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Fresnel, Fizeau, Hoek, Michelson-Morley, Michelson-Gale and Sagnac in Aetherless Galilean Space
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The experiments of Fizeau, et al, in the years 1851-1925 were designed to test for the motion of the Earth through the presumed aether, or to test for the extent to which the aether was constrained and carried in a moving, material medium. The results of these experiments resulted in the Fresnel coefficient of aether drag and the Lorentz transformations, each designed to explain the nature of the aether as evidenced by the data obtained. Building on these results (and much original thought), Einstein developed the special theory of relativity, keeping many of the results in form, but abandoning the aether. Analyzing the results of these experiments without the assumption of an aether eliminates the Fresnel aether drag coefficient, the Lorentz transformation, length contraction and time dilation, and, with this, the basis for special relativity. The correct form and value for the solutions are then derived utilizing Galilean transformations.
In this paper, we will survey chronologically the experiments and theories of Fresnel, et al, performed over a hundred year period, beginning in 1818. We will demonstrate the errors in base assumptions and result interpretations in each case, and show how these errors compounded with each additional test. At the same time, we will document the results assuming aetherless, Galilean space, and show that all experiments are compatible with these assumptions, without the need to invoke length contraction, time dilation or Fresnels "aether drag coefficient."
Fresnels Coefficient of Aether Drag
Fresnel was a firm believer in the concept of an aether, as it was "establishment" physics of the day. He proposed that the density of aether in a material body was different than that in the free aether. From an application of elastic waves to optics, Cauchy demonstrated that the ratio of the elastic constant, p, of a substance to r, the measure of its density, or mass per unit volume, is equal to c2, or:
Building on this, Fresnel imagined a bar of cross sectional area F moving through the aether with velocity v. This bar sweeps up aether, where it acquires a new density, r1, yet is only partially carried along by the bar. Thus the velocity of the bar with respect to the constrained aether, v1, is different than the velocity of the bar with respect to the outside aether. While the velocity and density of the constrained aether changes, its mass must remain constant. Thus, if the bar moves for a time, t, we can state:
Comparing (2) and (1), we see that the velocity of the constrained aether with respect to the bar is simply:
where the last term holds since the refractive index, h, is defined as c/c1, with c1 representing the velocity of light through the bar at rest.
Now, the velocity of light in the moving bar is simply the velocity of light through the aether in the bar minus the velocity of the constrained aether thought the bar, or:
Finally, the velocity of light as measured by an observer stationary in the lab frame of the moving bar is given by c1 plus the velocity of the bar through the lab frame, or:
This is the well known convection coefficient, or "aether drag" coefficient as derived by Fresnel. Most books on relativity praise Fresnel for his insight and ability in obtaining this important formula, even though they contend he got there by using bad assumptions. His theory is considered to be what Petr Beckmann refers to as a mere "equivalence."  A theory which produces the correct answer mathematically, but is based on the wrong underlying assumptions.
Special relativity derives this same result, based on the relativistic addition of velocities as obtained by invoking length contraction and time dilation. This equation has been "verified" experimentally, though, in all cases, the desired result was known in advance. We will now show that these "verifications" also support the Galilean view, well within the limits of experimental error, leading to the firm possibility that both Fresnels theory and Einsteins theory are mere equivalences.
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. From this, he assumed the validity of Fresnels formula on the 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 (5). 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."
Consider a tube of length l filled with water running through it with a velocity v. The velocity of light through the water, b, is less than c. The time to traverse the tube in moving water compared with air is given by:
where t1 is the time difference for water moving in the same direction of light travel. Similarly, for water moving against the direction of light travel the time difference, t2, is:
Next we must consider the 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--one a red shift, the other a blue shift:
We can now express the effective increase in path length as referenced to the frequency of the source traveling through air in numbers of wavelengths for each path. This is obtained by multiplying equations (6) and (7) by c and dividing by the appropriate wavelength of (8). Since each light beam covers the lab-frame distance l twice, the above expression is multiplied by 2 for each case. The total shift in fringe pattern in number of wavelengths, l, is then equal to the difference of these two expressions, or:
This solution is identical to that obtained by Michelson in determining the extent of "aether drag." Special relativity derives this result by applying the relativistic velocity addition rule. However, in so doing, relativity fails to take into account the Doppler shift experienced by light from a stationary source interacting with moving water. Using the relativistic addition rule in place of (b+v) would add an additional factor of (1+v/hc) to the final result of (9).
In 1868, Hoek performed an experiment similar in nature to that of Fizeau, but one which is much simpler in concept and easier to explain in the absence of an aether. Hoeks experiment lies about midway in simplicity between that of Fizeau and the later Michelson-Morley experiment.
As shown in figure 2, the Hoek interferometer passed a beam of light two ways around a loop where one leg passed through a tube filled with water. By rotating the apparatus through various angles, and observing the manner in which the interference patterns shift, one can determine the degree to which the aether is constrained by the water due to the motion of the earth in its orbit.
Figure 2. The experiment of Hoek
When Hoek carried out the experiment, the fringe pattern did not change at all for any orientation. This implies that each of the light beams take equal time to complete the circuit, regardless of the orientation of the equipment. Consulting figure 2, then, we obtain the following relation:
where f is the Fresnel convection coefficient. Solving (10) for f yields:
where Hoek neglected terms of second order and higher and obtained Fresnels formula. Hoeks analysis proceeds well if an aether is assumed. However, if we eliminate the aether, as we have done so far, we see that Hoeks experiment demonstrates nothing. The apparatus is small, thus, during the time of the experiment the motion of the equipment due to the earth may be considered as linear and inertial. Further, all components of the apparatus maintain the same speed and direction. Thus, in the frame of the apparatus, we can assume the experiment to be at rest, and no fringe shift is expected. If we rotate the equipment, it is still at rest in its own inertial frame, and, thus, there is no change in the interference pattern. Thus, in equation (10), we can simply eliminate v, (or set it equal to 0), and therefore eliminate f, which is dependent on v. We then obtain an identity, which is the aetherless solution Hoek neglected to consider. Had he considered this solution, the next test in this story might never have been performed.
The Michelson-Morley Experiment
The successful experiment of Michelson and Morley was performed in 1887. Utilizing an interferometer similar in principle to that depicted in figure 3 (but with many more reflections than indicated), they hoped to measure the velocity of the earth through the aether by measuring the degree to which the observed fringe pattern shifted as the apparatus was rotated through ninety degrees. The results expected were of the second order in v/c. Specifically, a fringe shift of 2lb2/l was expected, or a value of about 0.37. When the experiment was performed, no fringe shift was observed. This should have been expected, since Hoek had already demonstrated that there was no aether. In the absence of an aether, this experiment is even simpler than that of Hoek. Not only is there no relative motion between the various components of the apparatus, but there is also no material with a refractive index different from air placed in the path of the beam. If Hoek predicts no fringe shift, then clearly Michelson-Morley predicts no fringe shift. However, since the experiences of Fresnel, Fizeau and Hoek had all been interpreted in terms of an aether, this null result was entirely unexpected. But even at this point, the aether concept was not abandoned. Instead, Fitzgerald in 1892, and later Lorentz, came up with an explanationall material bodies contract in the direction of their motion with respect to the aether by the factor (1-v2/c2)1/2.
Rather than admitting the obvious, that there is no aether, Lorentz and Fitzgerald found a correction to the aether theory which allowed it to survivesimply change the length of every object placed in motion, and continue to change that length as the velocity continues to change. Of course, another part of this balancing equation indicates that time also changes with velocity, but Lorentz did not need this piece to explain Michelson-Morley, and considered the effect a mathematical oddity only.
Figure 3. The Michelson-Morley Experiment
Lorentz was trying to save the aether concept. Even though we, the observers, the lab and the entire interferometer set-up are all in the same IFR, the length contraction still occurs. The reason it occurs, according to Lorentz, is that the apparatus is moving with respect to the aether. It is this motion which causes the length contraction.
At this point, we see that Hoek created an equivalencedeveloping a value for f that supported the observed results, but was better represented by eliminating the term altogether. As experiments improved, and Michelson-Morley obtained second order results, the theory had to change to maintain equivalence and length contraction was added. The aether theory was saved.
In 1905, Einstein realized that, in order to maintain the mathematical validity of this equivalence theory, the time contractions could not be ignored. Playing with the math, he was able to show that by considering the time and length contractions as real, and abandoning the aether concept, all experimental results could be explained. Additionally, some purely imaginary thought experiments could be explained, not that any such results require any explanation at all.
It is important to realize that the length contraction proposed by Einstein is not the one proposed by Lorentz. In the Michelson-Morley experiment, length contraction occurs in Lorentz theory due to motion with respect to the aether. In Einsteins theory, no such length contraction exists, as there is no relative motion between apparatus and observer. Thus, Michelosn-Morley is not a test of special relativity.
In special relativity, Einstein effectively replaced the aether with any arbitrary observer, such that motion through the inertial frame of the observer obtained all the characteristics of what was formerly motion through the inertial frame of the aether. It was at this point that the mathematical time dilation became real and necessary.
We now have the final equivalence theory, one which has eliminated the aether that was never required in the first place, but kept all the ad-hoc corrections induced to save that theory from the contradictory results of real-world experiments. How much more appropriate it would have been to simply eliminate the aether at the time of Fizeaus experiment, and arrive at the correct Galilean results we have seen so farresults which fully support the observations of each of the three experiments documented. (There is a fourth experiment in this series, a test of aberration carried out by Arago using a telescope filled with water. However, the null result of this experiment is obviously supported by an aetherless theory. The effect has been addressed in  and adds no new insight here.)
We are now in a position to analyze experiments which prove problematical for special relativity. This non-applicability of SRT would be acceptable, if it were not that the domain of validity of special relativity is routinely applied to situations which are more out of range than the cases we are going to address. These include the Hafele-Keating round-the-world clocks experiment and milli-second pulsar timing algorithms  to name a few.
The Sagnac Experiment
The experiments of Sagnac in 1913 and Michelson-Gale in 1925, proved an embarrassment for special relativity, requiring the full force of general relativity to explain their results. That these solutions are complicated explains the experiments absence from most standard texts on Einstein's theories. Aether theories may be used to explain these experiments, but these theories, then, often fall short of explaining the experiment of Fizeau. However, if one abandons Einstein's second postulate, the results can be explained quite simply without the need for general relativity, and without resorting to concepts of aether or absolute space. This is true of Fizeau, Hoek, Michelson-Gale, and Sagnac.
The idea of abandoning SRTs second postulate begins again with what Petr Beckmann  referred to as an equivalencea mathematical description that, while it produces correct answers, is based on faulty assumptions. Einstein had one observationthat observers all recorded the same observed velocity of light, apparently regardless of their velocity with respect to the source. He also had Maxwells equations. His goal was to make these equations invariant under transformation between Inertial Reference Frames (IFRs), while keeping them compatible with the observed speed of light. He had at his disposal two concepts. The concept of space and time, and the concept of light velocity.
Einstein made the assumption that light velocity, even in transit, was absolutely consistent from all IFRs. From this assumption, he derived that lengths contract and time dilates due to the relative velocities of these IFRs. He could just as easily have assumed that lengths and time remain absolutely consistent between IFRs, and that light emerges from its source in a continuum of velocities, from zero to some undetermined upper limit. The two approaches are mathematically equivalent as far as any experiment dealing with both a light signal and a measure of length or time is concerned. Thus, Doppler equations and apparent increase in mass with velocity will appear in both approaches. Only when considering an experiment which covers only one issue, say determining the simultaneity of distant events, or the one-way speed of light for two distinct IFRs, will a difference arise between the two approaches. This concept was proposed in an earlier paper by this author , and was picked up again by Cynthia Whitney , in which she states: "We here adopt a more physically operational interpretation of the light-speed postulate. We require only that observable intensity pulses appear to propagate with isotropic speed c .we interpret "light velocity" not to mean abstract phase velocity, but rather to mean only observable velocity.
In the Sagnac experiment, we have a source, mirrors and screen all on a rotating platform (This experiment has been repeated using artificial satellites orbiting the earth to obtain the same results). The source light is split so that one beam travels clockwise and the other counter-clockwise around the apparatus. When the beams are brought together, an interference pattern forms and the fringe displacement is measured. The general relativistic solution to this problem is more complex than this paper deserves, and will not be presented. Instead, we derive the Galilean solution. There are actually two good approaches to this problem. One was addressed in , the other is presented here.
As with the experiment of Hoek, the motion of the beam-splitting and combining mirror may be considered linear during the term of the experiment. For example, even with a velocity of rw = 30 m/s, the difference between the linear approximation (sin q) and the actual distance traveled (rwt) is on the order of 10-12, as shown:
While it is not necessary for the current discussion, it is of value to note that the acceptable region of linearity can be extended indefinately. We are concerned only with the location and velocity of the mirror as a source compared with the location and velocity of the mirror as an observer at a later instant. The path taken by the mirror to get from its origin to its final destination is not important. We can always model the trip as having been along the most linear route, at a speed equal to the distance between these points divided by the time required to make the trip. We are then left with a small component of velocity at each mirror which is normal to our presumed linear route. However, this normal velocity component would be the same for light traveling each direction of the circuit, and would therefore have no effect on the final solution, whether considered under SRT or any competing theory ([1,2] for example). Now, back to Sagnac.
Having seen in (12) that the effective path length for the entire circuit changes by rwt, we must add this change to the light path moving in the direction of rotation, and subtract it from the light path moving counter to the rotation. Our linear approximation, accurate to within 10-12, demonstrates that this effect is experienced in the frame of the apparatus itself. Since this movement is rotational, unlike the Michelson-Morley or Hoek experiments, as the light moves through its circuit, various parts of the apparatus appear to be in motion with respect to each other. For example, in a four mirror system, while one mirror is moving east, the mirror to one side is moving south, while the mirror to the other side is moving north. The mirror on the opposite side of the wheel is moving west at a velocity of 2v. When integrated over the entire circuit, the effect is the addition and subtraction just addressed. This results in the following analysis for the Sagnac experiment:
This effective difference in path length results in a fringe shift as seen experimentally and given by:
The Michelson-Gale Experiment
This experiment utilized a large rectangular array of pipes and mirrors, with the legs lying in the direction of the earth's rotation having a length of 2010 feet, and the legs lying along longitudinal lines having a length of 1113 feet. A calibration loop had the same longitudinal length, but only a very short length in the direction of the earth's rotation, so that the effect of the earth's motion in the direction of the light traveling the 2010 foot legs could be compared to the effect in the calibration loop in which light traveled only a negligible distance in this direction. By comparing the fringe displacement of the large loop to that of the calibration loop, the effect of the earth's motion (through the aether) was to be discovered. This setup is depicted in figure 4.
Figure 4. The Michelson-Gale Experiment
Unlike the Michelson-Morley interferometer, this experiment did not produce null results--a displacement was observed, and that displacement was closely related to the rotational velocity of the earth, apparently lending support to aether theories. Here we will show that, without the initial assumption of an aether, the Michelson-Gale experiment proceeds quite naturally in a Galilean framework.
Since the equipment is very large, and is located on the earth, the long legs of the equipment are not equal (adjustments of the mirrors compensated for this), nor is their velocity equal. If f2 is the latitude of the top leg, and f1 is the latitude of the lower leg, then we have the following, where v0 is the rotational velocity of the earth at the equator:
Now, as light makes its way around the circuit, one beam clockwise and the other counter-clockwise, there is a difference in the time to complete these trips, due to the different velocities and lengths, even in the frame of the apparatus. As with the Sagnac experiment, if one is on the apparatus on a south-leg mirror, the north-leg mirror will appear to have a linear velocity with respect to the south leg mirror. This is again due to the rotational nature of the equipment, this time due to the unequal rotational velocities as seen from different latitudes. Once again, the second-order and higher effects are too far below the limits of experimental error to be determined.
The difference in travel time for each beam can be derived as below:
Substituting (15) into (16), we obtain:
We can simplify (17) utilizing the following relations, where h is the short (longitudinal) leg of the apparatus and R is the radius of the earth:
Substituting (18) into (17), we obtain:
Now, the shift in fringe pattern due to differing arrival times of (19), as compared with the control loop of the experiment, is given by:
This is the exact result obtained by Michelson and Gale, and is obtained without invoking an aether, time dilation, length contraction or general relativity. While the example could have been handled with identical results by simply projecting the area of the apparatus onto a plane normal to the rotation axis, it is better to show the analysis in terms of the appartus itself. In this way one eliminates claims that the projected area is in no way related to the actual apparatus, or that GRT is required to accurately describe the effects of the projection itself.
We have seen how the initial assumption of the existence of an aether led to more and more corrections to the theory to explain continually improved experiments. In the end, Einstein did away with the aether, and was left only with the "corrections" to Galilean theory. This paper, especially in connection with  and , has shown that none of these corrections are necessary. Thus, through a strange series of bad assumptions and faulty interpretations, Einstein was led to the special theory of relativity, a theory which provides a mathematical equivalence to the areas to which it is applied, but which is based on faulty underlying principles. How much simpler it is to go back to first principles when experiment contradicts theories, rather than to keep building "castles in the air" to rescue a doomed theory.
 Whitney, C., "Finding Absolution for Special Relativity Theory - Part I," Galilean Electrodynamics, vol. 7, no. 2, March/April 1996
 Renshaw, C., "Pulsar Timing and the Special Theory of Relativity," Galilean Electrodynamics, vol. 7, no. 2, March/April 1996
 Beckmann, P., "Einstein Plus Two," The Golem Press, Boulder, 1987
 Michelson, A., "Influence of Motion of the Medium on the Velocity of Light," American Journal of Science, 3rd, Vol. 31, No. 185, May 1886