A simple "classical" interpretation
of Fizeau's experiment
(Giuseppe Antoni - Umberto Bartocci)*
Abstract - As it is well known, one of the best classical aether theories, in agreement with almost all experimental results (see for instance R. Resnick's widespread textbook: Introduction to special relativity), it is Stokes aether dragged theory. This theory is usually dismissed on the ground of two natural phenomena, which are said to be unexplainable in Stokes context: Bradley's astronomical annual aberration, and light's speed in moving water (Fizeau's experiment). In this paper, a simple "time-delay" model for the behaviour of light in a transparent medium is given, which gets rid of the second of the previous objections.
- - - - -
It is well established that (in an "aether-frame" - as it could be a terrestrial laboratory, according with Stokes hypothesis!) the light's speed in vacuo c becomes c/n (n> 1), when light travels through a transparent medium, for instance water (the refraction index n depends on the light's wave length - we shall suppose from now on to speak of a monochromatic beam).
So, if light travels through a length L in water, instead of the time L/c , it would take a time L/(c/n) = nL/c .
We can then introduce the delay:
(1) nL/c - L/c = (n-1)L/c ,
and make the natural assumption that such a delay is due to the contribution of many single delays, due to the total number of obstacles that light meets during its travel. If we callt this single delay, and N the number of obstacles of the given medium for length unity, we can write:
(2) total delay = (n-1)L/c = NLt ,
from which we get:
(3)t = single delay = (n-1)/cN.
Well, let us suppose now that, in the given length L, the water moves with some (uniform) speed v (for instance in the same sense of light), with respect to the fixed aether-frame, and ask: what will be then the light's delay?
In 1817 Fresnel theorized that, because of the aether present in the water would have been "dragged", the light would have been dragged too, and that the delay would have then been less than the previous one. In 1851 Fizeau confirmed Fresnel's prevision, showing that the light's speed c(v) in this case is experimentally compatible with the expression:
(4) c(v) = c/n + v(1-1/n2) .
This would imply only a partial dragging of the aether, since one gets (4) instead of the (a priori more natural to be expected in an aether theory?!):
(5) c(v) = c/n + v .
Fizeau's result is nowadays used with two main different purposes:
(A) for showing that any aether theory must acknowledge the fact that the aether cannot be "completely dragged" by heavy bodies, and then to disprove Stokes theory with one more argument (another one: Bradley's aberration);
(B) to give another evidence in favour of the relativistic composition of velocities, since the "sum" of the two speeds c/n and v is, from the relativistic point of view, not equal to (5), but to:
(6) (v+c/n)/(1+vc/nc2) = c(1+nb )/n(1+b /n)
(where, as usual,b = v/c).
As far as (A) is concerning, we rather conjecture that the possibility that the aether is being dragged by the Earth, during its motion around the Sun, is quite unlikely, and that on the contrary one should perhaps better suppose that it is the aether to drag the Earth (Descartes-Leibniz vortex theory), and so Fizeau's experiment cannot say anything about this case.
As far as (B), one must indeed acknowledge that (6) is a rather good result in favour of Special Relativity (SR), since one can approximate that expression in order to get:
c/n times (1+nb )/(1+b /n) » c/n times (1+nb )(1-b /n) » c/n times (1+nb -b /n)
(up to higher order terms inb ),
which makes indeed c/n+v-v/n2 , surprisingly identical with the experimental datum (4)!
But the very important question to ask is: what would really be a good aether-theoretic prevision for the value of c(v), different from (5)?
As a matter of fact, one should perhaps conjecture that the aether is not dragged at all by the moving water, and that the only physical phenomenon we are facing of in this case, is that the light, during its travel through the moving water, say for a timeD t , simply meets less obstacles, and that the single delay for each obstacle is, in the case of water moving in the same direction of the light, less than (3).
Let us now try to compute the light's delay with these two further assumptions, obviously taking N(L-vD t) as the total number of obstacles, and as a possible correct value for each single delay the following expression:
(7)t (v) = (n-1)(c-v)/c2N .
This is in truth the simplest linear (in the parameter v) function oft (v), such that it does coincide with (3) when v = 0, and which is infinitesimal as v approaches c.
Well, with this value at our disposal fort (v), we have:
Dt = L/c + delay = L/c + N(L-vD t)t (v) =
= L/c + (n-1)(L-vD t)(c-v)/c2 ,
Dt[c2+v(n-1)(c-v)] = L[v+n(c-v)]
Dt = (L/c)*[b +n(1-b )]/[1+(n-1)b (1-b )]
c(v) = L/D t = c*[1+(n-1)b (1-b )]/[b +n(1-b )] =
= (c/n)*[1+(n-1)b (1-b )]/[1-(n-1)b /n] .
From this last identity one can deduce the following approximation, again up to higher order terms inb :
c(v)» (c/n)*[1+(n-1)b ]*[1+(n-1)b /n] » (c/n)*[1+(n2-1)b /n] =
= (c/n)*[1+nb -b /n] = c/n + v - v/n2 ,
which is exactly equal to the experimentally supported value (4)!
One could even notice that (7) is indeed an approximation, up to higher order terms inb , of the expression:
(8)t °(v) = (n-1)/(c+v)N ,
and that, if one makes use of this value, instead than (7), in the previous computation, then one would get as a final result exactly (6), that is to say the relativistic expectation, in place of (4).
Summing up, the previous "logical" argumentation shows, in one more case, that completely different theories (actually, SR and a very "natural" aether theory) can give, sometimes, the same experimental previsions.
- - - - -
Acknowledgement - The authors thank George Galeczki for some helpful suggestion.
Remark 1 - This paper is inspired to the ideas contained in G. Antoni's: "Una nuova interpretazione dell'esperienza di Fizeau, relativa al trascinamento della luce da parte del mezzo rifrangente in moto", Atti della Fondazione G. Ronchi, Anno VIII, N. 1, Pubblicazioni dell'Istituto Nazionale di Ottica, Serie IV, N. 143, Arcetri, Firenze, 1953.
Remark 2 - Information about Fizeau's experiment can be found, for instance, in E.T. Whittaker's: History of the Theories of Aether and Electricity, Dublin University Press Series, 1910, Chap. IV. Fizeau's experiment was shortly followed (1868) by an attempt of Hoek to detect, using the same "idea", a possible "absolute" Earth's speed, let us call it w. As Whittaker points out, if formula (4) holds, then Hoek's experiment should indeed give a "null result", as it did!, even if w was different from zero; but, of course, a null result for this experiment should be foreseen in the case w = 0 too, which is Stokes hypothesis. Once again, some experimental evidence can be not enough in order to discriminate between very different theoretical interpretations...
Perugia, July 1999 - Revised, August 2000
* Authors' address:
Dipartimento di Matematica
Università di Perugia - 06100 Perugia - Italy