July 21st 2016

Michou interferometer

Costel Munteanu Montreal, QC, Canada [email protected]

Abstract:

The speed of light c defined as a constant of nature is in fact the “two­way speed of light” ​ ​ calculated from the measured time of travel of light from a light source to a reflector and back to a detector situated next to the source. In a “two­way” experiment (like Fizeau­Foucault apparatus) the constant c is the harmonic mean of “forward” and “backward” speeds [13]. A simple ​ ​ ​ ​ Michelson­Morley interferometer with equal length arms detects a difference between the time of travel of light along its two perpendicular arms in certain frames of reference in motion or in gravitational field and this is interpreted as Lorentz–FitzGerald contraction of length in the direction of motion or along the gradient of gravitational field [5]. While no experiment seemed to allow the ​ ​ precise “one­way” measurement of speed of light, Lorentz­FitzGerald length contraction interpretation is preferred. In this article I present an interferometer that determines the difference between “forward” and “backward” speed of light in certain frames of reference ­ the anisotropy of speed of light along one direction ­ showing a flow of electromagnetic medium through the frame of reference of the device contradicting the Lorentz­FitzGerald length contraction hypothesis and bring proof for the first time of (local) preferred frame effects. The device determines the direction of the flow and allows to calculate the velocity of that flow using the measured (or calculated) value of canonical “two­way” speed of light c in that particular frame of reference and the anisotropy measured by the ​ ​ device presented herein. Experiments carried on with this device allow to bring clarifications on the nature of light, physical “vacuum”, inertia and gravitational field. Encouraging results seem to favor the concept of a dynamic massive “dark fluid” (a generalized Chaplygin gas model) [14], [15] that could replace the ​ ​ ​ ​ concept of “luminiferous aether” promoted by Hendrik Lorentz and narrow the direction of research for dark matter. This interferometer allows to bring direct experimental proof of equivalence principle between inertial and gravitational forces. According to Thanu Padmanabhan and Erik Verlinde, gravity might be emergent as hypothesized in “entropic gravity” theory [16] opening a new perspective on the thermodynamic ​ ​ aspect of gravity. The thermodynamic description of gravity has a history that goes back to the research on black hole thermodynamics by Bekenstein and Hawking in the mid­1970s. Thermodynamic analysis of gravitational field leads to the conclusion that on an arbitrary surface enclosing a certain mass we can define a temperature that is proportional to the mass enclosed. That means the surface radiates outwards as much power as the inwards power of

gravitational field establishing a thermodynamic equilibrium for masses remaining constant inside a surface. That inward flow of energy of the gravitational field might be a flow of a “dark fluid” (as discussed above) having a certain mass density (energy density), flowing (inwards) through the reference surface with a velocity equal to the “escape velocity” corresponding to the strength of the gravitational field (g) calculated at a point on the reference surface. In this case, the equivalence principle is demonstrated experimentally by measuring with the device presented herein the anisotropy of the speed of light along the gravitational field gradient (“on the vertical”) and comparing it with the anisotropy measured for a frame in constant motion (with the velocity equal to the corresponding “escape velocity”) in the absence of any gravitational field.

A century old controversy

In 1851, Hippolyte Fizeau carried on an experiment which proved that the medium of propagation of light is dragged by matter in motion somehow in accord with the theory of aether widespread at that time but only partially [1] . ​ ​ In 1881 and 1887, Albert A. Michelson and Edward W. Morley, carried on experiments to test the Lorentz hypothesis of the “luminiferous aether” according which the Earth moving through space should “feel” an “aether wind” [2] . ​ ​ The negative results of Michelson­Morley experiment lead on one side to the hypothesis of total drag of the “aether” by the moving Earth which was in contradiction with Fizeau’s experiment (showing only a partial drag), and on the other side lead to the concept of length contraction postulated by George FitzGerald in 1889 [3] and adopted by Hendrik Antoon Lorentz (1892) to ​ ​ explain the negative outcome of Michelson­Morley experiment [4], [5]. ​ ​ ​ ​ According to Lorentz, the “luminiferous aether” wind was not dragged by the Earth in motion and through the arm of the interferometer the “aether wind” effect was cancelled by the length contraction of the arm (in the direction of motion). Lorentz’s solution seemed to favor a static aether fixed in a “universal frame of reference” through which all massive bodies move without dragging it. But this contradicts the result of Fizeau experiment which proved that there is a drag. None of the three hypotheses related to aether (no­drag, complete drag or partial drag) seemed to fit the experimental data and this represented a conundrum in the new domain of electromagnetism. Electromagnetic phenomena seemed to prove an aether that is always static in the frame of the laboratory even when that frame is in motion. In 1905 Albert Einstein gave a solution to avoid this conundrum by proposing a simple algorithm and demonstrating that by taking into account only the relative motion between frames of reference and applying Lorentz transform between those frames, the effects of an eventual “aether flow” cancel out [6]. ​ ​ Still Einstein never gave up the idea of an “aether flow” through certain frames of reference but he never found a way to prove it. In one of lectures on general relativity and aether theory, an address delivered on May 5th, 1920, at the University of Leyden ­ "Aether and the theory of relativity" [7] , Einstein said that general relativity's gravitational field parameters could be said to ​ ​ ​ ​ have all the usual properties of an aether except one: it was not composed of particulate bodies that could be tracked over time, and so it could not be said to have the property of motion. In Einstein’s opinion, the FitzGerald length contraction was an artefact even if it made work the mathematical framework he proposed. But the time dilation, even if it has only a geometric meaning in the Relativity theory seemed to be a well real fact and it was only later proved by Ives­Stilwell (1938, 1941)[8], [9], Robert Pound and Glen A. Rebka (1959) [10] and Hafele­ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ Keating (1971) experiments [11] . Time dilation (or more precise”the slowing of the clock rate” in ​ ​ ​ certain frames of reference) has more profound causes being related to the “clocks” (like a

pendulum or the “quantum harmonic oscillators”) submitted to stresses generated by fields (inertial, gravitational and electromagnetic) ­ and those stresses shift the natural frequency of oscillation (the “clock rate”) to a lower frequency.

A simple solution older than the problem

In this article I present a simple interferometer that is a cross hybrid between the Michelson interferometer and an interferometer experimented by Martin Hoek in 1868 [12]. The interferometer ​ ​ I present ends a debate more than a century old by demonstrating “preferred frame” effects without equivoca. Michelson interferometer [2] uses isotropic refringent media (air or vacuum) through which ​ ​ light travels from a light source to a mirror and back towards the source on two equal length perpendicular arms. The difference in time of travel of light along the two perpendicular depends on the direction of motion of the reference frame of the device but not on the sense of motion. The device moving along the direction of one arm “forward” (for example) generates the same interference pattern as in the case it moves in opposite direction (backwards). Even if there is a difference between the speed of light traveling in one sense and the opposite sense (anisotropy of speed of light) the device could not measure it. Michelson interferometer seems to agree with FitzGerald length contraction hypothesis [5]. ​ ​ As long as no experiment could determine any anisotropy, the use of Lorentz transform(which is a symmetrical transform between inertial frames of reference) seemed to be safe and Lorentz symmetry largely accepted as a symmetry of nature. Even if Lorentz symmetry leads to paradoxes like “the twins paradox”, as long as there is no proof of light speed anisotropy the prefered way to deal with those paradoxes is to find even more complicated solutions [6] implying ​ ​ that the time dilation depends on the acceleration of the the frame of reference although everything indicates that it depends only on the instant value of the velocity. The interferometer described herein sheds light on all these controversies being able to measure the anisotropy of speed of light and brings the proof that Lorentz symmetry is broken in certain situations. More yet, this device is more efficient than Michelson interferometer, generating a difference in the travel time of the split beams of light proportional to the ratio v/c (ratio between the ​ ​ velocity of the device and the speed of light in vacuum) which for low velocities is many orders of magnitude larger than that generated by Michelson interferometer (which is proportional to the ratio 2 2 v /​ c )​ ​ ​ By measuring the anisotropy of the speed of light in the frame of the device, this interferometer demonstrates that the FitzGerald length contraction hypothesis is false and helps validate the equivalence principle ­ that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo­force experienced by an observer in a non­inertial (accelerated) frame of reference (or that the inertial mass is equivalent to the gravitational mass). This interferometer has even more applications allowing among others investigating the existence and behaviour of dark matter [14], [15] in the hypothesis that this concept is related to the ​ ​ ​ ​ concept of aether and is the medium of propagation of light. The difference between the two concepts is that dark matter has a certain mass density while the aether is considered massless . Part of the solution found existed for almost 150 years in Hoek’s device [12] which used ​ ​ different refringent media to differentiate between forward and backward speed of light on the arm aligned to the direction of motion. The original device designed by Hoek is a common path interferometer and while it has the advantage of canceling out the effect of vibrations and better performance (v/c dependence of the interference pattern) , it cannot distinguish between the motions in one sense or the opposite leaving room for FitzGerald “length contraction” interpretation. By splitting the two paths like in the case of Michelson interferometer and using optical medium with different indexes of refraction for forward and backward directions of propagation of

light on the arm oriented along the direction of motion together with a biased adjustment of the perpendicular arm’s length, the new device is capable of determining the anisotropy of the speed of light which is proof of local “preferred frame effects”.

Fizeau’s experiment and the aether drag

First glimpse at the anisotropy of the speed of light was allowed by an experiment carried out ​ by Hippolyte Fizeau in 1851 [1] and its schematic presentation is shown in Figure 1. ​ ​ ​ ​ ​ ​ Light from a source is split by a beam splitter and the two resulting beams are conducted on opposed directions on a common path which passes through a moving optical medium ­ here flowing water. The flowing water drags the medium through which light propagates and this generates different phase shifts of the two light beams which can be observed through the interference pattern at detector when the beams are merged. For clarity, the beams are shown on slightly offset paths but in reality they are on a common path.

Figure 1. Fizeau experiment ​ Assume that water flows through the pipes with speed v. According to the non­relativistic ​ ​ theory of the luminiferous aether, the speed of light should be increased when "dragged" along by the water, and decreased when "overcoming" the resistance of the water. The overall speed of a beam of light should be a simple additive sum of its speed through the water plus the speed of the water.

If n is the index of refraction of water, so that c/n is the velocity of light in stationary water, then the ​ ​ ​ ​ predicted speed of light w in the blue beam (light propagating downstream) would be ​ ​

(1)

Where f is the Fresnel drag coefficient for the medium through which light propagates. For the ​ ​ magenta beam (upstream):

(2)

Fizeau found that the drag coefficient (f) is dependent of the index of refraction of the medium (water in this case) and the relation is:

(3)

In 1895, Hendrik Lorentz predicted the existence of an extra term due to dispersion of light through a refractive medium.

From preliminary experiments I conclude that the drag has also a nonlinear dependence on the length of the refractive material segment. This dependence must be investigated further but for the presentation purpose of this article I will ignore it, considering the length of that segment large enough to render the value of drag coefficient close enough to that found by Fizeau.

The phase difference of the two beams will be:

(4)

We can make the observation that the phase difference is detectable even for low velocities of the flow, being proportional with the ratio v/c unlike the case of Michelson interferometer which ​ ​ 2 2 generates a phase difference proportional with v /​ c ​ (orders of magnitude smaller). ​ ​ ​ From this experiment we can conclude that the medium of propagation of the electromagnetic radiation (“the vacuum” or “the aether”) interacts with matter, being dragged by matter in motion and that leads to the anisotropy of speed of light in the frame of the moving matter and that the anisotropy could be measured. The common path configuration has the advantage of canceling external perturbations (vibrations) but it has the disadvantage of giving the illusion of preserving Lorentz symmetry: reversing the direction of flow gives the same interference pattern for the same values of flow velocity. Separating the two paths and passing only one path through moving optical medium would break the Lorentz symmetry.

Hoek experiment

In 1851, Martin Hoek carried on a different Earth bound experiment intended to detect an aether flow in the direction tangent to the surface of Earth [12]. The setup used in this experiment is ​ ​ presented schematically in Figure 2. This experiment is similar to Fizeau’s but instead of using a ​ ​ flow of water, it used a solid optical medium (Flint glass) in an asymmetrical configuration. In the hypothesis that the Earth moves through a light propagation medium fixed in a universal frame of reference, the apparatus would sense the anisotropy of the speed of light in the direction of motion due to a flow of the electromagnetic medium in the frame fixed to the surface of the Earth. If the light source is at West, and Earth surface moves through the medium (aether) in east direction, a flow of aether from East to West would slow the propagation of the magenta beam through the optical medium (segment AB) and speed it up in the segment CD passing through air. ​ ​ ​ ​

For the blue beam, light would slow down through air (segment DC) and speed up through ​ ​ glass. The difference of the index of refraction (hence the speed of propagation of light) on the two segments generates a phase difference when there is a flow of aether.

Figure 2. Hoek interferometer ​

Hoek considered that through the optical medium with index of refraction n, a flow of aether ​ ​ with velocity v would imprint a speed of propagation of light in that medium for the light propagating ​ ​ upstream (magenta beam on the segment AB) with the value: ​ ​

(Hoek ignored the drag coefficient found by Fizeau)

In his experiment Hoek did not find a flow of aether and he could not figure out if his formula is not accurate because of the Fresnel drag coefficient omission. But if the apparatus is put in motion relative to the surface of earth the experiment becomes equivalent to Fizeau’s experiment and for this reason I will use the formula corrected with Fresnel drag coefficient. From here on we use the speed of propagation of light relative to the frame of the (moving) apparatus. So, in the case of eastward motion of the apparatus with velocity v relative to the surface of ​ ​ Earth, the upstream speed of light relative to the frame of the apparatus would be (magenta beam on segment AB) ​ ​

For the blue beam propagating downstream through glass (segment BA) ​ ​

The time necessary for the magenta beam (ABCD path) to reach the detector (ignoring the other ​ ​ segments that would cancel out) would be

And for the blue beam (path DCBA) ​ ​

Formula for the upstream speed of light (2) can be written slightly different to facilitate the calculations

(5)

Where

(6)

And same for the downstream (1) and from now on we will use fM instead of fF as drag coefficient. ​ ​ ​ ​ ​ ​

The phase shift between the blue and magenta beams would be

(7)

Again we can conclude that this configuration is much more efficient than Michelson interferometer because it manifests a phase difference proportional to the ratio v/c and produces a ​ ​ visible interference pattern for even small dimensions of the apparatus.

To have an idea of the magnitude of the phase difference, for a length L = 1 m and index of ​ ​ refraction n = 1.5 (Flint glass) , the phase shift will be 5.6 nm (approx. 10% of the wavelength) for ​ ​ ​ ​ the velocity of apparatus of just v = 1 m/s. Compare to the best michelson interferometer which ​ ​ ­8 generates a phase shift 8 orders of magnitude smaller (x 10 )​ for the same velocity! ​

As a common path interferometer, this setup cancels out the effect of vibrations making it suitable for mobile experiments. The disadvantage of this setup is that it can not decide which is the direction of motion ­ forward and backward directions of motion will manifest the same interference pattern for the same magnitude of velocity vector. Still, it is a good proof that Lorentz symmetry [5] does not hold, mathematical framework of ​ ​ ​ Special Relativity failing to predict any phase difference, indifferent of the magnitude or direction of velocity vector of apparatus’ motion. Applying the Lorentz transform (the “length contraction” along the direction of motion) would shorten both paths equally.

Michou interferometer

Figure 3. Michou interferometer ​

To demonstrate without equivoc the anisotropy of the speed of light and breaking of Lorentz symmetry I propose a setup similar to Hoek’s but with separate paths for the split light beams. The sketch of this setup is presented in Figure 3. For simplicity and clarity we consider only ​ ​ the sections of length L1 and L2 ­ the other segments do not generate a sensible difference in the ​ ​ ​ ​ ​ ​ time of travel of light. This setup is basically a Michelson interferometer with one arm unbalanced by the difference of the index of refraction on the two opposed directions of propagation of light on the arm parallel to the direction of motion ­ like in the case of Hoek interferometer. This setup renders the Michelson setup more efficient ­ the phase difference between the two perpendicular arms is proportional to the ratio c/v and hence orders of magnitude greater than in the case of Michelson ​ ​ interferometer. Is more sensible to vibration than Hoek setup but could be built very rigid at the size of less than a metre.

The travel time of magenta beam from source to detector is:

(8)

For the perpendicular arm (blue beam) :

(9)

To have a readable interference pattern the travel time of the split beams on the two arms must be approximative equal when at rest so we chose L2 equal to ​ ​ ​

(10)

(11)

Phase shift will be

(12)

If the direction of motion is reversed, the speed of propagation through the glass segment is

(13)

(14)

(15)

(16)

So, when direction of motion is reversed, the sign of phase shift changes. The interference pattern will look the same for both directions of motion if the initial adjustment made with the apparatus at rest is made to show null shift. The solution is to use a biased initial adjustment and a limited length of the optical glass segment so the phase difference modulus is between null value for maximum velocity tested in one ​ ​ direction and π / 2 for the maximum velocity in the opposite direction of motion. ​ ​ The transversal arm length is adjustable and the segment L2 will be set slightly larger than ​ ​ ​ the value in formula (10). This way we can start measurements from a certain positive velocity v ​ ​ ​ and decrease the velocity in steps until the apparatus is motionless then continue into the negative domain for the velocity vector (­v). ​ ​

A small disadvantage of this interferometer compared to Hoek’s original one is that the phase shift is 2 times smaller for the same length of glass segment and same velocities.

The graph in Figure 4 shows the linear variation of phase shift as function of the vector ​ ​ velocity as recorded by Michou configuration (blue line) compared with a Hoek configuration (magenta). The graph shows the absolute value of the phase shift considering that the interference pattern looks the same for positive and negative difference of the travel times.

Figure 4. Comparison of Hoek (magenta) and Michou (blue) phase shifts (absolute values) function ​ of velocity vector

Conclusion

Anisotropy of speed of light can be measured with high accuracy and this fact opens a new perspective over the limits of the mathematical framework of Relativity theory, allowing to switch from speculative work to empirical evidence in the quest for understanding the nature of light and (possibly) dark matter and the coupling between inertia, gravity and quantum­mechanical phenomena. It allows to sort different hypothesis and channel narrower future inquiries. It offers new perspectives leading to new questions and reformulation of old unanswered ones. One of the most important questions: is dark matter the new “aether” ­ “The Aether 2.0” ? It might not be possible to find the particle that enters the composition of dark matter but at least can we detect its flow. It might be capable of detecting the gravitational waves and in this case not only it is a much more inexpensive alternative to LIGO experiment but using several of these oriented at different angles, such a device could actually find the direction of the source of gravitational waves.

References

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15. Alexandre Arbey (2006) "Dark Fluid: a complex scalar field to unify dark energy and dark matter http://arxiv.org/abs/astro­ph/0601274 ​

16. Erik P. Verlinde “On the Origin of Gravity and the Laws of Newton” (Submitted on 6 Jan 2010) http://arxiv.org/abs/1001.0785 ​

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