1967:). This requires that length contraction and time dilation have the exact relativistic values. Kennedy and Thorndike now argued that they could derive the complete Lorentz transformation solely from the experimental data of the Michelson–Morley experiment and the Kennedy–Thorndike experiment. But this is not strictly correct, since length contraction and time dilation having their exact relativistic values are sufficient but not necessary for the explanation of both experiments. This is because length contraction solely in the direction of motion is only one possibility to explain the Michelson–Morley experiment. In general, its null result requires that the
2012:
1983:, the following scheme can be used to describe the experiments: α represents time changes, β length changes in the direction of motion, and δ length changes perpendicular to the direction of motion. The Michelson–Morley experiment tests the relationship between β and δ, while the Kennedy–Thorndike experiment tests the relationship between α and β. So α depends on β which itself depends on δ, and only combinations of those quantities but not their individual values can be measured in these two experiments. Another experiment is necessary to
20:
213:
1797:
897:
2308:
of light on the observer’s velocity, as well as direction dependence of length contraction. However, no such oscillations were observed in either study, with a RMS velocity bound of ~10, comparable to the bounds set by Hils and Hall (1990). Hence both length contraction and time dilation must have the values predicted by relativity.
1476:
1987:
measure the value of one of these quantities. This was actually achieved with the Ives-Stilwell experiment, which measured α as having the value predicted by relativistic time dilation. Combining this value for α with the
Kennedy–Thorndike null result shows that β necessarily must assume the value of
1971:
between transverse and longitudinal lengths corresponds to the
Lorentz factor – which includes infinitely many combinations of length changes in the transverse and longitudinal direction. This also affects the role of time dilation in the Kennedy–Thorndike experiment, because its value depends on the
2307:
of reference and the speed of light depends on the observer's velocity, then anomalous oscillations should be observable in the Earth-Moon distance measurements. Since time dilation is already confirmed to high precision, the observance of such oscillations would demonstrate dependence of the speed
198:
took place, the interferometer was made extremely stable and the interference patterns were photographed for later comparison. The tests were done over a period of many months. As no significant fringe shift was found (corresponding to a velocity of 10±10 km/s within the margin of error), the
193:
By making one arm of the experiment much shorter than the other, a change in velocity of the Earth would cause changes in the travel times of the light rays, from which a fringe shift would result unless the frequency of the light source changed to the same degree. In order to determine if such a
1107:
110:
The principle on which this experiment is based is the simple proposition that if a beam of homogeneous light is split into two beams which after traversing paths of different lengths are brought together again, then the relative phases will depend on the velocity of the apparatus unless the
2050:
2002 repeat of the
Kennedy–Thorndike experiment. On the left, photodetectors (PD) monitor the resonance of a sapphire cryogenic optical resonator (CORE) length standard kept at liquid helium temperature to stabilize the frequency of a Nd:YAG laser to 1064 nm. On the right, the 532 nm
220:
Although
Lorentz–FitzGerald contraction (Lorentz contraction) by itself is fully able to explain the null results of the Michelson–Morley experiment, it is unable by itself to explain the null results of the Kennedy–Thorndike experiment. Lorentz–FitzGerald contraction is given by the formula:
1587:
2718:
Tobar, M. E.; Wolf, P.; Bize, S.; Santarelli, G.; Flambaum, V. (2010). "Testing local
Lorentz and position invariance and variation of fundamental constants by searching the derivative of the comparison frequency between a cryogenic sapphire oscillator and hydrogen maser".
1988:
relativistic length contraction. And combining this value for β with the
Michelson–Morley null result shows that δ must be zero. So the necessary components of the Lorentz transformation are provided by experiment, in agreement with the theoretical requirements of
637:
2043:(RMS), which indicates the relation between time dilation and length contraction, have been significantly improved. For instance, the original Kennedy–Thorndike experiment set bounds on RMS velocity dependence of ~10, but current limits are in the ~10 range.
1917:
1268:
2412:
Note: In contrast to the following demonstration, which is applicable only to light traveling along perpendicular paths, Kennedy and
Thorndike (1932) provided a general argument applicable to light rays following completely arbitrary
1930:
for the fringe shift to be independent of velocity or orientation of the apparatus, it is necessary that the frequency and thus the wavelength λ be modified by the
Lorentz factor. This is actually the case when the effect of
937:
523:
1213:
322:
1792:{\displaystyle \Delta N={\frac {\Delta L_{A}-\Delta L_{B}}{\lambda }}={\frac {2\left(L_{L}-L_{T}\right)}{\lambda }}\left({\frac {1}{\sqrt {1-v_{A}^{2}/c^{2}}}}-{\frac {1}{\sqrt {1-v_{B}^{2}/c^{2}}}}\right)}
39:. The modification is to make one arm of the classical Michelson–Morley (MM) apparatus shorter than the other one. While the Michelson–Morley experiment showed that the speed of light is independent of the
622:
892:{\displaystyle T_{L}=T_{1}+T_{2}={\frac {L_{L}/\gamma (v)}{c-v}}+{\frac {L_{L}/\gamma (v)}{c+v}}={\frac {2L_{L}/\gamma (v)}{c}}{\frac {1}{1-{\frac {v^{2}}{c^{2}}}}}={\frac {2L_{L}\gamma (v)}{c}}}
2286:
2215:
2123:
1813:
1471:{\displaystyle \Delta L_{A}={\frac {2\left(L_{L}-L_{T}\right)}{\sqrt {1-v_{A}^{2}/c^{2}}}},\qquad \Delta L_{B}={\frac {2\left(L_{L}-L_{T}\right)}{\sqrt {1-v_{B}^{2}/c^{2}}}}}
1935:
on the frequency is considered. Therefore, both length contraction and time dilation are required to explain the negative result of the
Kennedy–Thorndike experiment.
355:
431:
404:
2506:
381:
2867:
1102:{\displaystyle T_{T}={\frac {2L_{T}}{\sqrt {c^{2}-v^{2}}}}={\frac {2L_{T}}{c}}{\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {2L_{T}\gamma (v)}{c}}}
2665:
Wolf, P.; Tobar, M. E.; Bize, S.; Clairon, A.; Luiten, A. N.; Santarelli, G. (2004). "Whispering
Gallery Resonators and Tests of Lorentz Invariance".
2299:
In addition to terrestrial measurements, Kennedy–Thorndike experiments were carried out by Müller & Soffel (1995) and Müller et al. (1999) using
102:
only. Kennedy had already made several increasingly sophisticated versions of the MM experiment through the 1920s when he struck upon a way to test
2836:
533:
with respect to the hypothetical aether, the difference in time that it takes light to traverse the longitudinal and transverse arms is given by:
529:
Fig. 2 illustrates a Kennedy–Thorndike apparatus with perpendicular arms and assumes the validity of Lorentz contraction. If the apparatus is
3034:
2051:
absorbance line of a low pressure iodine reference is used as a time standard to stabilize the (doubled) frequency of a second Nd:YAG laser.
453:
3081:
55:
alone, the negative result of the Kennedy–Thorndike experiment requires time dilation in addition to length contraction to explain why no
2998:
2900:
1122:
227:
3060:
3029:
2860:
190:
allowed multiple exposures across the diameter of the rings to be recorded on a single photographic plate at different times of day.
2905:
2024:
1972:
value of length contraction used in the analysis of the experiment. Therefore, it's necessary to consider a third experiment, the
631:
The time it takes light to traverse back-and-forth along the Lorentz–contracted length of the longitudinal arm is given by:
2966:
931:
the length of the longitudinal interferometer arm. The time it takes light to go across and back the transverse arm is given by:
3003:
2961:
2936:
2001:
543:
3128:
3076:
2853:
3102:
2040:
1980:
1964:
1524:
are no longer equal. (The Michelson–Morley experiment isn't affected by velocity changes since the difference between
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2951:
2545:
2440:
2080:
2910:
127:
3013:
2890:
32:
2956:
2920:
2300:
79:
1556:
are different from the outset, so it is also capable of measuring the dependence of the speed of light on the
1496:=0. However, it can be seen that both formulas only cancel each other as long as the velocities are the same (
3008:
2163:
The frequency of a stationary cryogenic microwave oscillator, consisting of sapphire crystal operating in a
2993:
2946:
2941:
1973:
64:
2876:
1112:
The difference in time that it takes light to traverse the longitudinal and transverse arms is given by:
87:
1912:{\displaystyle \approx {\frac {L_{L}-L_{T}}{\lambda }}\left({\frac {v_{A}^{2}-v_{B}^{2}}{c^{2}}}\right)}
134:
kept the temperature regulated to within 0.001 °C. Monochromatic green light from a mercury source
2569:
Hils, Dieter; Hall, J. L. (1990). "Improved Kennedy–Thorndike experiment to test special relativity".
2258:
2187:
2095:
2182:
clocks. Changes during Earth's rotation have been searched for. Data between 2001–2002 was analyzed.
3138:
3133:
3107:
2370:
2327:
Kennedy, R. J.; Thorndike, E. M. (1932). "Experimental Establishment of the Relativity of Time".
2303:
data, in which the Earth-Moon distance is evaluated to an accuracy of centimeters. If there is a
2164:
1960:
2432:
2027:
as well as Kennedy–Thorndike type experiments have been repeated with increased precision using
2011:
3097:
2459:
Mansouri R.; Sexl R.U. (1977). "A test theory of special relativity: III. Second-order tests".
1952:
1262:
after rotation or velocity change due to Earth's own rotation or its rotation around the Sun):
68:
31:, first conducted in 1932 by Roy J. Kennedy and Edward M. Thorndike, is a modified form of the
156:
to prevent unwanted rear surface reflections. The two beams were directed towards two mirrors
2830:
47:
of the apparatus in different inertial frames. It also served as a test to indirectly verify
43:
of the apparatus, the Kennedy–Thorndike experiment showed that it is also independent of the
2234:(2003). An active temperature control was implemented. Data between 2002–2003 was analyzed.
2781:
2738:
2684:
2631:
2578:
2521:
2468:
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is zero. Therefore, the MM experiment only tests whether the speed of light depends on the
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8:
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415:
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179:
153:
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366:
99:
83:
52:
36:
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199:
experimenters concluded that time dilation occurs as predicted by Special relativity.
3050:
2915:
2793:
2758:
2647:
2612:
Wolf; et al. (2003). "Tests of Lorentz Invariance using a Microwave Resonator".
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2537:
2488:
2436:
2036:
2252:(2003). Data between 2002–2008 was analyzed for both sidereal and annual variations.
1944:
383:
is the length observed by an observer in relative motion with respect to the object,
2789:
2772:
Müller, J.; Soffel, M. H. (1995). "A Kennedy–Thorndike experiment using LLR data".
2746:
2704:
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2639:
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2529:
2505:
Braxmaier, C.; Müller, H.; Pradl, O.; Mlynek, J.; Peters, A.; Schiller, S. (2002).
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2344:
174:
of the 5461 Å mercury line (≈32 cm, allowing a difference in arm length Δ
171:
51:– while the negative result of the Michelson–Morley experiment can be explained by
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2533:
111:
frequency of the light depends on the velocity in the way required by relativity.
19:
2304:
2179:
1948:
212:
2590:
74:
Improved variants of the Kennedy–Thorndike experiment have been conducted using
2819:
Proceedings of the 11th International Workshop on Laser Ranging Instrumentation
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2168:
443:
434:
116:
75:
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2398:
1976:, in order to derive the Lorentz transformation from experimental data alone.
3122:
3055:
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1956:
1932:
358:
146:
103:
56:
48:
16:
Modified form of the Michelson–Morley experiment, testing special relativity
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2598:
2541:
2348:
2146:
1989:
195:
123:
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2504:
178: ≈ 16 cm). The reflected beams recombined to form circular
139:
1542:
of the apparatus.) But in the Kennedy–Thorndike experiment, the lengths
2664:
2480:
2679:
2626:
518:{\displaystyle \gamma (v)\equiv {\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\ }
406:
is the relative velocity between the observer and the moving object,
98:
The original Michelson–Morley experiment was useful for testing the
2810:
Müller, J., Nordtvedt, K., Schneider, M., Vokrouhlicky, D. (1999).
2176:
920:
is the velocity component with respect to the luminiferous aether,
2733:
2371:"Postulate versus Observation in the Special Theory of Relativity"
2431:(Reprint of 1968 ed.). Courier Dover Publications. pp.
2172:
2138:
Comparing the frequency of a cryogenic optical resonator with an
1563:
According to the previous formula, the travel length difference Δ
1208:{\displaystyle T_{L}-T_{T}={\frac {2(L_{L}-L_{T})\gamma (v)}{c}}}
317:{\displaystyle L=L_{0}{\sqrt {1-v^{2}/c^{2}}}=L_{0}/{\gamma (v)}}
67:. Combining the results of those three experiments, the complete
59:
will be detected while the Earth moves around the Sun. The first
115:
Referring to Fig. 1, key optical components were mounted within
2139:
2084:
216:
Figure 2. Kennedy–Thorndike light path using perpendicular arms
170:
which were set at distances as divergent as possible given the
2032:
2028:
2812:"Improved Determination of Relativistic Quantities from LLR"
2046:
Fig. 3 presents a simplified schematic diagram of Braxmaier
2507:"Tests of Relativity Using a Cryogenic Optical Resonator"
2771:
2568:
2326:
617:{\displaystyle T_{L}-T_{T}={\frac {2(L_{L}-L_{T})}{c}}}
145:
before entering the vacuum chamber, and was split by a
1482:
In order to obtain a negative result, we should have Δ
1234:, the following travel length differences are given (Δ
2261:
2190:
2098:
2039:. The bounds on velocity dependence according to the
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1271:
1125:
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640:
546:
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418:
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between the hypothetical aether and the moving object
391:
369:
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230:
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2424:
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2209:
2117:
1911:
1803:Neglecting magnitudes higher than second order in
1791:
1470:
1207:
1101:
891:
616:
517:
425:
398:
375:
349:
316:
63:confirmation of time dilation was achieved by the
207:
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2422:
2368:
23:Figure 1. The Kennedy–Thorndike experiment
2875:
1241:being the initial travel length difference and
1510:). But if the velocities are different, then Δ
2861:
2083:cavity with that of a laser stabilized to an
361:(the length of the object in its rest frame),
2835:: CS1 maint: multiple names: authors list (
1938:
1577:and consequently the expected fringe shift Δ
1248:the initial velocity of the apparatus, and Δ
2500:
2498:
909:is the travel time in direction of motion,
2868:
2854:
2805:
2803:
2015:Figure 3. Simplified diagram of Braxmaier
3030:Tests of relativistic energy and momentum
2732:
2678:
2625:
2397:
422:
395:
100:Lorentz–FitzGerald contraction hypothesis
2495:
2010:
1979:More precisely: In the framework of the
211:
18:
2967:Lorentz-violating neutrino oscillations
2800:
1581:are given by (λ being the wavelength):
3121:
2454:
2452:
2294:
2079:Comparing the frequency of an optical
3035:Kaufmann–Bucherer–Neumann experiments
3004:Experimental testing of time dilation
2962:Antimatter tests of Lorentz violation
2937:Modern searches for Lorentz violation
2849:
2364:
2362:
2360:
2358:
2002:Modern searches for Lorentz violation
1995:
82:. For a general overview of tests of
2611:
2322:
2320:
3103:Test theories of special relativity
2449:
2041:Robertson-Mansouri-Sexl test theory
1981:Robertson-Mansouri-Sexl test theory
13:
2697:10.1023/B:GERG.0000046188.87741.51
2667:General Relativity and Gravitation
2355:
1965:History of Lorentz transformations
1619:
1603:
1591:
1372:
1272:
14:
3150:
3061:Michelson–Gale–Pearson experiment
2972:Lorentz-violating electrodynamics
2952:Experiments of Rayleigh and Brace
2317:
93:
3014:Length contraction confirmations
2911:de Sitter double star experiment
2281:{\displaystyle \lesssim 10^{-8}}
2210:{\displaystyle \lesssim 10^{-7}}
2171:whose frequency was compared to
2118:{\displaystyle \lesssim 10^{-5}}
128:coefficient of thermal expansion
2765:
2711:
2006:
1371:
3082:Refutations of emission theory
2921:Measurements of neutrino speed
2658:
2605:
2562:
2416:
2406:
2145:frequency standard, using two
1943:In 1905, it had been shown by
1196:
1190:
1184:
1158:
1090:
1084:
880:
874:
804:
798:
754:
748:
707:
701:
605:
579:
466:
460:
310:
304:
208:Basic theory of the experiment
1:
2644:10.1103/PhysRevLett.90.060402
2534:10.1103/PhysRevLett.88.010401
2311:
2128:
106:as well. In their own words:
33:Michelson–Morley experimental
3077:Refutations of aether theory
2999:Moessbauer rotor experiments
2901:Moessbauer rotor experiments
2896:Kennedy–Thorndike experiment
2794:10.1016/0375-9601(94)01001-B
2025:Michelson–Morley experiments
29:Kennedy–Thorndike experiment
7:
3129:Tests of special relativity
2891:Michelson–Morley experiment
2877:Tests of special relativity
2591:10.1103/PhysRevLett.64.1697
924:is the speed of light, and
916:in the opposite direction,
182:which were photographed at
88:Tests of special relativity
10:
3155:
2957:Trouton–Rankine experiment
2751:10.1103/PhysRevD.81.022003
1999:
3090:
3069:
3043:
3022:
3009:Hafele–Keating experiment
2980:
2929:
2883:
2423:Albert Shadowitz (1988).
2399:10.1103/RevModPhys.21.378
2378:Reviews of Modern Physics
2369:Robertson, H. P. (1949).
2184:
2092:
1939:Importance for relativity
202:
3108:Standard-Model Extension
2994:Ives–Stilwell experiment
2947:Trouton–Noble experiment
2942:Hughes–Drever experiment
1974:Ives–Stilwell experiment
65:Ives–Stilwell experiment
2614:Physical Review Letters
2165:whispering gallery mode
1961:principle of relativity
3098:One-way speed of light
2349:10.1103/PhysRev.42.400
2282:
2211:
2119:
2020:
1953:Lorentz transformation
1913:
1793:
1472:
1209:
1103:
893:
618:
519:
427:
400:
377:
351:
318:
217:
140:Nicol polarizing prism
126:base of extremely low
113:
69:Lorentz transformation
24:
2906:Resonator experiments
2283:
2212:
2120:
2014:
2000:Further information:
1914:
1794:
1473:
1210:
1104:
894:
619:
520:
428:
401:
378:
352:
350:{\displaystyle L_{0}}
319:
215:
108:
22:
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2188:
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2068:velocity dependence
1814:
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1123:
938:
638:
544:
454:
416:
389:
367:
334:
228:
180:interference fringes
3023:Relativistic energy
2786:1995PhLA..198...71M
2743:2010PhRvD..81b2003T
2689:2004GReGr..36.2351W
2636:2003PhRvL..90f0402W
2583:1990PhRvL..64.1697H
2526:2001PhRvL..88a0401B
2473:1977GReGr...8..809M
2390:1949RvMP...21..378R
2341:1932PhRv...42..400K
2301:Lunar Laser Ranging
2295:Lunar laser ranging
2167:, is compared to a
1891:
1873:
1765:
1717:
1449:
1349:
426:{\displaystyle c\,}
399:{\displaystyle v\,}
80:Lunar Laser Ranging
35:procedure, testing
2986:Length contraction
2930:Lorentz invariance
2481:10.1007/BF00759585
2427:Special relativity
2278:
2207:
2115:
2037:optical resonators
2021:
1996:Recent experiments
1909:
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84:Lorentz invariance
53:length contraction
37:special relativity
25:
3116:
3115:
3056:Sagnac experiment
3051:Fizeau experiment
2916:Hammar experiment
2774:Physics Letters A
2721:Physical Review D
2673:(10): 2351–2372.
2577:(15): 1697–1700.
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376:{\displaystyle L}
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138:passed through a
130:. A water jacket
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770:
768:
757:
744:
739:
738:
728:
723:
721:
710:
697:
692:
691:
681:
676:
675:
663:
662:
650:
649:
623:
621:
620:
615:
613:
608:
604:
603:
591:
590:
574:
569:
568:
556:
555:
538:
537:
524:
522:
521:
516:
512:
511:
508:
507:
498:
493:
492:
477:
473:
432:
430:
429:
424:
405:
403:
402:
397:
382:
380:
379:
374:
356:
354:
353:
348:
346:
345:
323:
321:
320:
315:
313:
299:
294:
293:
281:
279:
278:
269:
264:
263:
248:
246:
245:
172:coherence length
154:Brewster's angle
76:optical cavities
71:can be derived.
3154:
3153:
3149:
3148:
3147:
3145:
3144:
3143:
3139:1932 in science
3134:Aether theories
3119:
3118:
3117:
3112:
3086:
3065:
3039:
3018:
2984:
2976:
2925:
2879:
2874:
2844:
2828:
2827:
2814:
2808:
2801:
2770:
2766:
2716:
2712:
2663:
2659:
2610:
2606:
2571:Phys. Rev. Lett
2567:
2563:
2554:
2552:
2548:
2514:Phys. Rev. Lett
2509:
2503:
2496:
2467:(10): 809–814.
2457:
2450:
2443:
2421:
2417:
2411:
2407:
2373:
2367:
2356:
2329:Physical Review
2325:
2318:
2314:
2305:preferred frame
2297:
2288:
2269:
2265:
2260:
2257:
2256:
2198:
2194:
2189:
2186:
2185:
2180:atomic fountain
2143:
2106:
2102:
2097:
2094:
2093:
2090:reference line.
2088:
2067:
2009:
2004:
1998:
1959:to satisfy the
1949:Albert Einstein
1941:
1897:
1893:
1886:
1881:
1868:
1863:
1858:
1856:
1852:
1840:
1836:
1827:
1823:
1822:
1820:
1815:
1812:
1811:
1775:
1771:
1766:
1760:
1755:
1739:
1727:
1723:
1718:
1712:
1707:
1691:
1690:
1686:
1669:
1665:
1656:
1652:
1651:
1647:
1643:
1641:
1626:
1622:
1610:
1606:
1602:
1600:
1589:
1586:
1585:
1575:
1568:
1555:
1548:
1537:
1530:
1522:
1515:
1508:
1501:
1494:
1487:
1459:
1455:
1450:
1444:
1439:
1416:
1412:
1403:
1399:
1398:
1394:
1390:
1388:
1379:
1375:
1359:
1355:
1350:
1344:
1339:
1316:
1312:
1303:
1299:
1298:
1294:
1290:
1288:
1279:
1275:
1270:
1267:
1266:
1260:
1253:
1246:
1239:
1231:
1227:
1178:
1174:
1165:
1161:
1154:
1152:
1143:
1139:
1130:
1126:
1124:
1121:
1120:
1075:
1071:
1067:
1065:
1051:
1047:
1041:
1037:
1035:
1023:
1011:
1007:
1003:
1001:
989:
985:
976:
972:
964:
960:
956:
954:
945:
941:
939:
936:
935:
929:
915:
908:
865:
861:
857:
855:
841:
837:
831:
827:
825:
818:
813:
790:
784:
780:
776:
774:
758:
740:
734:
730:
729:
727:
711:
693:
687:
683:
682:
680:
671:
667:
658:
654:
645:
641:
639:
636:
635:
599:
595:
586:
582:
575:
573:
564:
560:
551:
547:
545:
542:
541:
503:
499:
494:
488:
484:
472:
455:
452:
451:
417:
414:
413:
390:
387:
386:
368:
365:
364:
341:
337:
335:
332:
331:
300:
295:
289:
285:
274:
270:
265:
259:
255:
247:
241:
237:
229:
226:
225:
210:
205:
168:
161:
96:
17:
12:
11:
5:
3152:
3142:
3141:
3136:
3131:
3114:
3113:
3111:
3110:
3105:
3100:
3094:
3092:
3088:
3087:
3085:
3084:
3079:
3073:
3071:
3067:
3066:
3064:
3063:
3058:
3053:
3047:
3045:
3041:
3040:
3038:
3037:
3032:
3026:
3024:
3020:
3019:
3017:
3016:
3011:
3006:
3001:
2996:
2990:
2988:
2978:
2977:
2975:
2974:
2969:
2964:
2959:
2954:
2949:
2944:
2939:
2933:
2931:
2927:
2926:
2924:
2923:
2918:
2913:
2908:
2903:
2898:
2893:
2887:
2885:
2884:Speed/isotropy
2881:
2880:
2873:
2872:
2865:
2858:
2850:
2843:
2842:
2799:
2764:
2710:
2657:
2604:
2561:
2494:
2448:
2441:
2415:
2405:
2384:(3): 378–382.
2354:
2335:(3): 400–418.
2315:
2313:
2310:
2296:
2293:
2290:
2289:
2275:
2272:
2268:
2264:
2255:
2253:
2246:
2243:
2236:
2235:
2228:
2225:
2218:
2217:
2204:
2201:
2197:
2193:
2183:
2169:hydrogen maser
2161:
2158:
2151:
2150:
2141:
2136:
2133:
2126:
2125:
2112:
2109:
2105:
2101:
2091:
2086:
2077:
2074:
2070:
2069:
2064:
2061:
2058:
2008:
2005:
1997:
1994:
1945:Henri Poincaré
1940:
1937:
1922:For constant Δ
1920:
1919:
1907:
1900:
1896:
1889:
1884:
1880:
1876:
1871:
1866:
1862:
1855:
1849:
1843:
1839:
1835:
1830:
1826:
1819:
1801:
1800:
1787:
1778:
1774:
1769:
1763:
1758:
1754:
1750:
1747:
1743:
1738:
1730:
1726:
1721:
1715:
1710:
1706:
1702:
1699:
1695:
1689:
1683:
1678:
1672:
1668:
1664:
1659:
1655:
1650:
1646:
1640:
1635:
1629:
1625:
1621:
1618:
1613:
1609:
1605:
1599:
1596:
1593:
1573:
1566:
1553:
1546:
1535:
1528:
1520:
1513:
1506:
1499:
1492:
1485:
1480:
1479:
1462:
1458:
1453:
1447:
1442:
1438:
1434:
1431:
1425:
1419:
1415:
1411:
1406:
1402:
1397:
1393:
1387:
1382:
1378:
1374:
1370:
1362:
1358:
1353:
1347:
1342:
1338:
1334:
1331:
1325:
1319:
1315:
1311:
1306:
1302:
1297:
1293:
1287:
1282:
1278:
1274:
1258:
1251:
1244:
1237:
1229:
1225:
1220:
1219:
1216:
1215:
1202:
1198:
1195:
1192:
1189:
1186:
1181:
1177:
1173:
1168:
1164:
1160:
1157:
1151:
1146:
1142:
1138:
1133:
1129:
1110:
1109:
1096:
1092:
1089:
1086:
1083:
1078:
1074:
1070:
1064:
1054:
1050:
1044:
1040:
1034:
1031:
1027:
1020:
1014:
1010:
1006:
1000:
992:
988:
984:
979:
975:
967:
963:
959:
953:
948:
944:
927:
913:
906:
900:
899:
886:
882:
879:
876:
873:
868:
864:
860:
854:
844:
840:
834:
830:
824:
821:
817:
810:
806:
803:
800:
797:
793:
787:
783:
779:
773:
767:
764:
761:
756:
753:
750:
747:
743:
737:
733:
726:
720:
717:
714:
709:
706:
703:
700:
696:
690:
686:
679:
674:
670:
666:
661:
657:
653:
648:
644:
629:
628:
625:
624:
611:
607:
602:
598:
594:
589:
585:
581:
578:
572:
567:
563:
559:
554:
550:
527:
526:
506:
502:
497:
491:
487:
483:
480:
476:
471:
468:
465:
462:
459:
447:is defined as
444:Lorentz factor
439:
438:
435:speed of light
421:
411:
394:
384:
372:
362:
344:
340:
325:
324:
312:
309:
306:
303:
298:
292:
288:
284:
277:
273:
268:
262:
258:
254:
251:
244:
240:
236:
233:
209:
206:
204:
201:
166:
159:
117:vacuum chamber
95:
94:The experiment
92:
15:
9:
6:
4:
3:
2:
3151:
3140:
3137:
3135:
3132:
3130:
3127:
3126:
3124:
3109:
3106:
3104:
3101:
3099:
3096:
3095:
3093:
3089:
3083:
3080:
3078:
3075:
3074:
3072:
3068:
3062:
3059:
3057:
3054:
3052:
3049:
3048:
3046:
3044:Fizeau/Sagnac
3042:
3036:
3033:
3031:
3028:
3027:
3025:
3021:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2995:
2992:
2991:
2989:
2987:
2983:
2982:Time dilation
2979:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2934:
2932:
2928:
2922:
2919:
2917:
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2892:
2889:
2888:
2886:
2882:
2878:
2871:
2866:
2864:
2859:
2857:
2852:
2851:
2848:
2838:
2832:
2824:
2820:
2813:
2806:
2804:
2795:
2791:
2787:
2783:
2779:
2775:
2768:
2760:
2756:
2752:
2748:
2744:
2740:
2735:
2730:
2727:(2): 022003.
2726:
2722:
2714:
2706:
2702:
2698:
2694:
2690:
2686:
2681:
2680:gr-qc/0401017
2676:
2672:
2668:
2661:
2653:
2649:
2645:
2641:
2637:
2633:
2628:
2627:gr-qc/0210049
2623:
2620:(6): 060402.
2619:
2615:
2608:
2600:
2596:
2592:
2588:
2584:
2580:
2576:
2572:
2565:
2551:on 2021-03-23
2547:
2543:
2539:
2535:
2531:
2527:
2523:
2520:(1): 010401.
2519:
2515:
2508:
2501:
2499:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2455:
2453:
2444:
2442:0-486-65743-4
2438:
2434:
2429:
2428:
2419:
2409:
2400:
2395:
2391:
2387:
2383:
2379:
2372:
2365:
2363:
2361:
2359:
2350:
2346:
2342:
2338:
2334:
2330:
2323:
2321:
2316:
2309:
2306:
2302:
2273:
2270:
2266:
2262:
2254:
2251:
2247:
2244:
2242:
2238:
2237:
2233:
2229:
2226:
2224:
2220:
2219:
2202:
2199:
2195:
2191:
2181:
2178:
2174:
2170:
2166:
2162:
2159:
2157:
2153:
2152:
2148:
2147:Nd:YAG lasers
2144:
2137:
2134:
2132:
2127:
2110:
2107:
2103:
2099:
2089:
2082:
2078:
2075:
2073:Hils and Hall
2072:
2071:
2065:
2062:
2059:
2056:
2055:
2052:
2049:
2044:
2042:
2038:
2034:
2030:
2026:
2018:
2013:
2003:
1993:
1991:
1986:
1982:
1977:
1975:
1970:
1966:
1962:
1958:
1954:
1950:
1946:
1936:
1934:
1933:time dilation
1929:
1925:
1905:
1898:
1894:
1887:
1882:
1878:
1874:
1869:
1864:
1860:
1853:
1847:
1841:
1837:
1833:
1828:
1824:
1817:
1810:
1809:
1808:
1806:
1785:
1776:
1772:
1767:
1761:
1756:
1752:
1748:
1745:
1741:
1736:
1728:
1724:
1719:
1713:
1708:
1704:
1700:
1697:
1693:
1687:
1681:
1676:
1670:
1666:
1662:
1657:
1653:
1648:
1644:
1638:
1633:
1627:
1623:
1616:
1611:
1607:
1597:
1594:
1584:
1583:
1582:
1580:
1576:
1569:
1561:
1559:
1552:
1545:
1541:
1534:
1527:
1523:
1516:
1509:
1502:
1495:
1488:
1460:
1456:
1451:
1445:
1440:
1436:
1432:
1429:
1423:
1417:
1413:
1409:
1404:
1400:
1395:
1391:
1385:
1380:
1376:
1368:
1360:
1356:
1351:
1345:
1340:
1336:
1332:
1329:
1323:
1317:
1313:
1309:
1304:
1300:
1295:
1291:
1285:
1280:
1276:
1265:
1264:
1263:
1261:
1254:
1247:
1240:
1233:
1200:
1193:
1187:
1179:
1175:
1171:
1166:
1162:
1155:
1149:
1144:
1140:
1136:
1131:
1127:
1119:
1118:
1115:
1114:
1113:
1094:
1087:
1081:
1076:
1072:
1068:
1062:
1052:
1048:
1042:
1038:
1032:
1029:
1025:
1018:
1012:
1008:
1004:
998:
990:
986:
982:
977:
973:
965:
961:
957:
951:
946:
942:
934:
933:
932:
930:
923:
919:
912:
905:
884:
877:
871:
866:
862:
858:
852:
842:
838:
832:
828:
822:
819:
815:
808:
801:
795:
791:
785:
781:
777:
771:
765:
762:
759:
751:
745:
741:
735:
731:
724:
718:
715:
712:
704:
698:
694:
688:
684:
677:
672:
668:
664:
659:
655:
651:
646:
642:
634:
633:
632:
609:
600:
596:
592:
587:
583:
576:
570:
565:
561:
557:
552:
548:
540:
539:
536:
535:
534:
532:
504:
500:
495:
489:
485:
481:
478:
474:
469:
463:
457:
450:
449:
448:
446:
445:
436:
419:
412:
409:
392:
385:
370:
363:
360:
359:proper length
342:
338:
330:
329:
328:
307:
301:
296:
290:
286:
282:
275:
271:
266:
260:
256:
252:
249:
242:
238:
234:
231:
224:
223:
222:
214:
200:
197:
191:
189:
185:
181:
177:
173:
169:
162:
155:
151:
148:
147:beam splitter
144:
141:
137:
133:
129:
125:
121:
118:
112:
107:
105:
104:time dilation
101:
91:
89:
85:
81:
77:
72:
70:
66:
62:
58:
54:
50:
49:time dilation
46:
42:
38:
34:
30:
21:
3070:Alternatives
2895:
2831:cite journal
2822:
2818:
2780:(2): 71–73.
2777:
2773:
2767:
2724:
2720:
2713:
2670:
2666:
2660:
2617:
2613:
2607:
2574:
2570:
2564:
2553:. Retrieved
2546:the original
2517:
2513:
2464:
2460:
2426:
2418:
2408:
2381:
2377:
2332:
2328:
2298:
2249:
2240:
2231:
2222:
2155:
2130:
2047:
2045:
2022:
2016:
2007:Cavity tests
1990:group theory
1984:
1978:
1968:
1955:must form a
1942:
1927:
1923:
1921:
1804:
1802:
1578:
1571:
1564:
1562:
1557:
1550:
1543:
1539:
1532:
1525:
1518:
1511:
1504:
1497:
1490:
1483:
1481:
1256:
1249:
1242:
1235:
1223:
1221:
1111:
925:
921:
917:
910:
903:
901:
630:
530:
528:
442:
440:
407:
326:
219:
196:fringe shift
192:
187:
183:
175:
164:
157:
149:
142:
135:
131:
124:fused quartz
119:
114:
109:
97:
73:
60:
57:phase shifts
44:
40:
28:
26:
2081:Fabry–Pérot
2063:Description
1540:orientation
41:orientation
3123:Categories
2825:: 216–222.
2555:2012-07-21
2312:References
2129:Braxmaier
531:motionless
2759:119262822
2734:0912.2803
2489:121834946
2271:−
2263:≲
2248:See Wolf
2230:See Wolf
2200:−
2192:≲
2108:−
2100:≲
1951:that the
1875:−
1848:λ
1834:−
1818:≈
1749:−
1737:−
1701:−
1682:λ
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1985:directly
1558:velocity
441:and the
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3091:General
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2739:Bibcode
2705:8799879
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2579:Bibcode
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2386:Bibcode
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2173:caesium
2066:Maximum
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