Lorentz-Fitzgerald Contraction

In 1989, George FitzGerald, followed by Hendrik Lorentz in 1892, suggested that an object moving with speed $v$ with respect to the aether suffers a contraction in length by a factor $\sqrt{1-v^2/c^2}$ in the direction parallel to the motion, but suffer no contraction in the perpendicular directions. This so-called length contraction hypothesis explains the null result in the Michelson-Morley experiment. To be more exact, and referring to Figure 3.2, when the light traverses the leg of the apparatus that is parallel to its velocity with respect to the aether then it takes a time

$$t_1 = \frac{l_0\sqrt{1-v_e^{\,2}/c^2}}{c-v_e}+ \frac{l_0\sqrt{1-v_e^{\,2}/c^2}}{c+v_e}=\frac{2\,l_0}{c}\,\frac{1}{\sqrt{1-v_e^{\,2}/c^2}},$$ (3.60)

where $l_0$ is the uncontracted length of the leg. On the other hand, when the light traverses the leg of the apparatus that is perpendicular to its velocity with respect to the aether then it takes a time

$$t_2 = \frac{2\,l_0}{\sqrt{c^2-v_e^{\,2}}}= \frac{2\,l_0}{c}\,\frac{1}{\sqrt{1-v_e^{\,2}/c^2}}.$$ (3.61)

It can be seen that $t_1=t_2$, which explains the null result of the Michelson-Morley experiment.