Kennedy-Thorndike Experiment

Suppose that we were to perform a version of the Michelson-Morley experiment in which the two legs of the apparatus are of unequal (uncontracted) lengths $l_1$ and $l_2$. Taking length contraction into account, the time required for light to traverse the first leg of the apparatus is

$$t_1 = \frac{2\,l_1}{c}\,\frac{1}{\sqrt{1-v^{2}/c^2}},$$ (3.62)

where $v$ is the speed of the laboratory with respect to the aether rest frame. Likewise, the time required for light to traverse the second leg of the apparatus is

$$t_2 = \frac{2\,l_2}{c}\,\frac{1}{\sqrt{1-v^{2}/c^2}}.$$ (3.63)

Hence, the difference between these two times is

$$t_2-t_1 = \frac{2\,(l_2-l_1)}{c}\,\frac{1}{\sqrt{1-v^{2}/c^2}}\simeq \frac{2\,(l_2-l_1)}{c}+ \frac{l_2-l_1}{c}\,\frac{v^2}{c^2}.$$ (3.64)

Note that the time difference depends on $v$. Suppose that the laboratory is located on the Earth's equator. In this case, the actual speed of the laboratory with respect to the aether rest frame varies from $v=v_e-v_{\mit\Omega}$ to $v=v_e+v_{\mit\Omega}$, throughout the course of a day, where $v_e=2.977\times 10^{4}\,{\rm m\,s^{-1}}$ is the Earth's mean orbital velocity, specified in Equation (3.20), whereas

$$v_{\mit\Omega} = {\mit\Omega}\,R_e= 4.650\times 10^2\,{\rm m\,s^{-1}}$$ (3.65)

is the speed of the Earth's surface due its axial rotation. Here, ${\mit\Omega} = 7.292\times 10^{-5}\,{\rm rad\,s^{-1}}$ is the Earth's diurnal angular velocity [see Equation (1.351)], and $R_e= 6.378\times 10^3\,{\rm m}$ its equatorial radius. Thus, $v$ varies by about 3% during the course of the day. This variation leads to a variation in the time difference, (3.64), that should be easily measurable. However, when Roy Kennedy and Edward Thorndike performed this experiment in 1932 they observed no variation in the time difference.