das michelson morley experiment

In 1887, Albert A. Michelson and Edward W. Morley tried to measure the speed of the ether . The concept of the ether was made in analogy with other types of media in which different types of waves are able to propagate; sound waves can, for example, propagate in air or other materials. The result of the Michelson-Morley experiment was that the speed of the Earth through the ether (or the speed of the ether wind) was zero. Therefore, this experiment also showed that there is no need for any ether at all, and it appeared that the speed of light in vacuum was independent of the speed of the observer! Michelson and Morley repeated their experiment many times up until 1929, but always with the same results and conclusions. Michelson won the Nobel Prize in Physics in 1907.

Michelson-Morley Experiments: at the crossroads of Relativity, Cosmology and Quantum Physics

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.

1 Introduction

From the very beginning there are two interpretations of Relativity: Einstein’s Special Relativity [ 1 ] 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 Σ \Sigma roman_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 1 1 1 We 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 , 4 , 5 ] . , an intuitive representation is given in Fig.1.

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. 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 Σ \Sigma roman_Σ could not be detected, the usual attitude was to consider the difference between the two interpretations as a philosophical problem.

superscript 𝛽 ′′ subscript superscript 𝑥 ′′ 𝐵 𝑐 subscript superscript 𝑡 ′′ 𝐵 cT_{B}=\gamma^{\prime\prime}(\beta^{\prime\prime}x^{\prime\prime}_{B}+ct^{% \prime\prime}_{B}) italic_c italic_T start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ( italic_β start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT + italic_c italic_t start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) , with now 1 / γ ′′ = 1 − ( β ′′ ) 2 1 superscript 𝛾 ′′ 1 superscript superscript 𝛽 ′′ 2 1/\gamma^{\prime\prime}=\sqrt{1-(\beta^{\prime\prime})^{2}} 1 / italic_γ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT = square-root start_ARG 1 - ( italic_β start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG . Thus no ambiguity is possible, either T A > T B subscript 𝑇 𝐴 subscript 𝑇 𝐵 T_{A}>T_{B} italic_T start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT > italic_T start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT or viceversa so that the view in the preferred Σ − limit-from Σ \Sigma- roman_Σ - frame becomes the relevant one to decide on causal effects. .

But the mere logical possibility of Σ Σ \Sigma roman_Σ is not enough. For a full resolution of the paradox, the Σ − limit-from Σ \Sigma- roman_Σ - frame should show up through a determination of the kinematic parameters β ′ superscript 𝛽 ′ \beta^{\prime} italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , β ′′ superscript 𝛽 ′′ \beta^{\prime\prime} italic_β start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT … Thus, we arrive to the main point of this article: the prejudice that, even in a Lorentzian formulation of relativity, the individual β ′ superscript 𝛽 ′ \beta^{\prime} italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , β ′′ superscript 𝛽 ′′ \beta^{\prime\prime} italic_β start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT … 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 Σ \Sigma roman_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 ′ superscript 𝛽 ′ \beta^{\prime} italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , β ′′ superscript 𝛽 ′′ \beta^{\prime\prime} italic_β start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT … But the conspiracy works exactly only when the velocity of light c γ subscript 𝑐 𝛾 c_{\gamma} italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT propagating in the various interferometers coincides with the basic parameter c 𝑐 c italic_c entering Lorentz transformations. Therefore, one may ask, what happens if c γ ≠ c subscript 𝑐 𝛾 𝑐 c_{\gamma}\neq c italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ≠ italic_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 scheme [ 7 , 8 , 9 , 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 non-locality 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 , 12 , 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 v Q ⁢ I ≫ c much-greater-than subscript 𝑣 𝑄 𝐼 𝑐 v_{QI}\gg c italic_v start_POSTSUBSCRIPT italic_Q italic_I end_POSTSUBSCRIPT ≫ italic_c , could such extraordinary thing be so easily dismissed? Namely, could we ignore this ‘something’ just because, apparently, it cannot be efficiently controlled to send ‘messages’ 3 3 3 Experimental correlations between spacelike separated measurements can in principle be explained through hidden influences propagating at a finite speed v QI ≫ c much-greater-than subscript 𝑣 QI 𝑐 v_{\rm QI}\gg c italic_v start_POSTSUBSCRIPT roman_QI end_POSTSUBSCRIPT ≫ italic_c provided v QI subscript 𝑣 QI v_{\rm QI} italic_v start_POSTSUBSCRIPT roman_QI end_POSTSUBSCRIPT is large enough [ 15 ] . But in ref. [ 16 ] it is also shown that for any finite v QI subscript 𝑣 QI v_{\rm QI} italic_v start_POSTSUBSCRIPT roman_QI end_POSTSUBSCRIPT , with c < v QI < ∞ 𝑐 subscript 𝑣 QI c<v_{\rm QI}<\infty italic_c < italic_v start_POSTSUBSCRIPT roman_QI end_POSTSUBSCRIPT < ∞ , one can construct combined correlations to be used for faster-than-light communication. ? 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 contribution [ 6 ] 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 4 4 4 “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 ] . . 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 experimentally [ 18 , 19 , 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 − limit-from Σ \Sigma- roman_Σ - frame would free ourselves from the no-signalling argument to dispose of the causality paradox.

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 , 23 , 24 , 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 5 5 5 Before 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 ] . . 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 6 6 6 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 ] . . Thus, if the condensation process singles out a certain reference frame Σ Σ \Sigma roman_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 ) ⟩ ket superscript Ψ 0 |\Psi^{(0)}\rangle | roman_Ψ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ⟩ , appropriate to the observer at rest in the Σ − limit-from Σ \Sigma- roman_Σ - frame singled out by the condensation process, and the corresponding vacuum states | Ψ ′ ⟩ ket superscript Ψ ′ |\Psi^{\prime}\rangle | roman_Ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ , | Ψ ′′ ⟩ ket superscript Ψ ′′ |\Psi^{\prime\prime}\rangle | roman_Ψ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ⟩ ,.. appropriate to moving observers S ′ superscript 𝑆 ′ S^{\prime} italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , S ′′ superscript 𝑆 ′′ S^{\prime\prime} italic_S start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ,… By assuming that these different vacua are generated by Lorentz boost unitary operators U ′ superscript 𝑈 ′ U^{\prime} italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , U ′′ superscript 𝑈 ′′ U^{\prime\prime} italic_U start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT …acting on | Ψ ( 0 ) ⟩ ket superscript Ψ 0 |\Psi^{(0)}\rangle | roman_Ψ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ⟩ , say | Ψ ′ ⟩ = U ′ ⁢ | Ψ ( 0 ) ⟩ ket superscript Ψ ′ superscript 𝑈 ′ ket superscript Ψ 0 |\Psi^{\prime}\rangle=U^{\prime}|\Psi^{(0)}\rangle | roman_Ψ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ = italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT | roman_Ψ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ⟩ , | Ψ ′′ ⟩ = U ′′ ⁢ | Ψ ( 0 ) ⟩ ket superscript Ψ ′′ superscript 𝑈 ′′ ket superscript Ψ 0 |\Psi^{\prime\prime}\rangle=U^{\prime\prime}|\Psi^{(0)}\rangle | roman_Ψ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ⟩ = italic_U start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT | roman_Ψ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ⟩ … For any Lorentz-scalar operator G 𝐺 {G} italic_G , such that G = ( U ′ ) † ⁢ G ⁢ U ′ = ( U ′′ ) † ⁢ G ⁢ U ′′ ⁢ … 𝐺 superscript superscript 𝑈 ′ † 𝐺 superscript 𝑈 ′ superscript superscript 𝑈 ′′ † 𝐺 superscript 𝑈 ′′ … G=(U^{\prime})^{\dagger}GU^{\prime}=(U^{\prime\prime})^{\dagger}GU^{\prime% \prime}... italic_G = ( italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_G italic_U start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = ( italic_U start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_G italic_U start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT … , it follows trivially

𝑖 subscript 𝜂 𝜇 𝜌 subscript 𝐿 𝜈 𝜎 𝑖 subscript 𝜂 𝜇 𝜎 subscript 𝐿 𝜈 𝜌 𝑖 subscript 𝜂 𝜈 𝜎 subscript 𝐿 𝜇 𝜌 𝑖 subscript 𝜂 𝜈 𝜌 subscript 𝐿 𝜇 𝜎 [L_{\mu\nu},L_{\rho\sigma}]=-i\eta_{\mu\rho}L_{\nu\sigma}+i\eta_{\mu\sigma}L_{% \nu\rho}-i\eta_{\nu\sigma}L_{\mu\rho}+i\eta_{\nu\rho}L_{\mu\sigma} [ italic_L start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT , italic_L start_POSTSUBSCRIPT italic_ρ italic_σ end_POSTSUBSCRIPT ] = - italic_i italic_η start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_ν italic_σ end_POSTSUBSCRIPT + italic_i italic_η start_POSTSUBSCRIPT italic_μ italic_σ end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_ν italic_ρ end_POSTSUBSCRIPT - italic_i italic_η start_POSTSUBSCRIPT italic_ν italic_σ end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_μ italic_ρ end_POSTSUBSCRIPT + italic_i italic_η start_POSTSUBSCRIPT italic_ν italic_ρ end_POSTSUBSCRIPT italic_L start_POSTSUBSCRIPT italic_μ italic_σ end_POSTSUBSCRIPT where η μ ⁢ ν = diag ⁢ ( 1 , − 1 , − 1 , − 1 ) subscript 𝜂 𝜇 𝜈 diag 1 1 1 1 \eta_{\mu\nu}={\rm diag}(1,-1,-1,-1) italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = roman_diag ( 1 , - 1 , - 1 , - 1 ) is the Minkowski tensor. A Lorentz-invariant vacuum has to be annihilated by all 10 generators. is only known for the free-field case through the simple Wick-ordering prescription relatively to the free-field vacuum | 0 ⟩ ket 0 |0\rangle | 0 ⟩ . In an interacting theory, the construction is implemented order by order in perturbation theory. Therefore, in the presence of non-perturbative phenomena (such as Spontaneous Symmetry Breaking, chiral symmetry breaking, gluon condensation…) where the physical vacuum | Ψ ( 0 ) ⟩ ket superscript Ψ 0 |\Psi^{(0)}\rangle | roman_Ψ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ⟩ cannot be constructed from the free-field vacuum | 0 ⟩ ket 0 |0\rangle | 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 covariant but, with a non-Lorentz-invariant vacuum, there would be a preferred reference frame 8 8 8 To our knowledge,in four space-time dimensions, a non-perturbative analysis of a Lorentz-invariant vacuum has been attempted by very few authors. In the case of non-linear field theories with P ⁢ ( Φ ⁢ ( x ) ) 𝑃 Φ 𝑥 P(\Phi(x)) italic_P ( roman_Φ ( italic_x ) ) interactions, such as Φ 4 ⁢ ( x ) superscript Φ 4 𝑥 \Phi^{4}(x) roman_Φ start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_x ) , this was discussed by Segal [ 29 ] . He considered a suitable generalization of the standard Wick ordering : P ( Φ ) : :P(\Phi): : italic_P ( roman_Φ ) : relative to | 0 ⟩ ket 0 |0\rangle | 0 ⟩ , say : : P ( Φ ) : : ::P(\Phi):: : : italic_P ( roman_Φ ) : : , such that ⟨ Ψ ( 0 ) | : : P ( Φ ) : : | Ψ ( 0 ) ⟩ = 0 \langle\Psi^{(0)}|::P(\Phi)::|\Psi^{(0)}\rangle=0 ⟨ roman_Ψ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT | : : italic_P ( roman_Φ ) : : | roman_Ψ start_POSTSUPERSCRIPT ( 0 ) end_POSTSUPERSCRIPT ⟩ = 0 in the true vacuum state. His conclusion was that : : P ( Φ ) : : ::P(\Phi):: : : italic_P ( roman_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 ) : : ::P(\Phi):: : : italic_P ( roman_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. .

1.3 Relativity and the CMB

Finally, some remarks about the physical nature of the hypothetical Σ − limit-from Σ \Sigma- roman_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 dipole [ 30 ] vanishes. This dipole is in fact a direct consequence of the motion of the Earth ( β = V / c ) \beta=V/c) italic_β = italic_V / italic_c )

Accurate observations with satellites in space [ 31 ] have shown that the measured temperature variations correspond to a motion of the solar system described by an average velocity V ∼ 370 similar-to 𝑉 370 V\sim 370 italic_V ∼ 370 km/s, a right ascension α ∼ 168 o similar-to 𝛼 superscript 168 𝑜 \alpha\sim 168^{o} italic_α ∼ 168 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT and a declination γ ∼ − 7 o similar-to 𝛾 superscript 7 𝑜 \gamma\sim-7^{o} italic_γ ∼ - 7 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT , pointing approximately in the direction of the Leo constellation. This means that, if one sets T o ∼ similar-to subscript 𝑇 𝑜 absent T_{o}\sim italic_T start_POSTSUBSCRIPT italic_o end_POSTSUBSCRIPT ∼ 2.7 K and β ∼ 0.00123 similar-to 𝛽 0.00123 \beta\sim 0.00123 italic_β ∼ 0.00123 , there are angular variations of a few millikelvin

which represent by far the dominant contribution to the CMB anisotropy.

Could the reference system with vanishing CMB dipole represent the fundamental 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 Σ \Sigma roman_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 9 9 9 “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 ] . 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 Michelson-Morley 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-analysis [ 7 , 8 , 9 , 10 ] of the classical experiments 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 − limit-from Σ \Sigma- roman_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 𝑐 c italic_c of Lorentz transformations. For instance, when light propagates in a gas, the sought effect of a preferred system Σ Σ \Sigma roman_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 , 8 , 9 , 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 ) subscript ¯ 𝑐 𝛾 𝜃 \bar{c}_{\gamma}(\theta) over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_θ ) which is the only one that can be measured unambiguously

By introducing its anisotropy

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

1 italic-ϵ {\cal N}=1+\epsilon caligraphic_N = 1 + italic_ϵ , with ϵ ≪ 1 much-less-than italic-ϵ 1 \epsilon\ll 1 italic_ϵ ≪ 1 ). This gives directly the fringe patterns ( λ 𝜆 \lambda italic_λ is the light wavelength)

which were measured in the classical experiments.

In modern experiments, on the other hand, a possible anisotropy of c ¯ γ ⁢ ( θ ) subscript ¯ 𝑐 𝛾 𝜃 \bar{c}_{\gamma}(\theta) over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_θ ) would show up through the relative frequency shift, i.e. the beat signal, Δ ⁢ ν ⁢ ( θ ) Δ 𝜈 𝜃 \Delta\nu(\theta) roman_Δ italic_ν ( italic_θ ) of two orthogonal optical resonators, see Fig. 3 . Their frequency

is proportional to the two-way velocity of light within the resonator through an integer number m 𝑚 m italic_m , which fixes the cavity mode, and the length of the cavity L 𝐿 L italic_L as measured in the laboratory S ′ superscript 𝑆 ′ S^{\prime} italic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT frame. Therefore, once the length of a cavity in its rest frame does not depend on its orientation, one finds

where ν 0 subscript 𝜈 0 \nu_{0} italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the reference frequency of the two resonators.

2.2 The limit of refractive index 𝒩 = 1 + ϵ 𝒩 1 italic-ϵ {\cal N}=1+\epsilon caligraphic_N = 1 + italic_ϵ

𝜋 𝜃 \theta\to\pi+\theta italic_θ → italic_π + italic_θ , to lowest non-trivial level 𝒪 ⁢ ( ϵ ⁢ β 2 ) 𝒪 italic-ϵ superscript 𝛽 2 {\cal O}(\epsilon\beta^{2}) caligraphic_O ( italic_ϵ italic_β start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , one finds the general expression [ 7 , 24 ]

𝜋 𝜃 \theta\to\pi+\theta italic_θ → italic_π + italic_θ , the angular dependence has been given as an infinite expansion of even-order Legendre polynomials with arbitrary coefficients ζ 2 ⁢ n = 𝒪 ⁢ ( 1 ) subscript 𝜁 2 𝑛 𝒪 1 \zeta_{2n}={\cal O}(1) italic_ζ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT = caligraphic_O ( 1 ) . In Einstein’s Special Relativity, where there is no preferred reference frame, these ζ 2 ⁢ n subscript 𝜁 2 𝑛 \zeta_{2n} italic_ζ start_POSTSUBSCRIPT 2 italic_n end_POSTSUBSCRIPT coefficients should vanish identically. In a Lorentzian approach, on the other hand, there is no reason why they should vanish a priori 10 10 10 As 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 ] . .

By leaving out the first few ζ 𝜁 \zeta italic_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 → italic-ϵ 0 \epsilon\to 0 italic_ϵ → 0 limit with a preferred frame. This other argument is based on the effective space-time metric g μ ⁢ ν = g μ ⁢ ν ⁢ ( 𝒩 ) superscript 𝑔 𝜇 𝜈 superscript 𝑔 𝜇 𝜈 𝒩 g^{\mu\nu}=g^{\mu\nu}({\cal N}) italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ( caligraphic_N ) which, through the relation g μ ⁢ ν ⁢ p μ ⁢ p ν = 0 superscript 𝑔 𝜇 𝜈 subscript 𝑝 𝜇 subscript 𝑝 𝜈 0 g^{\mu\nu}p_{\mu}p_{\nu}=0 italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 0 , describes light propagation in a medium of refractive index 𝒩 𝒩 {\cal N} caligraphic_N . For the quantum theory, a derivation of this metric from first principles was given by Jauch and Watson [ 46 ] 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 Σ \Sigma roman_Σ .

In this perspective, let us consider two identical optical cavities, namely cavity 1, at rest in Σ Σ \Sigma roman_Σ , and cavity 2, at rest in S 𝑆 S italic_S , and denote by π μ ≡ ( E π c , π ) subscript 𝜋 𝜇 subscript 𝐸 𝜋 𝑐 𝜋 \pi_{\mu}\equiv({{E_{\pi}}\over{c}},{\bf\pi}) italic_π start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ≡ ( divide start_ARG italic_E start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT end_ARG start_ARG italic_c end_ARG , italic_π ) the light 4-momentum for Σ Σ \Sigma roman_Σ in his cavity 1 and by p μ ≡ ( E p c , 𝐩 ) subscript 𝑝 𝜇 subscript 𝐸 𝑝 𝑐 𝐩 p_{\mu}\equiv({{E_{p}}\over{c}},{\bf p}) italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ≡ ( divide start_ARG italic_E start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT end_ARG start_ARG italic_c end_ARG , bold_p ) the corresponding light 4-momentum for S 𝑆 S italic_S in his cavity 2. Let us also denote by g μ ⁢ ν superscript 𝑔 𝜇 𝜈 g^{\mu\nu} italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT the space-time metric that S 𝑆 S italic_S uses in the relation g μ ⁢ ν ⁢ p μ ⁢ p ν = 0 superscript 𝑔 𝜇 𝜈 subscript 𝑝 𝜇 subscript 𝑝 𝜈 0 g^{\mu\nu}p_{\mu}p_{\nu}=0 italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 0 and by

the metric used by Σ Σ \Sigma roman_Σ in the relation γ μ ⁢ ν ⁢ π μ ⁢ π ν = 0 superscript 𝛾 𝜇 𝜈 subscript 𝜋 𝜇 subscript 𝜋 𝜈 0 \gamma^{\mu\nu}\pi_{\mu}\pi_{\nu}=0 italic_γ start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_π start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_π start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = 0 and which gives an isotropic velocity c γ = E π / | π | = c 𝒩 subscript 𝑐 𝛾 subscript 𝐸 𝜋 𝜋 𝑐 𝒩 c_{\gamma}=E_{\pi}/|{\bf\pi}|={{c}\over{{\cal N}}} italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = italic_E start_POSTSUBSCRIPT italic_π end_POSTSUBSCRIPT / | italic_π | = divide start_ARG italic_c end_ARG start_ARG caligraphic_N end_ARG .

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 Σ \Sigma roman_Σ , as far as light propagation is concerned, this physical equivalence is only assumed in the ideal 𝒩 = 1 𝒩 1 {\cal N}=1 caligraphic_N = 1 limit. As anticipated, for 𝒩 ≠ 1 𝒩 1 {\cal N}\neq 1 caligraphic_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 Σ \Sigma roman_Σ and produce, in the frame S 𝑆 S italic_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 𝑆 S italic_S , which is in uniform translational motion with respect to Σ Σ \Sigma roman_Σ , no longer implies that the medium which stays inside, e.g. the gas, is in a state of thermodynamic equilibrium 11 11 11 Think 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. . Thus, one should keep an open mind and exploit the implications of the basic condition

where η μ ⁢ ν superscript 𝜂 𝜇 𝜈 \eta^{\mu\nu} italic_η start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT is the Minkowski tensor. This standard equality amounts to introduce a transformation matrix, say A ν μ subscript superscript 𝐴 𝜇 𝜈 A^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT , such that

This relation is strictly valid for 𝒩 = 1 𝒩 1 {\cal N}=1 caligraphic_N = 1 . However, by continuity, one is driven to conclude that an analogous relation between g μ ⁢ ν superscript 𝑔 𝜇 𝜈 g^{\mu\nu} italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT and γ μ ⁢ ν superscript 𝛾 𝜇 𝜈 \gamma^{\mu\nu} italic_γ start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT should also hold in the ϵ → 0 → italic-ϵ 0 \epsilon\to 0 italic_ϵ → 0 limit. The only subtlety is that relation ( 13 ) does not fix uniquely A ν μ subscript superscript 𝐴 𝜇 𝜈 A^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT . In fact, one can either choose the identity matrix, i.e. A ν μ = δ ν μ subscript superscript 𝐴 𝜇 𝜈 subscript superscript 𝛿 𝜇 𝜈 A^{\mu}_{\nu}=\delta^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = italic_δ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT , or a Lorentz transformation, i.e. A ν μ = Λ ν μ subscript superscript 𝐴 𝜇 𝜈 subscript superscript Λ 𝜇 𝜈 A^{\mu}_{\nu}=\Lambda^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = roman_Λ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT . Since for any finite v 𝑣 v italic_v these two matrices cannot be related by an infinitesimal transformation, it follows that A ν μ subscript superscript 𝐴 𝜇 𝜈 A^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT is a two-valued function in the ϵ → 0 → italic-ϵ 0 \epsilon\to 0 italic_ϵ → 0 limit. Therefore, in principle, there are two solutions. If A ν μ subscript superscript 𝐴 𝜇 𝜈 A^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT is the identity matrix, we find a first solution

while, if A ν μ subscript superscript 𝐴 𝜇 𝜈 A^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT is a Lorentz transformation, we find the other solution

v μ superscript 𝑣 𝜇 v^{\mu} italic_v start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT being the dimensionless S 𝑆 S italic_S 4-velocity, v μ ≡ ( v 0 , 𝐯 / c ) superscript 𝑣 𝜇 superscript 𝑣 0 𝐯 𝑐 v^{\mu}\equiv(v^{0},{\bf v}/c) italic_v start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ≡ ( italic_v start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT , bold_v / italic_c ) with v μ ⁢ v μ = 1 subscript 𝑣 𝜇 superscript 𝑣 𝜇 1 v_{\mu}v^{\mu}=1 italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_v start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = 1 .

Notice that with the former choice, implicitly adopted in Special Relativity to preserve isotropy in all reference systems also for 𝒩 ≠ 1 𝒩 1 {\cal N}\neq 1 caligraphic_N ≠ 1 , one is introducing a discontinuity in the transformation matrix for any ϵ ≠ 0 italic-ϵ 0 \epsilon\neq 0 italic_ϵ ≠ 0 . Indeed, the whole emphasis on Lorentz transformations depends on enforcing Eq.( 13 ) for A ν μ = Λ ν μ subscript superscript 𝐴 𝜇 𝜈 subscript superscript Λ 𝜇 𝜈 A^{\mu}_{\nu}=\Lambda^{\mu}_{\nu} italic_A start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT = roman_Λ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT so that Λ μ ⁢ σ ⁢ Λ σ ν = η μ ⁢ ν superscript Λ 𝜇 𝜎 subscript superscript Λ 𝜈 𝜎 superscript 𝜂 𝜇 𝜈 \Lambda^{\mu\sigma}\Lambda^{\nu}_{\sigma}=\eta^{\mu\nu} roman_Λ start_POSTSUPERSCRIPT italic_μ italic_σ end_POSTSUPERSCRIPT roman_Λ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT = italic_η start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT and the Minkowski metric applies to all equivalent frames.

On the other hand, with the latter solution, by replacing in the relation p μ ⁢ p ν ⁢ g μ ⁢ ν = 0 subscript 𝑝 𝜇 subscript 𝑝 𝜈 superscript 𝑔 𝜇 𝜈 0 p_{\mu}p_{\nu}g^{\mu\nu}=0 italic_p start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT = 0 , the photon energy now depends on the direction of propagation. Then, by defining the light velocity c γ ⁢ ( θ ) subscript 𝑐 𝛾 𝜃 c_{\gamma}(\theta) italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_θ ) from the ratio E p / | 𝐩 | subscript 𝐸 𝑝 𝐩 E_{p}/|{\bf p}| italic_E start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT / | bold_p | , one finds [ 7 , 24 ]

and a two-way velocity

here θ 𝜃 \theta italic_θ is the angle between 𝐯 𝐯 {\bf v} bold_v and 𝐩 𝐩 \bf p bold_p (as defined in the S 𝑆 S italic_S frame) and

1 3 2 𝒩 1 superscript 𝛽 2 {\cal N}_{\rm exp}\equiv\langle\bar{\cal N}(\theta)\rangle_{\theta}={\cal N}~{% }\left[1+{{3}\over{2}}({\cal N}-1)\beta^{2}\right] caligraphic_N start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT ≡ ⟨ over¯ start_ARG caligraphic_N end_ARG ( italic_θ ) ⟩ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = caligraphic_N [ 1 + divide start_ARG 3 end_ARG start_ARG 2 end_ARG ( caligraphic_N - 1 ) italic_β start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] . One can then determine the unknown value 𝒩 ≡ 𝒩 ⁢ ( Σ ) 𝒩 𝒩 Σ {\cal N}\equiv{\cal N}(\Sigma) caligraphic_N ≡ caligraphic_N ( roman_Σ ) (as if the container of the gas were at rest in Σ Σ \Sigma roman_Σ ), in terms of the experimentally known quantity 𝒩 exp ≡ 𝒩 ⁢ ( Earth ) subscript 𝒩 exp 𝒩 Earth {\cal N}_{\rm exp}\equiv{\cal N}({\rm Earth}) caligraphic_N start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT ≡ caligraphic_N ( roman_Earth ) and of v 𝑣 v italic_v . As discussed in refs. [ 7 ] - [ 10 ] , for v ∼ 370 similar-to 𝑣 370 v\sim 370 italic_v ∼ 370 km/s, the resulting difference | 𝒩 exp − 𝒩 | ≲ 10 − 9 less-than-or-similar-to subscript 𝒩 exp 𝒩 superscript 10 9 |{\cal N}_{\rm exp}-{\cal N}|\lesssim 10^{-9} | caligraphic_N start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT - caligraphic_N | ≲ 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT is well below the experimental accuracy on 𝒩 exp subscript 𝒩 exp {\cal N}_{\rm exp} caligraphic_N start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT and, for all practical purposes, can be ignored. .

3 A first look at the classical experiments

From Eq.( 17 ) we find a fractional anisotropy

which produces a fringe pattern

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 ] 0 𝜋 [0,\pi] [ 0 , italic_π ] , as in the classical theory (see e.g. [ 47 ] ). However, its amplitude

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

This difference could then be re-absorbed into an observable velocity which is related to the kinematical velocity v 𝑣 v italic_v through the gas refractive index

Thus, this v obs subscript 𝑣 obs v_{\rm obs} italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT is the small velocity traditionally extracted from the measured amplitude A 2 EXP subscript superscript 𝐴 EXP 2 A^{\rm EXP}_{2} italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in the classical analysis of the experiments

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 ⋅ 10 7 similar-to 𝐷 𝜆 ⋅ 2 superscript 10 7 {{D}\over{\lambda}}\sim 2\cdot 10^{7} divide start_ARG italic_D end_ARG start_ARG italic_λ end_ARG ∼ 2 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT , the classically expected amplitude was A 2 class ∼ similar-to subscript superscript 𝐴 class 2 absent A^{\rm class}_{2}\sim italic_A start_POSTSUPERSCRIPT roman_class end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.2. But the experimental amplitude measured in the six sessions was A 2 EXP = 0.01 ÷ 0.02 subscript superscript 𝐴 EXP 2 0.01 0.02 A^{\rm EXP}_{2}=0.01\div 0.02 italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.01 ÷ 0.02 . This corresponds to an average anisotropy | Δ ⁢ c ¯ θ | exp c ∼ 4 ⋅ 10 − 10 similar-to subscript Δ subscript ¯ 𝑐 𝜃 exp 𝑐 ⋅ 4 superscript 10 10 {{|\Delta\bar{c}_{\theta}|_{\rm exp}}\over{c}}\sim 4\cdot 10^{-10} divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT end_ARG start_ARG italic_c end_ARG ∼ 4 ⋅ 10 start_POSTSUPERSCRIPT - 10 end_POSTSUPERSCRIPT and was originally interpreted in terms of a velocity v obs ∼ 8 similar-to subscript 𝑣 obs 8 v_{\rm obs}\sim 8 italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT ∼ 8 km/s. However, for an experiment in air at room temperature and atmospheric pressure where ϵ ∼ 2.8 ⋅ 10 − 4 similar-to italic-ϵ ⋅ 2.8 superscript 10 4 \epsilon\sim 2.8\cdot 10^{-4} italic_ϵ ∼ 2.8 ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT , this observable velocity would now correspond to a true kinematic value v ∼ 340 similar-to 𝑣 340 v\sim 340 italic_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 apparatus [ 33 ] was extraordinary, about ± 0.004 plus-or-minus 0.004 \pm 0.004 ± 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 2 ⁢ A 2 class ∼ similar-to 2 subscript superscript 𝐴 class 2 absent 2A^{\rm class}_{2}\sim 2 italic_A start_POSTSUPERSCRIPT roman_class end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 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. [ 50 , 7 , 9 ] from the original article [ 33 ] , see Table 1. These data 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, see [ 35 , 51 ] ). One can thus extract amplitude and phase of the 2nd-harmonic component by fitting the even combination of fringe shifts

see Fig. 6 .

The fringe shifts Δ ⁢ λ ⁢ ( i ) λ Δ 𝜆 𝑖 𝜆 {{\Delta\lambda(i)}\over{\lambda}} divide start_ARG roman_Δ italic_λ ( italic_i ) end_ARG start_ARG italic_λ end_ARG for all noon (n.) and evening (e.) sessions of the Michelson-Morley experiment. The angle of rotation is defined as θ = i − 1 16 ⁢ 2 ⁢ π 𝜃 𝑖 1 16 2 𝜋 \theta={{i-1}\over{16}}~{}2\pi italic_θ = divide start_ARG italic_i - 1 end_ARG start_ARG 16 end_ARG 2 italic_π . The Table is taken from ref. [ 7 ] . \toprule 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 \botrule

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 A 2 EXP ∼ 0.016 ± 0.006 similar-to subscript superscript 𝐴 EXP 2 plus-or-minus 0.016 0.006 A^{\rm EXP}_{2}\sim 0.016\pm 0.006 italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.016 ± 0.006 . This value is consistent with an observable velocity

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 second-harmonic amplitude is non-zero at the 5 σ 𝜎 \sigma italic_σ level and in other two sessions (July 9 noon and July 8 evening) is non-zero at the 3 σ 𝜎 \sigma italic_σ level.

The 2nd-harmonic amplitudes for the six experimental sessions of the Michelson-Morley experiment. The table is taken from ref. [ 7 ] . \toprule SESSION        A 2 EXP subscript superscript 𝐴 EXP 2 A^{\rm EXP}_{2} italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT July 8 (noon) 0.010 ± 0.005 plus-or-minus 0.010 0.005 0.010\pm 0.005 0.010 ± 0.005 July 9 (noon) 0.015 ± 0.005 plus-or-minus 0.015 0.005 0.015\pm 0.005 0.015 ± 0.005 July 11 (noon) 0.025 ± 0.005 plus-or-minus 0.025 0.005 0.025\pm 0.005 0.025 ± 0.005 July 8 (evening) 0.014 ± 0.005 plus-or-minus 0.014 0.005 0.014\pm 0.005 0.014 ± 0.005 July 9 (evening) 0.011 ± 0.005 plus-or-minus 0.011 0.005 0.011\pm 0.005 0.011 ± 0.005 July 12 (evening) 0.024 ± 0.005 plus-or-minus 0.024 0.005 0.024\pm 0.005 0.024 ± 0.005 \botrule

As such, the average measured amplitude A 2 EXP ∼ 0.016 similar-to subscript superscript 𝐴 EXP 2 0.016 A^{\rm EXP}_{2}\sim 0.016 italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.016 , although much smaller than the classical expectation A 2 class ∼ similar-to subscript superscript 𝐴 class 2 absent A^{\rm class}_{2}\sim italic_A start_POSTSUPERSCRIPT roman_class end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.2, was not completely negligible. Thus it is natural to ask: should this 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 1933 [ 35 ] 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 magnitude of the amplitude but the sizeable changes in the ‘azimuth’, i.e in the phase θ 2 subscript 𝜃 2 \theta_{2} italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT 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 account that, in a 2nd-harmonic effect, the angle is always defined up to ± 180 o plus-or-minus superscript 180 𝑜 \pm 180^{o} ± 180 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT , one choice for the experimental θ 2 − limit-from subscript 𝜃 2 \theta_{2}- italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT - values is 357 o ± 14 o plus-or-minus superscript 357 𝑜 superscript 14 𝑜 357^{o}\pm 14^{o} 357 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT ± 14 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT , 285 o ± 10 o plus-or-minus superscript 285 𝑜 superscript 10 𝑜 285^{o}\pm 10^{o} 285 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT ± 10 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT and 317 o ± 8 o plus-or-minus superscript 317 𝑜 superscript 8 𝑜 317^{o}\pm 8^{o} 317 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT ± 8 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT respectively for the noon sessions of July 8th, 9th and 11th. For this assignment, the individual velocity vectors v obs ⁢ ( cos ⁡ θ 2 , − sin ⁡ θ 2 ) subscript 𝑣 obs subscript 𝜃 2 subscript 𝜃 2 v_{\rm obs}(\cos\theta_{2},-\sin\theta_{2}) italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT ( roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , - roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 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.

3.2 Further insights: Miller vs. Piccard-Stahel

To get further insights we have compared two other sets of measurements, namely Miller’s observations [ 35 ] 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”.

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 v obs ∼ 8.4 ± 2.2 similar-to subscript 𝑣 obs plus-or-minus 8.4 2.2 v_{\rm obs}\sim 8.4\pm 2.2 italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT ∼ 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 v obs ∼ ( 1.5 ÷ 1.7 ) similar-to subscript 𝑣 obs 1.5 1.7 v_{\rm obs}\sim(1.5\div 1.7) italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT ∼ ( 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 ( A 2 EXP , θ 2 EXP ) subscript superscript 𝐴 EXP 2 subscript superscript 𝜃 EXP 2 (A^{\rm EXP}_{2},\theta^{\rm EXP}_{2}) ( italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_θ start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) 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 % percent \% % CL, becomes now much larger, namely

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 of [ 35 ] 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 % percent \% % which is a quite high level. At the same time, since the agreement is restricted to the region v obs < 9 subscript 𝑣 obs 9 v_{\rm obs}<9 italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT < 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 v obs ∼ 6.0 ± 2.0 similar-to subscript 𝑣 obs plus-or-minus 6.0 2.0 v_{\rm obs}\sim 6.0\pm 2.0 italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT ∼ 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 similar-to 𝑣 plus-or-minus 250 80 v\sim 250\pm 80 italic_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.

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 v μ ⁢ ( t ) subscript 𝑣 𝜇 𝑡 v_{\mu}(t) italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) , with the corresponding projection of the global Earth motion, say v ~ μ ⁢ ( t ) subscript ~ 𝑣 𝜇 𝑡 \tilde{v}_{\mu}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_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 , 8 , 9 , 10 ] , we started from Maxwell’s original view [ 48 ] 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 fluid [ 52 , 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 v μ ⁢ ( t ) = v ~ μ ⁢ ( t ) subscript 𝑣 𝜇 𝑡 subscript ~ 𝑣 𝜇 𝑡 v_{\mu}(t)=\tilde{v}_{\mu}(t) italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) = over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) , there is also another solution where v μ ⁢ ( t ) subscript 𝑣 𝜇 𝑡 v_{\mu}(t) italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) becomes a continuous but nowhere differentiable velocity field [ 55 , 56 ] 13 13 13 The 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 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 Troshkin [ 59 ] (see also [ 60 ] and [ 61 ] ) within the Navier-Stokes equation, by Saul [ 62 ] within Boltzmann’s transport equation and in [ 63 ] within Landau’s hydrodynamics. The same picture of the vacuum (or ether) as a turbulent fluid was Nelson’s [ 64 ] 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 Schrödinger 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 ] . .

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 similar-to 𝑆 𝜔 superscript 𝜔 1.5 S(\omega)\sim\omega^{-1.5} italic_S ( italic_ω ) ∼ italic_ω start_POSTSUPERSCRIPT - 1.5 end_POSTSUPERSCRIPT of the 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 similar-to 𝑆 𝜔 superscript 𝜔 0.7 \sqrt{S(\omega)}\sim\omega^{-0.7} square-root start_ARG italic_S ( italic_ω ) end_ARG ∼ italic_ω start_POSTSUPERSCRIPT - 0.7 end_POSTSUPERSCRIPT 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 irregular 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 remains 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 parameters ( V , α , γ ) CMB subscript 𝑉 𝛼 𝛾 CMB (V,\alpha,\gamma)_{\rm CMB} ( italic_V , italic_α , italic_γ ) start_POSTSUBSCRIPT roman_CMB end_POSTSUBSCRIPT for the Earth motion in the CMB, a latitude ϕ italic-ϕ \phi italic_ϕ of the laboratory and a given sidereal time τ = ω sid ⁢ t 𝜏 subscript 𝜔 sid 𝑡 \tau=\omega_{\rm sid}t italic_τ = italic_ω start_POSTSUBSCRIPT roman_sid end_POSTSUBSCRIPT italic_t of the observations (with ω sid ∼ 2 ⁢ π 23 h ⁢ 56 ′ similar-to subscript 𝜔 sid 2 𝜋 superscript 23 ℎ superscript 56 ′ \omega_{\rm sid}\sim{{2\pi}\over{23^{h}56^{\prime}}} italic_ω start_POSTSUBSCRIPT roman_sid end_POSTSUBSCRIPT ∼ divide start_ARG 2 italic_π end_ARG start_ARG 23 start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT 56 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ). 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 trigonometry [ 71 ] gives the projections in the ( x , y ) 𝑥 𝑦 (x,y) ( italic_x , italic_y ) plane of the interferometer

with a magnitude

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

where v ⁢ ( t ) 𝑣 𝑡 v(t) italic_v ( italic_t ) and θ 2 ⁢ ( t ) subscript 𝜃 2 𝑡 \theta_{2}(t) italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) now indicate respectively the magnitude and direction of the local drift in the same ( x , y ) 𝑥 𝑦 (x,y) ( italic_x , italic_y ) plane of the interferometer. This can also be re-written as

and v x ⁢ ( t ) = v ⁢ ( t ) ⁢ cos ⁡ θ 2 ⁢ ( t ) subscript 𝑣 𝑥 𝑡 𝑣 𝑡 subscript 𝜃 2 𝑡 v_{x}(t)=v(t)\cos\theta_{2}(t) italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) = italic_v ( italic_t ) roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , v y ⁢ ( t ) = v ⁢ ( t ) ⁢ sin ⁡ θ 2 ⁢ ( t ) subscript 𝑣 𝑦 𝑡 𝑣 𝑡 subscript 𝜃 2 𝑡 v_{y}(t)=v(t)\sin\theta_{2}(t) italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t ) = italic_v ( italic_t ) roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) .

In an analogy with a turbulent flow, the requirement of statistical isotropy means that the local quantities v x ⁢ ( t ) subscript 𝑣 𝑥 𝑡 v_{x}(t) italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) and v y ⁢ ( t ) subscript 𝑣 𝑦 𝑡 v_{y}(t) italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_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 ) subscript ~ 𝑣 𝑥 𝑡 \tilde{v}_{x}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) and v ~ y ⁢ ( t ) subscript ~ 𝑣 𝑦 𝑡 \tilde{v}_{y}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t ) , and can only be simulated numerically. To this end, a representation in terms of random Fourier series [ 55 , 72 , 73 ] was adopted in refs. [ 7 , 8 , 9 , 10 ] in a simplest uniform-probability model, where the kinematic parameters of the global v ~ μ ⁢ ( t ) subscript ~ 𝑣 𝜇 𝑡 \tilde{v}_{\mu}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) are just used to fix the boundaries for the local random v μ ⁢ ( t ) subscript 𝑣 𝜇 𝑡 v_{\mu}(t) italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) . The basic ingredients are summarized in the Appendix.

In this model, the functions S ⁢ ( t ) 𝑆 𝑡 S(t) italic_S ( italic_t ) and C ⁢ ( t ) 𝐶 𝑡 C(t) italic_C ( italic_t ) have the characteristic behaviour of a white-noise signal with vanishing statistical averages ⟨ C ⁢ ( t ) ⟩ stat = 0 subscript delimited-⟨⟩ 𝐶 𝑡 stat 0 \langle C(t)\rangle_{\rm stat}=0 ⟨ italic_C ( italic_t ) ⟩ start_POSTSUBSCRIPT roman_stat end_POSTSUBSCRIPT = 0 and ⟨ S ⁢ ( t ) ⟩ stat = 0 subscript delimited-⟨⟩ 𝑆 𝑡 stat 0 \langle S(t)\rangle_{\rm stat}=0 ⟨ italic_S ( italic_t ) ⟩ start_POSTSUBSCRIPT roman_stat end_POSTSUBSCRIPT = 0 at any time t 𝑡 t italic_t and whatever the global cosmic motion of the Earth. One can then understand the observed irregular behaviour of the fringe shifts

In fact, their averages would be non vanishing just because the statistics is finite. Otherwise with more and more observations one would find

In particular, the direction θ 2 ⁢ ( t ) subscript 𝜃 2 𝑡 \theta_{2}(t) italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) of the local drift, defined by the relation tan ⁡ 2 ⁢ θ 2 ⁢ ( t ) = S ⁢ ( t ) / C ⁢ ( t ) 2 subscript 𝜃 2 𝑡 𝑆 𝑡 𝐶 𝑡 \tan 2\theta_{2}(t)=S(t)/C(t) roman_tan 2 italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = italic_S ( italic_t ) / italic_C ( italic_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 ) subscript ~ 𝐴 2 𝑡 \tilde{A}_{2}(t) over~ start_ARG italic_A end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) associated with the global motion

and then the amplitude A 2 ⁢ ( t ) subscript 𝐴 2 𝑡 A_{2}(t) italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) associated with the local non differentiable velocity components v x ⁢ ( t ) subscript 𝑣 𝑥 𝑡 v_{x}(t) italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) and v y ⁢ ( t ) subscript 𝑣 𝑦 𝑡 v_{y}(t) italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t ) , Eqs.( 83 ) and ( 84 ) of the Appendix

Clearly, the latter will exhibit sizeable fluctuations and be very different from the smooth A ~ 2 ⁢ ( t ) subscript ~ 𝐴 2 𝑡 \tilde{A}_{2}(t) over~ start_ARG italic_A end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) . However, as shown in the Appendix, the relation between A ~ 2 ⁢ ( t ) subscript ~ 𝐴 2 𝑡 \tilde{A}_{2}(t) over~ start_ARG italic_A end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) and the statistical average ⟨ A 2 ⁢ ( t ) ⟩ stat subscript delimited-⟨⟩ subscript 𝐴 2 𝑡 stat \langle A_{2}(t)\rangle_{\rm stat} ⟨ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ⟩ start_POSTSUBSCRIPT roman_stat end_POSTSUBSCRIPT is extremely simple

so that, by averaging the amplitude at different sidereal times, one can obtain the crucial information on the angular parameters α 𝛼 \alpha italic_α and γ 𝛾 \gamma italic_γ .

Altogether, the amplitudes of those old measurements can thus be interpreted in terms of three different velocities: a) as 6 ± 2 plus-or-minus 6 2 6\pm 2 6 ± 2 km/s in a classical picture b) as 250 ± 80 plus-or-minus 250 80 250\pm 80 250 ± 80 km/s, in a modern scheme and in a smooth picture of the drift c) as 340 ± 110 plus-or-minus 340 110 340\pm 110 340 ± 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 / π 2 ∼ 1.35 similar-to 18 superscript 𝜋 2 1.35 \sqrt{18/{\pi^{2}}}\sim 1.35 square-root start_ARG 18 / italic_π start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ∼ 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 book [ 9 ] where a detailed description is given of the experiments by Morley-Miller [ 34 ] , Miller [ 35 ] , Kennedy [ 36 ] , Illingworth [ 37 ] , Tomaschek [ 38 ] and Piccard-Stahel [ 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 Miller, were corresponding to maxima and minima of the ether-drift effect. Then, they were forming the difference

which are the only numbers reported by Pease. These δ − limit-from 𝛿 \delta- italic_δ - values have a maximal magnitude of ± 0.004 plus-or-minus 0.004 \pm 0.004 ± 0.004 and this is also the order of magnitude of the experimental amplitude A 2 EXP ∼ similar-to subscript superscript 𝐴 EXP 2 absent A^{\rm EXP}_{2}\sim italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.005 that is usually reported [ 74 ] for the MPP experiment when comparing with the much larger expected classical amplitudes A 2 class ∼ 0.45 similar-to subscript superscript 𝐴 class 2 0.45 A^{\rm class}_{2}\sim 0.45 italic_A start_POSTSUPERSCRIPT roman_class end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.45 or A 2 class ∼ 0.29 similar-to subscript superscript 𝐴 class 2 0.29 A^{\rm class}_{2}\sim 0.29 italic_A start_POSTSUPERSCRIPT roman_class end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 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 𝛿 \delta italic_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 Pease [ 42 ] (for optical path of fifty-five feet) which indicates a 2nd-harmonic amplitude A 2 EXP ∼ 0.006 similar-to subscript superscript 𝐴 EXP 2 0.006 A^{\rm EXP}_{2}\sim 0.006 italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.006 . By comparing with the classical prediction for 30 km/s, namely A 2 class ∼ 0.29 similar-to subscript superscript 𝐴 class 2 0.29 A^{\rm class}_{2}\sim 0.29 italic_A start_POSTSUPERSCRIPT roman_class end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.29 , this amplitude corresponds to an observable velocity v obs ∼ similar-to subscript 𝑣 obs absent v_{\rm obs}\sim italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT ∼ 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 understanding 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 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 ⟨ A 2 simul ⟩ ∼ similar-to delimited-⟨⟩ subscript superscript 𝐴 simul 2 absent \langle A^{\rm simul}_{2}\rangle\sim ⟨ italic_A start_POSTSUPERSCRIPT roman_simul end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⟩ ∼ 0.014. This corresponds to replace the value of the scalar velocity v ~ ⁢ ( t ) ∼ similar-to ~ 𝑣 𝑡 absent \tilde{v}(t)\sim over~ start_ARG italic_v end_ARG ( italic_t ) ∼ 370 km/s Eq.( 77 ), at the sidereal time of Pease’s observation, in the relation for the statistical average of the amplitude

In the above relation we have replaced D / λ ∼ 2.9 ⋅ 10 7 similar-to 𝐷 𝜆 ⋅ 2.9 superscript 10 7 D/\lambda\sim 2.9\cdot 10^{7} italic_D / italic_λ ∼ 2.9 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT (for optical path of fifty-five feet) and ϵ ∼ 2.8 ⋅ 10 − 4 similar-to italic-ϵ ⋅ 2.8 superscript 10 4 \epsilon\sim 2.8\cdot 10^{-4} italic_ϵ ∼ 2.8 ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT .

Note that the median of the amplitude distribution is about 0.007. As a consequence, the value A 2 EXP ∼ 0.006 similar-to subscript superscript 𝐴 EXP 2 0.006 A^{\rm EXP}_{2}\sim 0.006 italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.006 lies well within the 70 % percent \% % Confidence Limit. Also, the probability content becomes large at very small amplitudes 14 14 14 Strictly 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 Δ Δ \Delta roman_Δ of the apparatus. This smearing would force the curve to bend for A 2 → 0 → subscript 𝐴 2 0 A_{2}\to 0 italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT → 0 and tend to some limit which depends on Δ Δ \Delta roman_Δ . Nevertheless, this refinement should not modify substantially the probability content around the median which is very close to A 2 = 0.007 subscript 𝐴 2 0.007 A_{2}=0.007 italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.007 . and there is a long tail extending up to about A 2 ∼ 0.030 similar-to subscript 𝐴 2 0.030 A_{2}\sim 0.030 italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∼ 0.030 .

The wide interval of amplitudes corresponding to the 70 % percent \% % 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 𝑟 superscript subscript ~ 𝑣 max subscript ~ 𝑣 min 2 r=(\tilde{v}_{\rm max}/\tilde{v}_{\rm min})^{2} italic_r = ( over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT / over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT where v ~ max subscript ~ 𝑣 max \tilde{v}_{\rm max} over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT and v ~ min subscript ~ 𝑣 min \tilde{v}_{\rm min} over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT 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 similar-to 𝑟 2 r\sim 2 italic_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 particularly 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 experiment [ 43 ] and address to our book [ 9 ] 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 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, Swenson [ 76 , 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 ⋅ 10 7 𝐷 𝜆 ⋅ 3.75 superscript 10 7 D/\lambda=3.75\cdot 10^{7} italic_D / italic_λ = 3.75 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT and the refractive index 𝒩 helium ∼ similar-to subscript 𝒩 helium absent {\cal N}_{\rm helium}\sim caligraphic_N start_POSTSUBSCRIPT roman_helium end_POSTSUBSCRIPT ∼ 1.000033 for gaseous helium, an average daily projection of the cosmic Earth velocity v ~ ⁢ ( t ) = V ⁢ | sin ⁡ z ⁢ ( t ) | ∼ ~ 𝑣 𝑡 𝑉 𝑧 𝑡 similar-to absent \tilde{v}(t)=V|\sin z(t)|\sim over~ start_ARG italic_v end_ARG ( italic_t ) = italic_V | roman_sin italic_z ( italic_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 v obs ≲ 1.5 less-than-or-similar-to subscript 𝑣 obs 1.5 v_{\rm obs}\lesssim 1.5 italic_v start_POSTSUBSCRIPT roman_obs end_POSTSUBSCRIPT ≲ 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 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 ⋅ plus-or-minus 4.1 0.3 superscript 10 3 (4.1\pm 0.3)\cdot 10^{-3} ( 4.1 ± 0.3 ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT is abot ten times larger than the average amplitude ( 0.4 ± 0.3 ) ⋅ 10 − 3 ⋅ plus-or-minus 0.4 0.3 superscript 10 3 (0.4\pm 0.3)\cdot 10^{-3} ( 0.4 ± 0.3 ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT 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 similar-to subscript ~ 𝑣 min 250 (\tilde{v})_{\rm min}\sim 250 ( over~ start_ARG italic_v end_ARG ) start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT ∼ 250 km/s and ( v ~ ) max ∼ 370 similar-to subscript ~ 𝑣 max 370 (\tilde{v})_{\rm max}\sim 370 ( over~ start_ARG italic_v end_ARG ) start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ∼ 370 km/s. To understand these characteristic fluctuations, we have thus performed various numerical simulations of these amplitudes [ 7 , 9 ] in the stochastic model described in the Appendix and using the kinematical parameters ( V , α , γ ) CMB subscript 𝑉 𝛼 𝛾 CMB (V,\alpha,\gamma)_{\rm CMB} ( italic_V , italic_α , italic_γ ) start_POSTSUBSCRIPT roman_CMB end_POSTSUBSCRIPT 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 in [ 7 , 9 ] ).

85 100 \tilde{v}=305^{+85}_{-100} over~ start_ARG italic_v end_ARG = 305 start_POSTSUPERSCRIPT + 85 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 100 end_POSTSUBSCRIPT km/s. Second, when fitting with Eqs.( 76 ) and ( 77 ) the smooth black curve of the Joos data in Fig. 17 one finds [ 7 ] a right ascension α ⁢ ( fit − Joos ) = ( 168 ± 30 ) 𝛼 fit Joos plus-or-minus 168 30 \alpha({\rm fit-Joos})=(168\pm 30) italic_α ( roman_fit - roman_Joos ) = ( 168 ± 30 ) degrees and an angular declination γ ⁢ ( fit − Joos ) = ( − 13 ± 14 ) 𝛾 fit Joos plus-or-minus 13 14 \gamma({\rm fit-Joos})=(-13\pm 14) italic_γ ( roman_fit - roman_Joos ) = ( - 13 ± 14 ) degrees which are consistent with the present values α ⁢ ( CMB ) ∼ similar-to 𝛼 CMB absent \alpha({\rm CMB})\sim italic_α ( roman_CMB ) ∼ 168 degrees and γ ⁢ ( CMB ) ∼ − similar-to 𝛾 CMB \gamma({\rm CMB})\sim- italic_γ ( roman_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 4.3 .

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 ~ ≲ less-than-or-similar-to absent ~ 𝑣 less-than-or-similar-to absent \lesssim\tilde{v}\lesssim ≲ over~ start_ARG italic_v end_ARG ≲ 370 km/s of the Earth motion in the CMB and refractivities ϵ = 2.8 ⋅ 10 − 4 italic-ϵ ⋅ 2.8 superscript 10 4 \epsilon=2.8\cdot 10^{-4} italic_ϵ = 2.8 ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT for air and ϵ = 3.3 ⋅ 10 − 5 italic-ϵ ⋅ 3.3 superscript 10 5 \epsilon=3.3\cdot 10^{-5} italic_ϵ = 3.3 ⋅ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT 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 explicitly [ 42 ] and where the optical path was still fifty-five feet. The symbol ± … . plus-or-minus … \pm.... ± … . means that the experimental uncertainty cannot be determined from the available informations. The table is taken from ref. [ 10 ] . \toprule Experiment gas      A 2 EXP subscript superscript 𝐴 EXP 2 A^{\rm EXP}_{2} italic_A start_POSTSUPERSCRIPT roman_EXP end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT     2 ⁢ D λ 2 𝐷 𝜆 {{2D}\over{\lambda}} divide start_ARG 2 italic_D end_ARG start_ARG italic_λ end_ARG     ⟨ A 2 ⁢ ( t ) ⟩ stat subscript delimited-⟨⟩ subscript 𝐴 2 𝑡 stat \langle A_{2}(t)\rangle_{\rm stat} ⟨ italic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ⟩ start_POSTSUBSCRIPT roman_stat end_POSTSUBSCRIPT Michelson(1881) air ( 7.8 ± … . ) ⋅ 10 − 3 (7.8\pm....)\cdot 10^{-3} ( 7.8 ± … . ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT     4 ⋅ 10 6 ⋅ 4 superscript 10 6 4\cdot 10^{6} 4 ⋅ 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ( 0.7 ± 0.2 ) ⋅ 10 − 3 ⋅ plus-or-minus 0.7 0.2 superscript 10 3 (0.7\pm 0.2)\cdot 10^{-3} ( 0.7 ± 0.2 ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT Michelson-Morley(1887) air ( 1.6 ± 0.6 ) ⋅ 10 − 2 ⋅ plus-or-minus 1.6 0.6 superscript 10 2 (1.6\pm 0.6)\cdot 10^{-2} ( 1.6 ± 0.6 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT     4 ⋅ 10 7 ⋅ 4 superscript 10 7 4\cdot 10^{7} 4 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ( 0.7 ± 0.2 ) ⋅ 10 − 2 ⋅ plus-or-minus 0.7 0.2 superscript 10 2 (0.7\pm 0.2)\cdot 10^{-2} ( 0.7 ± 0.2 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT Morley-Miller(1902-1905) air ( 4.0 ± 2.0 ) ⋅ 10 − 2 ⋅ plus-or-minus 4.0 2.0 superscript 10 2 (4.0\pm 2.0)\cdot 10^{-2} ( 4.0 ± 2.0 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT     1.12 ⋅ 10 8 ⋅ 1.12 superscript 10 8 1.12\cdot 10^{8} 1.12 ⋅ 10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT ( 2.0 ± 0.7 ) ⋅ 10 − 2 ⋅ plus-or-minus 2.0 0.7 superscript 10 2 (2.0\pm 0.7)\cdot 10^{-2} ( 2.0 ± 0.7 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT Miller(1921-1926) air ( 4.4 ± 2.2 ) ⋅ 10 − 2 ⋅ plus-or-minus 4.4 2.2 superscript 10 2 (4.4\pm 2.2)\cdot 10^{-2} ( 4.4 ± 2.2 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT    1.12 ⋅ 10 8 ⋅ 1.12 superscript 10 8 1.12\cdot 10^{8} 1.12 ⋅ 10 start_POSTSUPERSCRIPT 8 end_POSTSUPERSCRIPT ( 2.0 ± 0.7 ) ⋅ 10 − 2 ⋅ plus-or-minus 2.0 0.7 superscript 10 2 (2.0\pm 0.7)\cdot 10^{-2} ( 2.0 ± 0.7 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT Tomaschek (1924) air ( 1.0 ± 0.6 ) ⋅ 10 − 2 ⋅ plus-or-minus 1.0 0.6 superscript 10 2 (1.0\pm 0.6)\cdot 10^{-2} ( 1.0 ± 0.6 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT     3 ⋅ 10 7 ⋅ 3 superscript 10 7 3\cdot 10^{7} 3 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ( 0.5 ± 0.2 ) ⋅ 10 − 2 ⋅ plus-or-minus 0.5 0.2 superscript 10 2 (0.5\pm 0.2)\cdot 10^{-2} ( 0.5 ± 0.2 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT Kennedy(1926) helium     < 0.002 absent 0.002 <0.002 < 0.002     7 ⋅ 10 6 ⋅ 7 superscript 10 6 7\cdot 10^{6} 7 ⋅ 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ( 1.4 ± 0.5 ) ⋅ 10 − 4 ⋅ plus-or-minus 1.4 0.5 superscript 10 4 (1.4\pm 0.5)\cdot 10^{-4} ( 1.4 ± 0.5 ) ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Illingworth(1927) helium ( 2.2 ± 1.7 ) ⋅ 10 − 4 ⋅ plus-or-minus 2.2 1.7 superscript 10 4 (2.2\pm 1.7)\cdot 10^{-4} ( 2.2 ± 1.7 ) ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT     7 ⋅ 10 6 ⋅ 7 superscript 10 6 7\cdot 10^{6} 7 ⋅ 10 start_POSTSUPERSCRIPT 6 end_POSTSUPERSCRIPT ( 1.4 ± 0.5 ) ⋅ 10 − 4 ⋅ plus-or-minus 1.4 0.5 superscript 10 4 (1.4\pm 0.5)\cdot 10^{-4} ( 1.4 ± 0.5 ) ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT Piccard-Stahel(1928) air ( 2.8 ± 1.5 ) ⋅ 10 − 3 ⋅ plus-or-minus 2.8 1.5 superscript 10 3 (2.8\pm 1.5)\cdot 10^{-3} ( 2.8 ± 1.5 ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT     1.28 ⋅ 10 7 ⋅ 1.28 superscript 10 7 1.28\cdot 10^{7} 1.28 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ( 2.2 ± 0.8 ) ⋅ 10 − 3 ⋅ plus-or-minus 2.2 0.8 superscript 10 3 (2.2\pm 0.8)\cdot 10^{-3} ( 2.2 ± 0.8 ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT Mich.-Pease-Pearson(1929) air ( 0.6 ± … ) ⋅ 10 − 2 ⋅ plus-or-minus 0.6 … superscript 10 2 (0.6\pm...)\cdot 10^{-2} ( 0.6 ± … ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT     5.8 ⋅ 10 7 ⋅ 5.8 superscript 10 7 5.8\cdot 10^{7} 5.8 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ( 1.0 ± 0.4 ) ⋅ 10 − 2 ⋅ plus-or-minus 1.0 0.4 superscript 10 2 (1.0\pm 0.4)\cdot 10^{-2} ( 1.0 ± 0.4 ) ⋅ 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT Joos(1930) helium ( 1.4 ± 0.8 ) ⋅ 10 − 3 ⋅ plus-or-minus 1.4 0.8 superscript 10 3 (1.4\pm 0.8)\cdot 10^{-3} ( 1.4 ± 0.8 ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT    7.5 ⋅ 10 7 ⋅ 7.5 superscript 10 7 7.5\cdot 10^{7} 7.5 ⋅ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT ( 1.5 ± 0.6 ) ⋅ 10 − 3 ⋅ plus-or-minus 1.5 0.6 superscript 10 3 (1.5\pm 0.6)\cdot 10^{-3} ( 1.5 ± 0.6 ) ⋅ 10 start_POSTSUPERSCRIPT - 3 end_POSTSUPERSCRIPT \botrule

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.

60 80 \langle\tilde{v}\rangle\sim 332^{+60}_{-80} ⟨ over~ start_ARG italic_v end_ARG ⟩ ∼ 332 start_POSTSUPERSCRIPT + 60 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 80 end_POSTSUBSCRIPT 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 𝑘 {}^{k} start_FLOATSUPERSCRIPT italic_k end_FLOATSUPERSCRIPT ), symmetry arguments can successfully describe a phenomenon regardless of the physical mechanisms behind it. The same is true here with our relation | Δ ⁢ c ¯ θ | c ∼ ϵ ⁢ ( v 2 / c 2 ) similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 italic-ϵ superscript 𝑣 2 superscript 𝑐 2 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim\epsilon(v^{2}/c^{2}) divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) 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 a strongly bound solid dielectric, where the refractivity is ( 𝒩 solid − 1 ) = O ⁢ ( 1 ) subscript 𝒩 solid 1 𝑂 1 ({\cal N}_{\rm solid}-1)=O(1) ( caligraphic_N start_POSTSUBSCRIPT roman_solid end_POSTSUBSCRIPT - 1 ) = italic_O ( 1 ) , and thus produce a much larger | Δ ⁢ c ¯ θ | c ∼ ( 𝒩 solid − 1 ) ⁢ ( v 2 / c 2 ) ∼ 10 − 6 similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 subscript 𝒩 solid 1 superscript 𝑣 2 superscript 𝑐 2 similar-to superscript 10 6 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim({\cal N}_{\rm solid}-1)(v^{2}/c^{2})% \sim 10^{-6} divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ ( caligraphic_N start_POSTSUBSCRIPT roman_solid end_POSTSUBSCRIPT - 1 ) ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∼ 10 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT . This is in contrast with the Shamir-Fox [ 78 ] experiment in perspex where the observed value was smaller by orders of magnitude.

We have thus re-considered [ 79 , 8 , 9 ] the traditional thermal interpretation [ 80 , 74 ] of the observed residuals. The idea was that, in a weakly bound system as a gas, a small temperature difference Δ ⁢ T gas ⁢ ( θ ) Δ superscript 𝑇 gas 𝜃 \Delta T^{\rm gas}(\theta) roman_Δ italic_T start_POSTSUPERSCRIPT roman_gas end_POSTSUPERSCRIPT ( italic_θ ) in the air of the two optical arms produces a difference in the refractive index and a ( Δ ⁢ c ¯ θ / c ) ∼ ϵ gas ⁢ Δ ⁢ T gas ⁢ ( θ ) / T similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 subscript italic-ϵ gas Δ superscript 𝑇 gas 𝜃 𝑇 (\Delta\bar{c}_{\theta}/c)\sim\epsilon_{\rm gas}\Delta T^{\rm gas}(\theta)/T ( roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT / italic_c ) ∼ italic_ϵ start_POSTSUBSCRIPT roman_gas end_POSTSUBSCRIPT roman_Δ italic_T start_POSTSUPERSCRIPT roman_gas end_POSTSUPERSCRIPT ( italic_θ ) / italic_T , where T ∼ similar-to 𝑇 absent T\sim italic_T ∼ 300 K is the absolute temperature of the laboratory 15 15 15 The 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), see [ 8 , 9 ] for the details. . Miller was aware [ 35 , 75 ] that his results could be due to a Δ ⁢ T gas ⁢ ( θ ) ≲ less-than-or-similar-to Δ superscript 𝑇 gas 𝜃 absent \Delta T^{\rm gas}(\theta)\lesssim roman_Δ italic_T start_POSTSUPERSCRIPT roman_gas end_POSTSUPERSCRIPT ( italic_θ ) ≲ 1 mK but objected that casual changes of the ambiance temperature would largely cancel when averaging over many measurements. Only temperature effects with a definite periodicity would survive. For a quantitative estimate, by averaging over all experiments in Table 4.3 we found ⟨ v ~ exp ⟩ ∼ 376 ± 46 similar-to delimited-⟨⟩ subscript ~ 𝑣 exp plus-or-minus 376 46 \langle\tilde{v}_{\rm exp}\rangle\sim 376\pm 46 ⟨ over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT ⟩ ∼ 376 ± 46 km/s. Therefore, by comparing Eq.( 40 ) with the corresponding form for a thermal light anisotropy, we find

and a value [ 8 , 9 ] | Δ ⁢ T gas ⁢ ( θ ) | ∼ ( 0.26 ± 0.07 ) similar-to Δ superscript 𝑇 gas 𝜃 plus-or-minus 0.26 0.07 |\Delta T^{\rm gas}(\theta)|\sim(0.26\pm 0.07) | roman_Δ italic_T start_POSTSUPERSCRIPT roman_gas end_POSTSUPERSCRIPT ( italic_θ ) | ∼ ( 0.26 ± 0.07 ) mK 16 16 16 Note that in Eq.( 43 ) the gas refractivity drops out. The old estimates [ 80 , 74 ] of about 1 mK, based on the relation ϵ gas ⁢ Δ ⁢ T gas ⁢ ( θ ) / T ∼ ( v Miller 2 / 2 ⁢ c 2 ) similar-to subscript italic-ϵ gas Δ superscript 𝑇 gas 𝜃 𝑇 subscript superscript 𝑣 2 Miller 2 superscript 𝑐 2 \epsilon_{\rm gas}\Delta T^{\rm gas}(\theta)/T\sim(v^{2}_{\rm Miller}/2c^{2}) italic_ϵ start_POSTSUBSCRIPT roman_gas end_POSTSUBSCRIPT roman_Δ italic_T start_POSTSUPERSCRIPT roman_gas end_POSTSUPERSCRIPT ( italic_θ ) / italic_T ∼ ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Miller end_POSTSUBSCRIPT / 2 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , with v Miller ∼ 10 similar-to subscript 𝑣 Miller 10 v_{\rm Miller}\sim 10 italic_v start_POSTSUBSCRIPT roman_Miller end_POSTSUBSCRIPT ∼ 10 km/s, were slightly too large. .

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 , 82 , 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 𝑘 {}^{k} start_FLOATSUPERSCRIPT italic_k end_FLOATSUPERSCRIPT 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 Δ ⁢ T CMB ⁢ ( θ ) Δ superscript 𝑇 CMB 𝜃 \Delta T^{\rm CMB}(\theta) roman_Δ italic_T start_POSTSUPERSCRIPT roman_CMB end_POSTSUPERSCRIPT ( italic_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 ϵ gas → 0 → subscript italic-ϵ gas 0 \epsilon_{\rm gas}\to 0 italic_ϵ start_POSTSUBSCRIPT roman_gas end_POSTSUBSCRIPT → 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 Δ ⁢ T Δ 𝑇 \Delta T roman_Δ italic_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 experiments [ 44 ] where Δ ⁢ c ¯ θ c ∼ Δ ⁢ ν ⁢ ( θ ) ν 0 similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 Δ 𝜈 𝜃 subscript 𝜈 0 {{\Delta\bar{c}_{\theta}}\over{c}}\sim{{\Delta\nu(\theta)}\over{\nu_{0}}} divide start_ARG roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_ARG start_ARG italic_c end_ARG ∼ divide start_ARG roman_Δ italic_ν ( italic_θ ) end_ARG start_ARG italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG 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 ] , see Black’s tutorial article [ 87 ] 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 CH 4 4 {}_{4} start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT reference laser (fixed in the laboratory) with the frequency of a cavity-stabilized He-Ne laser ( ν 0 ∼ 8.8 ⋅ 10 13 similar-to subscript 𝜈 0 ⋅ 8.8 superscript 10 13 \nu_{0}\sim 8.8\cdot 10^{13} italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∼ 8.8 ⋅ 10 start_POSTSUPERSCRIPT 13 end_POSTSUPERSCRIPT Hz) placed on a rotating table. Since the stabilizing optical cavity was placed inside a vacuum envelope, the measured shift Δ ⁢ ν ⁢ ( θ ) Δ 𝜈 𝜃 \Delta\nu(\theta) roman_Δ italic_ν ( italic_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 ± plus-or-minus \pm ± 20 Hz for a 1-second measurement, and comparable to the stability of the reference CH 4 4 {}_{4} start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT 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 | Δ ⁢ ν | ∼ similar-to Δ 𝜈 absent |\Delta\nu|\sim | roman_Δ italic_ν | ∼ 17 Hz (or a relative level 10 − 13 superscript 10 13 10^{-13} 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT ) and with a phase θ 2 ⁢ ( t ) subscript 𝜃 2 𝑡 \theta_{2}(t) italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_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 plus-or-minus 1 \pm 1 ± 1 Hz. Finally, by further averaging over a period of about 200 days, the residual ether-drift effect was an average frequency shift ⟨ Δ ⁢ ν ⟩ = delimited-⟨⟩ Δ 𝜈 absent \langle\Delta\nu\rangle= ⟨ roman_Δ italic_ν ⟩ = 0.13 ± plus-or-minus \pm ± 0.22 Hz, i.e. about 100 times smaller than the instantaneous | Δ ⁢ ν | Δ 𝜈 |\Delta\nu| | roman_Δ italic_ν | .

Since the 1979 Brillet-Hall article, substantial improvements have been introduced in the experiments. Just to have an idea, in present-day measurements [ 90 , 68 ] with vacuum cavities the typical magnitude of the instantaneous fractional signal | Δ ⁢ ν | / ν 0 Δ 𝜈 subscript 𝜈 0 |\Delta\nu|/\nu_{0} | roman_Δ italic_ν | / italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT has been lowered from 10 − 13 superscript 10 13 10^{-13} 10 start_POSTSUPERSCRIPT - 13 end_POSTSUPERSCRIPT to 10 − 15 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT , 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 ⟨ Δ ⁢ ν ν 0 ⟩ ≲ 10 − 18 less-than-or-similar-to delimited-⟨⟩ Δ 𝜈 subscript 𝜈 0 superscript 10 18 \langle{{\Delta\nu}\over{\nu_{0}}}\rangle\lesssim 10^{-18} ⟨ divide start_ARG roman_Δ italic_ν end_ARG start_ARG italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG ⟩ ≲ 10 start_POSTSUPERSCRIPT - 18 end_POSTSUPERSCRIPT [ 68 ] , i.e. about 1000 times smaller than the instantaneous 10 − 15 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT signal.

The assumptions behind the analysis of the data, however, are basically unchanged. 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 emphasized 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 superscript 10 9 10^{-9} 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT 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 ⟨ Δ ⁢ ν / ν 0 ⟩ ≲ 10 − 18 less-than-or-similar-to delimited-⟨⟩ Δ 𝜈 subscript 𝜈 0 superscript 10 18 \langle\Delta\nu/\nu_{0}\rangle\lesssim 10^{-18} ⟨ roman_Δ italic_ν / italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟩ ≲ 10 start_POSTSUPERSCRIPT - 18 end_POSTSUPERSCRIPT [ 68 ] . This just reflects the very irregular nature of the signal because its typical magnitude | Δ ⁢ ν | / ν 0 ∼ 10 − 15 similar-to Δ 𝜈 subscript 𝜈 0 superscript 10 15 |\Delta\nu|/\nu_{0}\sim 10^{-15} | roman_Δ italic_ν | / italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT is about 1000 times larger, see Fig. 19 or panel b) of Fig. 11 .

The most interesting aspect however is that this 10 − 15 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT 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 noise [ 93 ] ) remain the same for experiments operating in so different conditions, one can meaningfully explore the possibility that such an irregular 10 − 15 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT signal admits a physical interpretation.

1 subscript italic-ϵ 𝑣 {\cal N}_{v}=1+\epsilon_{v} caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = 1 + italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT 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 ϵ v subscript italic-ϵ 𝑣 \epsilon_{v} italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT should be at the 10 − 9 superscript 10 9 10^{-9} 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT level, in order to give | Δ ⁢ c ¯ θ | c ∼ ϵ v ⁢ ( v 2 / c 2 ) ∼ 10 − 15 similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 subscript italic-ϵ 𝑣 superscript 𝑣 2 superscript 𝑐 2 similar-to superscript 10 15 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim\epsilon_{v}~{}(v^{2}/c^{2})~{}\sim 10% ^{-15} divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∼ 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT and thus would fit with the original idea of [ 94 ] where, for an apparatus placed on the Earth’s surface, a vacuum refractivity ϵ v ∼ ( 2 ⁢ G N ⁢ M / c 2 ⁢ R ) ∼ 1.39 ⋅ 10 − 9 similar-to subscript italic-ϵ 𝑣 2 subscript 𝐺 𝑁 𝑀 superscript 𝑐 2 𝑅 similar-to ⋅ 1.39 superscript 10 9 \epsilon_{v}\sim(2G_{N}M/c^{2}R)\sim 1.39\cdot 10^{-9} italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∼ ( 2 italic_G start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT italic_M / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_R ) ∼ 1.39 ⋅ 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT was considered, G N subscript 𝐺 𝑁 G_{N} italic_G start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT being the Newton constant and M 𝑀 M italic_M and R 𝑅 R italic_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 scales much larger than the size of their elementary constituents. For this reason, several authors, see e.g. [ 95 , 96 , 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 s k ⁢ ( x ) subscript 𝑠 𝑘 𝑥 s_{k}(x) italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_x ) in flat space expressing the deviations of the effective metric g μ ⁢ ν ⁢ ( x ) subscript 𝑔 𝜇 𝜈 𝑥 g_{\mu\nu}(x) italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( italic_x ) from the Minkowski tensor η μ ⁢ ν subscript 𝜂 𝜇 𝜈 \eta_{\mu\nu} italic_η start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT , i.e.

𝛼 subscript 𝑠 0 t^{\mu}_{\nu}(s_{0})=-\partial^{\mu}s_{0}\partial_{\nu}s_{0}+1/2\delta^{\mu}_{% \nu}~{}\partial^{\alpha}s_{0}\partial_{\alpha}s_{0} italic_t start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ( italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) = - ∂ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 / 2 italic_δ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , to match the Einstein tensor, produces differences from the Schwarzschild metric which are beyond the present experimental accuracy. .

As an immediate consequence, suppose that the s k subscript 𝑠 𝑘 s_{k} italic_s start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ’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 understand [ 97 ] why the huge condensation energy of the unperturbed vacuum plays no role and thus obtain an intuitive solution of the cosmological-constant problem found in connection with the vacuum energy 18 18 18 This 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 ] . .

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 s 0 ⁢ ( x ) subscript 𝑠 0 𝑥 s_{0}(x) italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_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 level [ 100 ] ). How could the effects of s 0 ⁢ ( x ) subscript 𝑠 0 𝑥 s_{0}(x) italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_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 19 19 19 This 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 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 ] . .

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 c γ subscript 𝑐 𝛾 c_{\gamma} italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT is a universal constant, namely it remains the same, basic parameter c 𝑐 c italic_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 c γ subscript 𝑐 𝛾 c_{\gamma} italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT 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 d ⁢ T 𝑑 𝑇 dT italic_d italic_T and d ⁢ L 𝑑 𝐿 dL italic_d italic_L , are needed if all observers have to measure the same, universal speed of light (“Einstein postulate”). For a static metric, these definitions are d ⁢ T 2 = g 00 ⁢ d ⁢ t 2 𝑑 superscript 𝑇 2 subscript 𝑔 00 𝑑 superscript 𝑡 2 dT^{2}=g_{00}dt^{2} italic_d italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_g start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and d ⁢ L 2 = g i ⁢ j ⁢ d ⁢ x i ⁢ d ⁢ x j 𝑑 superscript 𝐿 2 subscript 𝑔 𝑖 𝑗 𝑑 superscript 𝑥 𝑖 𝑑 superscript 𝑥 𝑗 dL^{2}=g_{ij}dx^{i}dx^{j} italic_d italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_g start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_d italic_x start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT . Thus, in General Relativity, the condition d ⁢ s 2 = 0 𝑑 superscript 𝑠 2 0 ds^{2}=0 italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 , which governs the propagation of light, can be expressed formally as

and, by construction, yields the same universal speed c = d ⁢ L / d ⁢ T 𝑐 𝑑 𝐿 𝑑 𝑇 c=dL/dT italic_c = italic_d italic_L / italic_d italic_T .

For the same reason, however, if the physical units of time and space were instead defined to be d ⁢ T ^ 𝑑 ^ 𝑇 d\hat{T} italic_d over^ start_ARG italic_T end_ARG and d ⁢ L ^ 𝑑 ^ 𝐿 d\hat{L} italic_d over^ start_ARG italic_L end_ARG with, say, d ⁢ T = q ⁢ d ⁢ T ^ 𝑑 𝑇 𝑞 𝑑 ^ 𝑇 dT=q~{}d\hat{T} italic_d italic_T = italic_q italic_d over^ start_ARG italic_T end_ARG and d ⁢ L = p ⁢ d ⁢ L ^ 𝑑 𝐿 𝑝 𝑑 ^ 𝐿 dL=p~{}d\hat{L} italic_d italic_L = italic_p italic_d over^ start_ARG italic_L end_ARG , the same condition

would now be interpreted in terms of the different speed

The possibility of different standards for space-time measurements is thus a simple motivation for an effective vacuum refractive index 𝒩 v ≠ 1 subscript 𝒩 𝑣 1 {\cal N}_{v}\neq 1 caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≠ 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 c γ subscript 𝑐 𝛾 c_{\gamma} italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT , when measured in an inertial frame, coincides with the basic parameter c 𝑐 c italic_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 c γ = c subscript 𝑐 𝛾 𝑐 c_{\gamma}=c italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = italic_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 tidal effects of the external gravitational potential U ext ⁢ ( x ) subscript 𝑈 ext 𝑥 U_{\rm ext}(x) italic_U start_POSTSUBSCRIPT roman_ext end_POSTSUBSCRIPT ( italic_x ) can be ignored). Note that this is essential to obtain an operational definition of the otherwise unknown parameter c 𝑐 c italic_c .

As already discussed in ref. [ 94 ] , light propagation for an observer S 𝑆 S italic_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 𝑆 S italic_S is considered a freely falling frame. This amounts to assume c γ = c subscript 𝑐 𝛾 𝑐 c_{\gamma}=c italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = italic_c so that, given two events which, in terms of the local space-time units of S 𝑆 S italic_S , differ by ( d ⁢ x , d ⁢ y , d ⁢ z , d ⁢ t ) 𝑑 𝑥 𝑑 𝑦 𝑑 𝑧 𝑑 𝑡 (dx,dy,dz,dt) ( italic_d italic_x , italic_d italic_y , italic_d italic_z , italic_d italic_t ) , light propagation is described by the condition (ff=’free-fall’)

ii) To a closer look, however, an observer S 𝑆 S italic_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 𝑆 S italic_S as a freely-falling frame, in the same external gravitational field described by U ext ⁢ ( x ) subscript 𝑈 ext 𝑥 U_{\rm ext}(x) italic_U start_POSTSUBSCRIPT roman_ext end_POSTSUBSCRIPT ( italic_x ) , that however is also carrying on board a heavy object of mass M 𝑀 M italic_M (the Earth mass itself) which affects the local space-time structure, see Fig. 20 . To derive the required correction, let us denote by δ ⁢ U 𝛿 𝑈 \delta U italic_δ italic_U the extra Newtonian potential produced by the heavy mass M 𝑀 M italic_M at the experimental set up where one wants to describe light propagation. According to General Relativity, and to first order in δ ⁢ U 𝛿 𝑈 \delta U italic_δ italic_U , light propagation for the S 𝑆 S italic_S observer is now described by

𝑑 superscript 𝑥 2 𝑑 superscript 𝑦 2 𝑑 superscript 𝑧 2 dL^{2}=(1+2{{|\delta U|}\over{c^{2}}})(dx^{2}+dy^{2}+dz^{2}) italic_d italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( 1 + 2 divide start_ARG | italic_δ italic_U | end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ( italic_d italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_d italic_z start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) are the physical units of General Relativity in terms of which one obtains the universal value d ⁢ L / d ⁢ T = c γ = c 𝑑 𝐿 𝑑 𝑇 subscript 𝑐 𝛾 𝑐 dL/dT=c_{\gamma}=c italic_d italic_L / italic_d italic_T = italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = italic_c .

Though, to check experimentally the assumed identity c γ = c subscript 𝑐 𝛾 𝑐 c_{\gamma}=c italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = italic_c one should compare with a theoretical prediction for ( c − c γ ) 𝑐 subscript 𝑐 𝛾 (c-c_{\gamma}) ( italic_c - italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) 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 d ⁢ L ^ = d ⁢ L 𝑑 ^ 𝐿 𝑑 𝐿 d\hat{L}=dL italic_d over^ start_ARG italic_L end_ARG = italic_d italic_L but change the unit of time from d ⁢ T 𝑑 𝑇 dT italic_d italic_T to d ⁢ T ^ 𝑑 ^ 𝑇 d\hat{T} italic_d over^ start_ARG italic_T end_ARG . The reason derives from the observation that physical units of time scale as inverse frequencies and that the measured frequencies ω ^ ^ 𝜔 \hat{\omega} over^ start_ARG italic_ω end_ARG for δ ⁢ U ≠ 0 𝛿 𝑈 0 \delta U\neq 0 italic_δ italic_U ≠ 0 , when compared to the corresponding value ω 𝜔 \omega italic_ω for δ ⁢ U = 0 𝛿 𝑈 0 \delta U=0 italic_δ italic_U = 0 , are red-shifted according to

Therefore, rather than the natural unit of time d ⁢ T = ( 1 − | δ ⁢ U | c 2 ) ⁢ d ⁢ t 𝑑 𝑇 1 𝛿 𝑈 superscript 𝑐 2 𝑑 𝑡 dT=(1-{{|\delta U|}\over{c^{2}}})dt italic_d italic_T = ( 1 - divide start_ARG | italic_δ italic_U | end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) italic_d italic_t of General Relativity, one could consider the alternative, but natural (see our footnote s 𝑠 {}^{s} start_FLOATSUPERSCRIPT italic_s end_FLOATSUPERSCRIPT ), unit of time

Then, to reproduce d ⁢ s 2 = 0 𝑑 superscript 𝑠 2 0 ds^{2}=0 italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0 , we can introduce a vacuum refractive index

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

This gives d ⁢ L ^ / d ⁢ T ^ = c γ = c 𝒩 v 𝑑 ^ 𝐿 𝑑 ^ 𝑇 subscript 𝑐 𝛾 𝑐 subscript 𝒩 𝑣 d\hat{L}/d\hat{T}=c_{\gamma}={{c}\over{{\cal N}_{v}}} italic_d over^ start_ARG italic_L end_ARG / italic_d over^ start_ARG italic_T end_ARG = italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = divide start_ARG italic_c end_ARG start_ARG caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT end_ARG and, for an observer placed on the Earth’s surface, a refractivity

M 𝑀 M italic_M and R 𝑅 R italic_R being respectively the Earth mass and radius.

Notice that, with this natural definition d ⁢ T ^ 𝑑 ^ 𝑇 d\hat{T} italic_d over^ start_ARG italic_T end_ARG , the vacuum refractive index associated 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 deflection 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 permeability [ 105 ] ϵ i ⁢ k = μ i ⁢ k = − g ⁢ ( − g i ⁢ k ) g 00 subscript italic-ϵ 𝑖 𝑘 subscript 𝜇 𝑖 𝑘 𝑔 superscript 𝑔 𝑖 𝑘 subscript 𝑔 00 \epsilon_{ik}=\mu_{ik}=\sqrt{-g}~{}{{(-g^{ik})}\over{g_{00}}} italic_ϵ start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT = square-root start_ARG - italic_g end_ARG divide start_ARG ( - italic_g start_POSTSUPERSCRIPT italic_i italic_k end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_g start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_ARG . Indeed, in our case, from the relations g i ⁢ l ⁢ g l ⁢ k = δ k i superscript 𝑔 𝑖 𝑙 subscript 𝑔 𝑙 𝑘 subscript superscript 𝛿 𝑖 𝑘 g^{il}g_{lk}=\delta^{i}_{k} italic_g start_POSTSUPERSCRIPT italic_i italic_l end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_l italic_k end_POSTSUBSCRIPT = italic_δ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , ( − g i ⁢ k ) ∼ δ k i ⁢ g 00 similar-to superscript 𝑔 𝑖 𝑘 subscript superscript 𝛿 𝑖 𝑘 subscript 𝑔 00 (-g^{ik})\sim\delta^{i}_{k}~{}g_{00} ( - italic_g start_POSTSUPERSCRIPT italic_i italic_k end_POSTSUPERSCRIPT ) ∼ italic_δ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT , ϵ i ⁢ k = μ i ⁢ k = δ k i ⁢ 𝒩 v subscript italic-ϵ 𝑖 𝑘 subscript 𝜇 𝑖 𝑘 subscript superscript 𝛿 𝑖 𝑘 subscript 𝒩 𝑣 \epsilon_{ik}=\mu_{ik}=\delta^{i}_{k}{\cal N}_{v} italic_ϵ start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT = italic_μ start_POSTSUBSCRIPT italic_i italic_k end_POSTSUBSCRIPT = italic_δ start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT , we obtain

1 𝛿 𝑈 superscript 𝑐 2 {\cal N}_{v}\sim{{1}\over{\sqrt{g_{00}}}}\sim 1+{{|\delta U|}\over{c^{2}}} caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∼ divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_g start_POSTSUBSCRIPT 00 end_POSTSUBSCRIPT end_ARG end_ARG ∼ 1 + divide start_ARG | italic_δ italic_U | end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG . Concerning, these two possible definitions of 𝒩 v subscript 𝒩 𝑣 {\cal N}_{v} caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT , 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 c γ subscript 𝑐 𝛾 c_{\gamma} italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT , 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 superscript 10 9 10^{-9} 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT from the ideal value c 𝑐 c italic_c , as operationally defined with the same apparatus in a true freely-falling frame (panel (a) in our Fig. 20 ). In conclusion, this c γ − c subscript 𝑐 𝛾 𝑐 c_{\gamma}-c italic_c start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT - italic_c difference can be conveniently expressed through a vacuum refractivity of the form

𝜒 2 0.42 +{{\chi}\over{2}}(0.42) + divide start_ARG italic_χ end_ARG start_ARG 2 end_ARG ( 0.42 ) m/s with χ = 𝜒 absent \chi= italic_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 plus-or-minus ⋅ 4 superscript 10 9 \pm 4\cdot 10^{-9} ± 4 ⋅ 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT at the 3-sigma level or, equivalently, ± 1.2 plus-or-minus 1.2 \pm 1.2 ± 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 𝒩 v ≠ 1 subscript 𝒩 𝑣 1 {\cal N}_{v}\neq 1 caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≠ 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 Σ \Sigma roman_Σ while for a non-zero relative velocity there would be off diagonal elements g 0 ⁢ i ≠ 0 subscript 𝑔 0 𝑖 0 g_{0i}\neq 0 italic_g start_POSTSUBSCRIPT 0 italic_i end_POSTSUBSCRIPT ≠ 0 in the effective metric [ 105 ] . If Σ Σ \Sigma roman_Σ exists, we would then expect a light anisotropy | Δ ⁢ c ¯ θ | c ∼ ϵ v ⁢ ( v / c ) 2 ∼ 10 − 15 similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 subscript italic-ϵ 𝑣 superscript 𝑣 𝑐 2 similar-to superscript 10 15 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim\epsilon_{v}(v/c)^{2}\sim 10^{-15} divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_v / italic_c ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ∼ 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT , 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 experiment 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 ÷ \div ÷ 2 seconds, see Fig. 21 . For a very irregular signal, in fact, this typical variation, of about 10 − 15 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT , gives the magnitude of the instantaneous signal itself and, indeed, it is in good agreement with the mentioned room-temperature measurements.

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 obtained by sampling a function f = f ⁢ ( t ) 𝑓 𝑓 𝑡 f=f(t) italic_f = italic_f ( italic_t ) at steps of time τ 𝜏 \tau italic_τ . By defining

one generates a τ − limit-from 𝜏 \tau- italic_τ - dependent distribution of f ¯ i subscript ¯ 𝑓 𝑖 {\overline{f}}_{i} over¯ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT values. In a large time interval Λ = M ⁢ τ Λ 𝑀 𝜏 \Lambda=M\tau roman_Λ = italic_M italic_τ , the RAV is then defined as

and the factor of 2 is introduced to obtain the standard variance σ ⁢ ( f ) 𝜎 𝑓 \sigma(f) italic_σ ( italic_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 ⋅ 1.2 superscript 10 15 1.2\cdot 10^{-15} 1.2 ⋅ 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT . The lower solid, dashed and dot-dashed curves, denoted as ‘predicted newCORE’, indicate instead possible improved limits ( 2 ÷ 4 ) ⋅ 10 − 17 ⋅ 2 4 superscript 10 17 (2\div 4)\cdot 10^{-17} ( 2 ÷ 4 ) ⋅ 10 start_POSTSUPERSCRIPT - 17 end_POSTSUPERSCRIPT 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 ¯ ¯ 𝜏 \bar{\tau} over¯ start_ARG italic_τ end_ARG 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 ¯ ¯ 𝜏 \bar{\tau} over¯ start_ARG italic_τ end_ARG is typically 1 ÷ 2 1 2 1\div 2 1 ÷ 2 seconds so that one gets the relation with the average magnitude of the instantaneous signal

5.3 Comparing our model with experiments in vacuum

𝜃 𝜋 2 𝑡 {\Delta\nu(\theta;t)=\nu_{1}(\theta;t)-\nu_{2}(\theta+\pi/2;t)} roman_Δ italic_ν ( italic_θ ; italic_t ) = italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_θ ; italic_t ) - italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_θ + italic_π / 2 ; italic_t ) to the angular dependence of the velocity of light, namely see ( 33 )

where S ⁢ ( t ) 𝑆 𝑡 S(t) italic_S ( italic_t ) and C ⁢ ( t ) 𝐶 𝑡 C(t) italic_C ( italic_t ) are given in Eqs.( 34 ). As in the case of the classical experiments, the velocity components v x ⁢ ( t ) subscript 𝑣 𝑥 𝑡 v_{x}(t) italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) and v y ⁢ ( t ) subscript 𝑣 𝑦 𝑡 v_{y}(t) italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_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 2 ⁢ C ⁢ ( t ) 2 𝐶 𝑡 2C(t) 2 italic_C ( italic_t ) and 2 ⁢ S ⁢ ( t ) 2 𝑆 𝑡 2S(t) 2 italic_S ( italic_t ) is shown in Fig. 23 .

For a quantitative test, we concentrated on the observed value of the RAV of the frequency shift at the end point τ ¯ = 1 ÷ 2 ¯ 𝜏 1 2 \bar{\tau}=1\div 2 over¯ start_ARG italic_τ end_ARG = 1 ÷ 2 seconds of the white-noise branch of the spectrum, see Fig.3, bottom part of [ 68 ] . This has a value

or, in units of the reference frequency ν 0 = 2.8 ⋅ 10 14 subscript 𝜈 0 ⋅ 2.8 superscript 10 14 \nu_{0}=2.8\cdot 10^{14} italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 2.8 ⋅ 10 start_POSTSUPERSCRIPT 14 end_POSTSUPERSCRIPT Hz   [ 68 ]

As anticipated, our instantaneous, stochastic signal for Δ ⁢ ν ⁢ ( t ) Δ 𝜈 𝑡 \Delta\nu(t) roman_Δ italic_ν ( italic_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 ) ⟨ | Δ ⁢ ν | ⟩ stat → ⟨ | Δ ⁢ ν ⁢ ( t ) | ⟩ stat → subscript delimited-⟨⟩ Δ 𝜈 stat subscript delimited-⟨⟩ Δ 𝜈 𝑡 stat \langle|\Delta\nu|\rangle_{\rm stat}\to\langle|\Delta\nu(t)|\rangle_{\rm stat} ⟨ | roman_Δ italic_ν | ⟩ start_POSTSUBSCRIPT roman_stat end_POSTSUBSCRIPT → ⟨ | roman_Δ italic_ν ( italic_t ) | ⟩ start_POSTSUBSCRIPT roman_stat end_POSTSUBSCRIPT and evaluate the RAV in the corresponding temporal range. Therefore, from Δ ⁢ ν ν 0 = Δ ⁢ c ¯ θ c ∼ ϵ v ⋅ v 2 c 2 Δ 𝜈 subscript 𝜈 0 Δ subscript ¯ 𝑐 𝜃 𝑐 similar-to ⋅ subscript italic-ϵ 𝑣 superscript 𝑣 2 superscript 𝑐 2 {{\Delta\nu}\over{\nu_{0}}}={{\Delta\bar{c}_{\theta}}\over{c}}\sim\epsilon_{v}% \cdot{{v^{2}}\over{c^{2}}} divide start_ARG roman_Δ italic_ν end_ARG start_ARG italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = divide start_ARG roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ⋅ divide start_ARG italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , we arrive at our prediction

Then, by using Eq.( 56 ), for the projection v ~ ⁢ ( t ) = V ⁢ | sin ⁡ z ⁢ ( t ) | = ~ 𝑣 𝑡 𝑉 𝑧 𝑡 absent \tilde{v}(t)=V|\sin z(t)|= over~ start_ARG italic_v end_ARG ( italic_t ) = italic_V | roman_sin italic_z ( italic_t ) | = 250 ÷ 370 250 370 250\div 370 250 ÷ 370 Km/s used for the classical experiments, our prediction for the RAV can be expressed as

with χ = 𝜒 absent \chi= italic_χ = 1 or 2. By comparing with the experimental Eq.( 63 ), the data favour χ = 2 𝜒 2 \chi=2 italic_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 ⁢ sin 2 ⁡ z ⁢ ( t ) superscript 𝑉 2 superscript 2 𝑧 𝑡 V^{2}\sin^{2}z(t) italic_V start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_z ( italic_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 ¯ 𝜏 ¯ 𝜏 \tau=\bar{\tau} italic_τ = over¯ start_ARG italic_τ end_ARG should exhibit definite daily variations in the range (for χ = 2 𝜒 2 \chi=2 italic_χ = 2 )

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 refractive 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 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT vacuum signal considered above, with very precise measurements, should also become visible here. In particular, the large refractivity of the solid 𝒩 solid − 1 = subscript 𝒩 solid 1 absent {\cal N}_{\rm solid}-1= caligraphic_N start_POSTSUBSCRIPT roman_solid end_POSTSUBSCRIPT - 1 = O(1) should play no role.

1 subscript italic-ϵ 𝑣 {\cal N}_{v}=1+\epsilon_{v} caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = 1 + italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT there is a very tiny difference between the refractive index defined relatively to the ideal vacuum value c 𝑐 c italic_c and the refractive index relatively to the physical isotropic vacuum value c / 𝒩 v 𝑐 subscript 𝒩 𝑣 c/{\cal N}_{v} italic_c / caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT measured on the Earth surface. The relative difference between these two definitions is proportional to ϵ v ≲ 10 − 9 less-than-or-similar-to subscript italic-ϵ 𝑣 superscript 10 9 \epsilon_{v}\lesssim 10^{-9} italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ≲ 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT 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

and then use Eq.( 18 ) to replace 𝒩 v subscript 𝒩 𝑣 {\cal N}_{v} caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT in the denominator with its θ − limit-from 𝜃 \theta- italic_θ - dependent value

This is equivalent to define a θ − limit-from 𝜃 \theta- italic_θ - dependent refractive index for the solid dielectric

with an anisotropy

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

In ref. [ 10 ] , a detailed comparison with [ 70 ] 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 𝜔 0.5 \omega\geq 0.5 italic_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 similar-to ¯ 𝜏 1 2 \bar{\tau}\sim 1\div 2 over¯ start_ARG italic_τ end_ARG ∼ 1 ÷ 2 seconds. The data for the spectral amplitude were then fitted to an analytic, power-law form to describe the lower-frequency part 0.001 Hz ≤ ω ≤ 0.5 absent 𝜔 0.5 \leq\omega\leq 0.5 ≤ italic_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.

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 = 𝜏 absent \tau= italic_τ = 50 seconds. Reproducing the signal at larger τ 𝜏 \tau italic_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.

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 similar-to ¯ 𝜏 1 2 \bar{\tau}\sim 1\div 2 over¯ start_ARG italic_τ end_ARG ∼ 1 ÷ 2 second, is in the range ( 6.8 ÷ 8.6 ) ⋅ 10 − 16 ⋅ 6.8 8.6 superscript 10 16 (6.8\div 8.6)\cdot 10^{-16} ( 6.8 ÷ 8.6 ) ⋅ 10 start_POSTSUPERSCRIPT - 16 end_POSTSUPERSCRIPT , say   [ 70 ]

As such, it coincides with Eq.( 63 ) that we extracted from ref. [ 68 ] after normalizing their experimental result R ⁢ A ⁢ V ⁢ ( Δ ⁢ ν , τ ¯ ) exp = 0.20 ÷ 0.24 𝑅 𝐴 𝑉 subscript Δ 𝜈 ¯ 𝜏 exp 0.20 0.24 RAV(\Delta\nu,\bar{\tau})_{\rm exp}=0.20\div 0.24 italic_R italic_A italic_V ( roman_Δ italic_ν , over¯ start_ARG italic_τ end_ARG ) start_POSTSUBSCRIPT roman_exp end_POSTSUBSCRIPT = 0.20 ÷ 0.24 Hz to their laser frequency ν 0 = 2.8 ⋅ 10 14 subscript 𝜈 0 ⋅ 2.8 superscript 10 14 \nu_{0}=2.8\cdot 10^{14} italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 2.8 ⋅ 10 start_POSTSUPERSCRIPT 14 end_POSTSUPERSCRIPT Hz. At the same time, it is well consistent with our theoretical prediction Eq.( 65 ) for χ = 2 𝜒 2 \chi=2 italic_χ = 2 . Therefore this beautiful agreement, between ref. [ 68 ] (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 ϵ v subscript italic-ϵ 𝑣 \epsilon_{v} italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT Eq.( 56 ), for χ = 2 𝜒 2 \chi=2 italic_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 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT 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 Σ \Sigma roman_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.

𝜋 2 𝜃 subscript ¯ 𝑐 𝛾 𝜃 0 \Delta\bar{c}_{\theta}=\bar{c}_{\gamma}(\pi/2+\theta)-\bar{c}_{\gamma}(\theta)\neq 0 roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_π / 2 + italic_θ ) - over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_θ ) ≠ 0 . If an angular dependence can be detected, and correlated with the cosmic motion of the Earth, the long sought Σ Σ \Sigma roman_Σ tight to the CMB could finally emerge.

1 italic-ϵ {\cal N}=1+\epsilon caligraphic_N = 1 + italic_ϵ simple symmetry arguments lead to the relation | Δ ⁢ c ¯ θ | c ∼ ϵ ⁢ ( v 2 / c 2 ) similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 italic-ϵ superscript 𝑣 2 superscript 𝑐 2 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim\epsilon(v^{2}/c^{2}) divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . For a typical cosmic v ∼ similar-to 𝑣 absent v\sim italic_v ∼ 300 km/s and ϵ = 2.8 ⋅ 10 − 4 italic-ϵ ⋅ 2.8 superscript 10 4 \epsilon=2.8\cdot 10^{-4} italic_ϵ = 2.8 ⋅ 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT , for air, or ϵ = 3.3 ⋅ 10 − 5 italic-ϵ ⋅ 3.3 superscript 10 5 \epsilon=3.3\cdot 10^{-5} italic_ϵ = 3.3 ⋅ 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT , 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 character 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 physical 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 ) subscript ~ 𝑣 𝜇 𝑡 \tilde{v}_{\mu}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) of the global, cosmic velocity, at the site of the experiment, coincides with the local v μ ⁢ ( t ) subscript 𝑣 𝜇 𝑡 v_{\mu}(t) italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_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 turbulence that, as in most models, become statistically isotropic at small scales, the local v μ ⁢ ( t ) subscript 𝑣 𝜇 𝑡 v_{\mu}(t) italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_t ) would fluctuate randomly within boundaries fixed by the global v ~ μ ⁢ ( t ) subscript ~ 𝑣 𝜇 𝑡 \tilde{v}_{\mu}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_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 always 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 subscript 𝑉 𝛼 𝛾 CMB (V,\alpha,\gamma)_{\rm CMB} ( italic_V , italic_α , italic_γ ) start_POSTSUBSCRIPT roman_CMB end_POSTSUBSCRIPT to fix the boundaries of the local random v μ ⁢ ( t ) subscript 𝑣 𝜇 𝑡 v_{\mu}(t) italic_v start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ( italic_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 ) 𝛼 fit Joos plus-or-minus 168 30 \alpha({\rm fit-Joos})=(168\pm 30) italic_α ( roman_fit - roman_Joos ) = ( 168 ± 30 ) degrees and an angular declination γ ⁢ ( fit − Joos ) = ( − 13 ± 14 ) 𝛾 fit Joos plus-or-minus 13 14 \gamma({\rm fit-Joos})=(-13\pm 14) italic_γ ( roman_fit - roman_Joos ) = ( - 13 ± 14 ) degrees which are well consistent with the present values α ⁢ ( CMB ) ∼ similar-to 𝛼 CMB absent \alpha({\rm CMB})\sim italic_α ( roman_CMB ) ∼ 168 degrees and γ ⁢ ( CMB ) ∼ − similar-to 𝛾 CMB \gamma({\rm CMB})\sim- italic_γ ( roman_CMB ) ∼ - 7 degrees. The summary of all classical experiments given in Table 4.3 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 | Δ ⁢ c ¯ θ | c ∼ ϵ gas ⁢ ( v 2 / c 2 ) similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 subscript italic-ϵ gas superscript 𝑣 2 superscript 𝑐 2 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim\epsilon_{\rm gas}(v^{2}/c^{2}) divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ start_POSTSUBSCRIPT roman_gas end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) 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 explanation 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, | Δ ⁢ T gas ⁢ ( θ ) | = 0.2 ÷ 0.3 Δ superscript 𝑇 gas 𝜃 0.2 0.3 |\Delta T^{\rm gas}(\theta)|=0.2\div 0.3 | roman_Δ italic_T start_POSTSUPERSCRIPT roman_gas end_POSTSUPERSCRIPT ( italic_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 Δ ⁢ T CMB ⁢ ( θ ) = ± 3.3 Δ superscript 𝑇 CMB 𝜃 plus-or-minus 3.3 \Delta T^{\rm CMB}(\theta)=\pm 3.3 roman_Δ italic_T start_POSTSUPERSCRIPT roman_CMB end_POSTSUPERSCRIPT ( italic_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 irrelevant. 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 behind 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 deterministically 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 ⟨ Δ ⁢ c ¯ θ ⟩ c ≲ 10 − 18 less-than-or-similar-to delimited-⟨⟩ Δ subscript ¯ 𝑐 𝜃 𝑐 superscript 10 18 {{\langle\Delta\bar{c}_{\theta}\rangle}\over{c}}\lesssim 10^{-18} divide start_ARG ⟨ roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ⟩ end_ARG start_ARG italic_c end_ARG ≲ 10 start_POSTSUPERSCRIPT - 18 end_POSTSUPERSCRIPT , could just reflect the very irregular nature of the signal. Indeed, its typical instantaneous magnitude in vacuum | Δ ⁢ c ¯ θ | c ∼ 10 − 15 similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 superscript 10 15 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim 10^{-15} divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT is about 1000 times larger, see Fig. 19 or panel b) of Fig. 11 .

1 subscript italic-ϵ 𝑣 {\cal N}_{v}=1+\epsilon_{v} caligraphic_N start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT = 1 + italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT for the vacuum or, more precisely, for the physical vacuum established in an optical cavity placed on the Earth surface. The refractivity ϵ v subscript italic-ϵ 𝑣 \epsilon_{v} italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT should be at the 10 − 9 superscript 10 9 10^{-9} 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT level, in order to give | Δ ⁢ c ¯ θ | c ∼ ϵ v ⁢ ( v 2 / c 2 ) ∼ 10 − 15 similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 subscript italic-ϵ 𝑣 superscript 𝑣 2 superscript 𝑐 2 similar-to superscript 10 15 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim\epsilon_{v}~{}(v^{2}/c^{2})~{}\sim 10% ^{-15} divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∼ 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT 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 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 ϵ v ∼ ( 2 ⁢ G N ⁢ M / c 2 ⁢ R ) ∼ 1.39 ⋅ 10 − 9 similar-to subscript italic-ϵ 𝑣 2 subscript 𝐺 𝑁 𝑀 superscript 𝑐 2 𝑅 similar-to ⋅ 1.39 superscript 10 9 \epsilon_{v}\sim(2G_{N}M/c^{2}R)\sim 1.39\cdot 10^{-9} italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ∼ ( 2 italic_G start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT italic_M / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_R ) ∼ 1.39 ⋅ 10 start_POSTSUPERSCRIPT - 9 end_POSTSUPERSCRIPT , where G N subscript 𝐺 𝑁 G_{N} italic_G start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT is the Newton constant and M 𝑀 M italic_M and R 𝑅 R italic_R are the mass and radius of the Earth. This is the same type of refractivity considered 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 | Δ ⁢ c ¯ θ | c ∼ ϵ v ⁢ ( v 2 / c 2 ) ∼ 10 − 15 similar-to Δ subscript ¯ 𝑐 𝜃 𝑐 subscript italic-ϵ 𝑣 superscript 𝑣 2 superscript 𝑐 2 similar-to superscript 10 15 {{|\Delta\bar{c}_{\theta}|}\over{c}}\sim\epsilon_{v}~{}(v^{2}/c^{2})~{}\sim 10% ^{-15} divide start_ARG | roman_Δ over¯ start_ARG italic_c end_ARG start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT | end_ARG start_ARG italic_c end_ARG ∼ italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT ( italic_v start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∼ 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT , can also resolve this other ambiguity.

With this identification of ϵ v subscript italic-ϵ 𝑣 \epsilon_{v} italic_ϵ start_POSTSUBSCRIPT italic_v end_POSTSUBSCRIPT , 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 R ⁢ A ⁢ V ⁢ ( Δ ⁢ ν ν 0 , τ ) 𝑅 𝐴 𝑉 Δ 𝜈 subscript 𝜈 0 𝜏 RAV(\frac{\Delta\nu}{\nu_{0}},\tau) italic_R italic_A italic_V ( divide start_ARG roman_Δ italic_ν end_ARG start_ARG italic_ν start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG , italic_τ ) as function of the integration time τ 𝜏 \tau italic_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 R ⁢ A ⁢ V 𝑅 𝐴 𝑉 RAV italic_R italic_A italic_V measured at the end point of the white-noise branch of the spectrum. This is defined as the largest integration time τ ¯ ¯ 𝜏 \bar{\tau} over¯ start_ARG italic_τ end_ARG 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 ~ 𝑣 250 370 \tilde{v}=250\div 370 over~ start_ARG italic_v end_ARG = 250 ÷ 370 km/s used for the classical experiments, our theoretical prediction Eq.( 65 ) (for χ = 2 𝜒 2 \chi=2 italic_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 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT 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 confirmed 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 temperature. 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 experiments, 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 superscript 10 15 10^{-15} 10 start_POSTSUPERSCRIPT - 15 end_POSTSUPERSCRIPT value measured with the same interferometers on the Earth’s surface.

In this appendix, we will summarize the stochastic model used in refs. [ 7 , 8 , 9 , 10 ] to compare with experiments. To make explicit the time dependence of the signal let us first re-write Eq.( 19 ) as

where v ⁢ ( t ) 𝑣 𝑡 v(t) italic_v ( italic_t ) and θ 2 ⁢ ( t ) subscript 𝜃 2 𝑡 \theta_{2}(t) italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) indicate respectively the instantaneous magnitude and direction of the drift in the ( x , y ) 𝑥 𝑦 (x,y) ( italic_x , italic_y ) plane of the interferometer. This can also be re-written as

and v x ⁢ ( t ) = v ⁢ ( t ) ⁢ cos ⁡ θ 2 ⁢ ( t ) subscript 𝑣 𝑥 𝑡 𝑣 𝑡 subscript 𝜃 2 𝑡 v_{x}(t)=v(t)\cos\theta_{2}(t) italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) = italic_v ( italic_t ) roman_cos italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , v y ⁢ ( t ) = v ⁢ ( t ) ⁢ sin ⁡ θ 2 ⁢ ( t ) subscript 𝑣 𝑦 𝑡 𝑣 𝑡 subscript 𝜃 2 𝑡 v_{y}(t)=v(t)\sin\theta_{2}(t) italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t ) = italic_v ( italic_t ) roman_sin italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_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 𝑉 V italic_V , a right ascension α 𝛼 \alpha italic_α and an angular declination γ 𝛾 \gamma italic_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 ) 𝑣 𝑡 ~ 𝑣 𝑡 v(t)\equiv\tilde{v}(t) italic_v ( italic_t ) ≡ over~ start_ARG italic_v end_ARG ( italic_t ) and θ 2 ⁢ ( t ) ≡ θ ~ 2 ⁢ ( t ) subscript 𝜃 2 𝑡 subscript ~ 𝜃 2 𝑡 \theta_{2}(t)\equiv\tilde{\theta}_{2}(t) italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ≡ over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) where v ~ ⁢ ( t ) ~ 𝑣 𝑡 \tilde{v}(t) over~ start_ARG italic_v end_ARG ( italic_t ) and θ ~ 2 ⁢ ( t ) subscript ~ 𝜃 2 𝑡 \tilde{\theta}_{2}(t) over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) derive from the simple application of spherical trigonometry [ 71 ]

Here z = z ⁢ ( t ) 𝑧 𝑧 𝑡 z=z(t) italic_z = italic_z ( italic_t ) is the zenithal distance of 𝐕 𝐕 {\bf{V}} bold_V , ϕ italic-ϕ \phi italic_ϕ is the latitude of the laboratory, τ = ω sid ⁢ t 𝜏 subscript 𝜔 sid 𝑡 \tau=\omega_{\rm sid}t italic_τ = italic_ω start_POSTSUBSCRIPT roman_sid end_POSTSUBSCRIPT italic_t is the sidereal time of the observation in degrees ( ω sid ∼ 2 ⁢ π 23 h ⁢ 56 ′ similar-to subscript 𝜔 sid 2 𝜋 superscript 23 ℎ superscript 56 ′ \omega_{\rm sid}\sim{{2\pi}\over{23^{h}56^{\prime}}} italic_ω start_POSTSUBSCRIPT roman_sid end_POSTSUBSCRIPT ∼ divide start_ARG 2 italic_π end_ARG start_ARG 23 start_POSTSUPERSCRIPT italic_h end_POSTSUPERSCRIPT 56 start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG ) and the angle θ 2 subscript 𝜃 2 \theta_{2} italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is counted conventionally from North through East so that North is θ 2 = 0 subscript 𝜃 2 0 \theta_{2}=0 italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 and East is θ 2 = 90 o subscript 𝜃 2 superscript 90 𝑜 \theta_{2}=90^{o} italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 90 start_POSTSUPERSCRIPT italic_o end_POSTSUPERSCRIPT . With the identifications v ⁢ ( t ) ≡ v ~ ⁢ ( t ) 𝑣 𝑡 ~ 𝑣 𝑡 v(t)\equiv\tilde{v}(t) italic_v ( italic_t ) ≡ over~ start_ARG italic_v end_ARG ( italic_t ) and θ 2 ⁢ ( t ) ≡ θ ~ 2 ⁢ ( t ) subscript 𝜃 2 𝑡 subscript ~ 𝜃 2 𝑡 \theta_{2}(t)\equiv\tilde{\theta}_{2}(t) italic_θ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) ≡ over~ start_ARG italic_θ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , one thus arrives to the simple Fourier decomposition

where the C k subscript 𝐶 𝑘 C_{k} italic_C start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT and S k subscript 𝑆 𝑘 S_{k} italic_S start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT Fourier coefficients depend on the three parameters ( V , α , γ ) 𝑉 𝛼 𝛾 (V,\alpha,\gamma) ( italic_V , italic_α , italic_γ ) and are given explicitly in refs. [ 7 , 9 ] .

Though, the identification of the instantaneous quantities v x ⁢ ( t ) subscript 𝑣 𝑥 𝑡 v_{x}(t) italic_v start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) and v y ⁢ ( t ) subscript 𝑣 𝑦 𝑡 v_{y}(t) italic_v start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t ) with their counterparts v ~ x ⁢ ( t ) subscript ~ 𝑣 𝑥 𝑡 \tilde{v}_{x}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ( italic_t ) and v ~ y ⁢ ( t ) subscript ~ 𝑣 𝑦 𝑡 \tilde{v}_{y}(t) over~ start_ARG italic_v end_ARG start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t ) is not necessarily true. As anticipated in Sect.3, one could consider the alternative situation where the velocity field is a non-differentiable 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 , 8 , 9 , 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

where v μ ⁢ ( t ) superscript 𝑣 𝜇 𝑡 v^{\mu}(t) italic_v start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ( italic_t ) is a random 4-velocity field which describes the drift and whose boundaries depend on a smooth field v ~ μ ⁢ ( t ) superscript ~ 𝑣 𝜇 𝑡 \tilde{v}^{\mu}(t) over~ start_ARG italic_v end_ARG start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ( italic_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

MICHELSON-MORLEY EXPERIMENT

The MICHELSON-MORLEY EXPERIMENT was performed in the basement of a WESTERN RESERVE UNIV. dormitory in July 1887 by ALBERT A. MICHELSON of the Case School of Applied Science and EDWARD W. MORLEY of Western Reserve Univ. it was designed to detect the motion of the earth through the "luminiferous aether," a theoretical substance which, according to 19th century physicists, was essential to the transmission of light.

At the heart of the experiment was an interferometer—a device invented by Michelson—which utilized the interference of light waves to perform measurements of incredible precision. Although an identical experiment undertaken by Michelson at Potsdam in 1880-81 failed to detect any motion of the earth relative to the ether, leading physicists were anxious for a repetition. Thus encouraged, Michelson and Morley first undertook a series of preliminary measurements of the velocity of light in moving fluids. After these measurements were successfully completed, the two men constructed an interferometer which was larger and more sensitive than the original Potsdam interferometer. Earlier difficulties caused by extraneous vibration were resolved when Morley designed an ingenious mounting for the device, in which the interferometer rested on a large sandstone slab which rotated while floating in a pool of mercury. Despite these improvements and the collaboration of these two great experimental scientists, the experiment again failed to detect any motion of the expected magnitude.

This incongruous result puzzled the physicists of the world until 1905 when Einstein published his theory of relativity. Viewed in the light of Einstein's revolutionary work, the null results of the Michelson-Morley experiment were not only predictable, but provided experimental confirmation of Einstein's theory. In 1995 the American Chemical Society declared CWRU 's Adelbert Hall, the site of the experiment, a national historic chemical landmark, only the 4th location to be so designated in the country.

• Celestial Bodies

Michelson Morley Experiment

We know a medium is absolutely necessary for any transmission in science. The medium of transmission plays an important role in efficient channeling. Waves like sound waves use air for transmission. Then, how is the light transmitted?

Let us know more about the Michelson Morley Experiment in detail to know about light transmission and velocity of the earth.

Michelson Morley Experiment was performed by two eminent scientists Albert A. Michelson and Edward W. Morley in the year 1887 to explain and demonstrate the presence of luminiferous ether.

What Is Luminiferous Ether or Aether?

Luminiferous Ether is the theoretical substance that acts as the medium for the transmission of electromagnetic waves like light rays and X-rays. Ether was assumed to be a transmission medium for the propagation of light.

Luminiferous ether was believed to be a theoretical medium in the 19th century. These substances were assumed to be frictionless, weightless and transparent substances. When the special theory of relativity was developed, the concept of luminiferous ether lost significance gradually.

Michelson Morley Experiment compared the speed of light in perpendicular directions in an attempt to detect the relative motion of matter through the stationary luminiferous aether. But this experiment yielded no results to prove a significant difference between the speed of light in the direction of movement through the presumed aether, and the speed at right angles.

The Michelson Morley Experiment was one of the failed experiments that stands as proof against the existence of the luminiferous ether concept.

Michelson and Morley tried to explain that Earth moved around the sun on its orbit, and the flow of substances like ether across the Earth’s surface could produce a detectable “ether wind”.

They tried to demonstrate the concept that the speed of the light would depend on the magnitude of the ether wind and on the direction of the beam with respect to it when the light is emitted from a source on Earth. Ether was assumed to be stationary. The idea of the experiment was to measure the speed of light in different directions in order to measure the speed of the ether relative to Earth, thus establishing its existence.

Interferometer

To measure the velocity of the Earth with the help of ether and to measure the changing pattern of the light, Albert Michelson developed a device called an interferometer.

The interferometer features the following components

• beam splitter
• beam splitter reference mirror
• coherent light source
• movable mirror

Pictorial representation of the interferometer is as shown in the figure below.

The interferometer features a half-transparent mirror that is oriented at an angle of 45°. This mirror is used to divide the light beam into two equal parts. One part of the divided beam is transmitted towards a fixed mirror and part of the divided beam is reflected in a movable mirror. The half-transparent mirror has the same effect on the returning beams, splitting them into two beams. Thus, when two diminished light beams reach the screen, a constructive and destructive wave interference pattern is observed based on the length of the arms of the device.

The speed of light was measured in the experiment by analyzing the interference fringes pattern that resulted when the light had passed through the two perpendicular arms of the interferometer. Michelson and Morley observed that light traveled faster along an arm which was oriented in the same direction as the ether. The light traveled at a slower pace in the arm oriented in the opposite direction.

As shown in the figure above, the interferometer featured perpendicular arms. The split light would travel at the same speed in both arms and therefore arrive simultaneously at the screen if the instrument were motionless with respect to the ether.

When the orientation of the interferometer is changed, the crests and troughs of the light waves produced in the two arms would interfere slightly out of synchronization.

The two scientists Michelson and Morley were expecting light to have different speeds when they travel in different directions, but they found no significantly distinguishing fringes that specified a different speed in any orientation or at any position of the Earth.

Lorentz in 1895, concluded that the Michelson Morley experiment produced the null result. Einstein wrote that If the Michelson–Morley experiment had not brought us into serious embarrassment, no one would have regarded the relativity theory as a (halfway) redemption.

When the Michelson Morley experiment was performed with increasing sophistication, the existence of ether and velocity of earth could not be proved.

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Frequently Asked Questions on Michelson Morley Experiment

Name the device used in michelson morley experiment..

Interferometer.

What is an interferometer?

It is a device used to measure the changing pattern of the light.

What is Luminiferous Ether?

It is the theoretical substance that acts as the medium for the transmission of electromagnetic waves.

Name the scientists who performed the experiment.

Albert A. Michelson and Edward W. Morley.

Which theory superseded the Michelson Morley Experiment?

Special theory of relativity.

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AQA A-Level Physics Notes

12.3.1 the michelson-morley experiment, introduction.

In 1887, Albert A. Michelson and Edward W. Morley conducted an experiment that dramatically altered the course of theoretical physics. It directly challenged the then-prevailing concepts of absolute motion and aether, and provided crucial evidence for the invariance of the speed of light.

Principle of the Michelson-Morley Interferometer

Underlying theory.

The experiment was based on the hypothesis of the 'luminiferous aether', a medium thought necessary for light propagation.

It aimed to detect the Earth's motion through this aether, akin to detecting a breeze in still air.

Key Components and Function

Light Source: Typically, a coherent light source like a sodium flame.

Beam Splitter: A glass plate that divides the incoming light beam into two parts.

Mirrors: Positioned at right angles to each other, reflecting the beams back to converge.

Detector: A screen or an optical device to observe and analyse the interference pattern.

Detailed Experimental Setup

Precise configuration.

The apparatus was carefully aligned with Earth's presumed direction of travel around the Sun.

Each arm of the interferometer was equal in length, ensuring equal travel time for both light beams under the aether wind hypothesis.

Operational Mechanics

The beam splitter sends one light beam parallel and the other perpendicular to the presumed aether wind.

Upon recombination, any difference in speed caused by the aether wind would result in a shift in the interference pattern.

Significance in Challenging Absolute Motion

The concept of absolute motion.

Before the experiment, it was widely believed that the Earth moved through a stationary aether, creating an "aether wind".

This wind was expected to affect the speed of light depending on the Earth's direction of movement through the aether.

Experimental Outcome and Its Interpretation

Critical Observation: The experiment yielded a null result; no significant shift in the interference pattern was observed.

Implication: This outcome strongly suggested that the speed of light is unaffected by Earth’s motion, challenging the existence of aether and the concept of absolute motion.

Affirming the Invariance of Light Speed

Steadfast speed of light.

The null result implied that light's speed remains constant in all directions, contradicting the aether theory.

It was a significant deviation from Newtonian physics, which did not predict such constancy.

Influence on Physics and Thought

Theoretical Implications: It necessitated a rethinking of established physics, particularly concerning space and time.

Precursor to Special Relativity: The experiment's outcomes were integral in shaping Einstein’s postulates that the laws of physics are invariant in all inertial frames and that the speed of light in a vacuum is constant.

Experiment’s Role in the Genesis of Special Relativity

Paving the way for einstein’s theories.

The Michelson-Morley experiment's results were fundamental to Einstein’s formulation of special relativity in 1905.

The experiment negated the need for the aether concept and supported the idea of constant light speed across all frames of reference.

Transforming Scientific Understanding

This experiment marked a profound shift from classical mechanics to modern physics.

It altered the scientific perception of time, space, and relative motion, laying the groundwork for the theory of relativity.

Experiment’s Legacy in Modern Physics

Lasting impact.

The Michelson-Morley experiment is celebrated for its innovative approach and its significant role in advancing our understanding of the universe.

Its methodology and findings continue to influence modern scientific thought and experimentation.

Educational Importance

For A-level Physics students, understanding this experiment is crucial for grasping the foundational principles of special relativity and modern physics.

The Michelson-Morley experiment serves as a testament to the transformative power of empirical evidence in science. It not only refuted long-held beliefs about the aether and absolute motion but also laid the foundation for the revolutionary theory of special relativity. Its legacy continues to inspire and inform scientific inquiry into the fundamental laws of the universe.

How did the Michelson-Morley experiment contribute to the concept of spacetime in physics?

The Michelson-Morley experiment, by demonstrating the constancy of light speed, inadvertently set the stage for the development of the spacetime concept. Before this experiment, space and time were viewed as distinct and absolute entities. The experiment's null result, however, challenged the prevailing Newtonian mechanics, which could not account for the observed invariance of light speed in all frames of reference. This anomaly paved the way for Einstein's theory of special relativity, which introduced the revolutionary concept of spacetime – a unified four-dimensional continuum where space and time are intrinsically linked, rather than separate. In this new framework, the speed of light is constant, and the concepts of simultaneous events and fixed intervals of time and space become relative, depending on the observer's motion. This paradigm shift in understanding the nature of space and time was a direct consequence of the Michelson-Morley experiment’s findings.

What were the primary technical challenges faced in the Michelson-Morley experiment and how were they overcome?

The Michelson-Morley experiment faced significant technical challenges, primarily related to achieving the necessary sensitivity to detect minute differences in the speed of light. The accuracy required to measure such small variations was unprecedented. To overcome these challenges, Michelson and Morley used several innovative techniques:

Interferometer Design: They developed the Michelson interferometer, capable of measuring exceedingly small changes in light path lengths, enhancing the experiment's sensitivity.

Alignment Precision: Precise alignment of the mirrors and the interferometer arms was crucial. They achieved this through meticulous construction and adjustment techniques.

Minimising Environmental Factors: Vibrations and thermal expansions could affect the results. They isolated the apparatus from vibrations and conducted the experiment in a controlled environment to reduce thermal fluctuations.

Multiple Measurements: Repeated measurements were taken at different times of the day and year to ensure that any observed effects were not due to environmental or experimental errors.

Through these methods, Michelson and Morley significantly increased the precision of their experiment, allowing them to confidently interpret the null results they obtained.

What alternative explanations were proposed for the null result of the Michelson-Morley experiment?

Following the null result of the Michelson-Morley experiment, several alternative explanations were proposed within the scientific community. One of the most notable was the 'Lorentz-FitzGerald contraction hypothesis', proposed independently by Hendrik Lorentz and George FitzGerald. They suggested that objects moving through the aether would contract in the direction of motion, which could compensate for the expected change in the speed of light due to the aether wind. This hypothesis was an ad hoc explanation, lacking empirical evidence, but it did pave the way for the concept of length contraction in special relativity. Another explanation was that the Earth dragged the aether along with it, thereby nullifying the expected effect. However, subsequent experiments, such as the Hamar experiment and the Trouton-Noble experiment, also produced null results, lending support to the idea that the aether did not exist, rather than these alternative explanations.

How did the scientific community initially react to the Michelson-Morley experiment's findings?

The initial reaction of the scientific community to the Michelson-Morley experiment was mixed. While some physicists were intrigued by the null result, others were sceptical or dismissive, primarily because the outcome contradicted the well-established aether theory. Many scientists attempted to reconcile the experiment's findings with the aether theory, leading to various hypotheses like the Lorentz-FitzGerald contraction. The full significance of the experiment was not immediately recognised, and it took several years for its implications to be fully appreciated. It wasn't until the development of Einstein's theory of special relativity, which provided a theoretical framework consistent with the Michelson-Morley findings, that the experiment's true importance was acknowledged. Einstein's work shifted the focus away from the aether theory and towards a new understanding of space and time, with the Michelson-Morley experiment serving as a critical empirical foundation.

What advancements in experimental physics were influenced by the Michelson-Morley experiment?

The Michelson-Morley experiment had a profound influence on experimental physics, setting new standards for precision and measurement techniques. It inspired the development of more sophisticated and sensitive instrumentation, crucial for exploring the frontiers of physics. For instance, the interferometer designed by Michelson became a fundamental tool in various areas of physics and engineering, including the development of lasers, fibre optics, and holography. Additionally, the quest to measure minuscule effects as seen in the Michelson-Morley experiment led to advancements in experimental design, data analysis, and error reduction techniques. This legacy extended to modern experiments like the detection of gravitational waves using LIGO (Laser Interferometer Gravitational-Wave Observatory), which employs principles of interferometry akin to those used in the Michelson-Morley experiment. Thus, the experiment not only contributed to theoretical physics but also significantly advanced experimental methodologies.

Practice Questions

Explain why the Michelson-Morley experiment was significant in the context of the luminiferous aether theory and its impact on the development of special relativity.

The Michelson-Morley experiment was pivotal because it provided empirical evidence against the existence of the luminiferous aether, a concept widely accepted before the experiment. The experiment, designed to detect Earth's motion through the aether, yielded a null result, indicating that the speed of light remained constant regardless of the Earth's movement. This finding contradicted the aether theory, which posited that light's speed would vary with Earth's motion through the aether. The constancy of light speed, as demonstrated by the experiment, directly influenced Einstein's development of special relativity. Einstein's theory, which posits that the laws of physics are the same in all inertial frames and that the speed of light in a vacuum is constant, was built on the foundation laid by the Michelson-Morley experiment. Thus, this experiment not only challenged existing notions of motion and medium for light propagation but also catalysed a paradigm shift in physics, leading to the development of modern theories of space and time.

Describe the experimental setup of the Michelson-Morley experiment and explain how it was expected to detect the Earth's movement through the aether.

The Michelson-Morley experiment's setup involved an interferometer with two perpendicular arms of equal length. A light source emitted a beam that was split into two by a beam splitter, sending the beams along these arms. After reflection from mirrors at the arms' ends, the beams were recombined at the detector. The expectation was that if the Earth moved through the luminiferous aether, one light beam would travel faster than the other due to the aether wind, similar to how wind affects the speed of sound. This difference in speed would alter the interference pattern observed at the detector. However, the experiment consistently showed no significant difference in the interference patterns, regardless of the Earth's orientation or the time of year. This null result was significant as it disproved the aether theory and suggested the invariance of light speed, a key postulate in Einstein's theory of special relativity.

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The Michelson-Morley Experiment

Michael Fowler, University of Virginia

The Nature of Light

As a result of Michelson’s efforts in 1879, the speed of light was known to be 186,350 miles per second with a likely error of around 30 miles per second.  This measurement, made by timing a flash of light travelling between mirrors in Annapolis, agreed well with less direct measurements based on astronomical observations.  Still, this did not really clarify the nature of light.  Two hundred years earlier, Newton had suggested that light consists of tiny particles generated in a hot object, which spray out at very high speed, bounce off other objects, and are detected by our eyes.  Newton’s arch-enemy Robert Hooke, on the other hand, thought that light must be a kind of wave motion , like sound.  To appreciate his point of view, let us briefly review the nature of sound.

The Wavelike Nature of Sound

Actually, sound was already quite well understood by the ancient Greeks.  The essential point they had realized is that sound is generated by a vibrating material object, such as a bell, a string or a drumhead.  Their explanation was that the vibrating drumhead, for example, alternately pushes and pulls on the air directly above it, sending out waves of compression and decompression (known as rarefaction), like the expanding circles of ripples from a disturbance on the surface of a pond.  On reaching the ear, these waves push and pull on the eardrum with the same frequency (that is to say, the same number of pushes per second) as the original source was vibrating at, and nerves transmit from the ear to the brain both the intensity (loudness) and frequency (pitch) of the sound.

There are a couple of special properties of sound waves (actually any waves) worth mentioning at this point.  The first is called interference .  This is most simply demonstrated with water waves.  If you put two fingers in a tub of water, just touching the surface a foot or so apart, and vibrate them at the same rate to get two expanding circles of ripples, you will notice that where the ripples overlap there are quite complicated patterns of waves formed.  The essential point is that at those places where the wave-crests from the two sources arrive at the same time, the waves will work together and the water will be very disturbed, but at points where the crest from one source arrives at the same time as the wave trough from the other source, the waves will cancel each other out, and the water will hardly move.

There's a standard physics lab experiment illustrating this point, it's called two-slit interference. A monochromatic plane wave approaches a screen in which there are two narrow slit openings. If the slit opening width is smaller than the wavelength, a single slit will give approximately semicircular waves coming out the other side, the disturbance radiates out from the slit. The two slits together show the pattern generated by interference. It's best seen by clicking on this applet . Watch how the pattern changes with wavelength and with slit separation.

You can hear the interference effect for sound waves by playing a constant note through stereo speakers.  As you move around a room, you will hear quite large variations in the intensity of sound.  Of course, reflections from walls complicate the pattern.  This large variation in volume is not very noticeable when the stereo is playing music, because music is made up of many frequencies, and they change all the time.  The different frequencies, or notes, have their quiet spots in the room in different places.  The other point that should be mentioned is that high frequency tweeter-like sound is much more directional than low frequency woofer-like sound.  It really doesn’t matter where in the room you put a low-frequency woofer—the sound seems to be all around you anyway.  On the other hand, it is quite difficult to get a speaker to spread the high notes in all directions.  If you listen to a cheap speaker, the high notes are loudest if the speaker is pointing right at you.  A lot of effort has gone into designing tweeters, which are small speakers especially designed to broadcast high notes over a wide angle of directions.

Is Light a Wave?

Bearing in mind the above minireview of the properties of waves, let us now reconsider the question of whether light consists of a stream of particles or is some kind of wave.  The strongest argument for a particle picture is that light travels in straight lines.  You can hear around a corner, at least to some extent, but you certainly can’t see.  Furthermore, no wave-like interference effects are very evident for light.  Finally, it was long known, as we have mentioned, that sound waves were compressional waves in air.  If light is a wave, just what is waving?  It clearly isn’t just air, because light reaches us from the sun, and indeed from stars, and we know the air doesn’t stretch that far, or the planets would long ago have been slowed down by air resistance.

Despite all these objections, it was established around 1800 that light is in fact some kind of wave.  The reason this fact had gone undetected for so long was that the wavelength is really short, about one fifty-thousandth of an inch.  In contrast, the shortest wavelength sound detectable by humans has a wavelength of about half an inch.  The fact that light travels in straight lines is in accord with observations on sound that the higher the frequency (and shorter the wavelength) the greater the tendency to go in straight lines.  Similarly, the interference patterns mentioned above for sound waves or ripples on a pond vary over distances of the same sort of size as the wavelengths involved.  Patterns like that would not normally be noticeable for light because they would be on such a tiny scale.  In fact, it turns out, there are ways to see interference effects with light.  A familiar example is the many colors often visible in a soap bubble.  These come about because looking at a soap bubble you see light reflected from both sides of a very thin film of water—a thickness that turns out to be comparable to the wavelength of light.  The light reflected from the lower boundary has to go a little further to reach your eye, so that light wave must wave an extra time or two before getting to your eye compared with the light reflected from the top of the film.  What you actually see is the sum of the light reflected from the top and that reflected from the bottom.  Thinking of this now as the sum of two sets of waves, the light will be bright if the crests of the two waves arrive together, dim if the crests of waves reflected from the top layer arrive simultaneously with the troughs of waves reflected from the bottom layer.  Which of these two possibilities actually occurs for reflection from a particular bit of the soap film depends on just how much further the light reflected from the lower surface has to travel to reach your eye compared with light from the upper surface, and that depends on the angle of reflection and the thickness of the film.  Suppose now we shine white light on the bubble.  White light is made up of all the colors of the rainbow, and these different colors have different wavelengths, so we see colors reflected, because for a particular film, at a particular angle, some colors will be reflected brightly (the crests will arrive together), some dimly, and we will see the ones that win.

If Light is a Wave, What is Waving?

Having established that light is a wave, though, we still haven’t answered one of the major objections raised above.  Just what is waving?  We discussed sound waves as waves of compression in air.  Actually, that is only one case—sound will also travel through liquids, like water, and solids, like a steel bar.  It is found experimentally that, other things being equal, sound travels faster through a medium that is harder to compress: the material just springs back faster and the wave moves through more rapidly.  For media of equal springiness, the sound goes faster through the less heavy medium, essentially because the same amount of springiness can push things along faster in a lighter material.  So when a sound wave passes, the material—air, water or solid—waves as it goes through.  Taking this as a hint, it was natural to suppose that light must be just waves in some mysterious material, which was called the aether , surrounding and permeating everything.  This aether must also fill all of space, out to the stars, because we can see them, so the medium must be there to carry the light.  (We could never hear an explosion on the moon, however loud, because there is no air to carry the sound to us.)  Let us think a bit about what properties this aether must have.  Since light travels so fast, it must be very light, and very hard to compress.  Yet, as mentioned above, it must allow solid bodies to pass through it freely, without aether resistance, or the planets would be slowing down.  Thus we can picture it as a kind of ghostly wind blowing through the earth.  But how can we prove any of this? Can we detect it?

Detecting the Aether Wind: the Michelson-Morley Experiment

Detecting the aether wind was the next challenge Michelson set himself after his triumph in measuring the speed of light so accurately.  Naturally, something that allows solid bodies to pass through it freely is a little hard to get a grip on.  But Michelson realized that, just as the speed of sound is relative to the air, so the speed of light must be relative to the aether.  This must mean, if you could measure the speed of light accurately enough, you could measure the speed of light travelling upwind, and compare it with the speed of light travelling downwind, and the difference of the two measurements should be twice the windspeed.  Unfortunately, it wasn’t that easy.  All the recent accurate measurements had used light travelling to a distant mirror and coming back, so if there was an aether wind along the direction between the mirrors, it would have opposite effects on the two parts of the measurement, leaving a very small overall effect.  There was no technically feasible way to do a one-way determination of the speed of light.

At this point, Michelson had a very clever idea for detecting the aether wind.  As he explained to his children (according to his daughter), it was based on the following puzzle:

Suppose we have a river of width w (say, 100 feet), and two swimmers who both swim at the same speed v feet per second (say, 5 feet per second).  The river is flowing at a steady rate, say 3 feet per second.  The swimmers race in the following way: they both start at the same point on one bank.  One swims directly across the river to the closest point on the opposite bank, then turns around and swims back.  The other stays on one side of the river, swimming upstream a distance (measured along the bank) exactly equal to the width of the river, then swims back to the start.  Who wins?

Let’s consider first the swimmer going upstream and back.  Going 100 feet upstream, the speed relative to the bank is only 2 feet per second, so that takes 50 seconds.  Coming back, the speed is 8 feet per second, so it takes 12.5 seconds, for a total time of 62.5 seconds.

Figure 1:  In time t , the swimmer has moved ct relative to the water, and been carried downstream a distance vt .

The swimmer going across the flow is trickier.  It won’t do simply to aim directly for the opposite bank-the flow will carry the swimmer downstream.  To succeed in going directly across, the swimmer must actually aim upstream at the correct angle (of course, a real swimmer would do this automatically).  Thus, the swimmer is going at 5 feet per second, at an angle, relative to the river, and being carried downstream at a rate of 3 feet per second.  If the angle is correctly chosen so that the net movement is directly across, in one second the swimmer must have moved four feet across:  the distances covered in one second will form a 3,4,5 triangle.  So, at a crossing rate of 4 feet per second, the swimmer gets across in 25 seconds, and back in the same time, for a total time of 50 seconds.  The cross-stream swimmer wins.  This turns out to true whatever their swimming speed.  (Of course, the race is only possible if they can swim faster than the current!)

Figure 2:  This diagram is from the original paper. The source of light is at s , the 45 degree line is the half-silvered mirror, b and c are mirrors and d the observer.

Michelson’s great idea was to construct an exactly similar race for pulses of light, with the aether wind playing the part of the river.  The scheme of the experiment is as follows: a pulse of light is directed at an angle of 45 degrees at a half-silvered, half transparent mirror, so that half the pulse goes on through the glass, half is reflected.  These two half-pulses are the two swimmers.  They both go on to distant mirrors which reflect them back to the half-silvered mirror.  At this point, they are again half reflected and half transmitted, but a telescope is placed behind the half-silvered mirror as shown in the figure so that half of each half-pulse will arrive in this telescope.  Now, if there is an aether wind blowing, someone looking through the telescope should see the halves of the two half-pulses to arrive at slightly different times, since one would have gone more upstream and back, one more across stream in general.  To maximize the effect, the whole apparatus, including the distant mirrors, was placed on a large turntable so it could be swung around.

We have an animation, including the aether wind and the rotating turntable, here !

Let us think about what kind of time delay we expect to find between the arrival of the two half-pulses of light.  Taking the speed of light to be c  miles per second relative to the aether, and the aether to be flowing at v  miles per second through the laboratory, to go a distance w  miles upstream will take w / ( c − v )  seconds, then to come back will take w / ( c + v )  seconds.  The total roundtrip time upstream and downstream is the sum of these, which works out to be 2 w c / ( c 2 − v 2 ) ,  which can also be written

t upstream + downstream = 2 w c ⋅ 1 1 − ( v 2 / c 2 ) .  

Now, we can safely assume the speed of the aether is much less than the speed of light, otherwise it would have been noticed long ago, for example in timing of eclipses of Jupiter’s satellites.  This means v 2 / c 2  is a very small number, and we can use some handy mathematical facts to make the algebra a bit easier.  First, if x  is very small compared to 1 ,     1 / ( 1 − x )  is very close to 1 + x .   (You can check it with your calculator.)  Another fact we shall need in a minute is that for small x ,  the square root of 1 + x  is very close to 1 + x / 2.    

Putting all this together,

t upstream + downstream ≅ 2 w c × ( 1 + v 2 c 2 ) .

Figure 3 This is also from the original paper, and shows the expected path of light relative to the aether with an aether wind blowing.

Now, what about the cross-stream time?  The actual cross-stream speed must be figured out as in the example above using a right-angled triangle, with the hypoteneuse equal to the speed c ,  the shortest side the aether flow speed v ,  and the other side the cross-stream speed we need to find the time to get across.  From Pythagoras’ theorem, then, the cross-stream speed is c 2 − v 2 .

Since this will be the same both ways, the roundtrip cross-stream time will be

t cross and back = 2 w c 2 − v 2 .  

This can be written in the form

2 w c 1 1 − v 2 / c 2 ≅ 2 w c 1 1 − ( v 2 / 2 c 2 ) ≅ 2 w c ( 1 + v 2 2 c 2 )

where the two successive approximations, valid for v / c = x ≪ 1 ,  are   1 − x ≅ 1 − ( x / 2 )  and 1 / ( 1 − x ) ≅ 1 + x .

t cross and back   ≅ 2 w c × ( 1 + v 2 2 c 2 ) .

Looking at the two roundtrip times at the ends of the two paragraphs above, we see that they differ by an amount

t upstream + downstream  − t cross and back ≅ 2 w c ⋅ v 2 2 c 2 .  

Now, 2 w / c  is just the time the light would take if there were no aether wind at all, say, a few millionths of a second.  If we take the aether windspeed to be equal to the earth’s speed in orbit, for example, v / c  is about 1/10,000, so v 2 / c 2  is about 1/100,000,000.

This means the time delay between the pulses reflected from the different mirrors reaching the telescope is about one-hundred-millionth of a few millionths of a second.  

It seems completely hopeless that such a short time delay could be detected.  However, this turns out not to be the case, and Michelson was the first to figure out how to do it.  The trick is to use the interference properties of the light waves.  Instead of sending pulses of light, as we discussed above, Michelson sent in a steady beam of light of a single color.  This can be visualized as a sequence of ingoing waves, with a wavelength one fifty-thousandth of an inch or so.  Now this sequence of waves is split into two, and reflected as previously described.  One set of waves goes upstream and downstream, the other goes across stream and back.  Finally, they come together into the telescope and the eye.  If the one that took longer is half a wavelength behind, its troughs will be on top of the crests of the first wave, they will cancel, and nothing will be seen.  If the delay is less than that, there will still be some dimming.  However, slight errors in the placement of the mirrors would have the same effect.  This is one reason why the apparatus is built to be rotated—see the animation !  On turning it through 90 degrees, the upstream-downstream and the cross-stream waves change places.  Now the other one should be behind.  Thus, if there is an aether wind, if you watch through the telescope while you rotate the turntable, you should expect to see variations in the brightness of the incoming light.

To magnify the time difference between the two paths, in the actual experiment the light was reflected backwards and forwards several times, like a several lap race.

Michelson calculated that an aether windspeed of only one or two miles a second would have observable effects in this experiment, so if the aether windspeed was comparable to the earth’s speed in orbit around the sun, it would be easy to see.  In fact, nothing was observed.  The light intensity did not vary at all.  Some time later, the experiment was redesigned so that an aether wind caused by the earth’s daily rotation could be detected.  Again, nothing was seen.  Finally, Michelson wondered if the aether was somehow getting stuck to the earth, like the air in a below-decks cabin on a ship, so he redid the experiment on top of a high mountain in California.  Again, no aether wind was observed.  It was difficult to believe that the aether in the immediate vicinity of the earth was stuck to it and moving with it, because light rays from stars would deflect as they went from the moving faraway aether to the local stuck aether.

The only possible conclusion from this series of very difficult experiments was that the whole concept of an all-pervading aether was wrong from the start.  Michelson was very reluctant to think along these lines.  In fact, new theoretical insight into the nature of light had arisen in the 1860’s from the brilliant theoretical work of Maxwell, who had written down a set of equations describing how electric and magnetic fields can give rise to each other.  He had discovered that his equations predicted there could be waves made up of electric and magnetic fields, and the speed of these waves, deduced from electrostatic and magnetostatic experiments, was predicted to be 186,300 miles per second.   This is, of course, the speed of light, so it was natural to assume light to be made up of fast-varying electric and magnetic fields.  

But this led to a big problem: Maxwell’s equations predicted a definite speed for light, and it was the speed found by measurements.  But what was the speed to be measured relative to?  The whole point of bringing in the aether was to give a picture for light resembling the one we understand for sound, compressional waves in a medium.  The speed of sound through air is measured relative to air.  If the wind is blowing towards you from the source of sound, you will hear the sound sooner.  If there isn’t an aether, though, this analogy doesn’t hold up.  So what does light travel at 186,300 miles per second relative to?

There is another obvious possibility, which is called the emitter theory: the light travels at 186,300 miles per second relative to the source of the light.  The analogy here is between light emitted by a source and bullets emitted by a machine gun.  The bullets come out at a definite speed (the muzzle velocity) relative to the barrel of the gun.  If the gun is mounted on the front of a tank, which is moving forward, and the gun is pointing forward, then relative to the ground the bullets are moving faster than they would if shot from a tank at rest.  The simplest way to test the emitter theory of light, then, is to measure the speed of light emitted in the forward direction by a flashlight moving in the forward direction, and see if it exceeds the known speed of light by an amount equal to the speed of the flashlight.  Actually, this kind of direct test of the emitter theory only became experimentally feasible in the nineteen-sixties.  It is now possible to produce particles, called neutral pions, which decay each one in a little explosion, emitting a flash of light.  It is also possible to have these pions moving forward at 185,000 miles per second when they self destruct, and to catch the light emitted in the forward direction, and clock its speed.  It is found that, despite the expected boost from being emitted by a very fast source, the light from the little explosions is going forward at the usual speed of 186,300 miles per second.  In the last century, the emitter theory was rejected because it was thought the appearance of certain astronomical phenomena, such as double stars, where two stars rotate around each other, would be affected.  Those arguments have since been criticized, but the pion test is unambiguous.  The definitive experiment was carried out by Alvager et al., Physics Letters 12 , 260 (1964).

We have to mention here that the most spectacular realization (so far) of Michelson's interference techniques is the successful detection (first in 2016) of gravitational waves by a scaled-up version of the interferometer. The basic idea is that as a (transverse, polarized) gravitational wave passes the two arms have their lengths slightly altered by different amounts. The tiny effect is made detectable by having arms kilometers in length, and the beams go back and forth in the arms many times. 

Einstein’s Answer

The results of the various aether detection experiments discussed above seem to leave us really stuck.  Apparently light is not like sound, with a definite speed relative to some underlying medium.  However, it is also not like bullets, with a definite speed relative to the source of the light.  Yet when we measure its speed we always get the same result.  How can all these facts be interpreted in a simple consistent way?  We shall show how Einstein answered this question in the next lecture.

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November 1, 1964

The Michelson-Morley Experiment

Performed in Cleveland in 1887, this famous experiment disproved the hypothesis of a stationary "luminiferous ether." The problems it posed led indirectly to Einstein's special theory of relativity

By R. S. Shankland

Isaac Physics

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