Hafele - Keating experiment

The Hafele-Keating experiment is one of the tests of the theory of relativity . He directly demonstrated the reality of the twin paradox - the time dilation predicted by the theory of relativity for moving objects, as well as gravitational time dilation .

Experiment Description

One of the sets of cesium clocks HP 5061A Cesium Beam Frequency Standard used in the experiment

In October 1971, J. C. Hafele and Richard E. Keating traveled around the world twice, first east, then west, with four sets of cesium atomic clocks , after which they compared the “traveling” clocks with the same hours remaining at the US Naval Observatory (US WMO). Flights were carried out on regular airliners with regular commercial flights .

The eastward flight began at 19:30 UTC on October 4, 1971 and ended at 12:55 UTC on October 7, 1971 (duration 65.42 hours ); WMO route USA - Washington - London - Frankfurt - Istanbul - Beirut - Tehran - New Delhi - Bangkok - Hong Kong - Tokyo - Honolulu - Los Angeles - Dallas - Washington - WMO USA. The average speed relative to the surface of the earth was 243 m / s , the average height above sea level was 8.90 km , and the average latitude along the route was 34 ° s. w. ^[one]

The flight in a westerly direction began at 19:40 UTC on October 13, 1971, and ended after 80.33 hours at 04:00 UTC on October 17, 1971. Route: WMO USA - Washington - Los Angeles - Honolulu - Guam - Okinawa - Taipei - Hong Kong - Bangkok - Bombay - Tel Aviv - Athens - Rome - Paris - Shannon - Boston - Washington - WMO USA. In this direction, the average speed was 218 m / s , the average altitude was 9.36 km , and the average latitude along the route was 31 ° s. w. ^[one]

During flights, environmental conditions (temperature, humidity and air pressure) were monitored, and the magnetic field was measured. It was further demonstrated that a change in these conditions in the laboratory does not affect, within the limits of errors, the course of the hours used in the experiment ^[1] . It was also checked whether disconnecting one of the 4 used batteries does not affect the clock (such a loss of one of the batteries occurred during a western flight). Navigation information about the parameters of each flight was provided by the pilots.

For assembly from a set of watches and batteries, separate tickets were bought for two seats (in the name of Mr. Clock) ^[2] . The total price of tickets for the watch and two accompanying researchers was about $ 7,600 ; as a result, the Hafele - Keating experiment turned out to be one of the most inexpensive experiments performed to verify the theory of relativity ^[3] ^[4] .

Results

According to the special theory of relativity, the speed of the clock is greatest for the observer for whom they are at rest. In a frame of reference in which the clock does not rest, it goes slower, and this effect is proportional to the square of the speed. In the reference frame, which is at rest relative to the center of the Earth, the clock on board an airplane moving east (in the direction of the Earth’s rotation, the airplane’s speed is added to the rotational velocity of the Earth’s surface v _hours = R Ω + v of the _airplane ) are slower than the clock that remains on surface ( v _hours = R Ω ), and the clock on board a plane moving in a westerly direction (against the rotation of the Earth, the plane’s speed is subtracted from the rotational speed of the Earth’s surface v _hours = R Ω - v of the _plane ), go faster ^[5] ^[6] .

According to the general theory of relativity, another effect comes into play: a slight decrease (in absolute value) of the gravitational potential with increasing height again speeds up the clock. Since the planes flew at approximately the same altitude in both directions, this effect has little effect on the difference in the travel of the two “traveling” clocks, but it causes them to move away from the clock on the surface of the Earth.

The results were published in the journal Science in 1972 ^[5] :

The difference between the readings of traveling and remaining in place hours, nanoseconds
When moving	Calculated (predicted)			Actually measured
When moving	Gravity contribution ( GRT )	Kinematic contribution ( STO )	General contribution (GR + STO)	Actually measured
To the East	+ 144 ± 14	- 184 ± 18	- 40 ± 23	- 59 ± 10
To the west	+ 179 ± 18	+ 96 ± 10	+ 275 ± 21	+ 273 ± 7

The published experimental results were compatible with the predictions of the theory of relativity, and it was noted that the observed positive and negative differences in the course of the clock with a high confidence probability differ from zero.

One noteworthy approximate repetition of the original experiment took place on its 25th anniversary, using more accurate atomic clocks, and the results were verified with better error. ^[7] Currently, such relativistic effects are included in the calculations used for satellite global positioning systems - existing US GPS and Russian GLONASS and the developed European Galileo system ^[8] .

Equations

Equations and effects involved in the description of the experiment:

Total lag hours:

\tau =\Delta \tau _{v}+\Delta \tau _{g}+\Delta \tau _{s}.

{\ displaystyle \ tau = \ Delta \ tau _ {v} + \ Delta \ tau _ {g} + \ Delta \ tau _ {s}.}

Special relativistic contribution ( speed ):

\Delta \tau _{v}=-{\frac {1}{2c^{2}}}\sum _{i=1}^{k}v_{i}^{2}\Delta T_{i}.

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

General Relativistic Contribution ( Gravity ):

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

{\ displaystyle \ 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 }{c^{2}}}\sum _{i=1}^{k}R_{i}^{2}\cos ^{2}\varphi _{i}\Delta \lambda _{i}.

{\ displaystyle \ Delta \ tau _ {s} = - {\ frac {\ Omega} {c ^ {2}}} \ sum _ {i = 1} ^ {k} R_ {i} ^ {2} \ cos ^ {2} \ varphi _ {i} \ Delta \ lambda _ {i}.}

Here h is the height, v is the velocity relative to the center of the Earth, Ω is the angular velocity of the Earth, and $\Delta T_{i}$ ${\ displaystyle \ Delta T_ {i}}$ and $\Delta \lambda _{i}$ ${\ displaystyle \ Delta \ lambda _ {i}}$ represent the duration of the i- th flight section and the change in geographical longitude for it; $R_{i}$ ${\ displaystyle R_ {i}}$ - the distance from the center of the earth in this area, $\varphi _{i}$ ${\ displaystyle \ varphi _ {i}}$ - geographical latitude ; g is the acceleration of gravity , c is the speed of light . Effects are summarized throughout the flight, as the parameters change over time.

Notes

↑ ¹ ² ³ JC Hafele, “Performance and results of portable clocks in aircraft” PTTI, 3rd Annual Meeting, November 16-18, 1971.
↑ Martin Gardner, Relativity Simply Explained, Dover, 1997, p. 117.
↑ Time Magazine, October 18, 1971; http://www.time.com/time/magazine/article/0,9171,910115,00.html
↑ New Scientist, Feb 3, 1972, “The clock paradox resolved.”
↑ ¹ ² Hafele, J .; Keating, R. Around the world atomic clocks: predicted relativistic time gains (Eng.) // Science: journal. - 1972.- 14 July ( vol. 177 , no. 4044 ). - P. 166-168 . - DOI : 10.1126 / science.177.4044.166 . - PMID 17779917 .
↑ In these formulas, Ω is the angular velocity of the Earth's rotation in radians / s, R is the distance from the plane to the earth’s axis, and v is the _plane ’s speed relative to the earth’s surface; in addition, it is assumed that the linear velocity of a point on the Earth’s surface R Ω is greater than v of the _aircraft , so regardless of whether the plane flies east or west relative to the surface, it moves east relative to the center of the earth.
↑ Metromnia Issue 18 - Spring 2005.
↑ 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.

[HafeleTalk-1] ¹ ² ³ JC Hafele, “Performance and results of portable clocks in aircraft” PTTI, 3rd Annual Meeting, November 16-18, 1971.

[2] Martin Gardner, Relativity Simply Explained, Dover, 1997, p. 117.

[3] Time Magazine, October 18, 1971; http://www.time.com/time/magazine/article/0,9171,910115,00.html

[4] New Scientist, Feb 3, 1972, “The clock paradox resolved.”

[haf72-5] ¹ ² Hafele, J .; Keating, R. Around the world atomic clocks: predicted relativistic time gains (Eng.) // Science: journal. - 1972.- 14 July ( vol. 177 , no. 4044 ). - P. 166-168 . - DOI : 10.1126 / science.177.4044.166 . - PMID 17779917 .

[6] In these formulas, Ω is the angular velocity of the Earth's rotation in radians / s, R is the distance from the plane to the earth’s axis, and v is the _plane ’s speed relative to the earth’s surface; in addition, it is assumed that the linear velocity of a point on the Earth’s surface R Ω is greater than v of the _aircraft , so regardless of whether the plane flies east or west relative to the surface, it moves east relative to the center of the earth.

[7] Metromnia Issue 18 - Spring 2005.

[8] 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.

Hafele - Keating experiment

Experiment Description

Results

Equations

See also

Notes

More articles: