TheKennedy–Thorndike experiment, first conducted in 1932 by Roy J. Kennedy and Edward M. Thorndike, is a modified form of theMichelson–Morley experimental procedure, testingspecial relativity.^[1]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 theorientation of the apparatus, the Kennedy–Thorndike experiment showed that it is also independent of thevelocity of the apparatus in different inertial frames. It also served as a test to indirectly verifytime dilation – while the negative result of the Michelson–Morley experiment can be explained bylength contraction alone, the negative result of the Kennedy–Thorndike experiment requires time dilation in addition to length contraction to explain why nophase shifts will be detected while the Earth moves around the Sun. The firstdirect confirmation of time dilation was achieved by theIves–Stilwell experiment. Combining the results of those three experiments, the completeLorentz transformation can be derived.^[2]

Improved variants of the Kennedy–Thorndike experiment have been conducted usingoptical cavities orLunar Laser Ranging. For a general overview of tests ofLorentz invariance, seeTests of special relativity.

The original Michelson–Morley experiment was useful for testing theLorentz–FitzGerald contraction hypothesis only. Kennedy had already made several increasingly sophisticated versions of the MM experiment through the 1920s when he struck upon a way to testtime dilation as well. In their own words:^[1]

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 frequency of the light depends […] on the velocity in the way required by relativity.

Referring to Fig. 1, key optical components were mounted withinvacuum chamberV on afused quartz base of extremely lowcoefficient of thermal expansion. A water jacketW kept the temperature regulated to within 0.001 °C. Monochromatic green light from a mercury sourceHg passed through aNicol polarizing prismN before entering the vacuum chamber, and was split by abeam splitterB set atBrewster's angle to prevent unwanted rear surface reflections. The two beams were directed towards two mirrorsM₁ andM₂ which were set at distances as divergent as possible given thecoherence length of the 5461 Å mercury line (≈32 cm, allowing a difference in arm length ΔL ≈ 16 cm). The reflected beams recombined to form circularinterference fringes which were photographed atP. A slitS allowed multiple exposures across the diameter of the rings to be recorded on a single photographic plate at different times of day.

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 afringe shift 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 experimenters concluded that time dilation occurs as predicted by Special relativity.

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:

${\displaystyle L=L_0\sqrt{1-v^2/c^2}=L_0/\gamma (v)}$

where

${\displaystyle L_0}$ is theproper length (the length of the object in its rest frame),

${\displaystyle L}$ is the length observed by an observer in relative motion with respect to the object,

${\displaystyle v}$ is the relative velocity between the observer and the moving object,i.e. between the hypothetical aether and the moving object

${\displaystyle c}$ is thespeed of light,

and theLorentz factor is defined as

${\displaystyle \gamma (v)\equiv \frac{1}{\sqrt{1-v^2/c^2}}}$.

Fig. 2 illustrates a Kennedy–Thorndike apparatus with perpendicular arms and assumes the validity of Lorentz contraction.^[3] If the apparatus ismotionless with respect to the hypothetical aether, the difference in time that it takes light to traverse the longitudinal and transverse arms is given by:

${\displaystyle T_L-T_T=\frac{2(L_L-L_T)}{c}}$

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

${\displaystyle T_L=T_1+T_2=\frac{L_L/\gamma (v)}{c-v}+\frac{L_L/\gamma (v)}{c+v}=\frac{2L_L/\gamma (v)}{c}\frac{1}{1-\frac{v^2}{c^2}}=\frac{2L_L\gamma (v)}{c}}$

whereT₁ is the travel time in direction of motion,T₂ in the opposite direction,v is the velocity component with respect to the luminiferous aether,c is the speed of light, andL_L the length of the longitudinal interferometer arm. The time it takes light to go across and back the transverse arm is given by:

${\displaystyle T_T=\frac{2L_T}{\sqrt{c^2-v^2}}=\frac{2L_T}{c}\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{2L_T\gamma (v)}{c}}$

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

${\displaystyle T_L-T_T=\frac{2(L_L-L_T)\gamma (v)}{c}}$

Because ΔL=c(T_L-T_T), the following travel length differences are given (ΔL_A being the initial travel length difference andv_A the initial velocity of the apparatus, and ΔL_B andv_B after rotation or velocity change due to Earth's own rotation or its rotation around the Sun):^[4]

${\displaystyle \mathrm{\Delta }{L}_A=\frac{2\left(L_L-L_T\right)}{\sqrt{1-v_A^2/c^2}},\mathrm{\Delta }{L}_B=\frac{2\left(L_L-L_T\right)}{\sqrt{1-v_B^2/c^2}}}$.

In order to obtain a negative result, we should have ΔL_A−ΔL_B=0. However, it can be seen that both formulas only cancel each other as long as the velocities are the same (v_A=v_B). But if the velocities are different, then ΔL_A and ΔL_B are no longer equal. (The Michelson–Morley experiment isn't affected by velocity changes since the difference betweenL_L andL_T is zero. Therefore, the MM experiment only tests whether the speed of light depends on theorientation of the apparatus.) But in the Kennedy–Thorndike experiment, the lengthsL_L andL_T are different from the outset, so it is also capable of measuring the dependence of the speed of light on thevelocity of the apparatus.^[2]

According to the previous formula, the travel length difference ΔL_A−ΔL_B and consequently the expected fringe shift ΔN are given by (λ being the wavelength):

${\displaystyle \mathrm{\Delta }N=\frac{\mathrm{\Delta }{L}_A-\mathrm{\Delta }{L}_B}{\lambda }=\frac{2\left(L_L-L_T\right)}{\lambda }\left(\frac{1}{\sqrt{1-v_A^2/c^2}}-\frac{1}{\sqrt{1-v_B^2/c^2}}\right)}$.

Neglecting magnitudes higher than second order inv/c:

${\displaystyle \approx \frac{L_L-L_T}{\lambda }\left(\frac{v_A^2-v_B^2}{c^2}\right)}$

For constant ΔN,i.e. 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 oftime dilation on the frequency is considered. Therefore, both length contraction and time dilation are required to explain the negative result of the Kennedy–Thorndike experiment.

In 1905, it had been shown byHenri Poincaré andAlbert Einstein that theLorentz transformation must form agroup to satisfy theprinciple of relativity (seeHistory of Lorentz transformations). 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 theratio 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 value of length contraction used in the analysis of the experiment. Therefore, it's necessary to consider a third experiment, theIves–Stilwell experiment, in order to derive the Lorentz transformation from experimental data alone.^[2]

More precisely: In the framework of theRobertson-Mansouri-Sexl test theory,^[2][5] 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 todirectly 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 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 ofgroup theory.

In recent years,Michelson–Morley experiments as well as Kennedy–Thorndike type experiments have been repeated with increased precision usinglasers,masers, and cryogenicoptical resonators. The bounds on velocity dependence according to theRobertson-Mansouri-Sexl test theory (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 ~10⁻², but current limits are in the ~10⁻⁸ range.^[5]

Fig. 3 presents a simplified schematic diagram of Braxmaieret al.'s 2002 repeat of the Kennedy–Thorndike experiment.^[6] 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 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.

Author	Year	Description	Maximum velocity dependence
Hils and Hall[7]	1990	Comparing the frequency of an opticalFabry–Pérot cavity with that of a laser stabilized to anI2 reference line.	${\displaystyle \lesssim {10}^{-5}}$
Braxmaieret al.[6]	2002	Comparing the frequency of a cryogenic optical resonator with anI2 frequency standard, using twoNd:YAG lasers.
Wolfet al.[8]	2003	The frequency of a stationary cryogenic microwave oscillator, consisting of sapphire crystal operating in awhispering gallery mode, is compared to ahydrogen maser whose frequency was compared tocaesium andrubidiumatomic fountain clocks. Changes during Earth's rotation have been searched for. Data between 2001–2002 was analyzed.	${\displaystyle \lesssim {10}^{-7}}$
Wolfet al.[9]	2004	See Wolfet al. (2003). An active temperature control was implemented. Data between 2002–2003 was analyzed.
Tobaret al.[10]	2009	See Wolfet al. (2003). Data between 2002–2008 was analyzed for both sidereal and annual variations.	${\displaystyle \lesssim {10}^{-8}}$
Gurzadyan and Margaryan[11]	2018	Compton Edge data of GRAAL experiment at European Synchrotron Radiation Facility (ESRF, Grenoble) and of the calorimeter via the 1.27 MeV photons are analysed.	${\displaystyle \lesssim 7{10}^{-12}}$

In addition to terrestrial measurements, Kennedy–Thorndike experiments were carried out by Müller & Soffel (1995)^[12] and Müller et al. (1999)^[13] usingLunar Laser Ranging data, in which the Earth-Moon distance is evaluated to an accuracy of centimeters. If there is apreferred frame 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 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 ~10⁻⁵,^[13] comparable to the bounds set by Hils and Hall (1990). Hence both length contraction and time dilation must have the values predicted by relativity.

• Tests of special relativity
• Aether theories
• 1932 in science

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