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arXiv:2402.12421v1 [physics.gen-ph] 19 Feb 2024

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International Journal of Modern Physics A © World Scientific Publishing Company
Michelson-Morley Experiments: at the crossroads of Relativity, Cosmology and Quantum Physics
Maurizio Consoli Istituto Nazionale di Fisica Nucleare, Sezione di Catania
Via S.Sofia 64, 95123, Catania, Italy maurizio.consoli@ct.infn.it
Alessandro Pluchino Dipartimento di Fisica e Astronomia ’E.Majorana’, University of Catania
and Istituto Nazionale di Fisica Nucleare, Sezione di Catania Via S.Sofia 64, 95123, Catania, Italy alessandro.pluchino@ct.infn.it
Today, the original Michelson-Morley experiment and its early repetitions at the beginning of the 20th century are considered as a venerable historical chapter for which, at least from a physical point of view, there is nothing more to refine or clarify. The emphasis is now on the modern versions of these experiments, with lasers stabilized by optical cavities, that, apparently, have improved by many orders of magnitude on the limits placed by those original measurements. Though, in those old experiments light was propagating in gaseous systems (air or helium at atmospheric pressure) while now, in modern experiments, light propagates in a high vacuum or inside solid dielectrics. Therefore, in principle, the difference might not depend on the technological progress only but also on the different media that are tested by preventing a straightforward comparison. Starting from this observation, one can formulate a new theoretical scheme where the tiny, irregular residuals observed so far, from Michelson-Morley to the present experiments with optical resonators, point consistently toward the long sought preferred reference frame tight to the CMB. The existence of this scheme, while challenging the traditional ‘null interpretation’, presented in all textbooks and specialized reviews as a self-evident scientific truth, further emphasizes the central role of these experiments for Relativity, Cosmology and Quantum Physics. Keywords: Relativity; preferred reference system; quantum nonlocality
1

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1. Introduction From the very beginning there are two interpretations of Relativity: Einstein’s Special Relativity1 and the ‘Lorentzian’ formulation.2 Apart from all historical aspects, the difference could simply be phrased as follows. In a Lorentzian approach, the relativistic effects originate from the individual motion of each observer S’, S”...with respect to some preferred reference frame Σ, a convenient redefinition of Lorentz’ ether. Instead, according to Einstein, eliminating the concept of the ether leads to interpret the same effects as consequences of the relative motion of each pair of observers S’ and S”. This is possible because the basic quantitative ingredients, namely Lorentz Transformations, have a crucial group structure and are the same in both formulations. In the case of one-dimensional motion a, an intuitive representation is given in Fig.1.
Fig. 1. An intuitive representation of the two interpretations of Relativity.
For this reason, it has been generally assumed that there is a substantial phenomenological equivalence of the two formulations. This point of view was, for instance, already clearly expressed by Ehrenfest in his lecture ‘On the crisis of the light ether hypothesis’ (Leyden, December 1912) as follows: “So, we see that the ether-less theory of Einstein demands exactly the same here as the ether theory of Lorentz. It is, in fact, because of this circumstance, that according to Einstein’s theory an observer must observe exactly the same contractions, changes of rate, etc.
aWe ignore here the subtleties related to the Thomas-Wigner spatial rotation which is introduced when considering two Lorentz transformations along different directions, see e.g.3–5

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in the measuring rods, clocks, etc. moving with respect to him as in the Lorentzian theory. And let it be said here right away and in all generality. As a matter of principle, there is no experimentum crucis between the two theories”. Therefore, by assuming that, in a Lorentzian perspective, the motion with respect to Σ could not be detected, the usual attitude was to consider the difference between the two interpretations as a philosophical problem.
However, it was emphasized by Bell6 that adopting the Lorentzian point of view could be crucial to reconcile hypothetical faster-than-light signals with causality, as with the apparent non-local aspects of the Quantum Theory. Indeed, if all reference frames are placed on the same footing, as in Special Relativity, how to decide of the time ordering of two events A and B along the world line of a hypothetical effect propagating with speed > c? This ordering can be different in different frames, because in some frame S′ one could find t′A > t′B and in some other frame S′′ the opposite t′B′ > t′A′ . This causal paradox, which is the main reason why superluminal signals are not believed to exist, disappears in a Lorentzian formulation where the different views of the two observers become a sort of optical illusion, like an aberrationb.
But the mere logical possibility of Σ is not enough. For a full resolution of the paradox, the Σ− frame should show up through a determination of the kinematic parameters β′, β′′... Thus, we arrive to the main point of this article: the prejudice that, even in a Lorentzian formulation of relativity, the individual β′, β′′... cannot be experimentally determined. This belief derives from the assumption that the Michelson-Morley type of experiments, from the original 1887 trial to the modern versions with lasers stabilized by optical cavities, give ‘null results’, namely that the small residuals found in these measurements are just typical instrumental artifacts. We recall that in these precise interferometric experiments, one attempts to detect in laboratory an ‘ether-wind’, i.e. a small angular dependence of the velocity of light that might indicate the Earth motion with respect to the hypothetical Σ, e.g. the system where the Cosmic Microwave Background (CMB) is isotropic. While in Special Relativity, no ether wind can be observed by definition, in a Lorentzian perspective it is only a ‘conspiracy’ of relativistic effects which makes undetectable the individual velocity parameters β′, β′′... But the conspiracy works exactly only when the velocity of light cγ propagating in the various interferometers coincides with the basic parameter c entering Lorentz transformations. Therefore, one may
bIf S′ is connected to Σ by a Lorentz transformation with parameter β′ = v′/c, by the inverse transformation we can find the time coordinates in Σ starting from x′A, ct′A, x′B and ct′B , namely cTA = γ′(β′x′A + ct′A) and cTB = γ′(β′x′B + ct′B ), with 1/γ′ = 1 − (β′)2. Analogously, for S′′ and parameter β′′ = v′′/c, we will find the same values, i.e. cTA = γ′′(β′′x′A′ + ct′A′ ) and cTB = γ′′(β′′x′B′ + ct′B′ ), with now 1/γ′′ = 1 − (β′′)2. Thus no ambiguity is possible, either TA > TB or viceversa so that the view in the preferred Σ−frame becomes the relevant one to decide on causal effects.

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ask, what happens if cγ ̸= c, for instance when light propagates in air or in gaseous helium as in the old experiments? Starting from this observation, we have formulated a new theoretical scheme7–10 where the small residuals observed so far, from Michelson-Morley to the present experiments with optical resonators, point consistently toward the long sought preferred reference frame tight to the CMB. In this sense, our scheme is seriously questioning the standard null interpretation of these experiments which is presented in all textbooks and specialized reviews as a self-evident scientific truth. In this article we will review the main results of our extensive work and also propose further experimental tests.
We emphasize that, besides Relativity, our reinterpretation of the data intertwines with and influences other areas of contemporary physics, such as the nonlocality of the Quantum Theory, the current vision of the Vacuum State and Cosmology. These implications are so important to deserve a preliminary discussion in this Introduction.
1.1. Relativity and Quantum Non-Locality
The existence of intrinsically non-local aspects in the Quantum Theory and the relationship with relativity has been the subject of a countless number of books and articles, growing more and more rapidly in recent times, see e.g.11–13 for a list of references. The problem dates back to the very early days of Quantum Mechanics, even before the seminal work of Einstein-Podolski-Rosen (EPR).14 Indeed, the basic issue is already found in Heisenberg’s 1929 Chicago Lectures: “ We imagine a photon represented by a wave packet... By reflection at a semi-transparent mirror, it is possible to decompose into a reflected and a transmitted packet...After a sufficient time the two parts will be separated by any distance desired; now if by experiment the photon is found, say, in the reflected part of the packet, then the probability of finding the photon in the other part of the packet immediately becomes zero. The experiment at the position of the reflected packet thus exerts a kind of action (reduction of the wave packet) at the distant point and one sees that this action is propagated with a velocity greater than that of light”. After that, Heisenberg, almost frightened by his same words, feels the need to add the following remark: “However, it is also obvious that this kind of action can never be utilized for the transmission of signals so that it is not in conflict with the postulates of relativity”.
Heisenberg’s final observation is one of the first formulations of the so called ‘peaceful coexistence’. Actually, presenting as an ‘obvious’ fact that this type of effects can never be used to communicate between observers at a space-like separation sounds more as a way to avoid the causal paradox, which is present in Special Relativity, when dealing with faster than light signals. But, independently of that, this observation expresses a position that can hardly be considered satisfactory. In fact, if there were really some ‘Quantum Information’ which propagates with a speed vQI ≫ c, could such extraordinary thing be so easily dismissed? Namely, could we ignore this ‘something’ just because, apparently, it cannot be efficiently controlled

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to send ‘messages’c? After all, this explains why Dirac, more than forty years later, was still concluding that “The only theory which we can formulate at the present is a non-local one, and of course one is not satisfied with such a theory. I think one ought to say that the problem of reconciling quantum theory and relativity is not solved”.17
But only with Bell’s contribution6 the real terms of the problem were fully understood. He clearly spelled out the local, realistic point of view. If physical influences must propagate continuously through space, it becomes unavoidable to complete the quantum formalism by introducing additional ‘hidden’ variables associated with the space-time regions in question d. But, then, it is possible to derive a bound on the degree of correlation of physical systems that are no longer interacting but have interacted in their past. This bound has been used to rule out experimentally18–20 the class of local, hidden-variable theories which are based on causal influences propagating at subluminal speed. Experimentally excluding this class of theories means rejecting a familiar vision of reality. Thanks to Bell, “A seemingly philosophical debate about the nature of physical reality could be settled by an experiment! ...The conclusion is now clear: Einstein’s view of physical reality cannot be upheld”.21
Thus, the importance of Bell’s work cannot be underestimated: “Bell’s result combined with the EPR argument was not a ‘no hidden variables theorem’ but a non-locality theorem, the impossibility of hidden variables being only one step in a two-step argument...It means that some action at a distance exists in Nature, even though it does not say what this action consists of”.13 It was this awareness to give him the perception that “... we have an apparent incompatibility, at the deepest level, between the two fundamental pillars of contemporary theory”,6 namely Quantum Theory and Special Relativity. This inspired his view where the existence of the preferred Σ−frame would free ourselves from the no-signalling argument to dispose of the causality paradox.
cExperimental correlations between spacelike separated measurements can in principle be explained through hidden influences propagating at a finite speed vQI ≫ c provided vQI is large enough.15 But in ref.16 it is also shown that for any finite vQI, with c < vQI < ∞, one can construct combined correlations to be used for faster-than-light communication. d“In particular, Jordan had been wrong in supposing that nothing was real or fixed in the microscopic world before observation. For after observing only one of the two particles the result of subsequently observing the other (possibly at very remote place) is immediately predictable. Could it be that the first measurement somehow fixes what was unfixed or makes real what was unreal, not only for the near particle but also for the remote one? For EPR that would be an unthinkable ‘spooky action at distance’. To avoid such action at distance one has to attribute, to the space-time regions in question, real correlated properties in advance of the observation which predetermine the outcome of these particular observations. Since these real properties, fixed in advance of the observation, are not contained in the quantum formalism, that formalism for EPR is incomplete”.6

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1.2. Relativity and the Vacuum State
A frequent objection to the idea of relativity with a preferred frame is that, after all, Quantum Mechanics is not a fundamental description of the world. What about, if we started from a more fundamental Quantum Field Theory (QFT)? In this perspective, the issue of the preferred frame can be reduced to find a particular, logical step that prevents to deduce that Einstein Special Relativity, with no preferred frame, is the physically realized version of relativity. This is the version which is always assumed when computing S-matrix elements for microscopic processes. However, what one is actually using is the machinery of Lorentz transformations whose first, complete derivation dates back, ironically, to Larmor and Lorentz who were assuming the existence of a fundamental state of rest (the ether).
Our point, discussed in,22–25 is that there is indeed a particular element which has been missed so far and which concerns the assumed Lorentz invariance of the vacuum state. Even though one is still using the Latin word ‘vacuum’, which means empty, here we are dealing with the lowest energy state. According to the present view, this is not trivially empty but is determined by the condensation process of some elementary quanta e. Namely the energy is minimized when these quanta, such as Higgs particles, quark-antiquark pairs, gluons... macroscopically occupy the same quantum state, i.e. the zero-3-momentum state f . Thus, if the condensation process singles out a certain reference frame Σ, the fundamental question is how to reconcile this picture with the basic postulate of axiomatic QFT: the exact Lorentz invariance of the vacuum.28 This postulate, meaning that the vacuum state must remain unchanged under Lorentz boost, should not be confused with the condition that only local scalars (as the Higgs field, or the gluon condensate, or the chiral condensate...) acquire a non-zero vacuum expectation value.
To make this evident, let us introduce the reference vacuum state |Ψ(0)⟩, appropriate to the observer at rest in the Σ−frame singled out by the condensation process, and the corresponding vacuum states |Ψ′⟩, |Ψ′′⟩,.. appropriate to moving observers S′, S′′,... By assuming that these different vacua are generated by
eBefore our work, the idea that the phenomenon of vacuum condensation could produce ‘conceptual tensions’ with both Special and General Relativity, was discussed by Chiao:26 “The physical vacuum, an intrinsically nonlocal ground state of a relativistic quantum field theory, which possesses certain similarities to the ground state of a superconductor... This would produce an unusual ‘quantum rigidity’ of the system, associated with what London called the ‘rigidity of the macroscopic wave function’... The Meissner effect is closely analog to the Higgs mechanism in which the physical vacuum also spontaneously breaks local gauge invariance ”.26 f In the physically relevant case of the Standard Model, the phenomenon of vacuum condensation can be summarized by saying that “What we experience as empty space is nothing but the configuration of the Higgs field that has the lowest possible energy. If we move from field jargon to particle jargon, this means that empty space is actually filled with Higgs particles. They have Bose condensed”.27

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Lorentz boost unitary operators U ′, U ′′...acting on |Ψ(0)⟩, say |Ψ′⟩ = U ′|Ψ(0)⟩, |Ψ′′⟩ = U ′′|Ψ(0)⟩... For any Lorentz-scalar operator G, such that G = (U ′)†GU ′ = (U ′′)†GU ′′..., it follows trivially

⟨G⟩Ψ(0) = ⟨G⟩Ψ′ = ⟨G⟩Ψ′′ = ..

(1)

However, this by no means implies |Ψ(0)⟩ = |Ψ′⟩ = |Ψ′′⟩... To this end, one should

construct explicitly the three boost generators L0i (with i=1,2,3) and show that L0i|Ψ(0)⟩ = 0. But, in four space-time dimensions, the explicit construction of these operators, and of the corresponding Poincar´e algebra g is only known for the free-

field case through the simple Wick-ordering prescription relatively to the free-field

vacuum |0⟩. In an interacting theory, the construction is implemented order by

order in perturbation theory. Therefore, in the presence of non-perturbative phe-

nomena (such as Spontaneous Symmetry Breaking, chiral symmetry breaking, gluon condensation...) where the physical vacuum |Ψ(0)⟩ cannot be constructed from the

free-field vacuum |0⟩ order by order in perturbation theory, proving the Lorentz

invariance of the vacuum represents an insurmountable problem. In this situation,

with a Lorentz-invariant interaction, the resulting theory will still be Lorentz covari-

ant but, with a non-Lorentz-invariant vacuum, there would be a preferred reference frame h.

1.3. Relativity and the CMB

Finally, some remarks about the physical nature of the hypothetical Σ−frame. A natural candidate is the reference system where the temperature of the CMB looks exactly isotropic or, more precisely, where the CMB kinematic dipole30 vanishes. This dipole is in fact a direct consequence of the motion of the Earth (β = V /c)

T (θ) = To 1 − β2

(2)

1 − β cos θ

gThis means an operatorial representation of the 10 generators Pµ and Lµν (µ, ν= 0, 1, 2, 3), where Pµ describe the space-time translations and Lµν = −Lνµ the space rotations and Lorentz boosts, with commutation relations [Pµ, Pν ] = 0, [Lµν , Pρ] = iηνρPµ − iηµρPν and [Lµν , Lρσ] = −iηµρLνσ + iηµσLνρ − iηνσLµρ + iηνρLµσ where ηµν = diag(1, −1, −1, −1) is the Minkowski tensor. A Lorentz-invariant vacuum has to be annihilated by all 10 generators. hTo our knowledge,in four space-time dimensions, a non-perturbative analysis of a Lorentzinvariant vacuum has been attempted by very few authors. In the case of non-linear field theories with P (Φ(x)) interactions, such as Φ4(x), this was discussed by Segal.29 He considered a suitable generalization of the standard Wick ordering : P (Φ) : relative to |0⟩, say :: P (Φ) ::, such that ⟨Ψ(0)| :: P (Φ) :: |Ψ(0)⟩ = 0 in the true vacuum state. His conclusion was that :: P (Φ) :: is not well-defined until the physical vacuum is known, but, at the same time, the physical vacuum also depends on the definition given for :: P (Φ) ::. From this type of circularity Segal concluded that, in general, in such a nonlinear QFT, the physical vacuum will not be invariant under the full Lorentz symmetry of the underlying Lagrangian density.

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Accurate observations with satellites in space31 have shown that the measured temperature variations correspond to a motion of the solar system described by an average velocity V ∼ 370 km/s, a right ascension α ∼ 168o and a declination γ ∼ −7o, pointing approximately in the direction of the Leo constellation. This means that, if one sets To ∼ 2.7 K and β ∼ 0.00123, there are angular variations of a few millikelvin

∆T CMB(θ) ∼ Toβ cos θ = ±3.3 mK

(3)

which represent by far the dominant contribution to the CMB anisotropy. Could the reference system with vanishing CMB dipole represent the funda-
mental preferred frame for relativity as in the original Lorentzian formulation? The standard answer is that one should not confuse these two concepts. The CMB is a definite medium and, as such, sets a rest frame where the dipole anisotropy is zero. Our motion with respect to this system has been detected but, by itself, this is not in contradiction with Special Relativity. Though, to good approximation, this kinematic dipole arises by combining the various forms of peculiar motion which are involved (rotation of the solar system around the center of the Milky Way, motion of the Milky Way toward the center of the Local Group, motion of the Local Group of galaxies in the direction of the Great Attractor...)31 . Thus, if one could switch-off the local inhomogeneities that produce these peculiar forms of motion, it is natural to imagine a global frame of rest associated with the Universe as a whole. A vanishing CMB dipole could then just indicate the existence of this fundamental system Σ that we may conventionally decide to call ‘ether’ but the cosmic radiation itself would not coincide with this form of ether.
This is why, to discriminate between the two concepts, Michelson-Morley type of experiments become crucial. Detecting a small angular dependence of the velocity of light in the Earth laboratory, and correlating this angular dependence with the Earth cosmic motion, would provide the missing link with the logical arguments from Quantum Non-Locality i and with the idea of a condensed vacuum which selects a particular reference frame through the macroscopic occupation of the same zero 3-momentum state. More generally a non-null interpretation of the MichelsonMorley experiments would resolve the puzzle of a world endowed with a fundamental space and a fundamental time whose existence, otherwise, would remain forever hidden to us.
After this general Introduction, we will start by reviewing in Sect.2 the basic ingredients for a modern analysis of the Michelson-Morley experiments. Then we will summarize in Sects.3 and 4 our re-analysis7–10 of the classical experiments

i“Non-Locality is most naturally incorporated into a theory in which there is a special frame of reference. One possible candidate for this special frame of reference is the one in which the CMB is isotropic. However, other than the fact that a realistic interpretation of quantum mechanics requires a preferred frame and the CMB provides us with one, there is no readily apparent reason why the two should be linked”.32

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and in Sect.5 the corresponding treatment of the present experiments with optical resonators. As a matter of fact, once the small residuals are analyzed in our scheme, the long sought Σ−frame tight to the CMB is naturally emerging. Sect.6 will finally contain a summary and our conclusions.
2. A modern view of the ‘ether-drift’ experiments The Michelson-Morley experiments are also called ‘ether-drift’ experiments because they were designed to detect the drift of the Earth in the ether by observing a dragging of light associated with the Earth cosmic motion. Today, experimental evidence for both the undulatory and corpuscular aspects of radiation has substantially modified the consideration of an underlying ethereal medium, as support of the electromagnetic waves, and its logical need for the physical theory. Besides, Lorentz Transformations forbid dragging and the irregular behavior of the small observed residuals is inconsistent with the smooth time modulations that one would expect from the Earth rotation. Therefore, at first sight, the idea of detecting a non-null effect seems hopeless.
However, as anticipated, dragging is only forbidden if the velocity of light in the interferometers is the same parameter c of Lorentz transformations. For instance, when light propagates in a gas, the sought effect of a preferred system Σ could be due to the small fraction of refracted light. Obviously, this would be much smaller than classically expected but, in view of the extraordinary precision of the interferometers, it could still be measurable. In addition, the idea of smooth time modulations of the signal reflects the traditional identification of the local velocity field, which describes the drift, with the projection of the global Earth motion at the experimental site. This identification is equivalent to a form of regular, laminar flow where global and local velocity fields coincide. Instead, depending on the nature of the physical vacuum, the two velocity fields could only be related indirectly, as it happens in turbulent flows, so that numerical simulations would be needed for a consistent statistical description of the data.
In the following, we will summarize the scheme of refs.7–10 starting with the case of light propagating in gaseous media, as for the classical experiments.
2.1. Basics of the ether-drift experiments In the classical measurements in gases (Michelson-Morley, Miller, Tomaschek, Kennedy, Illingworth, Piccard-Stahel, Michelson-Pease-Pearson, Joos)33-,43 one was measuring the fringe shifts produced by a rotation of the interferometer. Instead, in modern experiments, with lasers stabilized by optical cavities, see e.g.44 for a review, one measures frequency shifts. The modern experiments adopt a different technology but, in the end, have exactly the same scope: searching for an anisotropy of the two-way velocity of light c¯γ(θ) which is the only one that can be measured

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Fig. 2. A schematic illustration of the Michelson interferometer. Note that, by computing the transit times as in Eq.(6), we are assuming the validity of Lorentz transformations so that the length of a rod does not depend on its orientation in the frame S where it is at rest.

unambiguously

c¯γ (θ)

=

2cγ(θ)cγ(π + θ) cγ(θ) + cγ(π + θ)

(4)

By introducing its anisotropy

∆c¯θ = c¯γ(π/2 + θ) − c¯γ(θ)

(5)

there is a simple relation with the time difference ∆t(θ) for light propagation back and forth along perpendicular rods of length D. In fact, by assuming the validity of Lorentz transformations, the length of a rod does not depend on its orientation, in the S frame where it is at rest, see Fig.2, and one finds,

2D

∆t(θ) =

−

2D

2D ∼

∆c¯θ

(6)

c¯γ(θ) c¯γ(π/2 + θ) c c

(where, in the last relation, we have assumed that light propagates in a medium of refractive index N = 1 + ϵ, with ϵ ≪ 1). This gives directly the fringe patterns (λ is the light wavelength)

∆λ(θ) ∼

2D

∆c¯θ

(7)

λ

λc

which were measured in the classical experiments.

In modern experiments, on the other hand, a possible anisotropy of c¯γ(θ) would show up through the relative frequency shift, i.e. the beat signal, ∆ν(θ) of two

orthogonal optical resonators, see Fig.3. Their frequency

ν(θ) = c¯γ(θ)m

(8)

2L

is proportional to the two-way velocity of light within the resonator through an

integer number m, which fixes the cavity mode, and the length of the cavity L as

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Fig. 3. The scheme of a modern ether-drift experiment. The light frequencies are first stabilized by coupling the lasers to Fabry-Perot optical resonators. The frequencies ν1 and ν2 of the resonators are then compared in the beat note detector which provides the frequency shift ∆ν(θ) = ν1(π/2 + θ) − ν2(θ). For a review, see e.g.44

measured in the laboratory S′ frame. Therefore, once the length of a cavity in its rest frame does not depend on its orientation, one finds

∆ν(θ) ∼ ∆c¯θ

(9)

ν0

c

where ν0 is the reference frequency of the two resonators.

2.2. The limit of refractive index N = 1 + ϵ

Let us consider light propagating in a medium which is close to the ideal vacuum limit, i.e. whose refractive index is N = 1 + ϵ with ϵ ≪ 1. The medium (e.g. a gas) fills an optical cavity at rest in a frame S which moves with velocity v with respect to the hypothetical Σ. Now, by assuming i) that the velocity of light is exactly isotropic when S ≡ Σ and ii) the validity of Lorentz transformations, then any anisotropy in S should vanish identically either for v = 0 or for the ideal vacuum case N = 1 when the velocity of light cγ coincides with the basic parameter c of Lorentz transformations. Thus, one can expand in powers of the two small parameters ϵ and β = v/c. By taking into account that, by its very definition, the two-way velocity c¯γ(θ) is invariant under the replacement β → −β and that, for any fixed β, is also invariant under the replacement θ → π + θ, to lowest non-trivial level O(ϵβ2), one finds the general expression7, 24

c c¯γ(θ) ∼ N

∞
1 − ϵ β2 ζ2nP2n(cos θ)

(10)

n=0

Here, to take into account invariance under θ → π + θ, the angular dependence

has been given as an infinite expansion of even-order Legendre polynomials with

arbitrary coefficients ζ2n = O(1). In Einstein’s Special Relativity, where there is

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no preferred reference frame, these ζ2n coefficients should vanish identically. In a

Lorentzian approach, on the other hand, there is no reason why they should vanish

a priori j.

By leaving out the first few ζ’s as free parameters in the fits, Eq.(10) could

already represent a viable form to compare with experiments. Still, one can further

sharpen the predictions by exploiting one more derivation of the ϵ → 0 limit with

a preferred frame. This other argument is based on the effective space-time metric

gµν = gµν (N ) which, through the relation gµν pµpν = 0, describes light propagation

in a medium of refractive index N . For the quantum theory, a derivation of this

metric from first principles was given by Jauch and Watson46 who worked out

the quantization of the electromagnetic field in a dielectric. They noticed that the

procedure introduces unavoidably a preferred reference frame, the one where the

photon energy spectrum does not depend on the direction of propagation, and which

is “usually taken as the system for which the medium is at rest”. However, such an

identification reflects the point of view of Special Relativity with no preferred frame.

Instead, one can adapt their results to the case where the angle-independence of

the photon energy is only valid when both medium and observer are at rest in some

particular frame Σ.

In this perspective, let us consider two identical optical cavities, namely cavity

1,

at

rest

in

Σ,

and

cavity

2,

at

rest

in

S,

and

denote

by

πµ

≡

(

Eπ c

,

π

)

the

light

4-momentum

for

Σ

in

his

cavity

1

and

by

pµ

≡

(

Ep c

,

p)

the

corresponding

light

4-momentum for S in his cavity 2. Let us also denote by gµν the space-time metric

that S uses in the relation gµνpµpν = 0 and by

γµν = diag(N 2, −1, −1, −1)

(11)

the metric used by Σ in the relation γµνπµπν = 0 and which gives an isotropic

velocity

cγ

=

Eπ /|π|

=

c N

.

Now, Special Relativity was formulated, more than a century ago, by assuming

the perfect equivalence of two reference systems in uniform translational motion.

Instead, with a preferred frame Σ, as far as light propagation is concerned, this

physical equivalence is only assumed in the ideal N = 1 limit. As anticipated, for

N ̸= 1, where light gets absorbed and then re-emitted, the fraction of refracted light

could keep track of the particular motion of matter with respect to Σ and produce,

in the frame S where matter is at rest, an angular dependence of the velocity of

light. Equivalently, assuming that the solid parts of cavity 2 are at rest in a frame

S, which is in uniform translational motion with respect to Σ, no longer implies

that the medium which stays inside, e.g. the gas, is in a state of thermodynamic

jAs anticipated, for Lorentz, only a conspiracy of effects prevents to detect the motion with respect to the ether, which, however different might be from ordinary matter, is nevertheless endowed with a certain degree of substantiality. For this reason, in his view, “it seems natural not to assume at starting that it can never make any difference whether a body moves through the ether or not”.45

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equilibriumk. Thus, one should keep an open mind and exploit the implications of the basic condition

gµν (N = 1) = γµν (N = 1) = ηµν

(12)

where ηµν is the Minkowski tensor. This standard equality amounts to introduce a transformation matrix, say Aµν , such that

gµν = Aµρ Aνσηρσ = ηµν

(13)

This relation is strictly valid for N = 1. However, by continuity, one is driven to
conclude that an analogous relation between gµν and γµν should also hold in the
ϵ → 0 limit. The only subtlety is that relation (13) does not fix uniquely Aµν . In fact, one can either choose the identity matrix, i.e. Aµν = δνµ, or a Lorentz transformation, i.e. Aµν = Λµν . Since for any finite v these two matrices cannot be related by an infinitesimal transformation, it follows that Aµν is a two-valued function in the ϵ → 0 limit. Therefore, in principle, there are two solutions. If Aµν is the identity matrix,
we find a first solution

[gµν (N )]1 = γµν ∼ ηµν + 2ϵδ0µδ0ν

(14)

while, if Aµν is a Lorentz transformation, we find the other solution

[gµν (N )]2 = Λµρ Λνσγρσ ∼ ηµν + 2ϵvµvν

(15)

vµ being the dimensionless S 4-velocity, vµ ≡ (v0, v/c) with vµvµ = 1. Notice that with the former choice, implicitly adopted in Special Relativity
to preserve isotropy in all reference systems also for N ̸= 1, one is introducing
a discontinuity in the transformation matrix for any ϵ ̸= 0. Indeed, the whole emphasis on Lorentz transformations depends on enforcing Eq.(13) for Aµν = Λµν so that ΛµσΛνσ = ηµν and the Minkowski metric applies to all equivalent frames.
On the other hand, with the latter solution, by replacing in the relation pµpνgµν = 0, the photon energy now depends on the direction of propagation. Then, by defining the light velocity cγ(θ) from the ratio Ep/|p|, one finds7, 24

c cγ(θ) ∼ N

1 − 2ϵβ cos θ − ϵβ2(1 + cos2 θ)

(16)

and a two-way velocity

c¯γ (θ)

=

2cγ(θ)cγ(π + θ) cγ(θ) + cγ(π + θ)

∼

c N

1 − ϵβ2 1 + cos2 θ

c ≡ N¯ (θ)

(17)

kThink for instance of the collective interaction of a gaseous medium with the CMB radiation or with hypothetical dark matter in the Galaxy. However weak this interaction may be, it would mimic a non-local thermal gradient that could bring the gas out of equilibrium. The advantage of the following analysis is that it only uses symmetry properties without requiring a knowledge of the underlying dynamical processes.

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here θ is the angle between v and p (as defined in the S frame) and

N¯ (θ) ∼ N [1 + ϵβ2(1 + cos2 θ)]

(18)

Eq.(17) corresponds to setting in Eq.(10) ζ0 = 4/3, ζ2 = 2/3 and all ζ2n = 0 for n > 1 and can be considered a modern version of Maxwell’s original calculation.48
It represents a definite, alternative model for the interpretation of experiments per-
formed close to the ideal vacuum limit ϵ = 0, such as in gaseous systems, and will be adopted in the following l.

3. A first look at the classical experiments

From Eq.(17) we find a fractional anisotropy

∆c¯θ = c¯γ(π/2 + θ) − c¯γ(θ) ∼ ϵ (v2/c2) cos 2θ

(19)

c

c

which produces a fringe pattern

∆λ(θ) λ

2D =
λ

∆c¯θ c

D ∼
λ

2ϵ

v2 c2 cos 2θ

(20)

The dragging of light in the Earth frame is then described as a pure 2nd-harmonic effect, which is periodic in the range [0, π], as in the classical theory (see e.g.47). However, its amplitude

D v2

A2 = λ 2ϵ c2

(21)

is now much smaller being suppressed by the factor 2ϵ relatively to the classical value. This was traditionally reported for the orbital velocity of 30 km/s as

Ac2lass

=

D 30 (
λ

km/s )2 c

(22)

This difference could then be re-absorbed into an observable velocity which is related

to the kinematical velocity v through the gas refractive index

vo2bs ∼ 2ϵv2

(23)

so that

D A2 = λ

vo2bs c2

(24)

lA conceptual detail concerns the relation of the gas refractive index N , as defined in the Σ−frame

through Eq.(11), to the experimental quantity Nexp which is extracted from measurements of the

two-way velocity in the Earth laboratory. By assuming a θ−dependent refractive index as in Eq.(18)

one should thus define Nexp by an angular average, i.e. Nexp ≡ ⟨N¯ (θ)⟩θ = N

1+

3 2

(N

− 1)β2

.

One can then determine the unknown value N ≡ N (Σ) (as if the container of the gas were at rest

in Σ), in terms of the experimentally known quantity Nexp ≡ N (Earth) and of v. As discussed in refs.7-,10 for v ∼ 370 km/s, the resulting difference |Nexp − N | ≲ 10−9 is well below the

experimental accuracy on Nexp and, for all practical purposes, can be ignored.

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Fig. 4. The observable velocity reported by Miller 35 for various experiments.

Thus, this vobs is the small velocity traditionally extracted from the measured amplitude AE2 XP in the classical analysis of the experiments

vobs ∼ 30 km/s

AE2 XP Ac2lass

(25)

see e.g. Fig.4.

We can now understand the pattern observed in the classical experiments. For

instance,

in the

original Michelson-Morley experiment, where

D λ

∼

2 · 107,

the

classically expected amplitude was Ac2lass ∼ 0.2. But the experimental amplitude

measured in the six sessions was AE2 XP = 0.01 ÷ 0.02. This corresponds to an

average

anisotropy

|∆c¯θ |exp c

∼

4 · 10−10

and

was

originally

interpreted

in

terms

of

a velocity vobs ∼ 8 km/s. However, for an experiment in air at room temperature

and atmospheric pressure where ϵ ∼ 2.8 · 10−4, this observable velocity would now

correspond to a true kinematic value v ∼ 340 km/s which would fit well with the

cosmic motion indicated by the CMB dipole anistropy. Therefore, the importance

of the issue requires to sharpen the analysis of the old experiments, starting from

the early 1887 trials.

3.1. The 1887 Michelson-Morley experiment in Cleveland
The precision of the Michelson-Morley apparatus33 was extraordinary, about ±0.004 of a fringe. For all details, we address the reader to our book.9 Here, we just limit ourselves to quote from Born.49 When discussing the classically expected fringe shift upon rotation of the apparatus by 90 degrees, namely 2Ac2lass ∼ 0.4, he says explicitly: “Michelson was certain that the one-hundredth part of this displacement would still be observable” (i.e. 0.004). As a check, the Michelson-Morley fringe shifts were recomputed in refs.7, 9, 50 from the original article,33 see Table 1. These data

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were then analyzed in a Fourier expansion, see e.g. Fig.5 (note that a 1st-harmonic has to be present in the data due to the arrangement of the mirrors needed to have fringes of finite width, see35, 51). One can thus extract amplitude and phase of the 2nd-harmonic component by fitting the even combination of fringe shifts

∆λ(θ) + ∆λ(π + θ)

B(θ) =

(26)

2λ

see Fig.6.

Table 1.

The

fringe

shifts

∆λ(i) λ

for

all

noon

(n.)

and

evening

(e.)

sessions

of

the

Michel-

son-Morley

experiment.

The

angle

of

rotation

is

defined

as

θ

=

i−1 16

2π.

The

Table

is

taken

from ref.7

i

July 8 (n.) July 9 (n.) July 11 (n.) July 8 (e.) July 9 (e.) July 12 (e.)

1

−0.001

+0.018

+0.016

−0.016

+0.007

+0.036

2

+0.024

−0.004

−0.034

+0.008

−0.015

+0.044

3 +0.053

−0.004

−0.038

−0.010

+0.006

+0.047

4

+0.015

−0.003

−0.066

+0.070

+0.004

+0.027

5

−0.036

−0.031

−0.042

+0.041

+0.027

−0.002

6

−0.007

−0.020

−0.014

+0.055

+0.015

−0.012

7

+0.024

−0.025

+0.000

+0.057

−0.022

+0.007

8

+0.026

−0.021

+0.028

+0.029

−0.036

−0.011

9

−0.021

−0.049

+0.002

−0.005

−0.033

−0.028

10 −0.022

−0.032

−0.010

+0.023

+0.001

−0.064

11 −0.031

+0.001

−0.004

+0.005

−0.008

−0.091

12 −0.005

+0.012

+0.012

−0.030

−0.014

−0.057

13 −0.024

+0.041

+0.048

−0.034

−0.007

−0.038

14 −0.017

+0.042

+0.054

−0.052

+0.015

+0.040

15 −0.002

+0.070

+0.038

−0.084

+0.026

+0.059

16 +0.022

−0.005

+0.006

−0.062

+0.024

+0.043

The 2nd-harmonic amplitudes for the six experimental sessions are reported in

Table 2. Due to their statistical consistency, one can compute the mean and variance

and obtain AE2 XP ∼ 0.016±0.006. This value is consistent with an observable velocity

vobs

∼

8.5

+1.7 −2.2

km/s

(27)

in complete agreement with Miller, ses Fig.4. In this sense, our re-analysis supports the claims of Hicks and Miller. The fringe shifts were much smaller than expected but in two experimental sessions (11 July noon and 12 July evening), the secondharmonic amplitude is non-zero at the 5σ level and in other two sessions (July 9 noon and July 8 evening) is non-zero at the 3σ level. As such, the average measured amplitude AE2 XP ∼ 0.016, although much smaller than the classical expectation Ac2lass ∼ 0.2, was not completely negligible. Thus it is natural to ask: should this

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Fig. 5. The fringe shifts for the session of July 9 evening. The fit is performed by including terms up to fourth harmonics. The figure is taken from ref.50
Fig. 6. A fit to the even combination B(θ) Eq.(26). The second harmonic amplitude is AE2 XP = 0.025 ± 0.005 and the fourth harmonic is AE4 XP = 0.004 ± 0.005. The figure is taken from ref.50
value be interpreted as a typical instrumental artifact (a “null result”) or could also indicate a genuine ether-drift effect? Of course, this question is not new and, in the past, greatest experts have raised objections to the standard null interpretation of the data. This point of view was well summarized by Miller in 193335 as follows: “The brief series of observations (by Michelson and Morley) was sufficient to show clearly that the effect did not have the anticipated magnitude. However, and this fact must be emphasized, the indicated effect was not zero”. The same conclusion had already been obtained by Hicks in 1902:51 “The data published by Michelson and Morley, instead of giving a null result, show distinct evidence for an effect of the kind to be expected”. There was a 2nd-harmonic effect whose amplitude, however, was substantially smaller than the classical expectation (see Fig.7).
Thus the real point about the Michelson-Morley data does not concern the small

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Table 2. The 2nd-harmonic amplitudes for the six experimental sessions of the Michelson-Morley experiment. The table is taken from ref.7

SESSION July 8 (noon) July 9 (noon) July 11 (noon) July 8 (evening) July 9 (evening) July 12 (evening)

AE2 XP 0.010 ± 0.005 0.015 ± 0.005 0.025 ± 0.005 0.014 ± 0.005 0.011 ± 0.005 0.024 ± 0.005

Fig. 7. The even combination of fringe shifts B(θ) for the Michelson-Morley data as reported by Hicks.51 Solid and dashed lines refer respectively to noon and evening observations. Compare the solid curve of July 11th with the analogous curve in Fig.6.
magnitude of the amplitude but the sizeable changes in the ‘azimuth’, i.e in the phase θ2 of the 2nd-harmonic which gives the direction of the drift in the plane of the interferometer. By performing observations at the same hour on consecutive days (so that variations in the orbital motion are negligible) one expects that this angle should remain the same within the statistical errors. Now, by taking into

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account that, in a 2nd-harmonic effect, the angle is always defined up to ±180o, one choice for the experimental θ2−values is 357o ± 14o, 285o ± 10o and 317o ± 8o respectively for the noon sessions of July 8th, 9th and 11th. For this assignment, the individual velocity vectors vobs(cos θ2, − sin θ2) and their mean are shown in Fig. 8. As a consequence, directly averaging the amplitudes of the individual sessions is considerably different from first performing the vector average of the data and then computing the resulting amplitude. In the latter case, the average amplitude is reduced from 0.016 to about 0.011, with a central value of the observable velocity which decreases from 8.5 km/s to 7 km/s.
This irregular character of the observations has always represented a strong argument to interpret the small observed residuals as typical instrumental effects.
Fig. 8. The observable velocities for the three noon sessions of the Michelson-Morley experiment and their mean. The x-axis corresponds to θ2 = 0o ≡ 360o and the y-axis to θ2 = 270o. Statistical uncertainties of the various determinations are not reported. All individual directions could also be reversed by 180o. The figure is taken from.7
3.2. Further insights: Miller vs. Piccard-Stahel To get further insights we have compared two other sets of measurements, namely Miller’s observations35 and those performed by Piccard and Stahel.39 Miller’s large interferometer had an optical path of about 32 metres and was installed on top of Mount Wilson. His most extensive observations were made in blocks of ten days around April 1, August 1, September 15, 1925, and later on around February 8, 1926, with a total number of 6402 turns of the interferometer.35 The result of his 1925 measurements, presented at the APS meeting in Kansas City on December 1925, was confirming his original claim of 1921, namely “there is a positive, systematic ether-drift effect, corresponding to a relative motion of the Earth and the ether, which at Mt. Wilson has an apparent velocity of 10 km/s”.

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Being aware that Miller’s previous 1921 announcement of a non-zero ether-drift of about 9 km/s, if taken seriously, could undermine the basis of Einstein’s relativity (Miller’s results “carried a mortal blow to the theory of relativity”), Piccard and Stahel realized a precise apparatus with a small optical path of 280 cm that could be carried on board of a free atmospheric balloon (at heights of 2500 and 4500 m) to check the dependence on altitude. In this first series of measurements thermal disturbances were so strong that they could only set an upper limit of about 9 km/s to the magnitude of any ether-drift. However, after this first series of trials, precise observations were performed on dry land in Brussels and on top of Mt.Rigi in Switzerland (at an height of 1800 m).
Despite the optical path was much shorter than the size of the instruments used at that times in the United States, Piccard and Stahel were convinced that the precision of their measurements was higher because spurious disturbances were less important. In particular, with respect to the traditional direct observation, the fringe shifts were registered by photographic recording. Also, for thermal insulation, the interferometer was surrounded either by a thermostat filled with ice or by an iron enclosure where it could be possible to create a vacuum. This last solution was considered after having understood that the main instability in the fringe system was due to thermal disturbances in the air of the optical arms (rather than to temperature differences in the solid parts of the apparatus). However, very often the interference fringes were put out of order after few minutes by the presence of residual bubbles of air in the vacuum chamber. For this reason, they finally decided to run the experiment at atmospheric pressure with the ice thermostat which, by its great heat capacity, was found to stabilize the temperature in a satisfactory way.
We have thus considered the compatibility of these two experiments. Miller was always reporting his observations by quoting separately the amplitude and the phase of the individual sessions. In this way, as shown in Fig.4, the average observable velocity, obtained from a classical interpretation of his data, was vobs ∼ 8.4 ± 2.2 km/s. Piccard and Stahel were instead first performing a vector average of the data and, since the phase was found to vary in a completely arbitrary way, were quoting the much smaller value vobs ∼ (1.5 ÷ 1.7) km/s. For this reason, their measurements are traditionally considered a definite refutation of Miller.
But suppose that the ether-drift phenomenon has an intrinsic non-deterministic nature, which induces random fluctuations in the direction of the local drift. In this case, a vector average of the data from various sessions would completely obscure the physical information contained in the individual observations. For this reason, a meaningful comparison with Miller requires to apply his same procedure to the Piccard-Stahel data. Namely, first summarizing each measurement into a definite pair (AE2 XP, θ2EXP) for amplitude and azimuth, and then computing the magnitude of the observable velocity from the measured amplitudes. With this different procedure, the Piccard and Stahel observable velocity, at the 75% CL, becomes now

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Fig. 9. We report in panel (a) the probability histogram W for the observable velocity obtained from the 24 individual amplitudes reported by Piccard and Stahel in.39 In panel (c) we report the analogous histogram obtained from Miller’s Figure 22d in.35 In both cases, the vertical normalization is to a unit area. Finally, in panel (b) we report the overlap of the two histograms. The area of the overlap is 0.645. This gives a consistency between the two experiments of about 64%. The figure is taken from.9

much larger, namely

vobs = 6.3−+12..50 km/s

(Piccard − Stahel)

(28)

and is now compatible with Miller’s results. For a more refined test, we constructed probability histograms by considering the large set of measurements reported by Miller in Figure 22d of35 and the 24 individual amplitudes reported by Piccard and Stahel in,39 see Fig.9. From the area of the overlap, the consistency of the two experiments can be estimated to be about 64% which is a quite high level. At the same time, since the agreement is restricted to the region vobs < 9 km/s, Miller’s higher values are likely affected by systematic disturbances. This would confirm Piccard and Stahel’s claim that their apparatus, although of a smaller size, was more precise.
Therefore, summarizing, there is a range of observable velocity, say vobs ∼ 6.0 ± 2.0 km/s, where the results of the three experiments we have considered, namely Michelson-Morley, Miller and Piccard and Stahel, overlap consistently. This common range is obtained from the 2nd-harmonic amplitudes measured in a plenty of experimental sessions performed at different sidereal times and in different laboratories. As such, to a large extent, it should also be independent of spurious systematic effects. On the basis of Eq.(23), this range corresponds to a true kinematic velocity v ∼ 250 ± 80 km/s which could reasonably fit with the projection of the Earth velocity within the CMB at the various laboratories. Truly enough, this is only a first, partial view which must be supplemented by a deeper understanding of the observed random variability of the phase.

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Fig. 10. The propagation of light in an optical cavity. It is emphasized that, independently of its particular name (physical vacuum, ether...) and differently from the solid parts of the apparatus, the underlying substrate is not completely entrained with the Earth motion. Thus, in general, its state of motion vµ(t) is different from v˜µ(t).
4. Going deeper into the ether-drift phenomenon The traditional expectation that an ether drift should precisely follow the smooth modulations induced by the Earth rotation, derives from the identification of the local velocity field which describes the drift in the plane of the interferometer, say vµ(t), with the corresponding projection of the global Earth motion, say v˜µ(t). By comparing with the motion of a macroscopic body in a fluid, this identification is equivalent to assume a form of regular, laminar flow, where global and local velocity fields coincide. Depending on the nature of the fluid, this assumption may be incorrect.
To formulate an alternative model of ether drift, in refs.,7–10 we started from Maxwell’s original view48 of light as a wave process which takes place in some substrate: “We are therefore obliged to suppose that the medium through which light is propagated is something distinct from the transparent media known to us”. He was calling the underlying substrate ‘ether’ while, today, we prefer to call it ‘physical vacuum’. However, this is irrelevant. The essential point for the propagation of light, e.g. inside an optical cavity, is that, differently from the solid parts of the apparatus, this physical vacuum is not completely entrained with the Earth motion see Fig.10.
Thus, to explain the irregular character of the data, one could try to model the state of motion of the vacuum substrate as in a turbulent fluid52, 53 or, more precisely, as in a fluid in the limit of zero viscosity. Then, the simple picture of a laminar flow is no more obvious due to the subtlety of the infinite-Reynolds-number limit, see e.g. Sect. 41.5 in Vol.II of Feynman’s lectures.54 In fact, beside the laminar regime where vµ(t) = v˜µ(t), there is also another solution where vµ(t) becomes a continuous but nowhere differentiable velocity field55, 56 m.
mThe idea of the physical vacuum as an underlying stochastic medium, similar to a turbulent fluid, is deeply rooted in basic foundational aspects of both quantum physics and relativity. For

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Instructions for typing manuscripts (paper’s title) 23
Fig. 11. We compare two different signals. Panel a) reports the turbulent velocity field measured in a wind tunnel.67 Panel b) reports the instantaneous frequency shift measured with vacuum optical resonators in ref.68 For the adopted laser frequency ν0 = 2.8 · 1014 Hz a ∆ν = ±1 Hz corresponds to a fractional value ∆ν/ν0 of about ±3.5 · 10−15.
Together with these theoretical arguments, the analogy with a turbulent flow finds support in modern ether drift experiments where one measures the frequency shifts of two optical resonators. To this end, consider Fig.11. Panel a) reports the turbulent velocity field measured in a wind tunnel.67 No doubt, this is a genuine signal, not noise. Panel b) reports instead the instantaneous frequency shift measured with vacuum optical cavities in ref.68 So far, this other signal is interpreted as spurious noise.
Consider now Fig.12. Panel a) shows the power spectrum S(ω) ∼ ω−1.5 of the
instance, at the end of XIX century, the last model of the ether was a fluid full of very small whirlpools (a ‘vortex-sponge’).57 The hydrodynamics of this medium was accounting for Maxwell equations and thus providing a model of Lorentz symmetry as emerging from a system whose elementary constituents are governed by Newtonian dynamics. In a different perspective, the idea of a quantum ether, as a medium subject to the fluctuations of the uncertainty relations, was considered by Dirac.58 More recently, the model of turbulent ether has been re-formulated by Troshkin59 (see also60 and61) within the Navier-Stokes equation, by Saul62 within Boltzmann’s transport equation and in63 within Landau’s hydrodynamics. The same picture of the vacuum (or ether) as a turbulent fluid was Nelson’s64 starting point. In particular, the zero-viscosity limit gave him the motivation to expect that “the Brownian motion in the ether will not be smooth” and, therefore, to conceive the particular form of kinematics at the base of his stochastic derivation of the Schr¨odinger equation. A qualitatively similar picture is also obtained by representing relativistic particle propagation from the superposition, at short time scales, of non-relativistic particle paths with different Newtonian mass.65 In this formulation, particles randomly propagate (as in a Brownian motion) in an underlying granular medium which replaces the trivial empty vacuum.66

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Fig. 12. We compare two different signals. Panel a) reports the power spectrum S(ω) ∼ ω−1.5 of the wind turbulence measured at the Florence Airport.69 Panel b) shows the spectral amplitude
S(ω) ∼ ω−0.7 of the frequency shift measured by Nagel et al.70 Above some minimal frequency the two curves reach a flat plateau. This corresponds to the maximum integration time beyond which the signal ceases to behave as a pure white-noise.

wind turbulence measured at the Florence Airport.69 No doubt, this is a physical signal. Panel b) shows the spectral amplitude S(ω) ∼ ω−0.7 of the frequency shift measured by Nagel et al.70 Again, this latter signal is interpreted as spurious noise.

Clearly these are just analogies but, very often, physical understanding proceeds

by analogies. We have thus exploited the idea that the irregular signal observed in

ether-drift experiments has a fundamental stochastic nature as when turbulence, at

small scales, becomes statistically homogeneous and isotropic. With such an irregu-

lar signal numerical simulations are needed for a consistent description of the data.

Therefore, for a check, one should first extract from the data the (2nd-harmonic)

phase and amplitude and concentrate on the latter which is positive definite and re-

mains non-zero under any averaging procedure. When measured at different times,

this amplitude will anyhow exhibit modulations that, though indirectly, can provide

information on the underlying cosmic motion.

To put things on a quantitative basis, let us assume the set of kinematic param-

eters (V, α, γ)CMB for the Earth motion in the CMB, a latitude ϕ of the laboratory

and

a

given

sidereal

time

τ

=

ωsidt

of

the

observations

(with

ωsid

∼

2π 23h 56′

).

Then,

for short-time observations of a few days, where the only time dependence is due

to the Earth rotation, a simple application of spherical trigonometry71 gives the

projections in the (x, y) plane of the interferometer

v˜x(t) = v˜(t) cos θ˜2(t) = V [sin γ cos ϕ − cos γ sin ϕ cos(τ − α)]

(29)

v˜y(t) = v˜(t) sin θ˜2(t) = V cos γ sin(τ − α)

(30)

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with a magnitude

v˜(t) = V | sin z(t)|

(31)

and

cos z(t) = sin γ sin ϕ + cos γ cos ϕ cos(τ − α)

(32)

As for the signal, let us also re-write Eq.(19) as

∆c¯θ (t) c

∼

v2(t) ϵ c2

cos 2(θ

−

θ2(t))

(33)

where v(t) and θ2(t) now indicate respectively the magnitude and direction of the local drift in the same (x, y) plane of the interferometer. This can also be re-written

as

∆c¯θ(t) ∼ 2S(t) sin 2θ + 2C(t) cos 2θ

(34)

c

with

2C (t)

=

ϵ

vx2(t) − vy2(t) c2

2S(t)

=

ϵ

2vx (t)vy (t) c2

(35)

and vx(t) = v(t) cos θ2(t), vy(t) = v(t) sin θ2(t). In an analogy with a turbulent flow, the requirement of statistical isotropy means

that the local quantities vx(t) and vy(t), which determine the observable properties

of the drift, are very irregular functions that differ non trivially from their smooth,

global counterparts v˜x(t) and v˜y(t), and can only be simulated numerically. To this end, a representation in terms of random Fourier series55, 72, 73 was adopted in

refs.7–10 in a simplest uniform-probability model, where the kinematic parameters

of the global v˜µ(t) are just used to fix the boundaries for the local random vµ(t). The basic ingredients are summarized in the Appendix.

In this model, the functions S(t) and C(t) have the characteristic behaviour of a

white-noise signal with vanishing statistical averages ⟨C(t)⟩stat = 0 and ⟨S(t)⟩stat =

0 at any time t and whatever the global cosmic motion of the Earth. One can then

understand the observed irregular behaviour of the fringe shifts

∆λ(θ; t) 2D

= [2S(t) sin 2θ + 2C(t) cos 2θ]

(36)

λ

λ

In fact, their averages would be non vanishing just because the statistics is finite.

Otherwise with more and more observations one would find

∆λ(θ; t)

2D

⟨ λ ⟩stat = λ [2 sin 2θ ⟨S(t)⟩stat + 2 cos 2θ ⟨C(t)⟩stat] → 0

(37)

In particular, the direction θ2(t) of the local drift, defined by the relation

tan 2θ2(t) = S(t)/C(t), would vary randomly with no definite limit.

We have then checked the model by comparing with the amplitudes. Here we have first to consider the theoretical amplitude A˜2(t) associated with the global
motion

A˜2(t)

∼

D λ

·

2ϵ

·

V

2

sin2 c2

z(t)

(38)

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and then the amplitude A2(t) associated with the local non differentiable velocity components vx(t) and vy(t), Eqs.(83) and (84) of the Appendix

A2(t)

∼

D λ

·

2ϵ

·

vx2(t) + c2

vy2(t)

(39)

Clearly, the latter will exhibit sizeable fluctuations and be very different from the smooth A˜2(t). However, as shown in the Appendix, the relation between A˜2(t) and
the statistical average ⟨A2(t)⟩stat is extremely simple

D ⟨A2(t)⟩stat = λ

· 2ϵ ·

⟨vx2(t) + vy2(t)⟩stat c2

∼

π2 18

· A˜2(t)

(40)

so that, by averaging the amplitude at different sidereal times, one can obtain the crucial information on the angular parameters α and γ.
Altogether, the amplitudes of those old measurements can thus be interpreted in terms of three different velocities: a) as 6 ± 2 km/s in a classical picture b) as 250 ± 80 km/s, in a modern scheme and in a smooth picture of the drift c) as 340 ± 110 km/s, in a modern scheme but now allowing for irregular fluctuations of the signal. In fact, by replacing Eq.(38) with Eq.(40), from the same data, one would now obtain kinematical velocities which are larger by a factor 18/π2 ∼ 1.35. In this third interpretation, the range of velocity agrees much better with the CMB value of 370 km/s.
To illustrate the agreement of our scheme with all classical measurements, we address to our book9 where a detailed description is given of the experiments by Morley-Miller,34 Miller,35 Kennedy,36 Illingworth,37 Tomaschek38 and PiccardStahel.39 Instead, here, we will only consider the two most precise experiments that, traditionally, have been considered as definitely ruling out Miller’s claims for a non-zero ether drift. Namely the Michelson-Pease-Pearson (MPP) observations at Mt. Wilson and the experiment performed in 1930 by Joos in Jena.43 In particular, the latter remains incomparable among the classical experiments. To have an idea, Sommerfeld, being aware that the residuals in the Michelson-Morley data were not entirely negligible, concluded that only “After its repetition at Jena by Joos, the negative result of Michelson’s experiment can be considered as definitely established” (A. Sommerfeld, Optics). However, there is again a subtlety because, as we shall see, Joos’ experiment was not performed in the same conditions as the other experiments we have previously considered.

4.1. Reanalysis of the MPP experiment
To re-analyze the Michelson-Pease-Pearson (MPP) experiment, we first observe that no numerical results are reported in the original articles.40, 41 Instead, for more precise indications, one should look at Pease’s paper.42 There, one learns that they concentrated on a purely differential type of measurements. Namely, they were first statistically averaging the fringe shifts at those sidereal times that, according to

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Fig. 13. The histogram W of a numerical simulation of 10,000 instantaneous amplitudes for the single session of January 13, 1928, reported by Pease42 . The vertical normalization is to a unit area. We show the median and the 70% CL. The limits on the random Fourier components Eqs.(83) and (84) of the Appendix were fixed by inserting the CMB kinematical parameters in Eq.(87).

Miller, were corresponding to maxima and minima of the ether-drift effect. Then, they were forming the difference

δ(θ)

=

⟨

∆λ(θ; tmax) λ

⟩stat

−

⟨

∆λ(θ; tmin) λ

⟩stat

(41)

which are the only numbers reported by Pease. These δ−values have a maximal

magnitude of ±0.004 and this is also the order of magnitude of the experimen-
tal amplitude AE2 XP ∼ 0.005 that is usually reported74 for the MPP experiment when comparing with the much larger expected classical amplitudes Ac2lass ∼ 0.45 or Ac2lass ∼ 0.29 for optical paths of eighty-five or fifty-five feet respectively. Now,
our stochastic, isotropic model predicts exactly zero statistical averages for vector

quantities such as the fringe shifts, see Eq(37). Therefore, it would be trivial to

reproduce the small δ-values in Eq.(41) in a numerical simulation with sufficiently

high statistics. We have thus decided to compare instead with the only basic experimental session reported by Pease42 (for optical path of fifty-five feet) which indicates a 2nd-harmonic amplitude AE2 XP ∼ 0.006. By comparing with the classical prediction for 30 km/s, namely Ac2lass ∼ 0.29, this amplitude corresponds to an observable velocity vobs ∼ 4.3 km/s but to a much larger value on the basis of
Eq.(39).

Since we are dealing with a single measurement, to obtain a better understand-

ing of its probability content, we have performed a direct numerical simulation by

generating 10,000 values of the amplitude at the same sidereal time 5:30 of the MPP

Mt. Wilson observation. The CMB kinematical parameters were used to bound the

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random Fourier components of the stochastic velocity field Eqs.(83) and (84) of the
Appendix. The resulting histogram is reported in Fig.13. From this histogram one obtains a mean simulated amplitude ⟨As2imul⟩ ∼ 0.014. This corresponds to replace the value of the scalar velocity v˜(t) ∼ 370 km/s Eq.(77), at the sidereal time of
Pease’s observation, in the relation for the statistical average of the amplitude

2ϵD ⟨A2(t)⟩stat = λ

⟨vx2(t) + vy2(t)⟩stat c2

∼

(1.6 · 104) ·

π2 18

·

v˜2(t) c2

∼

v˜2(t) 0.009 · (300 km/s)2

(42)

In the above relation we have replaced D/λ ∼ 2.9 · 107 (for optical path of fifty-five

feet) and ϵ ∼ 2.8 · 10−4.

Note that the median of the amplitude distribution is about 0.007. As a conse-

quence, the value AE2 XP ∼ 0.006 lies well within the 70% Confidence Limit. Also, the probability content becomes large at very small amplitudes n and there is a long

tail extending up to about A2 ∼ 0.030. The wide interval of amplitudes corresponding to the 70% C. L. indicates that,

in our stochastic model, one could account for single observations that differ by an

order of magnitude, say from 0.003 to 0.030. Thus, beside the statistical vanishing of

vector quantities, this is another crucial difference with a purely deterministic model

of the ether-drift. In this traditional view, in fact, within the errors, the amplitude

can vary at most by a factor r = (v˜max/v˜min)2 where v˜max and v˜min are respectively the maximum and minimum of the projection of the Earth velocity Eq.(31). Since,

for the known types of cosmic motion, one finds r ∼ 2, the observation of such large

fluctuations in the data would induce to conclude, in a deterministic model, that

there is some systematic effect which affects the measurements in an uncontrolled

way. With an ether drift of such irregular nature, it then becomes understandable

the MPP reluctance to quote the individual results and instead report those partic-

ularly small combinations in Eq.(41) obtained by averaging and further subtracting

large samples of data. This general picture of a highly irregular phenomenon is also

confirmed by our reanalysis of Joos’ experiment in the following subsection.

4.2. Joos’ experiment We will only give a brief description of Joos’ 1930 experiment43 and address to our book9 for more details. Its sensitivity was about 1/3000 of a fringe, the fringes were recorded photographically with an automatic procedure and the optical system was enclosed in a hermetic housing. As reported by Miller,35, 75 it has been traditionally
nStrictly speaking, for a more precise description of the data, one should fold the histogram with a smearing function which takes into account the finite resolution ∆ of the apparatus. This smearing would force the curve to bend for A2 → 0 and tend to some limit which depends on ∆. Nevertheless, this refinement should not modify substantially the probability content around the median which is very close to A2 = 0.007.

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Instructions for typing manuscripts (paper’s title) 29
Fig. 14. The fringe shifts reported by Joos.43 The yardstick corresponds to 1/1000 of a wavelength.
believed that the measurements were performed in vacuum. In his article, however, Joos is not clear on this particular aspect. Only when describing his device for electromagnetic fine movements of the mirrors, he refers to the condition of an evacuated apparatus.43 Instead, Swenson76, 77 declares that Joos’ fringe shifts were finally recorded with optical paths placed in a helium bath. Therefore, we have decided to follow Swenson’s explicit statements and assumed the presence of gaseous helium at atmospheric pressure.
From Eq.(40), by replacing D/λ = 3.75 · 107 and the refractive index Nhelium ∼ 1.000033 for gaseous helium, an average daily projection of the cosmic Earth velocity v˜(t) = V | sin z(t)| ∼ 330 km/s (appropriate for a Central-Europe laboratory) would provide the same amplitude as classically expected for the much smaller observable velocity of 2 km/s. We can thus understand the substantial reduction of the fringe shifts observed by Joos, with respect to the other experiments in air.
The data were taken at steps of one hour during the sidereal day and two observations (1 and 5) were finally deleted by Joos with the motivations that there were spurious disturbances, see Fig.14. From this picture, Joos adopted 1/1000 of a wavelength as upper limit and deduced the bound vobs ≲ 1.5 km/s. To this end, he was comparing with the classical expectation that, for his apparatus, a velocity of 30 km/s should have produced a 2nd-harmonic amplitude of 0.375 wavelengths. Though, since it is apparent that some fringe displacements were certainly larger than 1/1000 of a wavelength, we have performed 2nd-harmonic fits to Joos’ data, see Fig.15. The resulting amplitudes are reported in Fig.16.
We note that a 2nd-harmonic fit to the large fringe shifts in picture 11 has a

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30 Authors’ Names
Fig. 15. Some 2nd-harmonic fits to Joos’ data. The figure is taken from ref.9
Fig. 16. Joos’ 2nd-harmonic amplitudes, in units 10−3. The vertical band between the two lines corresponds to the range (1.4 ± 0.8) · 10−3. The uncertainty of each value is about ±3 · 10−4. The figure is taken from ref.7
very good chi-square, comparable and often better than other observations with smaller values, see Fig.15. Therefore, there is no reason to delete the observation n.11. Its amplitude, however, (4.1 ± 0.3) · 10−3 is abot ten times larger than the average amplitude (0.4 ± 0.3) · 10−3 from the observations 20 and 21. This difference cannot be understood in a smooth model of the drift where, as anticipated, the projected velocity squared at the observation site can at most differ by a factor of two, as for the CMB motion at typical Central-Europe latitude where (v˜)min ∼ 250 km/s and (v˜)max ∼ 370 km/s. To understand these characteristic fluctuations, we have thus performed various numerical simulations of these amplitudes7, 9 in the

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stochastic model described in the Appendix and using the kinematical parameters (V, α, γ)CMB to place the limits on the random velocity component Eqs.(83) and (84). Two simulations are shown in Figs.17 and 18 (the corresponding numerical values are reported in7, 9).

Fig. 17. Joos’ 2nd-harmonic amplitudes, in units 10−3 (black dots), are compared with a single simulation (red diamonds) at the same sidereal times of Joos’ observations. Two 5th-order polynomial fits to the two sets of values are also shown. The figure is taken from ref.7 .

We want to emphasize two aspects. First, Joos’ average amplitude ⟨AE2 XP⟩ =

(1.4 ± 0.8) · 10−3 when compared with the classical prediction for his interferometer

Ac2lass

=

D (30km/s)2

λ

c2

∼

0.375 gives indeed an observable velocity vobs

∼

(1.8 ±

0.5) km/s very close to the 1.5 km/s value quoted by Joos. But, when comparing

with our prediction in the stochastic model Eq.(40) one would now find a true
kinematical velocity v˜ = 305+−81500 km/s. Second, when fitting with Eqs.(76) and (77) the smooth black curve of the Joos data in Fig.17 one finds7 a right ascension

α(fit − Joos) = (168±30) degrees and an angular declination γ(fit − Joos) = (−13±

14) degrees which are consistent with the present values α(CMB) ∼ 168 degrees

and γ(CMB) ∼ −7 degrees. This confirms that, when studied at different sidereal

times, the measured amplitude can provide precious information on the angular

parameters.

4.3. Summary of all classical experiments A comparison with all classical experiments is finally shown in Table 3.

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Fig. 18. Joos’ 2nd-harmonic amplitudes in units 10−3 (black dots) are now compared with a simulation where one averages ten measurements, performed on 10 consecutive days, at the same sidereal times of Joos’ observations (red diamonds). The change of the averages observed by varying the parameters of the simulation was summarized into a central value and a symmetric error. The figure is taken from ref.7
Note the substantial difference with the analogous summary Table I of ref.74 where those authors were comparing with the classical amplitudes Eq.(22) and emphasizing the much smaller magnitude of the experimental fringes. Here, is just the opposite. In fact, our theoretical statistical averages are often smaller than the experimental results indicating, most likely, the presence of systematic effects in the measurements.
At the same time, by adopting Eq.(40), we find v˜exp ∼ 418 ± 62 km/s from all experiments in air and v˜exp ∼ 323 ± 70 km/s from the two experiments in gaseous helium, with a global average ⟨v˜exp⟩ ∼ 376±46 km/s which agrees well with the 370 km/s from the CMB observations. Even more, from the two most precise experiments of Piccard-Stahel (Brussels and Mt.Rigi) and Joos (Jena), we find two determinations, v˜exp = 360+−81510 km/s and v˜exp = 305+−81500 km/s respectively, whose average ⟨v˜⟩ ∼ 332+−6800 km/s reproduces to high accuracy the projection of the CMB velocity at a typical Central-Europe latitude.

4.4. The intriguing role of temperature

As anticipated in Sect.2 (see footnote k), symmetry arguments can successfully

describe a phenomenon regardless of the physical mechanisms behind it. The same

is

true

here

with

our

relation

|∆c¯θ | c

∼

ϵ(v2/c2)

in

Eq.(19).

It

works

but

does

not

explain the ultimate origin of the small effects observed in the gaseous systems. For

instance, as a first mechanism, we considered the possibility of different polarizations

in different directions in the dielectric, depending on its state of motion. But, if this

works in weakly bound gaseous matter, the same mechanism should also work in

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Table 3. The average 2nd-harmonic amplitudes of classical ether-drift experiments. These were extracted from the original papers by averaging the amplitudes of the individual observations and assuming the direction of the local drift to be completely random (i.e. no vector averaging of different sessions). These experimental values are then compared with the full statistical average Eq.(40) for a projection 250 km/s ≲ v˜ ≲ 370 km/s of the Earth motion in the CMB and refractivities ϵ = 2.8 · 10−4 for air and ϵ = 3.3 · 10−5 for gaseous helium. The experimental value for the Morley-Miller experiment is taken from the observed velocities reported in Miller’s Figure 4, here our Fig.4. The experimental value for the Michelson-Pease– Pearson experiment refers to the only known session for which the fringe shifts are reported explicitly42 and where the optical path was still fifty-five feet. The symbol ±.... means that the experimental uncertainty cannot be determined from the available informations. The table is taken from ref.10

Experiment Michelson(1881) Michelson-Morley(1887) Morley-Miller(1902-1905) Miller(1921-1926) Tomaschek (1924) Kennedy(1926) Illingworth(1927) Piccard-Stahel(1928) Mich.-Pease-Pearson(1929)
Joos(1930)

gas air air air air air helium helium air air helium

AE2 XP (7.8 ± ....) · 10−3 (1.6 ± 0.6) · 10−2 (4.0 ± 2.0) · 10−2 (4.4 ± 2.2) · 10−2 (1.0 ± 0.6) · 10−2
< 0.002 (2.2 ± 1.7) · 10−4 (2.8 ± 1.5) · 10−3 (0.6 ± ...) · 10−2 (1.4 ± 0.8) · 10−3

2D λ
4 · 106 4 · 107 1.12 · 108 1.12 · 108 3 · 107 7 · 106 7 · 106 1.28 · 107 5.8 · 107 7.5 · 107

⟨A2 (t)⟩stat (0.7 ± 0.2) · 10−3 (0.7 ± 0.2) · 10−2 (2.0 ± 0.7) · 10−2 (2.0 ± 0.7) · 10−2 (0.5 ± 0.2) · 10−2 (1.4 ± 0.5) · 10−4 (1.4 ± 0.5) · 10−4 (2.2 ± 0.8) · 10−3 (1.0 ± 0.4) · 10−2 (1.5 ± 0.6) · 10−3

a strongly bound solid dielectric, where the refractivity is (Nsolid − 1) = O(1), and

thus

produce

a

much

larger

|∆c¯θ | c

∼

(Nsolid

− 1)(v2/c2)

∼

10−6.

This

is

in

contrast

with the Shamir-Fox78 experiment in perspex where the observed value was smaller

by orders of magnitude.

We have thus re-considered8, 9, 79 the traditional thermal interpretation74, 80 of

the observed residuals. The idea was that, in a weakly bound system as a gas, a

small temperature difference ∆T gas(θ) in the air of the two optical arms produces

a difference in the refractive index and a (∆c¯θ/c) ∼ ϵgas∆T gas(θ)/T , where T ∼ 300 K is the absolute temperature of the laboratory o. Miller was aware35, 75 that

his results could be due to a ∆T gas(θ) ≲ 1 mK but objected that casual changes

of the ambiance temperature would largely cancel when averaging over many mea-

surements. Only temperature effects with a definite periodicity would survive. For

a quantitative estimate, by averaging over all experiments in Table 3 we found

⟨v˜exp⟩ ∼ 376 ± 46 km/s. Therefore, by comparing Eq.(40) with the corresponding

oThe starting point is the Lorentz-Lorenz equation for the molecular polarizability in the ideal-gas approximation (as for air or gaseous helium at atmospheric pressure), see8, 9 for the details.

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form for a thermal light anisotropy, we find

|∆c¯θ | c

∼

ϵgas

π2 18

⟨v˜exp⟩2 c2

∼

|∆T gas(θ)|

ϵgas

T

(43)

and a value8, 9 |∆T gas(θ)| ∼ (0.26 ± 0.07) mK p.

This motivates the following two observations. First, after a century from those

old measurements, in a typical room-temperature laboratory environment, a stability at the level of a fraction of millikelvin is still state of the art, see.81–83 This

would support the idea that we are dealing with a non-local effect that places a

fundamental limit. Second, as for possible dynamical explanations, we mentioned in footnote k a

collective interaction of the gaseous system with hypothetical dark matter in the

Galaxy or with the CMB radiation. For the consistency with the velocity of 370

km/s, the latter hypothesis seems now more plausible. In this interpretation, these

interactions would be so weak that, on average, the induced temperature differences in the optical paths are only 1/10 of the whole ∆T CMB(θ) in Eq.(3).

Nevertheless, whatever its precise origin, this thermal explanation can help intuition. In fact, it can explain the quantitative reduction of the effect in the vacuum

limit where ϵgas → 0 and the qualitative difference with solid dielectric media where temperature differences of a fraction of millikelvin cannot produce any appreciable

deviation from isotropy in the rest frame of the medium.

Admittedly, the idea that small modifications of gaseous matter, produced by the

tiny CMB temperature variations, can be detected by precise optical measurements

in a laboratory, while certainly unconventional, has not the same implications of a genuine preferred-frame effect due to the vacuum structure. Still, this thermal explanation of the small residuals in gases, very recently reconsidered by Manley,84

has a crucial importance. In fact, it implies that if a tiny, but non-zero, fundamental

signal could definitely be detected in vacuum then, with very precise measurements,

the same universal signal should also show up in a solid dielectric where a disturbing

∆T of a fraction of millikelvin becomes irrelevant. Detecting such ‘non-thermal’ light

anisotropy, for the same cosmic motion indicated by the CMB observations, is thus necessary to confirm the idea of a fundamental preferred frame.

5. The modern ether-drift experiments

Searching for a ‘non-thermal’ light anisotropy, which could be detected with light

propagating in vacuum and/or in solid dielectrics, we will now compare with the

modern experiments44

where

∆c¯θ c

∼

∆ν(θ) ν0

is now extracted from the frequency shift

of two optical resonators, see Fig.3. The particular type of laser-cavity coupling used

in the experiments is known in the literature as the Pound-Drever-Hall system,85, 86

pNote that in Eq.(43) the gas refractivity drops out. The old estimates74, 80 of about 1 mK, based on the relation ϵgas∆T gas(θ)/T ∼ (vM 2 iller/2c2), with vMiller ∼ 10 km/s, were slightly too large.

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see Black’s tutorial article87 for a beautiful introduction. A laser beam is sent into

a Fabry-Perot cavity which acts as a filter. Then, a part of the output of the cavity

is fed back to the laser to suppress its frequency fluctuations. This method provides

a very narrow bandwidth and has been crucial for the precision measurements we

are going to describe.

The first application to the ether-drift experiments was realized by Brillet and

Hall in 1979.88 They were comparing the frequency of a CH4 reference laser (fixed in the laboratory) with the frequency of a cavity-stabilized He-Ne laser (ν0 ∼ 8.8 · 1013
Hz) placed on a rotating table. Since the stabilizing optical cavity was placed inside

a vacuum envelope, the measured shift ∆ν(θ) was giving a measure of the anisotropy

of the velocity of light in vacuum. The short-term stability of the cavity-laser system

was found to be about ± 20 Hz for a 1-second measurement, and comparable to the

stability of the reference CH4 laser. It was also necessary to correct the data for a substantial linear drift of about 50 Hz/s.

By grouping the data in blocks of 10-20 rotations they found a signal with a

typical amplitude |∆ν| ∼ 17 Hz (or a relative level 10−13) and with a phase θ2(t) which was randomly varying. Therefore, by increasing the statistics and projecting

along the axis corresponding to the Earth cosmic velocity obtained from the first

CMB observations,89 the surviving average effect was substantially reduced down

to about ±1 Hz. Finally, by further averaging over a period of about 200 days, the

residual ether-drift effect was an average frequency shift ⟨∆ν⟩ =0.13 ± 0.22 Hz, i.e.

about 100 times smaller than the instantaneous |∆ν|.

Since the 1979 Brillet-Hall article, substantial improvements have been intro-

duced in the experiments. Just to have an idea, in present-day measurements68, 90

with vacuum cavities the typical magnitude of the instantaneous fractional signal

|∆ν|/ν0 has been lowered from 10−13 to 10−15, the linear drift from 50 Hz/s to about 0.05 Hz/s and, after averaging over many observations, the best limit which

is

reported

is

a

residual

⟨

∆ν ν0

⟩

≲

10−18,68

i.e.

about

1000

times

smaller

than

the

instantaneous 10−15 signal.

The assumptions behind the analysis of the data, however, are basically un-

changed. In fact, a genuine ether drift is always assumed to be a regular phenomenon

depending deterministically on the Earth cosmic motion and averaging more and

more observations is considered a way of improving the accuracy. But, as empha-

sized in Sect.4, the classical experiments indicate genuine physical fluctuations that

are not spurious noise but, instead, express how the cosmic motion of the Earth is

actually seen in a detector. For this reason, we will first consider the instantaneous

signal and try to understand if it can admit a physical interpretation.

5.1. A 10−9 refractivity for the vacuum (on the Earth surface)
As anticipated, after averaging many observations, the best limit which is reported for measurements with vacuum resonators is a residual ⟨∆ν/ν0⟩ ≲ 10−18.68 This just reflects the very irregular nature of the signal because its typical magnitude

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36 Authors’ Names
|∆ν|/ν0 ∼ 10−15 is about 1000 times larger, see Fig.19 or panel b) of Fig.11.

Fig. 19. The experimental frequency shift reported in Fig.9(a) of ref.90 (courtesy Optics Com-
munications). The black dots give the instantaneous signal, the red dots give the signal averaged over 1640 sequences. For a laser frequency ν0 = 2.8 · 1014 Hz a ∆ν = ±1 Hz corresponds to a fractional value ∆ν/ν0 of about ±3.5 · 10−15.

The most interesting aspect however is that this 10−15 instantaneous signal,

found in the room-temperature experiments of refs.90 and,68 was also found in ref.91

where the solid parts of the vacuum resonators were made of different material and

even in ref.92 were the apparatus was operating in the cryogenic regime. Since

it is very unlike that spurious effects (e.g. thermal noise93) remain the same for

experiments operating in so different conditions, one can meaningfully explore the

possibility that such an irregular 10−15 signal admits a physical interpretation.

In the same model discussed for the classical experiments, we are then lead to

the concept of a refractive index Nv = 1 + ϵv for the vacuum or, more precisely, for the physical vacuum established in an optical cavity, as in Fig.10, when this

is placed on the Earth surface. The refractivity ϵv should be at the 10−9 level, in

order

to

give

|∆c¯θ | c

∼

ϵv

(v2/c2)

∼ 10−15 and thus would fit with the original idea

of94 where, for an apparatus placed on the Earth’s surface, a vacuum refractivity

ϵv ∼ (2GN M/c2R) ∼ 1.39·10−9 was considered, GN being the Newton constant and M and R the mass and radius of the Earth. Since this idea will sound unconventional

to many readers, we have first to recall the main motivations.

An effective refractivity for the physical vacuum becomes a natural idea when

adopting a different view of the curvature effects observed in a gravitational field. In

General Relativity these curvature effects are viewed as a fundamental modification

of Minkowski space-time. However, it is an experimental fact that many physical

systems for which, at a fundamental level, space-time is exactly flat are nevertheless

described by an effective curved metric in their hydrodynamic limit, i.e. at length

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Instructions for typing manuscripts (paper’s title) 37

scales much larger than the size of their elementary constituents. For this reason, several authors, see e.g.,95–97 have explored the idea that Einstein gravity might represent an emergent phenomenon and started to considered those gravity-analogs (moving fluids, condensed matter systems with a refractive index, Bose-Einstein condensates,...) which are known in flat space.
The main ingredient of this approach consists in the introduction of some background fields sk(x) in flat space expressing the deviations of the effective metric gµν(x) from the Minkowski tensor ηµν, i.e.

gµν (x) − ηµν = δgµν [sk(x)]

(44)

with δgµν[sk = 0] = 0. In this type of approach, to (partially) fill the conceptual gap with classical General Relativity, as in the original Yilmaz derivation,98 one could impose that Einstein’s equations for the metric become algebraic identities which follow directly from the equations of motion for the sk’s in flat space, after introducing a suitable stress tensor tµν (sk) q.
As an immediate consequence, suppose that the sk’s represent excitations of the physical vacuum which therefore vanish identically in the equilibrium state. Then, if curvature effects are only due to departures from the lowest-energy state, one could immediately understand97 why the huge condensation energy of the unperturbed vacuum plays no role and thus obtain an intuitive solution of the cosmologicalconstant problem found in connection with the vacuum energy r.
Here, in our context of the ether-drift experiments, we will limit ourselves to explore some phenomenological consequence of this picture. To this end, let us assume a zeroth-order model of gravity with a scalar field s0(x) which behaves as the Newtonian potential (at least on some coarse-grained scale and consistently with the experimental verifications of the 1/r law at the sub-millimeter level100). How could the effects of s0(x) be effectively re-absorbed into a curved metric structure? At a pure geometrical level and regardless of the detailed dynamical mechanisms, the standard basic ingredients would be: 1) space-time dependent modifications of the physical clocks and rods and 2) space-time dependent modifications of the velocity of light s.

qIn the simplest, original Yilmaz approach98 there is only one inducing-gravity field s0(x) which plays the role of the Newtonian potential. Introducing its stress tensor tµν (s0) = −∂µs0∂ν s0 + 1/2δνµ ∂αs0∂αs0, to match the Einstein tensor, produces differences from the Schwarzschild metric which are beyond the present experimental accuracy. rThis is probably the simplest way to follow Feynman’s indication: “The first thing we should understand is how to formulate gravity so that it doesn’t interact with the vacuum energy”.99 sThis point of view can be well represented by some citations. For instance, “It is possible, on
the one hand, to postulate that the velocity of light is a universal constant, to define natural
clocks and measuring rods as the standards by which space and time are to be judged and then
to discover from measurement that space-time is really non-Euclidean. Alternatively, one can
define space as Euclidean and time as the same everywhere, and discover (from exactly the same

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38 Authors’ Names

Within this interpretation, one could thus try to check the fundamental assumption of General Relativity that, even in the presence of gravity, the velocity of light in vacuum cγ is a universal constant, namely it remains the same, basic parameter c entering Lorentz transformations. Notice that, here, we are not considering the so called coordinate-dependent speed of light. Rather, our interest is focused on the value of the true, physical cγ as, for instance, obtained from experimental measurements in vacuum optical cavities placed on the Earth surface.
To spell out the various aspects, a good reference is Cook’s article “Physical time and physical space in general relativity”.103 This article makes extremely clear which definitions of time and length, respectively dT and dL, are needed if all observers have to measure the same, universal speed of light (“Einstein postulate”). For a static metric, these definitions are dT 2 = g00dt2 and dL2 = gijdxidxj. Thus, in General Relativity, the condition ds2 = 0, which governs the propagation of light, can be expressed formally as

ds2 = c2dT 2 − dL2 = 0

(45)

and, by construction, yields the same universal speed c = dL/dT .
For the same reason, however, if the physical units of time and space were instead defined to be dTˆ and dLˆ with, say, dT = q dTˆ and dL = p dLˆ, the same condition

ds2 = c2q2dTˆ2 − p2dLˆ2 = 0

(46)

would now be interpreted in terms of the different speed

dLˆ q c

cγ = dTˆ = c p ≡ Nv

(47)

The possibility of different standards for space-time measurements is thus a simple

motivation for an effective vacuum refractive index Nv ̸= 1. With these premises, the unambiguous point of view of Special Relativity is that
the right space-time units are those for which the speed of light in the vacuum cγ, when measured in an inertial frame, coincides with the basic parameter c entering

Lorentz transformations. However, inertial frames are just an idealization. Therefore

the appropriate realization is to assume local standards of distance and time such

that the identification cγ = c holds as an asymptotic relation in the physical conditions which are as close as possible to an inertial frame, i.e. in a freely falling frame

(at least by restricting light propagation to a space-time region small enough that

measurements) how the velocity of light and natural clocks, rods and particle inertias really behave in the neighborhood of large masses”.101 Or “Is space-time really curved? Isn’t it conceivable that space-time is actually flat, but clocks and rulers with which we measure it, and which we regard as perfect, are actually rubbery? Might not even the most perfect of clocks slow down or speed up and the most perfect of rulers shrink or expand, as we move them from point to point and change their orientations? Would not such distortions of our clocks and rulers make a truly flat space-time appear to be curved? Yes.”102

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Instructions for typing manuscripts (paper’s title) 39

Fig. 20. A pictorial representation of the effect of a heavy mass M carried on board of a freelyfalling system, case (b). With respect to the ideal case (a), the mass M modifies the local space-time units and could introduce a vacuum refractivity so that now cγ ̸= c.

tidal effects of the external gravitational potential Uext(x) can be ignored). Note that this is essential to obtain an operational definition of the otherwise unknown parameter c.
As already discussed in ref.,94 light propagation for an observer S sitting on the Earth’s surface can then be described with increasing degrees of accuracy starting from step i), through ii) and finally arriving to iii):
i) S is considered a freely falling frame. This amounts to assume cγ = c so that, given two events which, in terms of the local space-time units of S, differ by (dx, dy, dz, dt), light propagation is described by the condition (ff=’free-fall’)

(ds2)ff = c2dt2 − (dx2 + dy2 + dz2) = 0

(48)

ii) To a closer look, however, an observer S placed on the Earth surface can only be considered a freely-falling frame up to the presence of the Earth gravitational field. Its inclusion can be estimated by considering S as a freely-falling frame, in the same external gravitational field described by Uext(x), that however is also carrying on board a heavy object of mass M (the Earth mass itself) which affects the local space-time structure, see Fig.20. To derive the required correction, let us denote by δU the extra Newtonian potential produced by the heavy mass M at the experimental set up where one wants to describe light propagation. According to General Relativity, and to first order in δU , light propagation for the S observer is

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40 Authors’ Names

now described by

ds2

=

c2dt2(1

−

2

|δU | c2 )

−

(dx2

+

dy2

+

dz2)(1

+

2

|δU | c2 )

≡

c2dT

2

−

dL2

=

0

(49)

where

dT 2

=

(1

−

2

|δU c2

|

)dt2

and

dL2

=

(1

+

2

|δU c2

|

)(dx2

+

dy2

+

dz2)

are

the

phys-

ical units of General Relativity in terms of which one obtains the universal value

dL/dT = cγ = c.

Though, to check experimentally the assumed identity cγ = c one should compare

with a theoretical prediction for (c − cγ) and thus necessarily modify some formal

ingredient of General Relativity. As a definite possibility, let us maintain the same definition of the unit of length dLˆ = dL but change the unit of time from dT to dTˆ.

The reason derives from the observation that physical units of time scale as inverse

frequencies and that the measured frequencies ωˆ for δU ̸= 0, when compared to the

corresponding value ω for δU = 0, are red-shifted according to

|δU |

ωˆ = (1 − c2 ) ω

(50)

Therefore,

rather

than

the

natural

unit

of

time

dT

=

(1 −

|δU c2

|

)dt

of

General

Rela-

tivity, one could consider the alternative, but natural (see our footnote s), unit of

time

dTˆ

=

(1

+

|δU | c2 )

dt

(51)

Then, to reproduce ds2 = 0, we can introduce a vacuum refractive index

|δU |

Nv ∼ 1 + 2 c2

(52)

so that the same Eq.(49) takes now the form

ds2

=

c2dTˆ2 Nv2

−

dLˆ2

=

0

(53)

This

gives dLˆ/dTˆ = cγ

=

c Nv

and,

for

an

observer

placed

on

the

Earth’s surface,

a

refractivity

ϵv

= Nv − 1 ∼

2GN M c2R

∼ 1.39 · 10−9

(54)

M and R being respectively the Earth mass and radius. Notice that, with this natural definition dTˆ, the vacuum refractive index asso-
ciated with a Newtonian potential is the same usually reported in the literature, at least since Eddington’s 1920 book,104 to explain in flat space the observed deflec-
tion of light in a gravitational field. The same expression is also suggested by the
formal analogy of Maxwell equations in General Relativity with the electrodynamics of a macroscopic medium with dielectric function and magnetic permeability105

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Instructions for typing manuscripts (paper’s title) 41

ϵik

=

µik

=

√ −g

. (−gik )
g00

Indeed,

in

our

case,

from

the

relations

gilglk

=

δki

,

(−gik) ∼ δki g00 , ϵik = µik = δki Nv , we obtain

√ Nv ∼ −g ∼

(1

−

2

|δU | c2 )(1

+

2

|δU c2

|

)3

∼

1

+

|δU | 2 c2

(55)

A difference is found with Landau’s and Lifshitz’ textbook107 where the vacuum

refractive index entering the constitutive relations is instead defined as Nv ∼ √g100 ∼

1

+

|δU c2

|

.

Concerning,

these

two

possible

definitions

of

Nv ,

we

address

the

reader

to Broekaert’s article,106 see his footnote 3, where a very complete set of references

for the vacuum refractive index in gravitational field is reported. However, this

difference of a factor of 2 is not really essential and can be taken into account as a

theoretical uncertainty. The main point is that cγ, as measured in a vacuum cavity on the Earth’s surface (panel (b) in our Fig.20), could differ at a fractional level

10−9 from the ideal value c, as operationally defined with the same apparatus in a

true freely-falling frame (panel (a) in our Fig.20). In conclusion, this cγ −c difference can be conveniently expressed through a vacuum refractivity of the form

ϵv

=

Nv

−1

∼

χ 2

1.39 · 10−9

(56)

where the factor χ/2 (with χ= 1 or 2) takes into account the mentioned theoretical uncertainty.

iii) Could one check experimentally if Nv ̸= 1? Today, the speed of light in

vacuum is assumed to be a fixed number with no error, namely 299 792 458 m/s.

Thus if, for instance, this estimate were taken to represent the value measured on the

Earth surface, in an ideal freely-falling frame there could be a slight increase, namely

+

χ 2

(0.42)

m/s

with

χ

=

1

or

2.

It

seems

hopeless

to

measure

unambiguously

such

a difference because the uncertainty of the last precision measurements performed

before the ‘exactness’ assumption had precisely this order of magnitude, namely

±4 · 10−9 at the 3-sigma level or, equivalently, ±1.2 m/s.108

Therefore, as pointed out in ref.,94 an experimental test cannot be obtained from

the value of the isotropic speed in vacuum but, rather, from its possible anisotropy.

In fact, with a preferred frame and for Nv ̸= 1, an isotropic propagation as in

Eq.(53) would only be valid for a special state of motion of the Earth laboratory.

This provides the definition of Σ while for a non-zero relative velocity there would

be off diagonal elements g0i ̸= 0 in the effective metric.105 If Σ exists, we would then

expect

a

light

anisotropy

|∆c¯θ | c

∼

ϵv (v/c)2

∼

10−15,

consistently

with

the

presently

measured value.

5.2. Some important technical aspects
Before considering the experiments, however, a rather technical discussion is necessary for an in-depth comparison with the data. In the mentioned cryogenic experi-

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42 Authors’ Names
Fig. 21. The stability of the fractional frequency shift of ref.92 at different integration times. The upper solid curve, denoted as ‘newCORE’, reports the actual measurements with the cryogenic apparatus in 2013. The lower solid, dashed and dot-dashed curves, denoted as ‘predicted newCORE’, indicate future stability limits (2 ÷ 4) · 10−17 that could be foreseen at that time.
ment of ref.,92 the instantaneous signal is not shown explicitly. However, its magnitude can be deduced from its typical variation observed over a characteristic time of 1÷2 seconds, see Fig.21. For a very irregular signal, in fact, this typical variation, of about 10−15, gives the magnitude of the instantaneous signal itself and, indeed, it is in good agreement with the mentioned room-temperature measurements.
Fig. 22. The typical trend of the RAV for a signal in various regimes. The minimum of the white-noise trend τ −0.5 defines the value τ = τ¯ indicated by the arrow.
The quantity which is reported in Fig.21 is the Root Square of the Allan Variance (RAV) of the fractional frequency shift. In general, the RAV describes the variation

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Instructions for typing manuscripts (paper’s title) 43

obtained by sampling a function f = f (t) at steps of time τ . By defining

1 ti+τ

f (ti; τ ) = τ ti dt f (t) ≡ f i

(57)

one generates a τ −dependent distribution of f i values. In a large time interval Λ = M τ , the RAV is then defined as

RAV (f, τ ) = RAV 2(f, τ )

(58)

where

RAV 2(f, τ ) =

1

M −1

2(M − 1)

f i − f i+1

2

i=1

(59)

and the factor of 2 is introduced to obtain the standard variance σ(f ) for uncorrelated data with zero mean, as for a pure white-noise signal.
Note that the actual measurements in Fig.21 are indicated by the upper solid curve denoted as ‘newCORE’. These were obtained with the cryogenic apparatus in 2013 (CORE=Cryogenic Optical REsonators) and were giving a stability at the level of about 1.2 · 10−15. The lower solid, dashed and dot-dashed curves, denoted as ‘predicted newCORE’, indicate instead possible improved limits (2 ÷ 4) · 10−17 that could be foreseen at that time. As a matter of fact, these limits have not yet been achieved because the highest stability limits are still larger by an order of magnitude. This persistent signal, which is crucial for our work, does not depend on the absolute temperature and/or the characteristics of the optical cavities.109
After this preliminaries, we then arrive at our main point. As anticipated, numerical simulations in our stochastic model indicate that our basic signal has the same characteristics as a universal white noise. This means that it should be compared with the frequency shift of two optical resonators at the largest integration time τ¯ where the pure white-noise component is as small as possible but other disturbances, that can affect the measurements, are not yet important, see Fig.22. In the experiments we are presently considering this τ¯ is typically 1 ÷ 2 seconds so that one gets the relation with the average magnitude of the instantaneous signal

RAV (∆ν, τ¯) ∼ σ(∆ν) ∼ ⟨|∆ν|⟩stat

(60)

5.3. Comparing our model with experiments in vacuum
We will now compare with the type of signal observed in68, 90 in vacuum at room temperature. To this end, we will use the relation which connects the frequency shift between two orthogonal resonators ∆ν(θ; t) = ν1(θ; t) − ν2(θ + π/2; t) to the angular dependence of the velocity of light, namely see (33)

∆ν(θ; t) = ∆c¯θ(t) = 2S(t) sin 2θ + 2C(t) cos 2θ

(61)

ν0

c

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44 Authors’ Names
where S(t) and C(t) are given in Eqs.(34). As in the case of the classical experiments, the velocity components vx(t) and vy(t) will be expressed as random Fourier series through the Eqs.(83) and (84) of the Appendix. A simulation of two short-time sequences of 2C(t) and 2S(t) is shown in Fig.23.

Fig. 23. For ϵv as in Eq.(56) and χ = 2, we report a simulation of two sequence of 45 seconds for the functions 2C(t) and 2S(t) Eqs.(34). Units are 10−15 and the two sets belong to the same random sequence for two sidereal times that differ by 6 hours. The boundaries of the stochastic velocity components, Eqs.(83) and (84) of the Appendix, are controlled by (V, α, γ)CMB through Eqs.(77) and (87). For a laser frequency of 2.8 · 1014 Hz, the range ±3.5 · 10−15 corresponds to a typical frequency shift ∆ν in the range ±1 Hz, as in our Fig.19.

For a quantitative test, we concentrated on the observed value of the RAV of the frequency shift at the end point τ¯ = 1 ÷ 2 seconds of the white-noise branch of the spectrum, see Fig.3, bottom part of68 . This has a value

[RAV (∆ν, τ¯)]exp = (0.20 ÷ 0.24) Hz

(62)

or, in units of the reference frequency ν0 = 2.8 · 1014 Hz 68

∆ν RAV ( , τ¯)

= (7.8 ± 0.7) · 10−16 Vacuum − room temperature (63)

ν0

exp

As anticipated, our instantaneous, stochastic signal for ∆ν(t) is, to very good

approximation, a pure white noise for which the RAV coincides with the standard

variance. At the same time, for a very irregular signal with zero mean of the type

shown in Fig. 23, but whose magnitude can have a long-term time dependence, one

should replace in Eq. (60) ⟨|∆ν|⟩stat → ⟨|∆ν(t)|⟩stat and evaluate the RAV in the

corresponding temporal range. Therefore, from

∆ν ν0

=

∆c¯θ c

∼

ϵv

·

v2 c2

,

we

arrive

at

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Instructions for typing manuscripts (paper’s title) 45

our prediction

∆ν(t)

RAV (

, τ¯)

ν0

theor

∼ ϵv

⟨vx2(t) + vy2(t)⟩stat c2

∼

π2 18

·

ϵv

·

V2 c2

sin2

z(t)

(64)

Then, by using Eq.(56), for the projection v˜(t) = V | sin z(t)| = 250 ÷ 370 Km/s

used for the classical experiments, our prediction for the RAV can be expressed as

∆ν RAV ( , τ¯)

∼ χ · (8.5 ± 3.5) · 10−16

(65)

ν0

theor

2

with χ = 1 or 2. By comparing with the experimental Eq.(63), the data favour

χ = 2, which is the only free parameter of our scheme. Also, the good agreement

with our theoretical value indicates that, at the end point of the white-noise part

of the signal, the corrections to our simplest model should be small.

Notice, however, that the range in Eq.(65) is not a theoretical uncertainty but reflects the daily variations of V 2 sin2 z(t) in Eq.(64). This means that, depending

on the sidereal time, the measurements of the RAV at the white-noise end point

τ = τ¯ should exhibit definite daily variations in the range (for χ = 2)

5 · 10−16 ≲

∆ν RAV ( , τ¯)
ν0

≲
theor

12 · 10−16

(66)

Thus it becomes crucial to understand whether these variations can be observed.

5.4. Comparing our model with experiments in solids

To consider modern experiments in solid dielectrics, we will compare with the very precise work of ref.70 This is a cryogenic experiment, with microwaves of 12.97 GHz,

where almost all electromagnetic energy propagates in a medium, sapphire, with re-

fractive index of about 3 (at microwave frequencies). As anticipated, with a thermal

interpretation of the residuals in gaseous media, we expect that the fundamental 10−15 vacuum signal considered above, with very precise measurements, should also

become visible here. In particular, the large refractivity of the solid Nsolid − 1 = O(1) should play no role.
Following refs.,8–10 we first observe that for Nv = 1 + ϵv there is a very tiny difference between the refractive index defined relatively to the ideal vacuum value

c and the refractive index relatively to the physical isotropic vacuum value c/Nv measured on the Earth surface. The relative difference between these two definitions is proportional to ϵv ≲ 10−9 and, for all practical purposes, can be ignored. All materials would now exhibit, however, the same background vacuum anisotropy. To

this end, let us replace the average isotropic value

c

c

→

Nsolid

Nv Nsolid

(67)

and then use Eq.(18) to replace Nv in the denominator with its θ−dependent value

N¯v(θ) ∼ 1 + ϵvβ2(1 + cos2 θ)

(68)

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46 Authors’ Names

This is equivalent to define a θ−dependent refractive index for the solid dielectric

N¯solid(θ) Nsolid

∼

1

+

ϵv β 2 (1

+

cos2

θ)

(69)

so that

c

c

[c¯γ (θ)]solid = N¯solid(θ) ∼ Nsolid

1 − ϵvβ2(1 + cos2 θ)

(70)

with an anisotropy

[∆c¯θ ]solid [c/Nsolid]

∼

ϵvβ2 cos 2θ

∼

10−15

(71)

In this way, a genuine 10−15 vacuum effect, if there, could also be detected in a solid dielectric thus implying the same prediction Eq.(65).

Fig. 24. We report two typical sets of 2000 seconds for our basic white-noise (WN) signal and its colored version obtained by Fourier transforming the spectral amplitude of ref.70 The boundaries of the random velocity components Eqs.(83) and (84) were defined by Eq.(87) by plugging in Eq.(77) the CMB kinematical parameters, for a sidereal time t = 4000 − 6000 seconds and for the latitude of Berlin-Duesseldorf, see the Appendix. The figure is taken from ref.10
In ref.,10 a detailed comparison with70 was performed. First, from Figure 3(c) of,70 see also panel b) of our Fig.12, it is seen that the spectral amplitude of this particular apparatus becomes flat at frequencies ω ≥ 0.5 Hz indicating that the end-point of the white-noise branch of the signal is at an integration time τ¯ ∼ 1 ÷ 2 seconds. The data for the spectral amplitude were then fitted to an analytic, powerlaw form to describe the lower-frequency part 0.001 Hz ≤ ω ≤ 0.5 Hz which reflects apparatus-dependent disturbances. This fitted spectrum was then used to generate a signal by Fourier transform. Finally, very long sequences of this signal were stored to produce “colored” version of our basic white-noise signal.

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Instructions for typing manuscripts (paper’s title) 47
To get a qualitative impression of the effect, we report in Fig.24 a sequence of our basic simulated white-noise signal and a sequence of its colored version. By averaging over many 2000-second sequences of this type, the corresponding RAV’s for the two simulated signals are then reported in Fig.25. The experimental RAV extracted from Figure 3(b) of ref.70 is also reported (for the non-rotating setup). At this stage, the agreement of our simulated, colored signal with the experimental data remains satisfactory only up τ = 50 seconds. Reproducing the signal at larger τ ’s would have required further efforts but this is not relevant, our scope being just to understand the modifications of our stochastic signal near the endpoint of the white-noise spectrum.

Fig. 25. We report the RAV for the fractional frequency shift obtained from many simulations of sequences of 2000 seconds for our basic white-noise (WN) signal (decreasing as τ −0.5) and for its colored version, see Fig.24. The direct experimental results of ref.,70 for the non-rotating setup, are also shown as red dots. The figure is taken from ref.10

As one can check from Fig.3(b) of ref.,70 see also the red dots in our Fig.25, the experimental RAV for the fractional frequency shift, at the white-noise end point τ¯ ∼ 1 ÷ 2 second, is in the range (6.8 ÷ 8.6) · 10−16, say 70

∆ν RAV ( , τ¯)

= (7.7 ± 0.9) · 10−16 Solid − cryogenic

(72)

ν0

exp

As such, it coincides with Eq.(63) that we extracted from ref.68 after normalizing
their experimental result RAV (∆ν, τ¯)exp = 0.20 ÷ 0.24 Hz to their laser frequency ν0 = 2.8 · 1014 Hz. At the same time, it is well consistent with our theoretical prediction Eq.(65) for χ = 2. Therefore this beautiful agreement, between ref.68

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48 Authors’ Names
(a vacuum experiment at room temperature) and ref.70 (a cryogenic experiment in a solid dielectric), on the one hand, and with our theoretical prediction Eq.(65), on the other hand, confirms our interpretation of the data in terms of a stochastic signal associated with the Earth cosmic motion within the CMB and determined by the vacuum refractivity ϵv Eq.(56), for χ = 2.
Two ultimate experimental checks still remain. First, as anticipated, one should try to detect our predicted, daily variations Eq.(66). Due to the excellent systematics, these variations should remain visible with both experimental setups. Second, one more complementary test should be performed by placing the vacuum (or solid dielectric) optical cavities on board of a satellite, as in the OPTIS proposal.110 In this ideal free-fall environment, as in panel (a) of our Fig.20, the typical instantaneous frequency shift should be much smaller (by orders of magnitude) than the corresponding 10−15 value measured with the same interferometers on the Earth surface.
6. Summary and outlook
In this paper, we started from the present, basic ambiguity concerning the version of relativity which is physically realized in nature, namely Einstein Special Relativity vs. a Lorentzian formulation with a preferred reference frame Σ. This ambiguity is usually presented by a two-step argument. First, the basic quantitative ingredients, namely Lorentz transformations, are the same in both formulations. Second, even in a Lorentzian formulation, Michelson-Morley experiments can only produce null results. Therefore, rather than introducing an experimentally unobservable and logically superfluous entity, it seemed more satisfactory to adopt the point of view of Special Relativity where those effects (length contraction and time dilation), that were at the base of the original Lorentzian formulation, so to speak, become part of the kinematics. In this way, relativity becomes axiomatic and extendable beyond the original domain of the electromagnetic phenomena. This wider perspective has been the main reason for the traditional supremacy given to Einstein’s view.
However, discarding all historical aspects, it was emphasized by Bell that a change of perspective, from Special Relativity to a Lorentzian formulation, could be crucial to reconcile hypothetical faster-than-light signals with causality, as with the apparent non-local aspects of the Quantum Theory. In addition, the present view of the lowest-energy state as a Bose condensate of elementary quanta (Higgs particles, quark-antiquark pairs, gluons...), indicates a vacuum structure with some degree of substantiality which could characterize non trivially the form of relativity which is physically realized in nature. So, there may be good reasons for a preferred reference frame but, without the possibility of detecting experimentally an ‘ether wind’ in laboratory, the difference between the two formulations remains a philosophical problem.
This impossibility-in-principle, however, is somewhat mysterious. While it is certainly true that evidence for both the undulatory and corpuscular aspects of radia-

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Instructions for typing manuscripts (paper’s title) 49

tion has substantially modified the consideration of an underlying ethereal medium,

yet, if Σ exists, only a ‘conspiracy’ of relativistic effects would make undetectable

our motion with respect to it. But this conspiracy works exactly if the velocity of

light cγ propagating in the various interferometers, or more precisely its two-way

combination c¯γ, coincides with the basic parameter c entering Lorentz transforma-

tions. Therefore if c¯γ ̸= c, as for instance in the presence of matter, where light gets

absorbed and then re-emitted, nothing would really prevent an angular dependence

∆c¯θ = c¯γ(π/2 + θ) − c¯γ(θ) ̸= 0. If an angular dependence can be detected, and cor-

related with the cosmic motion of the Earth, the long sought Σ tight to the CMB

could finally emerge.

We have thus recalled the two key points of our extensive work. First, one

should impose that all measurable effects vanish exactly in the c¯γ → c limit, i.e.

in the ideal vacuum limit of a refractive index N = 1. Instead, in the infinitesimal

region

N

=

1+ϵ

simple

symmetry

arguments

lead

to

the

relation

|∆c¯θ | c

∼

ϵ(v2/c2).

For a typical cosmic v ∼ 300 km/s and ϵ = 2.8 · 10−4 , for air, or ϵ = 3.3 · 10−5, for

gaseous helium, this reproduces the order of magnitude of the effects observed in

the classical experiments.

The other peculiar aspect of our analysis concerns the observed, irregular charac-

ter of the data that, giving often substantially different directions of the drift at the

same hour on consecutive days, were contradicting the traditional expectation of a

regular phenomenon completely determined by the cosmic motion of the Earth. As

we have emphasized, here again, there may be a logical gap. The relation between

the macroscopic motion of the Earth and the microscopic propagation of light in a

laboratory depends on a complicated chain of effects and, ultimately, on the physi-

cal nature of the vacuum. By comparing with the motion of a body in a fluid, the

standard view corresponds to a form of regular, laminar flow where the projection

v˜µ(t) of the global, cosmic velocity, at the site of the experiment, coincides with the

local vµ(t) that determines the signal in the plane of the interferometer. Instead,

some general arguments and some experimental analogies suggest that the physical

vacuum might rather resemble a turbulent fluid where large-scale and small-scale

flows are only related indirectly. In this different perspective, with forms of tur-

bulence that, as in most models, become statistically isotropic at small scales, the

local vµ(t) would fluctuate randomly within boundaries fixed by the global v˜µ(t)

(see the Appendix). Therefore, one should analyze the data in phase and amplitude

(giving respectively the instantaneous direction and magnitude of the drift) and

concentrate on the latter which is positive definite and remains non-zero under any

averaging procedure. In this way, by restricting to the amplitudes, experiments al-

ways believed in contradiction with each other, as Miller vs. Piccard-Stahel, become

consistent, see Fig.9. Most notably, by adopting the parameters (V, α, γ)CMB to fix

the boundaries of the local random vµ(t) in our stochastic model, one finds a good

description of the irregular behaviour of the amplitudes extracted from Joos’ very

precise observations (see Figs.17 and 18). Viceversa, by fitting Joos’ amplitudes

with Eqs.(76) and (77), one finds a right ascension α(fit − Joos) = (168 ± 30) de-

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50 Authors’ Names

grees and an angular declination γ(fit − Joos) = (−13 ± 14) degrees which are well

consistent with the present values α(CMB) ∼ 168 degrees and γ(CMB) ∼ −7 de-

grees. The summary of all classical experiments given in Table 3 shows the complete

consistency with our theoretical predictions.

To conclude our analysis of the classical experiments in gaseous systems, we

have

emphasized

that

our

basic

relation

|∆c¯θ | c

∼

ϵgas(v2/c2)

derives

from

general,

symmetry arguments and does not explain the ultimate origin of the tiny observed

residuals. Due to the consistency with the velocity of 370 km/s, a plausible explana-

tion consists in a collective interaction of gaseous matter with the CMB radiation.

This could bring the gas out of equilibrium as if there were an effective temperature

difference, |∆T gas(θ)| = 0.2 ÷ 0.3 mK, in the gas along the two optical paths. This

magnitude is slightly smaller than the value of about 1 mK considered by Joos

and Shankland and, being just a small fraction of the whole ∆T CMB(θ) = ±3.3

mK in Eq.(3), indicates the weakness of the collective gas-CMB interactions. Most

notably, the thermal interpretation leads to an important prediction. In fact, it

implies that if a physical signal could definitely be detected in vacuum then, with

very precise measurements, the same signal should also show up in a solid dielectric

where disturbing temperature differences of a fraction of millikelvin become irrele-

vant. Detecting such ‘non-thermal’ light anisotropy, through the combined analysis

of the modern experiments in vacuum and in solid dielectrics, for the same cosmic

motion indicated by the classical experiments, is thus necessary to confirm the idea

of a fundamental preferred frame.

Despite the much higher precision of modern experiments, the assumptions be-

hind the analysis of the data are basically the same as in the classical experiments.

A genuine signal is assumed to be a regular phenomenon, depending deterministi-

cally on the Earth cosmic motion, so that averaging more and more observations is

considered a way of improving the accuracy. But the classical experiments indicate

genuine physical fluctuations which are not spurious noise and, instead, express how

the cosmic motion of the Earth is actually seen in a detector. Therefore, the present

quoted

average,

namely

⟨∆c¯θ ⟩ c

≲

10−18,

could

just

reflect

the

very

irregular

nature

of

the

signal.

Indeed,

its

typical

instantaneous

magnitude

in

vacuum

|∆c¯θ | c

∼

10−15

is about 1000 times larger, see Fig.19 or panel b) of Fig.11.

To understand if this vacuum signal can admit a physical interpretation, a crucial

observation is that the same 10−15 magnitude is found in measurements where the

resonators are made of different materials, in measurements at room-temperature

and also in the cryogenic regime. Since it is very unlike that spurious effects remain

the same in so different conditions, in the same model used for the classical experi-

ments we are driven to the idea of a refractive index Nv = 1 + ϵv for the vacuum or,

more precisely, for the physical vacuum established in an optical cavity placed on

the Earth surface. The refractivity ϵv should be at the 10−9 level, in order to give

|∆c¯θ | c

∼

ϵv

(v2/c2)

∼ 10−15 and thus would fit with the original idea of ref.94 The

motivation was that, if Einstein’s gravity is a phenomenon which emerges, at some

February 21, 2024 2:5 WSPC/INSTRUCTION FILE Consoli˙review˙arxiv

Instructions for typing manuscripts (paper’s title) 51

small length scale, from a fundamentally flat space, for an apparatus placed on the

Earth surface (which is in free fall with respect to all masses in the Universe but not

with respect to the Earth, see Fig.20) there should be a tiny vacuum refractivity

ϵv ∼ (2GN M/c2R) ∼ 1.39 · 10−9, where GN is the Newton constant and M and R

are the mass and radius of the Earth. This is the same type of refractivity consid-

ered by Eddington, or much more recently by Broekaert, to explain in flat space the

deflection of light in a gravitational field. Therefore Michelson-Morley experiments,

by

detecting

a

light

anisotropy

|∆c¯θ | c

∼

ϵv

(v2/c2)

∼ 10−15, can also resolve this

other ambiguity.

With this identification of ϵv, we first compared qualitatively the observed signal,

in Fig.19 or in panel b) of Fig.11, with simulations in our stochastic model, see

Figs.23 and 24. For a more quantitative analysis, we then considered the value of a

particular statistical indicator which is used nowadays, namely the Allan Variance

of

the

fractional

frequency

shift

RAV

(

∆ν ν0

,

τ

)

as

function

of

the

integration

time

τ . Since the irregular signal of our stochastic model has the characteristics of a

universal white noise and should represent an irreducible component, we have thus

compared with the RAV measured at the end point of the white-noise branch of

the spectrum. This is defined as the largest integration time τ¯ where the white-

noise component is as small as possible but other spurious disturbances, that can

affect the measurements, are not yet important, see Fig.22. In this way, for the same

velocity range v˜ = 250÷370 km/s used for the classical experiments, our theoretical

prediction Eq.(65) (for χ = 2) is in very good agreement with the results of the most

precise experiment in vacuum Eq.(63).

But, then, the second crucial test. As anticipated, if this 10−15 signal observed

in vacuum has a real physical meaning, the same effect should also be detected with

a very precise experiment in a solid dielectric, see Eq.(71). This expectation is con-

firmed by the extraordinary agreement between Eq.(72) and Eq.(63). Note that the

two experiments are completely different because in ref.70 light propagates in a solid

in the cryogenic regime and in ref.68 light propagates in vacuum at room tempera-

ture. As such, there is a plenty of systematic differences. Yet, the two experiments

give exactly the same signal at the white-noise end point. Therefore, there must be

an ubiquitous form of white noise that admits a definite physical interpretation. Our

theoretical prediction Eq.(65) is, at present, the only existing explanation. Together

with the classical experiments, we thus conclude that there is now an alternative

scheme challenging the traditional ‘null interpretation’ of Michelson-Morley exper-

iments, always presented as a self-evident scientific truth.

We have also discussed two further experimental tests. First, one should try to

detect our predicted, daily variations Eq.(66). Second, one should also try to place

the optical cavities on a satellite, as in the OPTIS proposal.110 In this ideal free-fall

environment, as in panel (a) of our Fig.20, the typical instantaneous frequency shift

should be much smaller (by orders of magnitude) than the corresponding 10−15

value measured with the same interferometers on the Earth’s surface.

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52 Authors’ Names

Appendix

In this appendix, we will summarize the stochastic model used in refs.7–10 to compare with experiments. To make explicit the time dependence of the signal let us first re-write Eq.(19) as

∆c¯θ (t) c

∼

v2(t) ϵ c2

cos 2(θ

−

θ2(t))

(73)

where v(t) and θ2(t) indicate respectively the instantaneous magnitude and direction of the drift in the (x, y) plane of the interferometer. This can also be re-written as

∆c¯θ(t) ∼ 2S(t) sin 2θ + 2C(t) cos 2θ

(74)

c

with

2C (t)

=

ϵ

vx2(t) − vy2(t) c2

2S(t)

=

ϵ

2vx (t)vy (t) c2

(75)

and vx(t) = v(t) cos θ2(t), vy(t) = v(t) sin θ2(t) As anticipated in Sect.3, the standard assumption to analyze the data has always

been based on the idea of regular modulations of the signal associated with a cosmic

Earth velocity. In general, this is characterized by a magnitude V , a right ascension

α and an angular declination γ. These parameters can be considered constant for

short-time observations of a few days where there are no appreciable changes due

to the Earth orbital velocity around the sun. In this framework, where the only

time dependence is due to the Earth rotation, the traditional identifications are v(t) ≡ v˜(t) and θ2(t) ≡ θ˜2(t) where v˜(t) and θ˜2(t) derive from the simple application of spherical trigonometry71

cos z(t) = sin γ sin ϕ + cos γ cos ϕ cos(τ − α)

(76)

v˜(t) = V sin z(t)

(77)

v˜x(t) = v˜(t) cos θ˜2(t) = V [sin γ cos ϕ − cos γ sin ϕ cos(τ − α)]

(78)

v˜y(t) = v˜(t) sin θ˜2(t) = V cos γ sin(τ − α)

(79)

Here z = z(t) is the zenithal distance of V, ϕ is the latitude of the laboratory,

τ

=

ωsidt

is

the

sidereal

time

of

the

observation

in

degrees

(ωsid

∼

2π 23h 56′

)

and

the

angle θ2 is counted conventionally from North through East so that North is θ2 = 0

and East is θ2 = 90o. With the identifications v(t) ≡ v˜(t) and θ2(t) ≡ θ˜2(t), one

thus arrives to the simple Fourier decomposition

S(t) ≡ S˜(t) = S0 + Ss1 sin τ + Sc1 cos τ + Ss2 sin(2τ ) + Sc2 cos(2τ ) (80)

C(t) ≡ C˜(t) = C0 + Cs1 sin τ + Cc1 cos τ + Cs2 sin(2τ ) + Cc2 cos(2τ ) (81)
where the Ck and Sk Fourier coefficients depend on the three parameters (V, α, γ) and are given explicitly in refs.7, 9

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Instructions for typing manuscripts (paper’s title) 53

Though, the identification of the instantaneous quantities vx(t) and vy(t) with their counterparts v˜x(t) and v˜y(t) is not necessarily true. As anticipated in Sect.3, one could consider the alternative situation where the velocity field is a nondifferentiable function and adopt some other description, for instance a formulation in terms of random Fourier series.55, 72, 73 In this other approach, the parameters of the macroscopic motion are used to fix the typical boundaries for a microscopic velocity field which has an intrinsic non-deterministic nature.
The model adopted in refs.7–10 corresponds to the simplest case of a turbulence which, at small scales, appears homogeneous and isotropic. The analysis can then be embodied in an effective space-time metric for light propagation

gµν (t) ∼ ηµν + 2ϵvµ(t)vν (t)

(82)

where vµ(t) is a random 4-velocity field which describes the drift and whose boundaries depend on a smooth field v˜µ(t) determined by the average Earth motion.
For homogeneous turbulence a series representation, suitable for numerical simulations of a discrete signal, can be expressed in the form

∞

vx(tk) = [xn(1) cos ωntk + xn(2) sin ωntk]

(83)

n=1

∞

vy(tk) = [yn(1) cos ωntk + yn(2) sin ωntk]

(84)

n=1

Here ωn = 2nπ/T and T is the common period of all Fourier components. Furthermore, tk = (k − 1)∆t, with k = 1, 2..., and ∆t is the sampling time. Finally, xn(i = 1, 2) and yn(i = 1, 2) are random variables with the dimension of a velocity and vanishing mean. In our simulations, the value T = Tday= 24 hours and a sampling step ∆t = 1 second were adopted. However, the results would remain unchanged by any rescaling T → sT and ∆t → s∆t.
In general, we can denote by [−dx(t), dx(t)] the range for xn(i = 1, 2) and by [−dy(t), dy(t)] the corresponding range for yn(i = 1, 2). Statistical isotropy would require to impose dx(t) = dy(t). However, to illustrate the more general case, we will first consider dx(t) ̸= dy(t).
If we assume that the random values of xn(i = 1, 2) and yn(i = 1, 2) are chosen with uniform probability, the only non-vanishing (quadratic) statistical averages are

⟨x2n(i

=

1, 2)⟩stat

=

d2x(t) 3 n2η

⟨yn2 (i

=

1, 2)⟩stat

=

d2y (t) 3 n2η

(85)

Here, the exponent η ensures finite statistical averages ⟨vx2(t)⟩stat and ⟨vy2(t)⟩stat for an arbitrarily large number of Fourier components. In our simulations, between the two possible alternatives η = 5/6 and η = 1 of ref.,73 we have chosen η = 1 that
corresponds to the Lagrangian picture in which the point where the fluid velocity
is measured is a wandering material point in the fluid.

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54 Authors’ Names

Finally, the connection with the Earth cosmic motion is obtained by identifying dx(t) = v˜x(t) and dy(t) = v˜y(t) as given in Eqs. (76)−(79). If, however, we require statistical isotropy, the relation

v˜x2(t) + v˜y2(t) = v˜2(t)

(86)

requires the identification

v˜(t)

dx(t) = dy(t) =

√ 2

(87)

For such isotropic model, by combining Eqs.(83)−(87) and in the limit of an infinite statistics, one gets

⟨vx2 (t)⟩stat

=

⟨vy2(t)⟩stat

=

v˜2(t) 2

1 3

∞

1 n2

v˜2(t) =
2

π2 18

n=1

⟨vx(t)vy(t)⟩stat = 0

(88)

and vanishing statistical averages

⟨C(t)⟩stat = 0

⟨S(t)⟩stat = 0

(89)

at any time t, see Eqs.(75). Therefore, by construction, this model gives a definite non-zero signal but, if the same signal were fitted with Eqs.(80) and (81), it would also give average values (Ck)avg = 0, (Sk)avg = 0 for the Fourier coefficients.
To understand how radical is the modification produced by Eqs.(89), we recall the traditional procedure adopted in the classical experiments. One was measuring the fringe shifts at some given sidereal time on consecutive days so that changes of the orbital velocity were negligible. Then, see Eqs.(20) and (74), the measured shifts at the various angle θ were averaged

∆λ(θ; t)

2D

⟨ λ ⟩stat = λ [2 sin 2θ ⟨S(t)⟩stat + 2 cos 2θ ⟨C(t)⟩stat]

(90)

and finally these average values were compared with models for the Earth cosmic

motion.

However if the signal is so irregular that, by increasing the number of mea-

surements, ⟨C(t)⟩stat → 0 and ⟨S(t)⟩stat → 0 the averages Eq.(90) would have no meaning. In fact, these averages would be non vanishing just because the statis-

tics is finite. In particular, the direction θ2(t) of the drift (defined by the relation tan 2θ2(t) = S(t)/C(t)) would vary randomly with no definite limit.
This is why one should concentrate the analysis on the 2nd-harmonic amplitudes

2D A2(t) = λ 2

S2(t) + C2(t)

∼

2D λ

ϵ

vx2

(t)

+ c2

vy2

(t)

(91)

which are positive-definite and remain non-zero under the averaging procedure.

Moreover, these are rotational-invariant quantities and their statistical average

2D π2 V 2 sin2 z(t)

⟨A2(t)⟩stat ∼

λ

· ·ϵ· 18

c2

(92)

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Instructions for typing manuscripts (paper’s title) 55

would remain unchanged in the isotropic model Eq.(87) or with the alternative

choice dx(t) ≡ v˜x(t) and dy(t) ≡ v˜y(t). Analogous considerations hold for the mod-

ern

experiments

where

∆c¯θ (t) c

is

extracted

from

the

frequency

shift

of

two

optical

resonators. Again, the C(t) and S(t) obtained, through Eq.(74), from the very ir-

regular, measured signal (see e.g. Fig.19), are compared with the slowly varying

parameterizations Eqs.(80) and (81) to extract the Ck and Sk Fourier coefficients.

Then, by comparing with our simulation of the C(t) and S(t) in Fig.23, it is no

surprise if the average values (Ck)avg → 0, (Sk)avg → 0 by simply increasing the

number of observations.

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