The Experiment of Fizeau as a Test of Relativistic Simultaneity
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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 ones 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 SRTrelative simultaneityhas 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 Fresnels 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 Fresnels formula based on the results of experiment, rather than the other way around. Experimental results, within the level of error available in the mid-1800s, are not sufficient to derive Fresnels formula. These results can only confirm that, within error limits, the formula provides answers consistent with experiment. In fact, Fizeaus 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 Fresnels formula on the partial convection of the aether.
Fizeaus 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 . Working backwards from the observed fringe shift, Michelson was able to calculate an apparent convection coefficient equivalent to Fresnels 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 Fresnels 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. Michelsons 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:
The phase shift for light in such a medium is:
The optical path length is defined from (2) as lh. The optical path difference between the medium and air is then:
The phase difference compared with the same path in air is:
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 (l1). Light moving with the flow will be red shifted (l2):
To see why the Doppler shift cannot be ignored in Fizeaus 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) M1 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 M1 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 M1, 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:
The optical path length is defined from the above as:
The optical path difference between the medium and air is then:
The phase difference compared with the same path in air is:
For light traveling different paths and experiencing different Doppler effects, the total phase shift is given by:
In the Fizeau experiment, l1 and l2 are given by (8). The path lengths l1 and l2 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.
Substituting these values into (13) for each path gives the following results:
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:
As a result, the path lengths derived in (14) become:
The derivation of the total phase shift then becomes:
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.
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 simultaneitythat 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 experimentan experiment performed almost ten years before the development of SRT.