Fizeau experiment

The Fizeau experiment was carried out by Hippolyte Fizeau in 1851 to measure the relative speeds of light in moving water. Albert Einstein later pointed out the importance of the experiment for special relativity.

Although it is referred to as the Fizeau experiment, Fizeau was an active experimenter who carried out a wide variety of different experiments involving measuring the speed of light in different situations.

Fresnel drag coefficient

Fizeau's test was designed to evaluate the prediction by Augustin Fresnel that a moving dispersive medium should create a partial offset in the speed of any light moving through it, because the refractive index, n, would be dependent on the entrainment of the luminiferous aether. The Fresnel drag coefficient is

$1 - \frac{1}{n^2}$.

When light is transmitted through a moving medium, its speed relative to a stationary observer is:

$V = \frac {c}{n} + v \left(1 - \frac{1}{n^2} \right)$

This was confirmed by Fizeau's experiment in 1851 and Michelson/Morley in 1886.

In 1895, Lorentz predicted the existence of an extra term due to dispersion:

$V = \frac {c}{n} + v \left(1 - \frac{1}{n^2} - \frac{\lambda}{n} \! \cdot \! \frac{ \mathrm{d} n }{ \mathrm{d} \lambda } \right)$

This was confirmed by Zeeman in 1914.

The experiment

Fizeau Experiment: A light ray emanated from the source S is reflected by a glass plate G and is led parallel by lens L. After passing the slits O¹ and O², two light rays travel through the tubes A¹ and A², whereby water is streaming in opposite directions through the tubes. The rays are directed by a mirror m at the focus of lens L', so that one always propagates in this direction, the other opposite to the direction of the water stream. After passing the tube twice, both rays will be united at S', where they produce interference fringes.

It was shown by Hendrik Lorentz (1892, 1895) that the experiment can be explained by the reaction of the moving water upon the interfering waves without the need of any aether entrainment. On this occasion, Lorentz introduced a different time coordinate for moving bodies within the aether, the so called Local time (an early form of the Lorentz transformation for small velocities compared to the speed of light). In 1895, Lorentz went a step further and explained the coefficient by local time alone and without mentioning any interaction of light and matter.

Within special relativity, Lorentz's formalism was simplified with the aid of the velocity addition formula, which was derived by Albert Einstein (1905) from the complete Lorentz transformation. Thus it was shown by Max von Laue (1907) that the Fresnel drag coefficient can be easily explained by that formula and so the experiment is supporting evidence for the colinear case of Einstein's velocity addition formula. Einstein later emphasized the importance of the experiment for developing and confirming the theory.

Fizeau's result was replicated and improved by Albert Michelson and Edward Morley (1886), and by Pieter Zeeman (1914). Another experiment was the one of Franz Harress (1910). Contrary to Fizeau, he used a rotating device and actually confirmed Fresnel's dragging coefficient. However, he additionally found a "systematic bias" in the data, which later turned out to be the Sagnac effect.

Such tests of special relativity can now be routinely repeated in undergraduate laboratories.

Derivation in special relativity

The speed of light in immobile water is c/n. From the velocity composition law it follows that the speed of light observed in the laboratory, where water is flowing with speed v (in the same direction as light) is

$V_{LAB}=\frac{\frac{c}{n}+v}{1+\frac{\frac{c}{n}v}{c^2}}=\frac{\frac{c}{n}+v}{1+\frac{v}{cn}}$

Thus the difference in speed is (assuming v is small comparing to c, approximating to the first non-trivial correction)

$V_{LAB}-\frac{c}{n}=\frac{\frac{c}{n}+v}{1+\frac{v}{cn}}-\frac{c}{n}=\frac{\frac{c}{n}+v-\frac{c}{n}(1+\frac{v}{cn})}{1+\frac{v}{cn}}=\frac{v\left(1-\frac{1}{n^2}\right)}{1+\frac{v}{cn}}\approx v\left(1-\frac{1}{n^2}\right)$

See also

• Tests of special relativity
• Aether drag hypothesis
• History of special relativity

References

Secondary sources

Primary sources