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 and flew twice around the world, first eastward, then westward, and compared the clocks against those of the United States Naval Observatory.

Overview

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 slower, expressed by the Lorentz factor. In a frame of reference at rest with respect to the center of the earth, the clock aboard the plane moving eastward, in the direction of the Earth's rotation, has a greater velocity (resulting in a relative time loss) than a clock that remains on the ground, while the clock aboard the plane moving westward, against the Earth's rotation, has a lower velocity than the one on the ground, resulting in a relative time gain.

According to general relativity, another effect comes into play: the slight increase in gravitational potential due to altitude that speeds the clocks back up. Since the aircraft are flying at roughly the same altitude in both directions, this effect is more "constant" between the two clocks, but nevertheless it causes a difference in comparison to the clock on the ground.

The results were published in Science in 1972:

	nanoseconds gained
	predicted			measured
	gravitational (general relativity)	kinematic (special relativity)	total
eastward	144±14	−184 ± 18	−40 ± 23	−59 ± 10
westward	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

The results were verified in an improved experiment in 1976 by the University of Maryland, this time agreeing with the relativistic predictions to a precision of about 1%. Versions of the experiment have also been done in which the only effect was gravitational and in which the only effect was kinematic.

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. 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 was 230 ± 20 ns.

Nowadays such relativistic effects are, for example, routinely incorporated into the calculations used for the Global Positioning System. Because the 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. 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.

Equations

The equations and effects involved in the experiment are:

Total time dilation

Τ = Δτ_v + Δτ_g + Δτ_s

Velocity

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

Gravitation

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

Sagnac effect

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

Where c = speed of light, h = height, g=acceleration of gravity, v = velocity, ω = angular velocity of Earth's rotation and τ represents the duration/distance of a section of the flight. 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, 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. Even as late as 1970, physicists such as Herbert Dingle and Mendel Sachs maintained vocally that the twin paradox was an erroneous prediction of special relativity, and that a null result would be observed with clocks.

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. 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.

Hafele and Keating obtained $8000 in funding from the Office of Naval Research 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, 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, including one with a photo showing a stewardess ironically checking her wristwatch while standing behind the instruments.

See also

• Tests of general relativity
• Twin paradox
• Time dilation
• GPS Time Dilation