# Hafele-Keating experiment

One of the atomic clocks used in the experiment, type HP 5061A (today a museum piece)

The Hafele-Keating experiment was a test of the time dilation following from the theory of relativity . Joseph C. Hafele and Richard E. Keating took 1,971 four cesium - atomic clocks on board a commercial airliner flew twice around the world, first eastward, then westward, and compared the clocks with those of the United States Naval Observatory .

According to the special theory of relativity , a clock moves fastest for an observer who is at rest relative to it. In a system that moves relative to it, the clock runs more slowly ( time dilation ); this effect is proportional to the square of the speed. It has meanwhile been proven in numerous tests of the special theory of relativity , see Ives-Stilwell experiment and time dilation of moving particles .

According to the general theory of relativity , clocks in the higher gravitational potential run faster at higher altitudes than in the lower gravitational potential near the earth's surface. This effect has also been confirmed in numerous tests of general relativity such as the Pound-Rebka experiment .

In the Hafele-Keating experiment, both effects are demonstrated at the same time. Similar experiments have now been repeated several times with increased precision, for example in the Maryland experiment (see below). The functioning of the GPS navigation system also confirms the theory.

## Hafele-Keating experiment

In the reference system, which is at rest with respect to the earth's center, the on-board clock moves eastward in the direction of the earth's rotation and has a greater speed than a clock on the earth's surface. According to the special theory of relativity, the on-board clock runs more slowly than the floor clock, so it loses time. On the other hand, the on-board clock, which moves westwards and thus against the rotation of the earth, has a lower speed than the floor clock, so it gains time. According to the general theory of relativity , the slight increase in the gravitational potential at greater heights also comes into play, so that due to the gravitational time dilation, both on-board clocks go faster than the floor clocks to the same extent.

The results of the time gains and losses, published in 1972, confirmed the relativistic predictions.

predicted measured
Gravitation according to ART Speed ​​according to SRT total
east 144 ± 14 ns −184 ± 18 ns −40 ± 23 ns −59 ± 10 ns
westward 179 ± 18 ns 96 ± 10 ns 275 ± 21 ns 273 ± 7 ns

## Repetitions

Replicas of the original experiment were carried out by the National Physical Laboratory (NPL) in 1996 with a higher degree of accuracy, on a flight from London to Washington, DC and back again. The on-board clocks measured an action of 39 ± 2 ns, in good agreement with the relativistic value of 39.8 ns. In June 2010, NPL carried out the experiment again, this time around the entire globe (London - Los Angeles - Auckland - Hong Kong - London). The relativistic value was 246 ± 3 ns, measured 230 ± 20 ns, again in good agreement.

## Maryland experiment

A more complex experiment of a similar nature was carried out from 1975 to 1976 by researchers at the University of Maryland , USA. Three atomic clocks were transported by aircraft to an altitude of about 10,000 m above Chesapeake Bay in Maryland , and three atomic clocks were on the ground. Special containers protect the watches from external influences such as vibrations, magnetic fields, temperature and air pressure fluctuations. Was used turboprop machines that barely 500 km / h managed to keep small around the speed effect. The aircraft were on a fixed course and were constantly monitored by radar. Initially, several test flights were completed and finally five main flights, each lasting 15 hours. Position and speed were determined every second.

On the one hand, the time difference was measured by direct comparison of clocks on the ground before and after the flight over about 20 hours. On the other hand, the time difference was read during the flight by laser light pulses of 0.1 ns duration by sending a signal to the aircraft, which was reflected by the aircraft and picked up again at the ground station. The difference increased steadily during the flight. Due to the gravitational effect, the aircraft clocks run faster and faster during the flight. A deviation of 47.1 ± 1.5 ns, consisting of −5.7 ns of slowing down due to the velocity effect, and 52.8 ns due to gravity was observed . This agrees very well with the value of 47.1 ± 0.25 ns predicted by the theory of relativity . The error calculation showed an accuracy of 1.6%.

## More experiments

Iijima & Fujiwara carried out measurements of the gravitational time dilation between 1975 and 1977 by alternately transporting a commercial cesium clock from the National Astronomical Observatory of Japan in Mitaka at 58 m above sea level to Mount Norikura at 2876 m above sea level. The corresponding difference in altitude was 2818 m. During the stay in Mitaka, the clock was compared with another cesium clock stationary there. The calculated blue shift of the transported watch due to gravity was 30.7 × 10 −14 , the measured value was (29 ± 1.5) × 10 −14 in agreement with the theoretical value. The ratio between the two values ​​was 0.94 ± 0.05.

In 1976 Briatore & Leschiutta compared the rate of two cesium clocks, one in Turin at 250 m and the second on the Plateau Rosa at 3500 m above sea level. The comparison was carried out by evaluating the arrival times of VHF television synchronization pulses and LORAN -C chains. The predicted difference was 30.6 ns per day. Using two surgical criteria, differences of 33.8 ± 6.8 ns / d and 36.5 ± 5.8 ns / d were measured, in agreement with the predicted value.

In 2010, Chou u. a. Tests carried out with which both gravitational and speed-related effects were measured at far lower distances and speeds. In this aluminum ions used as extremely precise clocks. The time dilation due to the speed was measured with an accuracy of approx. 10 −16 at speeds of approx. 36 km / h. The gravitational time dilation was also confirmed by raising the clocks by only 33 cm.

Other precise confirmation of the gravitational time dilation are the pound-rebka experiment and Gravity Sample A . Today, both speed-related and gravitation-related time dilation must be taken into account, for example in the calculations of the GPS navigation system . Because of this and a number of other high-precision experiments, the existence of relativistic time dilation is undisputed among experts. See Tests of the Special Theory of Relativity and Tests of the General Theory of Relativity .

## Equations

The equations of the effects relevant to the Hafele-Keating experiment have the following form:

The time dilation results from the sum of three contributions:

${\ displaystyle \ mathrm {T} = \ Delta \ tau _ {v} + \ Delta \ tau _ {g} + \ Delta \ tau _ {s}}$

Contribution of the speed according to the SRT:

${\ displaystyle \ Delta \ tau _ {v} = - {\ frac {1} {2c ^ {2}}} \ sum _ {i = 1} ^ {k} v_ {i} ^ {2} \ Delta \ dew _ {i}}$

Contribution of gravity according to the ART:

${\ displaystyle \ Delta \ tau _ {g} = {\ frac {g} {c ^ {2}}} \ sum _ {i = 1} ^ {k} (h_ {i} -h_ {0}) \ Delta \ tau _ {i}}$

Contribution from the Sagnac effect :

${\ displaystyle \ Delta \ tau _ {s} = - {\ frac {\ omega} {c ^ {2}}} \ sum _ {i = 1} ^ {k} R_ {i} ^ {2} \ cos ^ {2} \ phi _ {i} \ Delta \ lambda _ {i}}$

with c = speed of light, h = altitude, g = gravitational acceleration, v = speed, = angular velocity of the earth's rotation, τ = duration / length of a flight segment. The effects were integrated over the entire flight, as the parameters change over time. ${\ displaystyle \ omega}$

## Individual evidence

1. ^ Sexl, Roman & Schmidt, Herbert K .: Space-Time-Relativity . Vieweg, Braunschweig 1979, ISBN 3-528-17236-3 , pp. 39-43.
2. ^ J. Hafele, R. Keating: Around the world atomic clocks: predicted relativistic time gains . In: Science . 177, No. 4044, July 14, 1972, pp. 166-168. bibcode : 1972Sci ... 177..166H . doi : 10.1126 / science.177.4044.166 . PMID 17779917 . Retrieved September 18, 2006.
3. ^ J. Hafele, R. Keating: Around the world atomic clocks: observed relativistic time gains . In: Science . 177, No. 4044, July 14, 1972, pp. 168-170. bibcode : 1972Sci ... 177..168H . doi : 10.1126 / science.177.4044.168 . PMID 17779918 . Retrieved September 18, 2006.
4. NPL Metromnia: Issue 18 - Spring 2005 (PDF; 1.0 MB).
5. ^ NPL news: Time flies, Feb. 1, 2011.
6. ^ Roman Sexl, Herbert K. Schmidt: Space-Time-Relativity . Vieweg, Braunschweig 1979, ISBN 3-528-17236-3 , pp. 37-39.
7. ^ CO Alley: Relativity and Clocks . In: Proceedings of the 33rd Annual Symposium on Frequency Control . 1979, pp. 4-39. doi : 10.1109 / FREQ.1979.200296 .
8. ^ CO Alley, CO: Introduction to some fundamental concepts of general relativity and to their required use in some modern timekeeping systems Archived from the original on August 26, 2012. In: Proceedings of the Precise Time And Time Interval systems and applications meeting . 13, 1981, pp. 687-727.
9. ^ S. Iijima, K. Fujiwara: An experiment for the potential blue shift at the Norikura Corona Station . In: Annals of the Tokyo Astronomical Observatory . 17, 1978, pp. 68-78. bibcode : 1978AnTok..17 ... 68I .
10. L. Briatore, p Leschiutta: Evidence for the earth gravitational shift by direct atomic-time-scale comparison . In: Nuovo Cimento B . 37, No. 2, 1977, pp. 219-231. doi : 10.1007 / BF02726320 .
11. CW Chou, DB Hume, T. Rosenband, DJ Wineland: Optical Clocks and Relativity . In: Science . 329, No. 5999, 2010, pp. 1630-1633. bibcode : 2010Sci ... 329.1630C . doi : 10.1126 / science.1192720 . PMID 20929843 .
12. Yours: 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.