formulasearchengine - User contributions [en]

Beltrami vector field

2014-02-17T22:52:40Z

146.201.196.196:

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Louvre Pyramid

2014-01-24T14:58:32Z

146.201.213.39: Changed height from 20.6m/70 ft (an incorrect conversion) to 21.6m/71ft, which matches the referenced source and other external sources that do not cite Wikipedia. 20.6 was probably a typographical error.

An '''exponential factorial''' is a positive integer ''n'' [[exponentiation|raised to the power]] of ''n'' − 1, which in turn is raised to the power of ''n'' − 2, and so on and so forth, that is,

: <math>n^{(n - 1)^{(n - 2) \cdots }}.\,</math>

The exponential factorial can also be defined with the [[recurrence relation]]

: <math>a_0 = 1,\quad a_n = n^{a_{n - 1}}.\,</math>

The first few exponential factorials are [[1 (number)|1]], [[1 (number)|1]], [[2 (number)|2]], [[9 (number)|9]], 262144, etc. {{OEIS|id=A049384}}. So, for example, 262144 is an exponential factorial since

: <math>262144 = 4^{3^{2^{1}}}.\,</math>

The exponential factorials grow much more quickly than regular [[factorial]]s or even [[Factorial#Hyperfactorial|hyperfactorial]]s. The exponential factorial of 5 is 5262144 which is approximately 6.206069878660874 × 10183230.

The sum of the reciprocals of the exponential factorials from 1 onwards is the [[transcendental number]] 1.6111149258083767361111... {{OEIS2C|id=A080219}}.

Like [[tetration]], there is currently no accepted method of extension of the exponential factorial function to [[real number|real]] and [[complex number|complex]] values of its argument, unlike the [[factorial]] function, for which such an extension is provided by the [[gamma function]].

==References==
*Jonathan Sondow, "[http://mathworld.wolfram.com/ExponentialFactorial.html Exponential Factorial]" From [[Mathworld]], a Wolfram Web resource

[[Category:Factorial and binomial topics]]
[[Category:Integer sequences]]
[[Category:Large integers]]

{{Numtheory-stub}}

Airy disk

2013-11-12T16:59:41Z

146.201.201.19: /* Cameras */

[[File:Hafele–Keating experiment.jpg|thumb|right|Hafele and Keating aboard a commercial airliner, with two of the atomic clocks and a stewardess.]]
[[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 October 1971, [[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 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 (journal)|Science]]'' in 1972:<ref>{{cite doi|10.1126/science.177.4044.166}}</ref><ref>{{cite doi|10.1126/science.177.4044.168}}</ref>

{| class="wikitable"
| rowspan=3 |
! colspan=4 | [[nanosecond]]s gained
|-
! colspan=3 | predicted
! rowspan=2 | 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==
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%.<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}}</ref>

A reenactment of the original experiment by the [[National Physical Laboratory (United Kingdom)|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.<ref>NPL Metromnia, [http://resource.npl.co.uk/docs/publications/newsletters/metromnia/issue18_einstein.pdf Issue 18 - Spring 2005]</ref> 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.<ref>NPL news, [http://www.npl.co.uk/news/time-flies Time flies, 1 Feb. 2011]</ref>

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.<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 58 m above sea level, to [[Mount Norikura|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.<ref>{{Cite journal|author=Iijima, S.; Fujiwara, K.; H. Kobayashi; 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]] 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 [[Very high frequency|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.<ref>{{Cite journal|author=Briatore, L.; 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 }}</ref> 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.<ref>Nontechnical explanation at http://www.scientificamerican.com/article.cfm?id=time-dilation</ref><ref>{{cite journal|author=Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. | 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}}</ref>

Nowadays both gravitational and velocity effects are, for example, routinely incorporated into the calculations used for the [[GPS|Global Positioning System]].<ref>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.</ref>

== Equations ==
{{unsourced|section|date=October 2013}}
The equations and effects involved in the experiment are:

Total [[time dilation]]
:<math>\Tau = \Delta\tau_v + \Delta\tau_g + \Delta\tau_s</math>

[[Special relativity|Velocity]]
:<math>\Delta\tau_v = - \frac{1}{2c^2} \sum_{i=1}^{k}v_i^2 \Delta t_i</math>

[[General relativity|Gravitation]]
:<math>\Delta\tau_g = \frac{g}{c^2} \sum_{i=1}^{k} (h_i - h_0) \Delta t_i</math>

[[Sagnac effect]]
:<math>\Delta\tau_s = - \frac{\omega^2}{2c^2} \sum_{i=1}^{k} R_i^2 \cos^2 \phi_i \Delta t_i</math>

Where c = speed of light, h = height, g=acceleration of gravity, v = velocity, <math>\omega</math> = angular velocity of Earth's rotation and <math>\Delta \tau</math> represents the proper duration/distance of a section of the flight while <math>\Delta t</math> 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]],<ref>A. Einstein, "On the electrodynamics of moving bodies," Annalen der Physik 17 (10): 891, tr. W. Perrett and G.B. Jeffery, 1923</ref> 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 relativity|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|gravitational]] time dilation would precisely cancel. The kinematic effect was verified in the 1938 [[Ives–Stilwell experiment]] and in the 1940 [[Time dilation of moving particles|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.

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">New Scientist, February 3, 1972, "The clock paradox resolved"</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>Hafele, "Performance and results of portable clocks in aircraft," PTTI, 3rd Annual Meeting, 1971; http://www.pttimeeting.org/archivemeetings/ptti1971.html</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, there were several accounts published in the popular press and other publications,<ref name="New Scientist 1972"/><ref>Time Magazine, October 18, 1971; http://www.time.com/time/magazine/article/0,9171,910115,00.html</ref> including one with a photo showing a stewardess ironically checking her wristwatch while standing behind the instruments.<ref>John Pearson, "[http://books.google.com/books?id=iNQDAAAAMBAJ&lpg=PP1&lr&rview=1&pg=PA30 Science Worldwide]", ''Popular Mechanics'', January 1972, p. 30.</ref>

==See also==
* [[Tests of general relativity]]
* [[Twin paradox]]
* [[Time dilation]]
* [[Error_analysis_for_the_Global_Positioning_System#Relativity|GPS Time Dilation]]
{{Tests of special relativity}}

==References==
<references/>

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