By the early 1800s experiments had shown that light slows down when passing through liquids. A. J. Fresnel suggested in 1818 that there would be a partial drag on light by the medium through which the light was passing. Fresnel’s suggestion explained the problem of stellar aberration if the Earth was at rest in the ether. In a famous experiment in 1851, H. L. Fizeau measured the “ether” drag coefficient for
light passing in opposite directions through flowing water. Let a moving system K’ be at rest in the flowing water and let v be the speed of the flowing water with respect to a fixed observer in K (see Figure 2.16). The speed of light in the water at rest (that is, in system K’) is u’, and the speed of light as measured in K is u. If the index of refraction of the water is n, Fizeau found experimentally that
which was in agreement with Fresnel’s prediction. This result was considered an affirmation of the ether concept. The factor 1-1 / n^{2} became known as Fresnel’s drag coefficient. Show that this result can be explained using relativistic velocity addition without the ether concept.
Strategy We note from introductory physics that the velocity of light in a medium of index of refraction n is u^{\prime}=c/n. We use Equation (2.23a) to solve for u.
u_{x}=\frac{d x}{d t}=\frac{\gamma\left(d x^{\prime}+v d t^{\prime}\right)}{\gamma\left[d t^{\prime}+\left(v / c^{2}\right) d x^{\prime}\right]}=\frac{u_{x}^{\prime}+v}{1+\left(v / c^{2}\right) u_{x}^{\prime}} (2.23a)
