Kennedy–Thorndike experiment

thumb|250px|Figure 1. The Kennedy–Thorndike experiment

The Kennedy–Thorndike experiment, first conducted in 1932 by Roy J. Kennedy and Edward M. Thorndike, is a modified form of the Michelson–Morley experimental procedure, testing special relativity.

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 orientation of the apparatus, the Kennedy–Thorndike experiment showed that it is also independent of the velocity of the apparatus in different inertial frames. It also served as a test to indirectly verify time dilation – while the negative result of the Michelson–Morley experiment can be explained by length contraction alone, the negative result of the Kennedy–Thorndike experiment requires time dilation in addition to length contraction to explain why no phase shifts will be detected while the Earth moves around the Sun. The first direct confirmation of time dilation was achieved by the Ives–Stilwell experiment. Combining the results of those three experiments, the complete Lorentz transformation can be derived.

Improved variants of the Kennedy–Thorndike experiment have been conducted using optical cavities or Lunar Laser Ranging. For a general overview of tests of Lorentz invariance, see Tests of special relativity.

The experiment

The original Michelson–Morley experiment was useful for testing the Lorentz–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 test time dilation as well. In their own words: If the apparatus is motionless with respect to the hypothetical aether, the difference in time that it takes light to traverse the longitudinal and transverse arms is given by:

:{| class="wikitable" style="border: 1px solid darkgray;"

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! <math>T_{L} - T_{T} = \frac{2 (L_{L} - L_{T}) }{c} </math>

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

:<math>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}</math>

where T₁ 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, and L_L the length of the longitudinal interferometer arm. The time it takes light to go across and back the transverse arm is given by:

:<math>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}</math>

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

:{| class="wikitable" style="border: 1px solid darkgray;"

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! <math>T_{L} - T_{T} = \frac{2 (L_{L} - L_{T}) \gamma (v)}{c} </math>

|-

|}

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

:<math>\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}</math>.

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 between L_L and L_T is zero. Therefore, the MM experiment only tests whether the speed of light depends on the orientation of the apparatus.) But in the Kennedy–Thorndike experiment, the lengths L_L and L_T are different from the outset, so it is also capable of measuring the dependence of the speed of light on the velocity of the apparatus.

Fig. 3 presents a simplified schematic diagram of Braxmaier et al.'s 2002 repeat of the Kennedy–Thorndike experiment.|| 1990 || Comparing the frequency of an optical Fabry–Pérot cavity with that of a laser stabilized to an I₂ reference line.||rowspan=2 style="text-align:center;" |<math>\lesssim10^{-5}</math>

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| nowrap="nowrap" | Braxmaier et al.|| 2002 || Comparing the frequency of a cryogenic optical resonator with an I₂ frequency standard, using two Nd:YAG lasers.

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|Wolf et al.|| 2003 || The frequency of a stationary cryogenic microwave oscillator, consisting of sapphire crystal operating in a whispering gallery mode, is compared to a hydrogen maser whose frequency was compared to caesium and rubidium atomic fountain clocks. Changes during Earth's rotation have been searched for. Data between 2001–2002 was analyzed.||rowspan=2 style="text-align:center;" |<math>\lesssim10^{-7}</math>

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|Wolf et al.|| 2004 || See Wolf et al. (2003). An active temperature control was implemented. Data between 2002–2003 was analyzed.

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|Tobar et al.|| 2009 || See Wolf et al. (2003). Data between 2002–2008 was analyzed for both sidereal and annual variations.||

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|Gurzadyan and Margaryan || 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.||

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Lunar laser ranging

In addition to terrestrial measurements, Kennedy–Thorndike experiments were carried out by Müller & Soffel (1995) and Müller et al. (1999) using Lunar Laser Ranging data, in which the Earth-Moon distance is evaluated to an accuracy of centimeters. If there is a preferred 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,