Kennedy–Thorndike experiment

Figure 1. The Kennedy–Thorndike experiment

The Kennedy–Thorndike experiment, first conductedin 1932, is a modified form of the Michelson–Morleyexperimental procedure, testing special relativity.[1] Themodification is to make one arm of the classicalMichelson–Morley (MM) apparatus shorter than theother one. While the Michelson–Morley experimentshowed that the speed of light is independent of the ori-entation of the apparatus, the Kennedy–Thorndike exper-iment showed that it is also independent of the velocity ofthe apparatus in different inertial frames. It also servedas a test to indirectly verify time dilation – while the neg-ative result of the Michelson–Morley experiment can beexplained by length contraction alone, the negative resultof the Kennedy–Thorndike experiment requires time di-lation in addition to length contraction to explain why nophase shifts will be detected while the earthmoves aroundthe sun. The first direct confirmation of time dilation wasachieved by the Ives–Stilwell experiment. Combining theresults of those three experiments, the complete Lorentztransformation can be derived.[2]

Improved variants of the Kennedy–Thorndike experi-ment have been conducted using optical cavities or LunarLaser Ranging. For a general overview of tests of Lorentzinvariance, see Tests of special relativity.

1 The experiment

The original Michelson–Morley experiment was usefulfor testing the Lorentz–FitzGerald contraction hypothe-sis only. Kennedy had already made several increasinglysophisticated versions of the MM experiment through the1920s when he struck upon a way to test time dilation as

well. In their own words:[1]

The principle on which this experiment isbased is the simple proposition that if a beamof homogeneous light is split […] into twobeams which after traversing paths of differ-ent lengths are brought together again, then therelative phases […] will depend […] on the ve-locity of the apparatus unless the frequency ofthe light depends […] on the velocity in the wayrequired by relativity.

Referring to Fig. 1, key optical components weremounted within vacuum chamber V on a fused quartzbase of extremely low coefficient of thermal expansion. Awater jacketW kept the temperature regulated to within0.001 °C. Monochromatic green light from a mercurysource Hg passed through a Nicol polarizing prism Nbefore entering the vacuum chamber, and was split bya beam splitter B set at Brewster’s angle to prevent un-wanted rear surface reflections. The two beams were di-rected towards two mirrors M1 and M2 which were setat distances as divergent as possible given the coherencelength of the 5461 Å mercury line (≈32 cm, allowinga difference in arm length ΔL ≈ 16 cm). The reflectedbeams recombined to form circular interference fringeswhich were photographed at P. A slit S allowed multipleexposures across the diameter of the rings to be recordedon a single photographic plate at different times of day.By making one arm of the experiment much shorter thanthe other, a change in velocity of the earth would causechanges in the travel times of the light rays, from which afringe shift would result unless the frequency of the lightsource changed to the same degree. In order to deter-mine if such a fringe shift took place, the interferome-ter was made extremely stable and the interference pat-terns were photographed for later comparison. The testswere done over a period of many months. As no signif-icant fringe shift was found (corresponding to a velocityof 10±10 km/s within the margin of error), the experi-menters concluded that time dilation occurs as predictedby Special relativity.

2 Theory

1

2 2 THEORY

Figure 2. Kennedy–Thorndike light path using perpendiculararms

2.1 Basic theory of the experiment

Although Lorentz–FitzGerald contraction (Lorentz con-traction) by itself is fully able to explain the null resultsof the Michelson–Morley experiment, it is unable by it-self to explain the null results of the Kennedy–Thorndikeexperiment. Lorentz–FitzGerald contraction is given bythe formula:

L = L0

√1− v2/c2 = L0/γ(v)

where

L0 is the proper length (the length of the objectin its rest frame),L is the length observed by an observer in rel-ative motion with respect to the object,v is the relative velocity between the observerand the moving object, i.e. between the hypo-thetical aether and the moving objectc is the speed of light,

and the Lorentz factor is defined as

γ(v) ≡ 1√1− v2/c2

Fig. 2 illustrates a Kennedy–Thorndike apparatus withperpendicular arms and assumes the validity of Lorentzcontraction.[3] If the apparatus is motionless with respectto the hypothetical aether, the difference in time that ittakes light to traverse the longitudinal and transverse armsis given by:

The time it takes light to traverse back-and-forth alongthe Lorentz–contracted length of the longitudinal arm isgiven by:

TL = T1 + T2 = LL/γ(v)c−v + LL/γ(v)

c+v =2LL/γ(v)

c1

1− v2

c2

= 2LLγ(v)c

where T1 is the travel time in direction of motion, T2 inthe opposite direction, v is the velocity component withrespect to the luminiferous aether, c is the speed of light,and LL the length of the longitudinal interferometer arm.The time it takes light to go across and back the transversearm is given by:

TT = 2LT√c2−v2

= 2LT

c1√

1− v2

c2

= 2LT γ(v)c

The difference in time that it takes light to traverse thelongitudinal and transverse arms is given by:

Because ΔL=c(TL-TT), the following travel length differ-ences are given (ΔLA being the initial travel length differ-ence and vA the initial velocity of the apparatus, and ΔLBand vB after rotation or velocity change due to Earth’sown rotation or its rotation around the Sun):[4]

∆LA =2 (LL − LT )√

1− v2A/c2, ∆LB =

2 (LL − LT )√1− v2B/c

2

In order to obtain a negative result, we should haveΔLA−ΔLB=0. However, it can be seen that both formu-las only cancel each other as long as the velocities are thesame (vA=vB). But if the velocities are different, thenΔLA and ΔLB are no longer equal. (The Michelson–Morley experiment isn't affected by velocity changessince the difference between LL and LT is zero. There-fore the MM experiment only tests whether the speed oflight depends on the orientation of the apparatus.) But inthe Kennedy–Thorndike experiment, the lengths LL andLT are different from the outset, so it is also capable ofmeasuring the dependence of the speed of light on thevelocity of the apparatus.[2]

According to the previous formula, the travel length dif-ference ΔLA−ΔLB and consequently the expected fringeshift ΔN are given by (λ being the wavelength):

∆N = ∆LA−∆LB

λ =

2(LL−LT )λ

(1√

1−v2A/c2

− 1√1−v2

B/c2

).

Neglecting magnitudes higher than second order in v/c:

≈ LL − LT

λ

(v2A − v2B

c2

)For constant ΔN, i.e. for the fringe shift to be independentof velocity or orientation of the apparatus, it is necessarythat the frequency and thus the wavelength λ be modi-fied by the Lorentz factor. This is actually the case whenthe effect of time dilation on the frequency is considered.Therefore both length contraction and time dilation arerequired to explain the negative result of the Kennedy–Thorndike experiment.

3.1 Cavity tests 3

2.2 Importance for relativity

In 1905, it had been shown by Henri Poincaré and AlbertEinstein that the Lorentz transformation must form agroup to satisfy the principle of relativity (see Historyof Lorentz transformations). This requires that lengthcontraction and time dilation have the exact relativisticvalues. Kennedy and Thorndike now argued that theycould derive the complete Lorentz transformation solelyfrom the experimental data of the Michelson–Morley ex-periment and the Kennedy–Thorndike experiment. Butthis is not strictly correct, since length contraction andtime dilation having their exact relativistic values are suf-ficient but not necessary for the explanation of both ex-periments. This is because length contraction solely inthe direction of motion is only one possibility to explaintheMichelson–Morley experiment. In general, its null re-sult requires that the ratio between transverse and longi-tudinal lengths corresponds to the Lorentz factor – whichincludes infinitely many combinations of length changesin the transverse and longitudinal direction. This also af-fects the role of time dilation in the Kennedy–Thorndikeexperiment, because its value depends on the value oflength contraction used in the analysis of the experi-ment. Therefore it’s necessary to consider a third experi-ment, the Ives–Stilwell experiment, in order to derive theLorentz transformation from experimental data alone.[2]

More precisely: In the framework of the Robertson-Mansouri-Sexl test theory,[2][5] the following scheme canbe used to describe the experiments: α represents timechanges, β length changes in the direction of motion, andδ length changes perpendicular to the direction of mo-tion. The Michelson–Morley experiment tests the rela-tionship between β and δ, while the Kennedy–Thorndikeexperiment tests the relationship between α and β. So αdepends on β which itself depends on δ, and only combi-nations of those quantities but not their individual valuescan be measured in these two experiments. Another ex-periment is necessary to directlymeasure the value of oneof these quantities. This was actually achieved with theIves-Stilwell experiment, which measured α as having thevalue predicted by relativistic time dilation. Combiningthis value for α with the Kennedy–Thorndike null resultshows that β necessarily must assume the value of rel-ativistic length contraction. And combining this valuefor β with the Michelson–Morley null result shows thatδ must be zero. So the necessary components of theLorentz transformation are provided by experiment, inagreement with the theoretical requirements of group the-ory.

3 Recent experiments

Further information: Modern searches for Lorentzviolation

3.1 Cavity tests

Figure 3. Simplified diagram of Braxmaier et al. 2002

In recent years, Michelson–Morley experiments as wellas Kennedy–Thorndike type experiments have been re-peated with increased precision using lasers, masers, andcryogenic optical resonators. The bounds on velocity de-pendence according to the Robertson-Mansouri-Sexl testtheory (RMS), which indicates the relation between timedilation and length contraction, have been significantlyimproved. For instance, the original Kennedy–Thorndikeexperiment set bounds on RMS velocity dependence of~10−2, but current limits are in the ~10−8 range.[5]

Fig. 3 presents a simplified schematic diagram of Brax-maier et al.'s 2002 repeat of the Kennedy–Thorndikeexperiment.[6] On the left, photodetectors (PD) monitorthe resonance of a sapphire cryogenic optical resonator(CORE) length standard kept at liquid helium tempera-ture to stabilize the frequency of a Nd:YAG laser to 1064nm. On the right, the 532 nm absorbance line of a lowpressure iodine reference is used as a time standard tostabilize the (doubled) frequency of a second Nd:YAGlaser.

3.2 Lunar laser ranging

In addition to terrestrial measurements, Kennedy–Thorndike experiments were carried out by Müller &Soffel (1995)[11] and Müller et al. (1999)[12] using LunarLaser Ranging data, in which the Earth-Moon distanceis evaluated to an accuracy of centimeters. If there isa preferred frame of reference and the speed of lightdepends on the observer’s velocity, then anomalous os-cillations should be observable in the Earth-Moon dis-tance measurements. Since time dilation is already con-firmed to high precision, the observance of such oscilla-tions would demonstrate dependence of the speed of lighton the observer’s velocity, as well as direction dependenceof length contraction. However, no such oscillations wereobserved in either study, with a RMS velocity bound of~10−5,[12] comparable to the bounds set by Hils and Hall(1990). Hence both length contraction and time dilationmust have the values predicted by relativity.

4 4 REFERENCES

4 References[1] Kennedy, R. J.; Thorndike, E. M. (1932). “Experimental

Establishment of the Relativity of Time”. Physical Re-view 42 (3): 400–418. Bibcode:1932PhRv...42..400K.doi:10.1103/PhysRev.42.400.

[2] Robertson, H. P. (1949). “Postulate versusObservation in the Special Theory of Rela-tivity”. Reviews of Modern Physics 21 (3):378–382. Bibcode:1949RvMP...21..378R.doi:10.1103/RevModPhys.21.378.

[3] Note: In contrast to the following demonstration, whichis applicable only to light traveling along perpendicularpaths, Kennedy and Thorndike (1932) provided a generalargument applicable to light rays following completely ar-bitrary paths.

[4] Albert Shadowitz (1988). Special relativity (Reprint of1968 ed.). Courier Dover Publications. p. 161. ISBN0-486-65743-4.

[5] Mansouri R., Sexl R.U. (1977). “A test theory of specialrelativity: III. Second-order tests”. General. Relat. Gravit.8 (10): 809–814. Bibcode:1977GReGr...8..809M.doi:10.1007/BF00759585.

[6] Braxmaier, C.; Müller, H.; Pradl, O.; Mlynek, J.; Pe-ters, A.; Schiller, S. (2002). “Tests of Relativity Us-ing a Cryogenic Optical Resonator” (PDF). Phys. Rev.Lett. 88 (1): 010401. Bibcode:2002PhRvL..88a0401B.doi:10.1103/PhysRevLett.88.010401. PMID 11800924.

[7] Hils, Dieter; Hall, J. L. (1990). “ImprovedKennedy–Thorndike experiment to test spe-cial relativity”. Phys. Rev. Lett. 64 (15):1697–1700. Bibcode:1990PhRvL..64.1697H.doi:10.1103/PhysRevLett.64.1697. PMID 10041466.

[8] Wolf; et al. (2003). “Tests of Lorentz Invari-ance using a Microwave Resonator”. Physi-cal Review Letters 90 (6): 060402. arXiv:gr-qc/0210049. Bibcode:2003PhRvL..90f0402W.doi:10.1103/PhysRevLett.90.060402. PMID 12633279.

[9] Wolf, P.; Tobar, M. E.; Bize, S.; Clairon, A.; Luiten,A. N.; Santarelli, G. (2004). “Whispering GalleryResonators and Tests of Lorentz Invariance”. Gen-eral Relativity and Gravitation 36 (10): 2351–2372.arXiv:gr-qc/0401017. Bibcode:2004GReGr..36.2351W.doi:10.1023/B:GERG.0000046188.87741.51.

[10] Tobar, M. E.; Wolf, P.; Bize, S.; Santarelli, G.; Flam-baum, V. (2010). “Testing local Lorentz and positioninvariance and variation of fundamental constantsby searching the derivative of the comparison fre-quency between a cryogenic sapphire oscillator andhydrogen maser”. Physical Review D 81 (2): 022003.arXiv:0912.2803. Bibcode:2010PhRvD..81b2003T.doi:10.1103/PhysRevD.81.022003.

[11] Müller, J.; Soffel, M. H. (1995). “A Kennedy–Thorndike experiment using LLR data”. Physics Let-ters A 198 (2): 71–73. Bibcode:1995PhLA..198...71M.doi:10.1016/0375-9601(94)01001-B.

[12] Müller, J., Nordtvedt, K., Schneider, M., Vokrouhlicky,D.: (1999). “Improved Determination of RelativisticQuantities from LLR” (PDF). Proceedings of the 11th In-ternational Workshop on Laser Ranging Instrumentation10: 216–222.

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