Editing Hafele–Keating experiment

{{Short description|Test of relativistic time dilation}}
[[File:Hafele–Keating experiment.jpg|thumb|right|Hafele and Keating aboard a commercial airliner, with two of the atomic clocks]]
[[File:HP 5061A Cesium Beam Frequency Standard.JPG|thumb|right|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 1971,<ref>The flights were completed October 7. ({{cite web|work=[[The New York Times]] |date= October 8, 1971|url= https://www.nytimes.com/1971/10/08/archives/they-beat-the-clock-checking-on-einstein.html |title=They beat the clock checking on Einstein| access-date=2020-05-03}})</ref> [[Joseph C. Hafele]], a physicist, and [[Richard E. Keating]], an astronomer, took four [[cesium]]-beam [[atomic clock]]s aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks in motion to stationary clocks 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 relativity|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 [[experimental testing of time dilation|others]].<ref>{{cite book |title=Astrophysical Formulae: Space, Time, Matter and Cosmology |edition=3rd, illustrated |first1=Kenneth |last1=Lang |publisher=Springer |year=2013 |isbn=978-3-662-21639-2 |page=152 |url=https://books.google.com/books?id=Nq_1CAAAQBAJ}} [https://books.google.com/books?id=Nq_1CAAAQBAJ&pg=PA152 Extract of page 152]</ref> Considering the Hafele–Keating experiment in a frame of reference at rest with respect to the center of the Earth (because this is an inertial frame<ref>{{cite book |title=Relativity in Rotating Frames: Relativistic Physics in Rotating Reference Frames |edition=illustrated |first1=Guido |last1=Rizzi |first2=Matteo Luca |last2=Ruggiero |publisher=Springer Science & Business Media |year=2003 |isbn=978-1-4020-1805-3 |page=25 |url=https://books.google.com/books?id=DH7jEf48KjgC}} [https://books.google.com/books?id=DH7jEf48KjgC&pg=PA25 Extract of page 25]</ref>), 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.<ref name="donkoks">{{cite book |title=Explorations in Mathematical Physics: The Concepts Behind an Elegant Language |edition=illustrated |first1=Don |last1=Koks |publisher=Springer Science & Business Media |year=2006 |isbn=978-0-387-30943-9 |page=500 |url=https://books.google.com/books?id=ytTFazXPy6oC}} [https://books.google.com/books?id=ytTFazXPy6oC&pg=PA500 Extract of page 500]</ref>

===Gravitational time dilation===
{{main|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.<ref name="donkoks" />

===Results===
The results were published in ''[[Science (journal)|Science]]'' in 1972:<ref name=paper1>{{Cite journal | last1 = Hafele | first1 = J. C. | author-link1 = Joseph C. Hafele | last2 = Keating | first2 = R. E. | author-link2 = Richard E. Keating | doi = 10.1126/science.177.4044.166 | title = Around-the-World Atomic Clocks: Predicted Relativistic Time Gains | journal = [[Science (journal)|Science]] | volume = 177 | issue = 4044 | pages = 166–168 | date = July 14, 1972 | pmid = 17779917 | bibcode = 1972Sci...177..166H | s2cid = 10067969 | ref = {{harvid|Hafele|Keating|1972a}} | url = http://www.personal.psu.edu/rq9/HOW/Atomic_Clocks_Predictions.pdf | access-date = March 30, 2017 | archive-date = March 31, 2017 | archive-url = https://web.archive.org/web/20170331121014/http://www.personal.psu.edu/rq9/HOW/Atomic_Clocks_Predictions.pdf | url-status = dead }}</ref><ref name=paper2>{{Cite journal | last1 = Hafele | first1 = J. C. | author-link1 = Joseph C. Hafele | last2 = Keating | first2 = R. E. | author-link2 = Richard E. Keating | doi = 10.1126/science.177.4044.168 | title = Around-the-World Atomic Clocks: Observed Relativistic Time Gains | journal = Science | volume = 177 | issue = 4044 | pages = 168–170 | date = July 14, 1972 | pmid = 17779918 | bibcode = 1972Sci...177..168H | s2cid = 37376002 | ref = {{harvid|Hafele|Keating|1972b}} | url = http://www.personal.psu.edu/rq9/HOW/Atomic_Clocks_Experiment.pdf | access-date = March 30, 2017 | archive-date = March 31, 2017 | archive-url = https://web.archive.org/web/20170331121107/http://www.personal.psu.edu/rq9/HOW/Atomic_Clocks_Experiment.pdf | url-status = dead }}</ref>

{| class="wikitable" style="text-align: center"
| rowspan=2 | 
! colspan=3 | [[nanoseconds]] gained, predicted
! rowspan=2 | nanoseconds<br/>gained,<br />measured 
! rowspan=2 | difference
|- 
! gravitational
([[general relativity]])
! kinematic
([[special relativity]])
! total 
|- 
! eastward
| +144 ±14
| −184 ±18
| −40 ±23
| −59 ±10
| {{#expr:(-40 - -59)/sqrt(23*23 + 10*10) round 2}} σ
|- 
! westward
| +179 ±18
| +96 ±10
| +275 ±21
| +273 ±7
| {{#expr:(275 - 273)/sqrt(21*21 + 7*7) round 2}} σ
|}

The published outcome of the experiment was consistent with both special and general relativity. The observed time gains and losses were in agreement with relativistic predictions to within the ~10% precision expected of the experiment.<ref>{{cite book |title=Fundamentals of Physics, Part 4 |edition=7th, illustrated |first1=David |last1=Halliday |first2=Robert |last2=Resnick |first3=JearlWalker |last3=Walker |publisher=Wiley |year=2004 |isbn=978-0-471-42964-7 |page=1030 |url=https://archive.org/details/fundamentalsofph0000hall|url-access=registration }}</ref>

==Historical and scientific background==
In his original 1905 paper on [[special relativity]],<ref>{{cite journal |first=A. |last=Einstein |author-link=Albert Einstein |title=On the electrodynamics of moving bodies |journal=[[Annalen der Physik]] |volume=17 |issue=10 |page=891 |translator1-first=W. |translator1-last=Perrett |translator2-first=G.B. |translator2-last=Jeffery |date=1923 |publication-date=1905 |url=https://www.fourmilab.ch/etexts/einstein/specrel/www/|bibcode=1905AnP...322..891E |doi=10.1002/andp.19053221004 |doi-access=free }}</ref> [[Albert 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." Actually, it is now known that all clocks located at sea level on the Earth's surface tick at the same rate, regardless of latitude, because kinematic and gravitational time dilation effects cancel out (assuming that Earth's surface is an [[equipotential]] one).<ref>{{Cite web|url=https://physics.stackexchange.com/q/126919 |title=Does time move slower at the equator?|website=physics.stackexchange.com|access-date=2018-03-04}}</ref><ref>{{Cite journal|last=Drake|first=S.P.|date=January 2006|title=The equivalence principle as a stepping stone from special to general relativity|url=http://newt.phys.unsw.edu.au/einsteinlight/jw/2006AJP.pdf|journal=[[American Journal of Physics]] |volume=74|issue=1|pages=22–25|arxiv=gr-qc/0501034|bibcode=2006AmJPh..74...22D|doi=10.1119/1.2135316|s2cid=119539826}}</ref> The kinematic effect was verified in the 1938 [[Ives–Stilwell experiment]] and in the 1940 [[experimental testing of time dilation|Rossi–Hall experiment]]. General relativity's prediction of the gravitational effect was confirmed in 1959 by [[Pound–Rebka experiment|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.{{citation needed|date=June 2017}}

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.<ref name="New Scientist 1972">{{cite journal |last=Wick |first=Gerald |date=February 3, 1972 |title=The clock paradox resolved |url=https://books.google.com/books?id=BAIlXPgcsHwC&pg=PA261 |journal=[[New Scientist]] |pages=261&ndash;263}}{{dead link|date=November 2023}}</ref> 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.<ref name="New Scientist 1972"/>

Hafele and Keating obtained $8000 in funding from the [[Office of Naval Research]]<ref>{{cite conference |last=Hafele |first=J. C. |author-link=Joseph C. Hafele |title=Performance and results of portable clocks in aircraft |conference=PTTI, 3rd Annual Meeting |year=1971 |conference-url=http://tycho.usno.navy.mil/ptti/ptti1971.html |url=http://tycho.usno.navy.mil/ptti/1971papers/Vol%2003_17.pdf |access-date=2017-03-31 |archive-url=https://web.archive.org/web/20170126213131/http://tycho.usno.navy.mil/ptti/1971papers/Vol%2003_17.pdf |archive-date=2017-01-26 |url-status=dead }}</ref> 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,<ref>Martin Gardner, Relativity Simply Explained, Dover, 1997, p. 117</ref> 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 (journal)|Science]]'',<ref name=paper1/><ref name=paper2/> there were several accounts published in the popular press and other publications.<ref name="New Scientist 1972"/><ref>{{cite news |magazine=[[Time (magazine)|Time]] |date=October 18, 1971 |title=Science: A Question of Time |url=http://www.time.com/time/magazine/article/0,9171,910115,00.html |archive-url=https://web.archive.org/web/20081221210138/http://www.time.com/time/magazine/article/0,9171,910115,00.html |url-status=dead |archive-date=December 21, 2008 |url-access=subscription}}</ref><ref>{{cite news |first=John |last=Pearson |url=https://books.google.com/books?id=iNQDAAAAMBAJ&pg=PA30 |title=Science Worldwide |magazine=[[Popular Mechanics]] |date=January 1972 |page=30}}</ref>

==Repetitions==
A more complex and precise experiment of this kind was performed by a research group at the [[University of Maryland]] between September 1975 and January 1976. Three atomic clocks were brought to an altitude of 10&nbsp;km above [[Chesapeake Bay]] in Maryland, and three other atomic clocks were at the ground. A [[turboprop]] plane was used, flying at only 500&nbsp;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&nbsp;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&nbsp;ns was measured, which consisted of the velocity effect of −5.7&nbsp;ns and a gravitational effect of 52.8&nbsp;ns. This agrees with the relativistic predictions to a precision of about 1.6%.<ref>{{Cite journal|author=Alley, C.O.|title=Relativity and Clocks |journal=Proceedings of 33rd Annual Symposium on Frequency Control|year= 1979 |pages=4–39|doi=10.1109/FREQ.1979.200296}}</ref><ref>{{Cite journal|author=Alley, C.O.|title=Introduction to some fundamental concepts of general relativity and to their required use in some modern timekeeping systems|journal=Proceedings of the Precise Time and Time Interval Systems and Applications Meeting|volume=13|year=1981|pages=687–727|url=http://www.pttimeeting.org/archivemeetings/ptti1981.html|url-status=dead|archive-url=https://web.archive.org/web/20120826103801/http://www.pttimeeting.org/archivemeetings/ptti1981.html|archive-date=2012-08-26}}</ref>

A reenactment of the original experiment by the [[National Physical Laboratory (United Kingdom)|National Physical Laboratory]] 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 {{val|39|2|u=ns}} was observed, compared to a relativistic prediction of 39.8&nbsp;ns.<ref>NPL Metromnia, [http://resource.npl.co.uk/docs/publications/newsletters/metromnia/issue18_einstein.pdf Issue 18 - Spring 2005]</ref> In June 2010, the National Physical Laboratory again repeated the experiment, this time around the globe (London - [[Los Angeles]] - [[Auckland]] - [[Hong Kong]] - London). The predicted value was {{val|246|3|u=ns}}, the measured value {{val|230|20|u=ns}}.<ref>NPL news, [http://www.npl.co.uk/news/time-flies Time flies, 1 Feb. 2011]</ref>

Because the Hafele–Keating experiment has been 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.<ref name="Wolfgang Rindler 1979, p. 45">Wolfgang Rindler, Essential Relativity: Special, General, and Cosmological, Springer-Verlag, 1979, p. 45</ref> 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.<ref>Roberts and Schleif, [http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#Twin_paradox What is the experimental basis of Special Relativity?]</ref>

== 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, Tokyo|Mitaka]], at {{convert|58|m|abbr=on}} above sea level, to [[Mount Norikura|Norikura corona station]], at {{convert|2876|m|abbr=on}} above sea level, corresponding to an altitude difference of {{convert|2818|m|abbr=on}}. 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<sup>−14</sup>, consistent with the result of 30.7×10<sup>−14</sup> predicted by general relativity.<ref>{{Cite journal|author1=Iijima, S. |author2=Fujiwara, K. |author3=H. Kobayashi |author4=T. Kato |title=An experiment for the potential blue shift at the Norikura Corona Station|journal=Annals of the Tokyo Astronomical Observatory|year=1978|volume=17|pages=68–78|bibcode=1978AnTok..17...68I}}</ref>

In 1976, Briatore and Leschiutta compared the rates of two cesium clocks, one in [[Turin]] {{convert|250|m|abbr=on}} above sea level, the other at [[Plateau Rosa]] {{convert|3500|m|abbr=on}} above sea level. The comparison was conducted by evaluating the arrival times of [[Very high frequency|VHF]] television synchronization pulses and of a [[LORAN]]-C chain. The predicted difference was 30.6&nbsp;ns/d. Using two different operating criteria, they found differences of 33.8±6.8&nbsp;ns/d and 36.5±5.8&nbsp;ns/d, respectively, in agreement with general relativity.<ref>{{Cite journal|author1=Briatore, L. |author2=Leschiutta, S. |title= Evidence for the earth gravitational shift by direct atomic-time-scale comparison|journal=[[Il Nuovo Cimento B]]|volume=37|issue=2|pages=219–231|year=1977|doi=10.1007/BF02726320|bibcode = 1977NCimB..37..219B |s2cid=124043777 }}</ref> Environmental factors were controlled far more precisely than in the Iijima experiment, in which many complicated corrections had to be applied.{{citation needed|date=June 2017}}

In 2005, van Baak measured the gravitational time dilation of a weekend at 5400' ASL on [[Mount Rainier]] using two ensembles of three HP&nbsp;5071A cesium beam clocks.<ref>{{cite magazine |title=Amateur Time Hackers Play With Atomic Clocks at Home |first=Quinn |last=Norton |magazine=[[Wired (magazine)|Wired]] |date=12 November 2007 |url=https://www.wired.com/2007/12/time-hackers/}}</ref><ref>{{cite conference |conference=38th Annual Precise Time and Time Interval Meeting 2006 |conference-url=http://tycho.usno.navy.mil/ptti/ptti2006.html |date=6 December 2006 |first=Tom |last=van Baak |title=Project GREAT: General Relativity Einstein/Essen Anniversary Test |type=Conference presentation |url=http://leapsecond.com/ptti2006/tvb-project-great-ptti-ppt.pdf}}<!-- This ref primarily for notability: recognition by a professional society.--></ref><ref>{{cite web|url=http://leapsecond.com/great2005/|title=Project GREAT: General Relativity Einstein/Essen Anniversary Test|last=van Baak|first=Tom|date=25 September 2005|access-date=2017-03-30|quote=It was the best extra 22 nanoseconds I've ever spent with the kids.}}<!--This ref self-published, but very informative--></ref>  He repeated the experiment in 2016 on [[Mount Lemmon]] for the television show ''[[Genius by Stephen Hawking]]''.<ref>{{cite web |first=Tom |last=van Baak |title=Project GREAT 2016a &mdash; Hawking, Einstein, and Time Dilation on Mt Lemmon |url=http://www.leapsecond.com/great2016a |date=17 May 2016 |access-date=2017-03-30}}</ref>

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<sup>−16</sup> level at speeds below 36&nbsp;km/h. Also, gravitational time dilation was measured from a difference in elevation between two clocks of only {{convert|33|cm|abbr=on}}.<ref>Nontechnical explanation at http://www.scientificamerican.com/article.cfm?id=time-dilation</ref><ref>{{cite journal|author1=Chou, C. W. |author2=Hume, D. B. |author3=Rosenband, T. |author4=Wineland, D. J. |author-link4=David J. Wineland | year=2010 | title=Optical Clocks and Relativity | journal=Science | volume=329 | issue=5999 |  pages=1630–1633 | doi=10.1126/science.1192720 |bibcode=2010Sci...329.1630C | pmid=20929843 |s2cid=206527813 | url=http://ws680.nist.gov/publication/get_pdf.cfm?pub_id=905055 |url-access=subscription }}</ref>

Presently both gravitational and velocity effects are routinely incorporated, for example, into the calculations used for the [[GPS|Global Positioning System]].<ref>{{Cite conference |last=Deines |first=S.D. |year=1992 |title=Uncompensated relativity effects for a ground-based GPSA receiver |url=https://ieeexplore.ieee.org/document/185812 |conference=IEEE PLANS 92 Position Location and Navigation Symposium |publisher=IEEE |pages=7–11 |doi=10.1109/PLANS.1992.185812 |isbn=978-0-7803-0468-0|url-access=subscription }}</ref>

==See also==
* [[Twin paradox]]
* [[Error analysis for the Global Positioning System#Relativity|GPS time dilation]]

==References==
{{reflist}}

{{Tests of special relativity}}

{{DEFAULTSORT:Hafele-Keating Experiment}}
[[Category:Tests of general relativity]]
[[Category:Tests of special relativity]]
[[Category:Albert Einstein]]
[[Category:1971 in science]]