in fruitless attempts to get funding for such an experiment, until he was approached after a talk on the topic by Keating, an astronomer at the United States Naval Observatory who worked with atomic clocks.[15]

Hafele and Keating obtained $8000 in funding from the Office of Naval Research[16] for one of the most inexpensive tests ever conducted of general relativity. Of this amount,$7600 was spent on the eight round-the-world plane tickets,[17] including two seats on each flight for "Mr. Clock." They flew eastward around the world, ran the clocks side by side for a week, and then flew westward. The crew of each flight helped by supplying the navigational data needed for the comparison with theory. In addition to the scientific papers published in Science, there were several accounts published in the popular press and other publications,[15][18] including one with a photo showing a stewardess ironically checking her wristwatch while standing behind the instruments.[19]

## References

1. ^ Hafele, J. C.; Keating, R. E. (July 14, 1972). "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains". Science 177 (4044): 166–168. Bibcode:1972Sci...177..166H. doi:10.1126/science.177.4044.166. PMID 17779917. edit
2. ^ Hafele, J. C.; Keating, R. E. (July 14, 1972). "Around-the-World Atomic Clocks: Observed Relativistic Time Gains". Science 177 (4044): 168–170. Bibcode:1972Sci...177..168H. doi:10.1126/science.177.4044.168. PMID 17779918. edit
3. ^ Alley, C.O. (1979). "Relativity and Clocks". Proceedings of 33rd Annual Symposium on Frequency Control: 4–39. doi:10.1109/FREQ.1979.200296.
4. ^ Alley, C.O. (1981). "Introduction to some fundamental concepts of general relativity and to their required use in some modern timekeeping systems". Proceedings of the Precise Time And Time Interval systems and applications meeting 13: 687–727.
5. ^ NPL Metromnia, Issue 18 - Spring 2005
6. ^ NPL news, Time flies, 1 Feb. 2011
7. ^ Wolfgang Rindler, Essential Relativity: Special, General, and Cosmological, Springer-Verlag, 1979, p. 45
8. ^ Roberts and Schleif, What is the experimental basis of Special Relativity?
9. ^ Iijima, S.; Fujiwara, K.; H. Kobayashi; T. Kato (1978). "An experiment for the potential blue shift at the Norikura Corona Station". Annals of the Tokyo Astronomical Observatory 17: 68–78. Bibcode:1978AnTok..17...68I.
10. ^ Briatore, L.; Leschiutta, S. (1977). "Evidence for the earth gravitational shift by direct atomic-time-scale comparison". Il Nuovo Cimento B 37 (2): 219–231. Bibcode:1977NCimB..37..219B. doi:10.1007/BF02726320.
11. ^ Nontechnical explanation at http://www.scientificamerican.com/article.cfm?id=time-dilation
12. ^ Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843.
13. ^ Deines, "Uncompensated relativity effects for a ground-based GPSA receiver", Position Location and Navigation Symposium, 1992. Record. '500 Years After Columbus - Navigation Challenges of Tomorrow'. IEEE PLANS '92.
14. ^ A. Einstein, "On the electrodynamics of moving bodies," Annalen der Physik 17 (10): 891, tr. W. Perrett and G.B. Jeffery, 1923
15. ^ a b c New Scientist, February 3, 1972, "The clock paradox resolved"
16. ^ Hafele, "Performance and results of portable clocks in aircraft," PTTI, 3rd Annual Meeting, 1971; http://www.pttimeeting.org/archivemeetings/ptti1971.html
17. ^ Martin Gardner, Relativity Simply Explained, Dover, 1997, p. 117
18. ^ Time Magazine, October 18, 1971; http://www.time.com/time/magazine/article/0,9171,910115,00.html
19. ^ John Pearson, "Science Worldwide", Popular Mechanics, January 1972, p. 30.