Kennedy Thorndike experiment

The Kennedy Thorndike experiment. Hg = mercury vapor lamp, P = photo plate with exposure slit S, V = vacuum chamber, W = water jacket for temperature stabilization, N = Nicol's prism , B = beam splitter at Brewster's angle , M1, M2 mirror

The Kennedy-Thorndike experiment (1932) was intended to show whether the change in the speed of the observer in different inertial systems has an influence on the propagation of light. It is a variant of the Michelson-Morley experiment in which the side arms of the interferometer used are of different lengths. In accordance with the special theory of relativity , the experiment carried out delivered a zero result and confirmed that, in addition to the length contraction , the time dilation must also be assumed. After this indirect proof, the first direct proof of time dilation was made by the Ives-Stilwell experiment . From the combination of the results of these three experiments (the Michelson-Morley experiment for measuring the dependence of the speed of light on the observer orientation, the Kennedy-Thorndike experiment for measuring the dependence of the speed of light on the observer speed and the Ives-Stilwell experiment for measuring the Time dilation) the Lorentz transformation can be derived.

Modern Kennedy Thorndike experiments, with which the same relationships between observer speed and the speed of light or between time dilation and length contraction are investigated, are carried out with significantly greater accuracy using optical resonators and lunar laser ranging and confirm the original result. (See tests of special relativity for further experiments.)

history

→ Main article: History of special relativity

The Kennedy-Thorndike experiment is a variant of the Michelson-Morley experiment (1887). With the latter, the existence of a relative movement between earth and ether (ether wind) should be determined, but the result was negative. To explain this negative outcome, George Francis FitzGerald (1889) and Hendrik Antoon Lorentz (1892) postulated the existence of a length contraction . This original contraction hypothesis was extended by Hendrik Antoon Lorentz on the basis of the experiments of Rayleigh and Brace (1902, 1904) and the Trouton Noble experiment (1903) by adding the time dilation to the Lorentz transformation and resulted in that formulated by Albert Einstein in 1905 The special theory of relativity still valid today . This has been confirmed by the Kennedy-Thorndike experiment (which determines the relationship between length contraction and time dilation) and the Ives-Stilwell experiment (which measures time dilation directly). Apart from the special theory of relativity, this experiment can also be explained by a completely carried ether or the emission theory, but these two theories have already been refuted by other experiments and no longer need to be considered.

The experiment

To prove whether time dilation is necessary in addition to length contraction, Kennedy and Thorndike (1932) modified the Michelson-Morley experiment by making one arm of the interferometer much shorter than the other. The difference in length was 16 cm; a larger difference was ruled out for experimental reasons (the coherence of the light source had to be ensured). In addition, Kennedy and Thorndike did not rotate the interferometer, but instead observed possible changes in the interference patterns due to a change in the speed of the earth in the respective arms of the interferometer (as the direction and magnitude of the orbital speed around the sun changes) over a number of months. If there were only the contraction in length, the interference fringes would also shift and the result would be positive. If the speed in the longer arm were changed, there would be a relative shift in the interference pattern due to the length contraction. Only if the frequency changed in the same ratio in terms of time dilation would the result be negative, that is, no shift would be observed.

In order to be able to determine such a shift over several months, the interferometer was built to be extremely stable and the interference patterns were photographed for later comparisons. The tests were carried out over several months. When no significant shift could be determined (upper limit for speed changes of 10 ± 10 km / s), the experimenters concluded that the original contraction hypothesis alone was not sufficient and that the time dilation actually exists.

theory

General theory of the experiment

The distances covered by light

If an interferometer with two mutually perpendicular arms of lengths L _L and L _{T is} used, as in the Michelson-Morley experiment or the Kennedy-Thorndike experiment, the following distance differences result (where Δ L _A is the original difference and v _A the original speed of the apparatus; and Δ L _B and v _B after a rotation or a change in speed of the apparatus due to the rotation of the earth or the orbit of the sun):

${\displaystyle \Delta L_{A}={\frac {2L_{L}}{1-v_{A}^{2}/c^{2}}}-{\frac {2L_{T}}{\sqrt {1-v_{A}^{2}/c^{2}}}},}$

${\displaystyle \Delta L_{B}={\frac {2L_{L}}{\sqrt {1-v_{B}^{2}/c^{2}}}}-{\frac {2L_{T}}{1-v_{B}^{2}/c^{2}}}}$.

Δ L _A and Δ L _B are not the same, from which it follows that the speed of light depends on the orientation of the apparatus. However, this difference disappears when the apparatus experiences a length contraction in the direction of movement, i.e.:

${\displaystyle \Delta L_{A}={\frac {2\left(L_{L}-L_{T}\right)}{\sqrt {1-v_{A}^{2}/c^{2}}}},\qquad \Delta L_{B}={\frac {2\left(L_{L}-L_{T}\right)}{\sqrt {1-v_{B}^{2}/c^{2}}}}}$.

However, this means that the route differences are only the same if the two speeds v _A and v _B match. Different speeds would namely differences between Δ L _A and Δ L _B lead. (This does not apply to the Michelson-Morley experiment, because here L _L = L _T and thus Δ L _A = Δ L _B = 0 from the start. Therefore, this test only measures the dependence of the speed of light on the orientation of the apparatus, The length contraction alone is sufficient for this.) In contrast, in the Kennedy-Thorndike experiment, the interferometer arms are of different lengths, which is why this test can be used to check the dependence of the speed of light on the speed of the apparatus.

According to the above formula, the distance difference Δ L _A −Δ L _B and consequently the expected fringe displacement (where λ is the wavelength) results with:

${\displaystyle \Delta N={\frac {\Delta L_{A}-\Delta L_{B}}{\lambda }}={\frac {2\left(L_{L}-L_{T}\right)}{\lambda }}\left({\frac {1}{\sqrt {1-v_{A}^{2}/c^{2}}}}-{\frac {1}{\sqrt {1-v_{B}^{2}/c^{2}}}}\right)}$

or if only second order quantities are taken into account in v / c :

${\displaystyle \approx {\frac {L_{L}-L_{T}}{\lambda }}\left({\frac {v_{A}^{2}-v_{B}^{2}}{c^{2}}}\right)}$

To prevent this fringe shift, the frequency and thus the wavelength λ must be modified by a factor that corresponds exactly to the time dilation. This means that both length contraction and time dilation together can explain the results of this and similar experiments.

Significance for the theory of relativity

As early as 1905, Henri Poincaré and Albert Einstein showed that the Lorentz transformations only form a group if length contraction and time dilation have the exact relativistic values. The group property is in turn a necessary requirement of the principle of relativity, which has been repeatedly confirmed in experience.

Now Kennedy and Thorndike believed that it was also possible to derive the Lorentz transformation only from the results of the Michelson-Morley experiment and the Kennedy-Thorndike experiment. Strictly speaking, this is not correct, since length contraction and time dilation in their exactly relativistic form are sufficient, but not necessary, to explain the experiments. This follows from the fact that the length contraction in the direction of movement of the body is only a special case of possible explanations for the Michelson-Morley experiment. In general, it must only be assumed that the dimensions of moving bodies in the longitudinal and transversal direction are related to one another by the Lorentz factor - this includes an infinite number of possible combinations of shortening and stretching. This also influences the role of time dilation in the Kennedy-Thorndike experiment, since its value in the analysis of the experiment depends on the value of the length contraction. It is therefore necessary to consider a third experiment, the Ives-Stilwell experiment , in order to derive the Lorentz transformation exactly.

More precisely: Howard P. Robertson and others showed that the following scheme (see Robertson-Mansouri-Sexl test theory ) can be used to analyze the experiments: α corresponds to time changes, β to changes in length in the direction of movement and δ to changes in length perpendicular to the direction of movement. The Michelson-Morley experiment tests the combination of β and δ and the Kennedy-Thorndike experiment the combination of α and β. Thus α depends on β, which in turn depends on δ. Since only the mentioned combinations of these expressions can be determined by these two experiments, but not their individual values, it is necessary to determine one of these expressions directly experimentally. This was achieved using the Ives-Stilwell experiment, which measured α in accordance with the relativistic time dilation. If this value of α is combined with the Kennedy-Thorndike zero result, the value of the relativistic length contraction necessarily results for β. If this value of β is again combined with the Michelson-Morley zero result, it results that δ must be zero. As a result, the exact properties of the Lorentz transformation are experimentally given, as they also result theoretically from the requirements of group theory .

Modern experiments

See also : Modern Tests of Lorentz invariance

Resonator experiments

Experiments of the Kennedy-Thorndike type (as well as of the Michelson-Morley type) have been carried out to this day in different variations, using lasers , maser or optical resonators . Anisotropies in the sense of a dependence on the speed of the observer, and thus tests of the relationship between length contraction and time dilation in the sense of the Robertson-Mansouri-Sexl test theory (RMS), were excluded with considerably increased precision. For comparison: In the context of RMS, the original experiment achieved an accuracy of ~ 10 ⁻² .

author	year	description	Maximum speed dependency
Hils and Hall	1990	The frequencies of an optical Fabry-Pérot resonator are compared with those of a laser that has been stabilized to an I 2 line.	${\displaystyle \lesssim 10^{-5}}$
Braxmaier et al.	2002	The frequencies of a cryogenic optical resonator compared to an I 2 line using two Nd: YAG lasers .
Wolf et al.	2003	The frequency of a cryogenic microwave oscillator is compared to a hydrogen maser. Data between 2001 and 2002 were analyzed.	${\displaystyle \lesssim 10^{-7}}$
Wolf et al.	2004	See Wolf et al. (2003). Data between 2002 and 2003 were analyzed.
Tobar et al.	2009	See Wolf et al. (2003). Data between 2002 and 2008 were analyzed.	${\displaystyle \lesssim 10^{-8}}$

Lunar laser ranging

In addition to these terrestrial variants, Müller and Soffel (1995) and Müller et al . (1999) carried out a kind of Kennedy – Thorndike experiment with lunar laser ranging , that is, on the basis of earth-moon distance measurements. Any dependence of the speed of light on the speed of the observer relative to a preferred reference system would have to lead to changes in the transit time over the course of the year and thus to variations in the measured distance from the earth to the moon. In order to compensate for this (as in all other Kennedy-Thorndike experiments) both length contraction and time dilation must assume the exact relativistic values. Since the time dilation has already been proven very precisely experimentally, a positive result would have meant, in addition to a dependence of the speed of light on the observer, also a variability of the length contraction. The result was negative, with a maximum velocity dependence of ∼10 ⁻⁵ , which is comparable to the terrestrial experiments by Hils and Hall (1990).

Single receipts

1. ↑ RJ Kennedy, Thorndike, EM: Experimental Establishment of the Relativity of Time . In: Physical Review . 42, No. 3, 1932, pp. 400-418. bibcode : 1932PhRv ... 42..400K . doi : 10.1103 / PhysRev.42.400 .
2. ↑ ^{a b c d} Robertson, HP: Postulates versus Observation in the Special Theory of Relativity . In: Reviews of Modern Physics . 21, No. 3, 1949, pp. 378-382. doi : 10.1103 / RevModPhys.21.378 .
3. ↑ ^{a b} Albert Shadowitz: Special relativity , Reprint edition of the 1968th Edition, Courier Dover Publications, 1988, ISBN 0-486-65743-4 , p. 161.
4. ^ Robert Resnick, Introduction to Special Relativity , Wiley 1968
5. ↑ ^{a b} R.Mansouri, RU Sexl: A test theory of special relativity: III. Second-order tests . In: General. Relat. Gravit. . 8, No. 10, 1977, pp. 809-814. bibcode : 1977GReGr ... 8..809M . doi : 10.1007 / BF00759585 .
6. ↑ Dieter Hils, JL Hall: Improved Kennedy-Thorndike experiment to test special relativity . In: Phys. Rev. Lett. . 64, No. 15, 1990, pp. 1697-1700. bibcode : 1990PhRvL..64.1697H . doi : 10.1103 / PhysRevLett.64.1697 . PMID 10041466 .
7. ↑ C. Braxmaier et al. : Tests of Relativity Using a Cryogenic Optical Resonator . In: Phys. Rev. Lett. . 88, No. 1, 2002, p. 010401. bibcode : 2002PhRvL..88a0401B . doi : 10.1103 / PhysRevLett.88.010401 . PMID 11800924 .
8. ↑ Wolf et al. : Tests of Lorentz Invariance using a Microwave Resonator . In: Physical Review Letters . 90, No. 6, 2003, p. 060402. arxiv : gr-qc / 0210049 . bibcode : 2003PhRvL..90f0402W . doi : 10.1103 / PhysRevLett.90.060402 . PMID 12633279 .
9. ↑ P. Wolf et al. : Whispering Gallery Resonators and Tests of Lorentz Invariance . In: General Relativity and Gravitation . 36, No. 10, 2004, pp. 2351-2372. arxiv : gr-qc / 0401017 . bibcode : 2004GReGr..36.2351W . doi : 10.1023 / B: GERG.0000046188.87741.51 .
10. ↑ ME Tobar et al. : 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 . In: Physical Review D . 81, No. 2, 2010, p. 022003. arxiv : 0912.2803 . bibcode : 2010PhRvD..81b2003T . doi : 10.1103 / PhysRevD.81.022003 .
11. ↑ Müller, J .; Soffel, MH: A Kennedy-Thorndike experiment using LLR data . In: Physics Letters A . 198, 1995, pp. 71-73. doi : 10.1016 / 0375-9601 (94) 01001-B .
12. ↑ ^{a b} Müller, J. et al. : Improved Determination of Relativistic Quantities from LLR . (PDF) In: Proceedings of the 11th International Workshop on Laser Ranging Instrumentation . 10, 1999, pp. 216-222.