Editing Kennedy–Thorndike experiment (section)

== Theory ==

=== Basic theory of the experiment ===
[[File:Kennedy-Thorndike calculations.svg|thumb|300px|Figure 2. Kennedy–Thorndike light path using perpendicular arms]]
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:

:<math>L = L_{0}\sqrt{1-v^{2}/c^{2}} = L_{0}/{\gamma(v)}</math>

where
:<math>L_0</math> is the [[proper length]] (the length of the object in its rest frame),
:<math>L</math> is the length observed by an observer in relative motion with respect to the object,
:<math> v \,</math> is the relative velocity between the observer and the moving object, ''i.e.'' between the hypothetical aether and the moving object
:<math> c \,</math> is the [[speed of light]],

and the ''[[Lorentz factor]]'' is defined as

:<math>\gamma (v) \equiv \frac{1}{\sqrt{1-v^2/c^2}} \ </math>.

Fig. 2 illustrates a Kennedy–Thorndike apparatus with perpendicular arms and assumes the validity of Lorentz contraction.<ref>Note: In contrast to the following demonstration, which is applicable only to light traveling along perpendicular paths, Kennedy and Thorndike (1932) provided a general argument applicable to light rays following completely arbitrary paths.</ref> 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;"
|-
! <math>T_{L} - T_{T} = \frac{2 (L_{L} - L_{T}) }{c}  </math>
|-
|}

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 ''T''<sub>1</sub> is the travel time in direction of motion, ''T''<sub>2</sub> in the opposite direction, ''v'' is the velocity component with respect to the luminiferous aether, ''c'' is the speed of light, and ''L<sub>L</sub>'' 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;"
|-
! <math>T_{L} - T_{T} = \frac{2 (L_{L} - L_{T}) \gamma (v)}{c}  </math>
|-
|}

Because Δ''L=c(T<sub>L</sub>-T<sub>T</sub>)'', the following travel length differences are given (Δ''L<sub>A</sub>'' being the initial travel length difference and ''v<sub>A</sub>'' the initial velocity of the apparatus, and Δ''L<sub>B</sub>'' and ''v<sub>B</sub>'' after rotation or velocity change due to Earth's own rotation or its rotation around the Sun):<ref>{{cite book |author=Albert Shadowitz  |title=Special relativity |url=https://archive.org/details/specialrelativit0000shad  |url-access=registration  |isbn=0-486-65743-4 |publisher=Courier Dover Publications |edition=Reprint of 1968 |year=1988|pages=[https://archive.org/details/specialrelativit0000shad/page/161 161]}}</ref>

:<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 Δ''L<sub>A</sub>''−Δ''L<sub>B</sub>''=0. However, it can be seen that both formulas only cancel each other as long as the velocities are the same (''v<sub>A</sub>''=''v<sub>B</sub>''). But if the velocities are different, then Δ''L<sub>A</sub>'' and Δ''L<sub>B</sub>'' are no longer equal. (The Michelson–Morley experiment isn't affected by velocity changes since the difference between ''L''<sub>L</sub> and ''L''<sub>T</sub> 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''<sub>L</sub> and ''L''<sub>T</sub> 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.<ref name=rob />

According to the previous formula, the travel length difference Δ''L<sub>A</sub>''−Δ''L<sub>B</sub>'' and consequently the expected fringe shift Δ''N'' are given by (λ being the wavelength):

:<math>\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)</math>.

Neglecting magnitudes higher than second order in ''v/c'':

:<math>\approx\frac{L_{L}-L_{T}}{\lambda}\left(\frac{v_{A}^{2}-v_{B}^{2}}{c^{2}}\right)</math>

For constant Δ''N'', ''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 of [[time dilation]] on the frequency is considered. Therefore, both length contraction and time dilation are required to explain the negative result of the Kennedy&ndash;Thorndike experiment.

=== Importance for relativity ===
In 1905, it had been shown by [[Henri Poincaré]] and [[Albert Einstein]] that the [[Lorentz transformation]] must form a [[Lorentz group|group]] to satisfy the [[principle of relativity]] (see [[History 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 the ''ratio'' 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, the [[Ives–Stilwell experiment]], in order to derive the Lorentz transformation from experimental data alone.<ref name=rob />

More precisely: In the framework of the [[Test theories of special relativity|Robertson-Mansouri-Sexl test theory]],<ref name=rob /><ref name=sexl /> 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 to ''directly'' 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 of [[group theory]].