Einstein on Fizeau's Experiment
Lorentz Transformation applied to Fizeau's Experiment (1851)
Among the experimental works, in the pre-relativistic period, to measure the velocity of light in various situations, Fiseau's experiment (1851) for determining the velocity of light in a moving liquid is one of the most famous (for historical details, see Stachel 1995). He found that the velocity of light in a liguid is smaller than that in vacuum, depending on how dense that liguid is.
Einstein, in his popular book on relativity (Relativity: the special and the general theory, 1st ed., 1916; English translation, Three Rivers Press, 1961), refers to this experiment, and analyzes its results as follows:
The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity w. How quickly does it travel in the direction of the arrow in the tube T (see the accompanying diagram, Fig. 3 [replaced by Uchii's figure]) when the liquid above mentioned is flowing through the tube with a velocity v?
[Original Fig. 3 is replaced by this]
In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube. (45)
Now, Einstein already has the equations for such a velocity, according to the Galiean transformation (A) and the Lorentz transformation (B), respectively. Here, W signifies the velocity relative to the tube. Notice that (B)---the relativistic rule for compound velocity---is obtained by expressing x' (= wt') and t' of the liquid system in terms of x and t of the tube system, according to the Lorentz transformation (c is the velocity of light in vacuum, and the square is expressed by 'cc' for typographical reason).
(A) W = w + v
(B) W = (w + v) / (1 + wv/cc).
How can we obtain (B)? Recall that the velocity of light seen from the Tube system is x/t; and this ratio is obtained in the following manner.
But since (wv/cc) is very small compared with 1, (B) may be replaced by an approximation
(B') W = w + v(1 - wv/cc).
Now, Fizeau's experiment gave the results expressed by
(C) W = w + v(1 - ww/cc).
Thus, (A) is refuted by (C) and the relativistic account is in conformity with the results, to the same order of approximation as Fizeau's formula (46).
Recall that Fizeau's experiment was made in the context of the theory of ether: whether or not ether is dragged by moving bodies (Stachel 1995, 258-260; translation 267-269), and how the velocity of light is affected in a moving medium. Referring to Fresnel's formula which correctly predicted the results of Fizeau's 1851 experiment, Stachel says: "in retrospect, the formula can be seen as the first indication of a breakdown of the Galilean law of addition for velocities close to c." (Stachel 1995, 260; translation, 269.)
Reference
Stachel, John (1995, Japanese tr., 1999) 「相対性理論の歴史」『20世紀の物理学』vol. 1、pp. 257-366、丸善。
Last modified August 20, 2002. (c) Soshichi Uchii
