**The Experiment
of Fizeau as a Test of Relativistic Simultaneity**

Curt Renshaw

680 America’s Cup Cove, Alpharetta, Georgia 30202

Email: *crenshaw@telinc.com
*Web Site: *http://renshaw.teleinc.com
*phone: (770) 751-9481 fax: (770) 751-9829

Special relativity (SRT) was born on the basis of a gedanken experiment involving the relative simultaneity of distant events as perceived by observers with different inertial velocities. It is this assumed aspect of special relativity that is must troubling to one’s intuition, accustomed as we are to living in a world of absolute, not relative, simultaneity. Regardless of the adequacy of special relativity to present a true model of the nature of space and time, the theory at least presents a mathematical equivalence to most problems to which it is applied. Such tests include Doppler effects, clock retardation and apparent mass increase with velocity. As such, further tests of these effects to ever greater precision are not likely to produce any new insights into the validity of special relativity. Surprisingly, however, an actual test of the most troubling aspect of SRT—relative simultaneity—has already been performed, and demonstrates that relativistic simultaneity, in the form of the relativistic velocity addition formula, is incorrect.

**The Experiment of Fizeau**

In 1851, Fizeau carried out an experiment which tested for the aether convection coefficient. This was the first such test of Fresnel’s formula, derived without experimental evidence, over twenty years earlier. Fresnel, in fact, had died more than twenty years before this experiment took place, a point of interest only because many texts derive Fresnel’s formula based on the results of experiment, rather than the other way around. Experimental results, within the level of error available in the mid-1800’s, are not sufficient to derive Fresnel’s formula. These results can only confirm that, within error limits, the formula provides answers consistent with experiment. In fact, Fizeau’s experimental results were so course that the only conclusion he could draw was that the displacement was less than should have been produced by the motion of the liquid if light were completely convected by the medium. From this, he assumed the validity of Fresnel’s formula on the partial convection of the aether.

Fizeau’s experiment involved
passing light two ways through moving water (*v* ~ 7 m/s)
and observing the interference pattern obtained, as illustrated
in figure 1. The experiment was repeated by Michelson in 1886
with much more rigor, and quantitative results were obtained [5].
Working backwards from the observed fringe shift, Michelson was
able to calculate an apparent convection coefficient equivalent
to Fresnel’s formula. Varying the velocity and direction of
the flow allowed for a variety of test points. By observing the
change in interference pattern, the effective velocity of light
through the moving medium, as measured in the lab frame, was
calculated. Within experimental limits, the results obtained by
measuring the fringe shift agreed with the results predicted by
Fresnel’s formula. However, Michelson neglected to take into
account the Doppler effect of light from a stationary source
interacting with moving water, and therefore concluded that the
aether convection concept of Fresnel was essentially correct.

**Figure 1.**
The experiment of Fizeau.

We now examine this experiment in
a purely Galilean environment, taking into account the Doppler
shift (change in wavelength) experienced by each beam of light.
Michelson’s paper gives an excellent analysis whereby the
retarded velocity, *b*, seen in the water may be considered
as due to the number of collisions with atoms, the "velocity
of light through the atoms," and the width of the atoms.
Since there will likely be objections to that analysis based on
current understandings of the microscopic world, we present a
more general approach. In what follows, the retarded velocity is
again considered as due to the "collisions"
(absorptions and re-emissions) of the photons in the medium, as
it must be, but we do not require any assumptions as to
"atom width," or "velocity through the atom."

For light traveling through a medium, the effective wavelength changes:

(1)

The phase shift for light in such a medium is:

(2)

The optical path length is
defined from (2) as *l**h**. *The optical path difference
between the medium and air is then:

(3)

The phase difference compared with the same path in air is:

(4)

In the Fizeau experiment we must
consider Doppler effects. Since the water is moving with respect
to the source, the two paths of light will experience Doppler
shifts upon entering the water. Light moving in the opposite
direction to the flow of water will be blue-shift (l_{1}).
Light moving with the flow will be red shifted (l_{2}):

(8)

To
see why the Doppler shift cannot be ignored in Fizeau’s
experiment, imagine the apparatus depicted in figure 2. All
mirrors, the source and the observing screen are sealed in water
filled containers. The water is not flowing, but is stationary in
the containers. Alternatively, the containers could be made of
solid glass, so long as the refractive index is different than
air. The entire apparatus, with the exception of mirror
(detector) M_{1} moves through the lab frame at a
velocity of *v*. Thus, air is moving through the gap, *l*,
at a velocity of *v* in the equipment frame. To first order
in *v/c*, the wavelengths of the light detected at M_{1}
is given by equation (8).

We now fill the apparatus
containers with air and pass the entire apparatus through water.
In the equipment frame, water is moving through the gap at a
velocity *v.* The motion induced Doppler in the water,
experienced by M_{1}, remains unchanged. If we, the
observers, move along with the apparatus, this setup is
indistinguishable from the actual Fizeau experiment. From our
frame of reference, the equipment is at rest, water is moving
through the gap at a velocity *v*, and the image on the
screen reflects the fringe shift due to that motion. Thus we can
replace the gap with a tube of flowing water, hold the rest of
the apparatus stationary in the lab frame, and obtain a one-sided
Fizeau experiment. Clearly, whatever analysis one uses to derive
the formulas for the observed fringe shift, one must take into
account the fact that the wavelength of the light in the moving
medium is different from that of the source due to the motion
induced Doppler effect of (8).

Substituting (8) into (2), we see that the phase shift including Doppler effects becomes:

(9)

The optical path length is defined from the above as:

(10)

The optical path difference between the medium and air is then:

(11)

The phase difference compared with the same path in air is:

(12)

For light traveling different paths and experiencing different Doppler effects, the total phase shift is given by:

(13)

In the Fizeau experiment, l_{1}
and l_{2} are given by (8). The path lengths *l*_{1}
and *l*_{2} are respectively given below, where the
factor of two is included because the light travels through two
tubes of length *l*, and *b* is the velocity of light
in the reference frame of the liquid.

(14)

Substituting these values into (13) for each path gives the following results:

(15)

(16)

Notice how these results were obtained without invoking "aether" drag, or relativistic velocity addition.

In the special relativistic
analysis of this experiment, the velocity of light in the moving
liquid as measured in the lab frame is no longer *b + v*,
but is given by the relativistic velocity addition formula:

(17)

As a result, the path lengths derived in (14) become:

(18)

The derivation of the total phase shift then becomes:

(19)

(20)

The two results, (16) and (20),
differ in the exponent of the last h term. When
Michelson and Morley performed the experiment, they obtained
sixty one trials, using three different combinations of water
velocity and tube length. The graph below shows the distribution
of these results, normalized to a tube length of ten meters and a
water velocity of one meter per second. The line marked RCM
represents the value obtained from equation (16). The line marked
SRT reflects the value obtained from (20). While there is a
distribution of results, owing to experimental error, Michelson
claimed an overall shift of 0.184 __+__ 0.02 fringe. This is
completely consistent with (16), but eliminates the special
relativistic result, with a value of 0.247, from consideration.

**Summary**

It is very difficult to find adequate tests between special relativity and other competing theories. Most theories overlap with SRT on a vast majority of the prediction made by each, yet are based on different underlying physical principles. Ultimately one must find a test that checks not only the results of the application of the mathematical theory, but also the underlying assumptions. The major conceptual difference between SRT and most competing theories is the idea of relative simultaneity—that distant events that are simultaneous for one observer will not be simultaneous for and observer in motion relative to the first. The relativistic velocity addition rule is a direct consequence of relativistic simultaneity, and the Fizeau experiment represents a direct test of the velocity addition formula. Regardless of what the correct theory is or may be, it is clear that SRT fails to give predictions consistent with results in this experiment—an experiment performed almost ten years before the development of SRT.