Ives–Stilwell experiment

In physics, the Ives–Stilwell experiment tested the contribution of relativistic time dilation to the Doppler shift of light. The result was in agreement with the formula for the transverse Doppler effect and was the first direct, quantitative confirmation of the time dilation factor. Since then many Ives–Stilwell type experiments have been performed with increased precision. Together with the Michelson–Morley and Kennedy–Thorndike experiments it forms one of the fundamental tests of special relativity theory. Other tests confirming the relativistic Doppler effect are the Mössbauer rotor experiment and modern Ives–Stilwell experiments. Both time dilation and the relativistic Doppler effect were predicted by Albert Einstein in his seminal 1905 paper. Einstein subsequently (1907) suggested an experiment based on the measurement of the relative frequencies of light perceived as arriving from "canal rays" (positive ion beams created by certain types of gas-discharge tubes) in motion with respect to the observer, and he calculated the additional Doppler shift due to time dilation. This effect was later called "transverse Doppler effect" (TDE), since such experiments were initially imagined to be conducted at right angles with respect to the moving source, in order to avoid the influence of the longitudinal Doppler shift. Eventually, Herbert E. Ives and G. R. Stilwell (referring to time dilation as following from the theory of Lorentz and Larmor) gave up the idea of measuring this effect at right angles. They used rays in longitudinal direction and found a way to separate the much smaller TDE from the much bigger longitudinal Doppler effect. The experiment was performed in 1938 and was reprised several times. Similar experiments were conducted several times with increased precision, for example, by Otting (1939), Mandelberg et al. (1962), Hasselkamp et al. (1979), and Botermann et al.

Experiments with "canal rays" In these experiments, the large relative velocities needed to exhibit relativistic effects are between the experimental apparatus (laboratory) and positive ions accelerated in a discharge tube, the streams of such ions being the canal rays. One observes the emission spectra of these ions, and in particular how the spectra change depending on the ion velocity, which can be varied via the voltage used to accelerate them.

Experimental challenges Initial attempts to measure the second order transverse Doppler effect in canal rays completely failed. For example, Stark's 1906 measurements showed systematic errors ten times the predicted effect. The maximum speed achievable in early gas-discharge tubes was about 0.005 c, which implied a transverse Doppler shift of only about 1.25×10−5. The small TDE achievable was considerably less than the width of the emission lines, which were relatively diffuse due to the Doppler line-broadening resulting from non-uniformity of ion speeds. By the 1930s, improvements in canal-ray tubes allowed for considerable sharpening of the emission lines. Even with these improvements, however, performing the experiment as usually imagined (with the observation being made at right angles to the beam) would be extremely difficult since small errors in the angle of observation would result in line-shifts of magnitude comparable to the magnitude of the anticipated effect.

To avoid the issues associated with observing the beam at right angles, Ives and Stilwell used a small mirror within the canal ray tube (See Fig. 1 and Fig. 3) to observe the beam simultaneously in two directions both with and against the motions of the particles. The TDE would manifest itself as a shift of the center of gravity of the simultaneously red- and blue-shifted spectral lines.

Theory In 1937, Ives performed a detailed analysis of the spectral shifts to be expected of particle beams observed at different angles following a "test theory" which was consistent with the Michelson–Morley experiment (MMX) and the Kennedy–Thorndike experiment (KTX), but which differed from special relativity (and the mathematically equivalent theory of Lorentz and Larmor) in including a parameter n {\displaystyle n} whose value can not be determined by MMX and KTX alone. Various values of n {\displaystyle n} would correspond to various combinations of length contraction, width expansion, and time dilation, where n = 1 {\displaystyle n=1} would be the value predicted by special relativity. Ives proposed the optical experiment described in this article to determine the precise value of n . {\displaystyle n.}

We will not present Ives's 1937 analysis, but instead will compare the predictions of special relativity against the predictions of "classical" aether theory with the apparatus stationary in the hypothetical aether, even though the classical aether had already long been ruled out by MMX and KTX.

Classical analysis In the classical Doppler effect, the wavelength of light observed by a stationary observer of light emitted by a source moving at speed v {\displaystyle v} away from or towards the observer is given by

λ o b s = λ ⋅ ( 1 ± β ) {\displaystyle \lambda _{obs}=\lambda \cdot (1\pm \beta )} where β = v / c {\displaystyle \beta =v/c}

The top sign is used if the source is receding, and the bottom sign if it is approaching the observer.

We note that the magnitude of the wavelength shift for the source moving away from the observer exactly equals the magnitude of the wavelength shift for the source moving towards the observer The average of the observed wavelengths for a source moving away from the observer and the source moving towards the observer at the same speed exactly equals the wavelength of the source.

Relativistic analysis In the relativistic longitudinal Doppler effect, the observed wavelength with source and observer moving away from each other at speed v {\displaystyle v} is given by

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