The classical ether-drift experiments: A modern re-interpretation

Maurizio Consoli

The classical ether-drift experiments: A modern re-interpretation

Maurizio Consoli

2013, European Physical Journal Plus

20 Views70 Pages

1 File ▾

Physics

Show more ▾

Translate PDF

Original PDF 140 minute read

Summary

scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd.

scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd. scribd.

a r X i v : 1 3 0 2 . 3 5 0 8 v 2 [ p h y s i c s . g e n - p h ] 6 J u n 2 0 1 3

The classical ether-drift experiments:a modern re-interpretation

M. Consoli

(

)

, C. Matheson

(

)

and A. Pluchino

(

a,c

)

a) Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Italyb) Selwyn College, Cambridge, United Kingdomc) Dipartimento di Fisica e Astronomia dell’Universit`a di Catania, Italy

Abstract

The condensation of elementary quanta and their macroscopic occupation of the same quan-tum state, say

= 0 in some reference frame Σ, is the essential ingredient of the degeneratevacuum of present-day elementary particle physics. This represents a sort of ‘quantum ether’which characterizes the physically realized form of relativity and could play the role of pre-ferred reference frame in a modern re-formulation of the Lorentzian approach. In spite of this,the so called ‘null results’ of the classical ether-drift experiments, traditionally interpreted asconﬁrmations of Special Relativity, have so deeply inﬂuenced scientiﬁc thought as to preventa critical discussion on the real reasons underlying its alleged supremacy. In this paper, weargue that this traditional null interpretation is far from obvious. In fact, by using Lorentztransformations to connect the Earth’s frame to Σ, the small observed eﬀects point to anaverage Earth’s velocity of about 300 km/s, as in most cosmic motions. A common featureis the irregular behaviour of the data. While this has motivated, so far, their standard inter-pretation as instrumental artifacts, our new re-analysis of the very accurate Joos experimentgives clear indications for the type of Earth’s motion associated with the CMB anisotropy andleaves little space for this traditional interpretation. The new explanation requires instead aview of the vacuum as a stochastic medium, similar to a ﬂuid in a turbulent state of motion, inagreement with basic foundational aspects of both quantum physics and relativity. The over-all consistency of this picture with the present experiments with vacuum optical resonatorsand the need for a new generation of dedicated ether-drift experiments are also emphasized.PACS: 03.30.+p; 01.55.+b; 11.30.Cp

1. Introduction

An analysis of the ether-drift experiments, starting from the original Michelson-Morley exper-iment of 1887, should be suitably framed within a general discussion of the basic diﬀerencesbetween Einstein’s Special Relativity [1] and the Lorentzian point of view [2, 3, 4]. There is no doubt that the former interpretation is today widely accepted. However, in spite of the deep conceptual diﬀerences, it is not obvious how to distinguish experimentally betweenthe two formulations. This type of conclusion was, for instance, already clearly expressed byEhrenfest 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 hereas the ether theory of Lorentz. It is, in fact, because of this circumstance, that according toEinstein’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 Lorentziantheory. 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”. This can be understood since,independently of all interpretative aspects, the basic quantitative ingredients, namely Lorentztransformations, are the same in both formulations. Their validity will be assumed in thefollowing to discuss the possible existence of a preferred reference frame.For a modern presentation of the Lorentzian philosophy one can then refer to Bell [5,6, 7]. In this alternative approach, diﬀerently from the usual derivations, one starts from physical modiﬁcations of matter (namely Larmor’s time dilation and Lorentz-Fitzgerald lengthcontraction in the direction of motion) to deduce Lorentz transformations. In this way, due tothe fundamental group properties, the relation between two observers

′

and

′′

, individuallyrelated to the preferred frame Σ by Lorentz transformations with dimensionless parameters

′

and

′′

, is also a Lorentz transformation with relative velocity parameter

rel

ﬁxed by the relativistic composition rule

rel

′

−

′′

−

′

′′

(1)(for simplicity we restrict to the case of one-dimensional motion). This produces a substan-tial quantitative equivalence with Einstein’s formulation for most standard experimental testswhere one just compares the relative measurements of a pair of observers. Hence the impor-tance of the ether-drift experiments where one attempts to measure an absolute velocity.At the same time, if the velocity of light

propagating in the various interferometerscoincides with the basic parameter

entering Lorentz transformations, relativistic eﬀectsconspire to make undetectable the individual

′

′′

,...This means that a null result of theether-drift experiments should

not

be automatically interpreted as a conﬁrmation of Special1

Relativity. As stressed by Ehrenfest, the motion with respect to Σ might remain unobservable,yet one could interpret relativity ‘ `a la Lorentz’. This could be crucial, for instance, to reconcilefaster-than-light signals with causality [8] and thus provide a diﬀerent view of the apparentnon-local aspects of the quantum theory [9].However, to a closer look, is it really impossible to detect the motion with respect to Σ?This possibility, which was implicit in Lorentz’ words [4] “...it seems natural not to assumeat starting that it can never make any diﬀerence whether a body moves through the ether ornot..”, may induce one to re-analyze the classical ether-drift experiments. Let us ﬁrst givesome general theoretical arguments that could motivate this apparently startling idea.A possible observation is that Lorentz symmetry might not be an exact symmetry. Inthis case, one could conceivably detect the eﬀects of absolute motion. For instance Lorentzsymmetry could represent an ‘emergent’ phenomenon and thus reﬂect the existence of someunderlying form of ether. This is an interesting conceptual possibility which, in many diﬀerentforms, objectively reﬂects the fast growing interest of part of the physics community, a partiallist including i) the idea of the vacuum as a quantum liquid [10, 11] (which can explain in a natural way the huge diﬀerence between the typical vacuum-energy scales of modern particlephysics and the cosmological term needed in Einstein’s equations to ﬁt the observations)ii) the idea of Lorentz symmetry as associated with an infrared ﬁxed point [12, 13] in non-symmetric quantum ﬁeld theories iii) the quantum-gravity literature which, by starting fromthe original concept [14] of ‘space-time foam’, explicitly models the vacuum as a turbulentﬂuid [15, 16, 17] iv) the idea of deformations of Lorentz symmetry in a theoretical scheme (‘Doubly Special Relativity’) [18, 19, 20] where besides an invariant speed there is also an invariant length associated with the Planck scale v) the representation of relativistic particlepropagation from the superposition, at very short time scales, of non-relativistic particle pathswith diﬀerent Newtonian mass [21].Here, however, we shall adopt a diﬀerent perspective and concentrate our analysis ona peculiar aspect of today’s quantum ﬁeld theories: the representation of the vacuum as a‘condensate’ of elementary quanta. These condense because their trivially empty vacuum isa meta-stable state and not the true ground state of the theory. In the physically relevantcase of the Standard Model of electroweak interactions, this situation can be summarizedby saying [22] 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 Bosecondensed”. The explicit translation from ﬁeld jargon to particle jargon, with the substantialequivalence between the eﬀective potential of quantum ﬁeld theory and the energy density of 2

a dilute particle condensate, can be found for instance in ref.[23].The trivial empty vacuum will eventually be re-established by heating the system abovea critical temperature

where the condensate ‘evaporates’. This temperature in theStandard Model is so high that one can safely approximate the ordinary vacuum as a zero-temperature system (think of

He at a temperature 10

−

K). This observation allows oneto view the physical vacuum as a superﬂuid medium [10] where bodies can ﬂow without anyapparent friction, consistently with the experimental results. Clearly, this form of quantumvacuum is not the kind of ether imagined by Lorentz. However, if possible, this modern viewof the vacuum state is even more diﬀerent from the empty space-time of Special Relativitythat Einstein had in mind in 1905. Therefore, one might ask [24] if Bose condensation, i.e. themacroscopic occupation of the same quantum state, say

= 0 in some reference frame Σ, canrepresent the operative construction of a ‘quantum ether’. This characterizes the

physically realized form of relativity

and could play the role of the preferred reference frame in a modernLorentzian approach.Usually this possibility is not considered with the motivation, perhaps, that the averageproperties of the condensed phase are summarized into a single quantity which transformsas a world scalar under the Lorentz group, for instance, in the Standard Model, the vacuumexpectation value





of the Higgs ﬁeld. However, this does not imply that the vacuum stateitself has to be

Lorentz invariant

. Namely, Lorentz transformation operators ˆ

′

, ˆ

′′

,..mighttransform non trivially the reference vacuum state

(0)



(appropriate to an observer at restin Σ) into

′



′′



,.. (appropriate to moving observers

′

′′

,..) and still, for any Lorentz-invariant operator ˆ

, one would ﬁnd





(0)





′





′′

(2)Here, we are assuming the existence of a suitable operatorial representation of the Poincar´ealgebra for the quantum theory in terms of 10 generators

α,β

(

=0, 1, 2, 3) where

are the 4 generators of the space-time translations and

αβ

−

βα

are the 6 generatorsof the Lorentzian rotations with commutation relations[

] = 0 (3)[

αβ

] =

βγ

−

αγ

(4)[

αβ

γδ

] =

αγ

βδ

αγ

−

βγ

αδ

−

αδ

βγ

(5)where

αβ

= diag(1

−

1).With these premises, the possibility of a Lorentz-non-invariant vacuum state was addressedin refs.[25, 26] by comparing two basically diﬀerent approaches. In the ﬁrst description, as in 3

the axiomatic approach to quantum ﬁeld theory [27], one could describe the physical vacuumas an eigenstate of the energy-momentum vector. This physical vacuum state

(0)



wouldmaintain both zero momentum and zero angular momentum, i.e. (i,j=1,2,3)ˆ

(0)



= ˆ

(0)



= 0 (6)but, at the same time, be characterized by a non-vanishing energyˆ

(0)



(0)



(7)This vacuum energy might have diﬀerent explanations. Here, we shall limit ourselves to explor-ing the physical implications of its existence by just observing that, in interacting quantumﬁeld theories, there is no known way to ensure consistently the condition

= 0 withoutimposing an unbroken supersymmetry, which is not phenomenologically acceptable. In thisframework, by using the Poincar´e algebra of the boost and energy-momentum operators, onethen deduces that the physical vacuum cannot be a Lorentz-invariant state and that, in anymoving frame, there should be a non-zero vacuum spatial momentum





′



= 0 along thedirection of motion. In this way, for a moving observer

′

the physical vacuum would look likesome kind of ethereal medium for which, in general, one can introduce a momentum density





′

through the relation (i=1,2,3)





′

≡







′



= 0 (8)On the other hand, there is an alternative approach where one tends to consider the vacuumenergy

as a spurious concept and only concentrate on an energy-momentum tensor of thefollowing form [28, 29]



µν



(0)

µν

(9)(

being a space-time independent constant). In this case, one is driven to completelydiﬀerent conclusions since, by introducing the Lorentz transformation matrices Λ

µν

to anymoving frame

′

, deﬁning



µν



′

through the relation



µν



′

= Λ

σµ

ρν



σρ



(0)

(10)and using Eq.(9), it follows that the expectation value of ˆ

in any boosted vacuum state

′



vanishes, just as it vanishes in

(0)



, i.e.







′

≡



′

= 0 (11)

We ignore here the problem of vacuum degeneracy by assuming that any overlapping among equivalentvacua vanishes in the inﬁnite-volume limit of quantum ﬁeld theory (see e.g. S. Weinberg,

The Quantum Theory of Fields

, Cambridge University press, Vol.II, pp. 163-167).

As discussed in ref.[25], both alternatives have their own good motivations and it is not soobvious how to decide between Eq.(8) and Eq.(11) on purely theoretical grounds. For instance, in a second-quantized formalism, single-particle energies

(

) are deﬁned as the energies of the corresponding one-particle states



minus the energy of the zero-particle, vacuum state.If

is considered a spurious concept,

(

) will also become an ill-deﬁned quantity. At adeeper level, one should also realize that in an approach based solely on Eq.(9) the propertiesof

(0)



under a Lorentz transformation are not well deﬁned. In fact, a transformed vacuumstate

′



is obtained, for instance, by acting on

(0)



with the boost generator ˆ

. Once

(0)



is considered an eigenstate of the energy-momentum operator, one can deﬁnitely show[25] that, for



= 0,

′



and

(0)



diﬀer non-trivially. On the other hand, if

= 0 thereare only two alternatives: either ˆ

(0)



= 0, so that

′



(0)



, or ˆ

(0)



is a statevector proportional to

(0)



, so that

′



and

(0)



diﬀer by a phase factor.Therefore, if the structure in Eq.(9) were really equivalent to the exact Lorentz invarianceof the vacuum, it should be possible to show similar results, for instance that such a

(0)



state can remain invariant under a boost, i.e. be an eigenstate of ˆ

−



(

−

) (12)with zero eigenvalue. As far as we can see, there is no way to obtain such a result by just starting from Eq.(9) (this only amounts to the weaker condition





(0)

= 0). Thus,independently of the ﬁniteness of

, it should not come as a surprise that one can run intocontradictory statements once

(0)



is instead characterized by means of Eqs.(6)

−

(7). Forthese reasons, it is not obvious that the local relations (9) represent a more fundamentalapproach to the vacuum.Alternatively, one could argue that a satisfactory solution of the vacuum-energy prob-lem lies deﬁnitely beyond ﬂat space. A non-zero

, in fact, should induce a cosmologicalterm in Einstein’s ﬁeld equations and a non-vanishing space-time curvature which anyhowdynamically breaks global Lorentz symmetry. Nevertheless, in our opinion, in the absence of a consistent quantum theory of gravity, physical models of the vacuum in ﬂat space can beuseful to clarify a crucial point that, so far, remains obscure: the huge renormalization eﬀectwhich is seen when comparing the typical vacuum-energy scales of modern particle physicswith the experimental value of the cosmological term needed in Einstein’s equations to ﬁtthe observations. For instance, as anticipated, the picture of the vacuum as a superﬂuidcan explain in a natural way why there might be no non-trivial macroscopic curvature inthe equilibrium state where any liquid is self-sustaining [10]. In any liquid, in fact, curva-ture requires

deviations

from the equilibrium state. The same happens for a crystal at zero5

temperature where all lattice distortions vanish and electrons can propagate freely as in aperfect vacuum. In such representations of the lowest energy state, where large condensationenergies (of the liquid and of the crystal) play no observable role, one can intuitively under-stand why curvature eﬀects can be orders of magnitude smaller than those naively expectedby solving Einstein’s equations with the full



µν



(0)

as a cosmological term. In this per-spective, ‘emergent-gravity’ approaches [30, 31, 32], where gravity somehow arises from the same physical ﬂat-space vacuum, may become natural

and, to ﬁnd the eﬀective form forthe cosmological term to be inserted in Einstein’s ﬁeld equations, we are lead to sharpen ourunderstanding of the vacuum structure and of its excitation mechanisms by starting from thephysical picture of a superﬂuid medium. To decide between Eqs.(8) and (11), one could then work out the possible observable consequences and check experimentally the existence of afundamental energy-momentum ﬂow.

2. Vacuum energy-momentum ﬂow as an ether drift

To explore the idea of a non-zero vacuum energy-momentum ﬂow, one can adopt a phe-nomenological model [25] where the physical vacuum is described as a relativistic ﬂuid [34]. In this representation, a non-zero





′

gives rise to a tiny heat ﬂow and an eﬀective ther-mal gradient in a moving frame

′

. This would represent a fundamental perturbation which,if present, is likely too small to be detectable in most experimental conditions by standardcalorimetric devices. However, it could eventually be detected through very accurate ether-drift experiments performed in forms of matter that react by producing convective currentsin the presence of arbitrarily small thermal gradients, i.e. in gaseous systems.To better explain this possibility, let us ﬁrst recall that in the modern version of theseexperiments one looks for a possible anisotropy of the two-way velocity of light through therelative frequency shift ∆

(

) of two orthogonal optical cavities [35, 36]. Their frequency

(

) = ¯

(

)

(

) (13)is proportional to the two-way velocity of light ¯

(

) within the cavity through an integernumber

, which ﬁxes the cavity mode, and the length of the cavity

(

) as measured inthe laboratory. In principle, by ﬁlling the resonating cavities with some gaseous medium, theexistence of a vacuum energy-momentum ﬂow could produce two basically diﬀerent eﬀects:

In this sense, by exploring emergent-gravity approaches based on an underlying superﬂuid medium, oneis taking seriously Feynman’s indication : ”...the ﬁrst thing we should understand is how to formulate gravityso that it doesn’t interact with the energy in the vacuum” [33].

a) modiﬁcations of the solid parts of the apparatus. These can change the cavity lengthupon active rotations of the apparatus or under the Earth’s rotation.b) convective currents of the gas molecules

inside

the optical cavities. These can producean anisotropy of the two-way velocity of light. In this sense, the reference frame

′

where thesolid container of the gas is at rest would not deﬁne a true state of rest.Now, an anisotropy of the cavity length, in the laboratory frame, would amount to ananisotropy of the basic atomic parameters, a possibility which is severely limited experimen-tally. In fact, in the most recent versions of the original Hughes-Drever experiment [37, 38], where one measures the atomic energy levels as a function of their orientation with respect tothe ﬁxed stars, possible deviations from isotropy have been found below the 10

−

level [39].This is incomparably smaller than any other eﬀect on the velocity of light that we are goingto discuss. Therefore, mechanism a), if present, is completely negligible and, from now on, weshall assume

(

) =

=constant. In this way, one re-obtains the standard relation adoptedin the analysis of the experiments∆

phys

(

)

= ¯

(

π/

2 +

)

−

(

)

≡

∆¯

(14)where

is the reference frequency of the two optical resonators and the suﬃx “phys” indicatesa hypothetical physical part of the frequency shift after subtraction of all spurious eﬀects.Let us now estimate the possible eﬀects of mechanism b) by ﬁrst recalling that rigoroustreatments of light propagation in dielectric media are based on the extinction theory [40].This was originally formulated for continuous media where the inter-particle distance is smallerthan the light wavelength. In the opposite case of an isotropic, dilute random medium [41]as a gas, it is relatively easy to compute the scattered wave in the forward direction andobtain the refractive index. However, the presence of convective currents would produce ananisotropy of the velocity of refracted light.To derive the relevant relations, let us introduce from scratch the refractive index

of thegas. By assuming isotropy, the time

spent by refracted light to cover some given distance

within the medium is

L/c

. This can be expressed as the sum of

L/c

and

= (

N −

L/c

where

is the same time as in the vacuum and

represents the additional,average time by which refracted light is slowed down by the presence of matter. If there areconvective currents, due to the motion of the laboratory with respect to a preferred referenceframe Σ, then

will be diﬀerent in diﬀerent directions, and there will be an anisotropy of the velocity of light proportional to (

N −

1). In fact, let us consider light propagating in a2-dimensional plane and express

Lc f

(

,θ,β

) (15)7

with

V/c

being (the projection on the considered plane of) the relevant velocity withrespect to Σ where the isotropic form

(

,θ,

0) =

N −

1 (16)is assumed. By expanding around

= 1 where, whatever

vanishes by deﬁnition, oneﬁnds for gaseous systems (where

N −

≪

1) the universal trend

(

,θ,β

)

∼

(

N −

(

θ,β

) (17)with

(

θ,β

)

≡

(

∂f/∂

)

(18)and

(

θ,

0) = 1. Therefore, by introducing the one-way velocity of light

(

,θ,β

) =

(

,θ,β

)

∼

(

N −

(

θ,β

) (19)one gets

(

,θ,β

)

∼

−

(

N −

1) (

(

θ,β

)

−

1)] (20)Analogous relations hold for the two-way velocity ¯

(

,θ,β

)¯

(

,θ,β

) = 2

(

,θ,β

)

(

,π

θ,β

)

(

,θ,β

) +

(

,π

θ,β

)

∼



−

(

N −



(

θ,β

) +

(

θ,β

−



(21)A more explicit expression can be obtained by exploring some general properties of the func-tion

(

θ,β

). By expanding in powers of

β F

(

θ,β

)

−

1 =

βF

(

) +

(

) +

...

(22)and taking into account that, by the very deﬁnition of two-way velocity, ¯

(

,θ,β

) =¯

(

,θ,

−

), it follows that

(

) =

−

(

). Therefore, to

(

), we get the generalstructure [26]¯

(

,θ,β

)

∼



−

(

N −

∞



(cos

)



(23)in which we have expressed the combination

(

) +

(

) as an inﬁnite expansion of even-order Legendre polynomials with unknown coeﬃcients

(1) which depend on thecharacteristics of the induced convective motion of the gas molecules inside the cavities.Eq.(23), in principle, is exact to the given accuracy but it is of limited utility if one wantsto compare with real experiments. In fact, it would require the complete control of all possiblemechanisms that can produce the gas convective currents by starting from scratch with the8

macroscopic Earth’s motion in the physical vacuum. This general structure can, however, becompared with the particular form (see Eq.(109) of the Appendix) obtained by using Lorentztransformations to connect

′

to the preferred frame¯

(

,θ,β

)

∼

−

(

N −

1)(

sin

)] (24)with

= 2 and

−

1 which corresponds to setting

= 4

= 2

3 and all

= 0for

n >

1 in Eq.(23). Eq.(24) represents a deﬁnite realization of the general structure in (23) and a particular case of the Robertson-Mansouri-Sexl (RMS) scheme [42, 43] for anisotropy parameter

|B|

N −

1 (see the Appendix). In this sense, it provides a partial answer to theproblems posed by our limited knowledge of the electromagnetic properties of gaseous systemsand will be adopted in the following as a tentative model for the two-way velocity of light

.Summarizing: in this scheme, the theoretical estimate for a possible anisotropy of thetwo-way velocity of light is



∆

ν ν



Theorgas



∆¯



Theorgas

∼

(

gas

−

(26)Then, by assuming the typical velocity of most Earth’s cosmic motions

∼

300 km/s, onewould expect

∆¯



−

for experiments performed in air at atmospheric pressure, where

N ∼

00029, or

∆¯



−

for experiments performed in helium at atmospheric pressure,where

N ∼

000035. Therefore these potential eﬀects are much larger than those possiblyassociated with vacuum cavities. In fact, from experiments one ﬁnds [44]

−

[50]



∆

ν ν



EXPvacuum



∆¯



EXPvacuum

∼

−

(27)

One conceptual detail concerns the gas refractive index whose reported values are experimentally measuredon the Earth by two-way measurements. For instance for air, the most precise determinations are at the level10

−

, say

air

= 1

0002926

at STP (Standard Temperature and Pressure). By assuming a non-zero anisotropyin the Earth’s frame, one should interpret the isotropic value

air

as an angular average of Eq.(24), i.e.

air

≡

( ¯

air

,θ,β

)



air

−

32( ¯

air

−

] (25)From this relation, one can determine in principle the unknown value ¯

air

≡ N

(Σ) (as if the gas were atrest in Σ), in terms of the experimentally known quantity

air

≡ N

(

Earth

) and of

. In practice, for thestandard velocity values involved in most cosmic motions, say

∼

300 km/s, the diﬀerence between

(Σ)and

(

Earth

) is at the level 10

−

and thus completely negligible. The same holds true for the other gaseoussystems at STP (say nitrogen, carbon dioxide, helium,..) for which the present experimental accuracy in therefractive index is, at best, at the level 10

−

. Finally, the isotropic two-way speed of light is better determinedin the low-pressure limit where (

N −

→

0. In the same limit, for any given value of

, the approximation

(Σ) =

(

Earth

) becomes better and better.

or smaller and thus completely negligible when compared with those of Eq.(26).On the other hand, if one were considering light propagation in a strongly bound system,such as a solid or liquid transparent medium, the small energy ﬂow generated by the motionwith respect to the vacuum condensate should mainly dissipate by heat conduction with noappreciable particle ﬂow and no light anisotropy in the rest frame of the container of themedium. This conclusion is in agreement with the experiments [7, 51] that seem to indicate the existence of

two

regimes. A former region of gaseous systems where

N ∼

1 and there aresmall residuals which are roughly consistent with Eq.(26). A latter region where the diﬀerenceof

from unity is substantial, (e.g.

N ∼

5 as with perspex in the experiment by Shamirand Fox [52]), where light propagation is seen isotropic in the rest frame of the medium (i.e.in the Earth’s frame). Although it would be diﬃcult to describe in a fully quantitative waythe transition between the two regimes, some simple arguments can be given along the linessuggested by de Abreu and Guerra (see pages 165-170 of ref.[53]).For this reason, it was proposed in refs.[7, 25, 26] that one should design a new class of dedicated experiments in gaseous systems. Such a type of ‘non-vacuum’ experiment would bealong the lines of ref.[54] where just the use of optical cavities ﬁlled with diﬀerent materialswas considered as a useful complementary tool to study deviations from exact Lorentz invari-ance. In the meantime, due to the heuristic nature of our approach, and to further motivatethis new series of experiments, one could try to obtain quantitative checks by applying thesame interpretative scheme to the classical ether-drift experiments (Michelson-Morley, Miller,Illingworth, Joos,...). These old experiments were performed with interferometers where lightwas propagating in air or helium at atmospheric pressure. In this regime, where (

N −

1) isa very small number, the theoretical fringe shifts expected on the basis of Eqs.(23) and (24) are much smaller than the classical prediction

(

) and it becomes conceivable that tinynon-zero eﬀects might have been erroneously interpreted as ‘null results’.To make this more evident, let us adopt Eq.(24). Then, an anisotropy of the two-wayvelocity of light could be measured by rotating a Michelson interferometer. As anticipated,in the rest frame

′

of the apparatus, the length

of its arms does not depend on theirorientation so that the interference pattern between two orthogonal beams of light dependson the time diﬀerence∆

(

) = 2

(

,θ,β

)

−

(

,π/

2 +

θ,β

) (28)In this way, by introducing the wavelength

of the light source and the projection

of therelative velocity in the plane of the interferometer, one ﬁnds to order

the fringe shift∆

(

)

∼

∆

(

)

∼

Lλv

2obs

cos2(

−

) (29)10

In the above equation the angle

(

) indicates the apparent direction of the ether-driftin the plane of the interferometer (the ‘azimuth’) and the square of the

observable

velocity

2obs

(

)

∼

N −

(

) (30)is re-scaled by the tiny factor 2(

N −

1) with respect to the true

kinematical

velocity

(

).We emphasize that

obs

is just a short-hand notation to summarize into a single quantity thecombined eﬀects of a given kinematical

and of the gas refractive index

. In this sense, onecould also avoid its introduction altogether. However, in our opinion, it is a useful, compactparametrization since, in this way, relation (29) is formally identical to the classical predictionof a second-harmonic eﬀect with the only replacement

→

obs

. For this reason, as we shallsee in the following sections, it is in terms of

obs

, rather than in terms of the true kinematical

, that one can more easily compare with the original analysis of the classical ether-driftexperiments.In conclusion, in this scheme, the interpretation of the experiments is transparent. Ac-cording to Special Relativity, there can be no fringe shift upon rotation of the interferometer.In fact, if light propagates in a medium, the frame of isotropic propagation is always assumedto coincide with the laboratory frame

′

, where the container of the medium is at rest, andthus one has

obs

= 0. On the other hand, if there were fringe shifts, one could try todeduce the existence of a preferred frame Σ



′

provided the following minimal requirementsare fulﬁlled : i) the fringe shifts exhibit an angular dependence of the type in Eq.(29) ii) byusing gaseous media with diﬀerent refractive index one gets consistency with Eq.(30) in sucha way that diﬀerent

obs

correspond to the same kinematical

.Before starting with the analysis of the classical experiments, one more remark is in order.In principle, even a

single

observation, within its experimental accuracy, can determine theexistence of an ether-drift. However interpretative models are required to compare resultsobtained at diﬀerent times and in diﬀerent places. In the scheme of Eqs.(29) and (30), the crucial information is contained in the two time-dependent functions

(

) and

(

),respectively the magnitude of the velocity and the apparent direction of the azimuth in theplane of the interferometer. For their determination, the standard assumption is to considera cosmic Earth’s velocity with well deﬁned magnitude

, right ascension

and angulardeclination

that can be considered constant for short-time observations of a few days wherethere are no appreciable changes due to the Earth’s orbital velocity around the Sun. In thisframework, where the only time dependence is due to the Earth’s rotation, one identiﬁes

(

)

≡

(

) and

(

)

≡

(

) where ˜

(

) and ˜

(

) derive from the simple application of 11

spherical trigonometry [55]cos

(

) = sin

sin

+ cos

cos

cos(

−

) (31)˜

(

)

≡

sin

(

)cos ˜

(

) = sin

cos

−

cos

sin

cos(

−

) (32)˜

(

)

≡

sin

(

)sin ˜

(

) = cos

sin(

−

) (33)˜

(

)

≡



(

) + ˜

(

) =

sin

(

)

(34)Here

(

) is the zenithal distance of

is the latitude of the observatory,

sid

is the sidereal time of the observation in degrees (

sid

∼

′

) and the angle

is countedconventionally from North through East so that North is

= 0 and East is

= 90

.To explore the observable implications, let us ﬁrst re-write the basic Eq.(29) as∆

(

)

∼

(

N −

λv

(

)

cos2(

−

(

))

≡

(

)cos2

+ 2

(

)sin2

(35)where

(

) =

(

N −

λv

(

)

cos2

(

)

(

) =

(

N −

λv

(

)

sin2

(

) (36)Then Eqs. (31)

−

(34) amount to the structure

(

)

≡

(

) =

sin

cos

sin(2

) +

cos(2

) (37)

(

)

≡

(

) =

sin

cos

sin(2

) +

cos(2

) (38)with Fourier coeﬃcients (

R≡

(

N−

λV

)

−

(3cos2

−

1)cos

(39)

−

sin

sin2

φ C

−

cos

sin2

(40)

= 12

sin2

cos

(1 + sin

)

= 12

cos2

cos

(1 + sin

) (41)and

−

sin

φ S

sin

(42)

−

2sin

1 + sin

φC

= 2sin

1 + sin

φC

(43)These standard forms are nowadays adopted in the analysis of the data of the ether-driftexperiments [46]. However, one should not forget that Eq.(24) represents only an

approxi-mation

for the full structure Eq.(23). Therefore, even for short-time observations, one might12

not obtain from the data completely consistent determinations of the kinematical parameters(

V,α,γ

). In addition, by using a physical analogy, and by representing the Earth’s motion inthe physical vacuum as the motion of a body in a ﬂuid, the scheme Eqs.(37)

(38) of smoothsinusoidal variations associated with the Earth’s rotation corresponds to the conditions of a pure laminar ﬂow associated with a simple regular motion. Instead, the physical vacuummight behave as a turbulent ﬂuid, where large-scale and small-scale ﬂows are only

indirectly

related.In this modiﬁed perspective, which ﬁnds motivations in some basic foundational aspectsof both quantum physics and relativity [56, 57, 58, 59, 60] and in those representations of the vacuum as a form of ‘space-time foam’ which indeed resembles a turbulent ﬂuid [14, 15, 16, 17], the ether-drift might exhibit forms of time modulations that do

not

ﬁt in the scheme of Eqs.(37)

(38). To evaluate the potential eﬀects, and by still retaining the functional formEq.(35), one could ﬁrst re-write Eqs.(36) as

(

) =

(

N −

λv

(

)

−

(

)

(

) =

(

N −

(

)

(

)

(44)where

(

) =

(

)cos

(

) and

(

) =

(

)sin

(

). Then, by exploiting the turbulencescenario, one could model the two velocity components

(

) and

(

) as stochastic ﬂuctua-tions. In this diﬀerent scheme, where now

(

)



= ˜

(

) and

(

)



= ˜

(

), experimental resultswhich, on consecutive days and at the same sidereal time, deviate from Eqs.(31)

−

(34) do

not

necessarily represent spurious eﬀects. Equivalently, if data collected at the same sidereal timeaverage to zero this does

not

necessarily mean that there is no ether-drift. This particularaspect will be discussed at length in the rest of the paper.After this important premise, we shall now proceed in Sects. 3-8 with our re-analysis of the classical experiments. In the end, Sect.9 will contain a summary, a brief discussion of themodern experiments and our conclusions.

3. The original Michelson-Morley experiment

The Michelson-Morley experiment [61] is probably the most celebrated experiment in thehistory of physics. Its result and its interpretation have been (and are still) the subject of endless controversies. For instance, for some time there was the idea [62] that, by taking intoaccount the reﬂection from a moving mirror and other eﬀects, the predicted shifts would belargely reduced and become unobservable. These points of view are summarized in Hedrick’scontribution to the ‘Conference on the Michelson-Morley experiment’ [63] (Pasadena, Febru-ary 1927) which was attended by the greatest experts of the time, in particular Lorentz and13

Michelson. The arguments presented by Hedrick were, however, refuted by Kennedy [64] in apaper of 1935 where, by using Huygens principle, he re-obtained to order

the classicalresult of Eq.(29) (with the identiﬁcation

obs

).Figure 1:

The Michelson-Morley fringe shifts as reported by Hicks [ 66 ]. Solid and dashed lines refer respectively to noon and evening observations.

In this framework, the fringe shift is a second-harmonic eﬀect, i.e. periodic in the range[0

,π

], whose amplitude

is predicted diﬀerently by using the classical formulas or Lorentztransformations (29)

class2

Lλv

rel2

Lλv

obs2

∼

N −

class2

(45)Notice also that upon rotation of

π/

2 with respect to

the predicted fringe shift is 2

.Now, for the Michelson-Morley interferometer the whole eﬀective optical path was about

= 11 meters, or about 2

in units of light wavelengths, so for a velocity

∼

30 km/s(the Earth’s orbital velocity about the Sun, and consequently the minimum anticipated drift14

velocity) the expected classical 2nd-harmonic amplitude was

class2

∼

2. This value can thusbe used as a reference point to obtain an observable velocity, in the plane of the interferometer,from the actual measured value of

through the relation

obs

∼



2 km

s (46)Michelson and Morley performed their six observations in 1887, on July 8th, 9th, 11th and12th, at noon and in the evening, in the basement of the Case Western University of Cleveland.Each experimental session consisted of six turns of the interferometer performed in about 36minutes. As well summarized by Miller in 1933 [65], “The brief series of observations wassuﬃcient 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 [66]: ”..the data publishedby Michelson and Morley, instead of giving a null result, show distinct evidence for an eﬀectof the kind to be expected”. Namely, there was a second-harmonic eﬀect. But its amplitudewas substantially smaller than the classical expectation (see Fig.1).Quantitatively, the situation can be summarized in Figure 2, taken from Miller [65], wherethe values of the eﬀective velocity measured in various ether-drift experiments are reportedand compared with a smooth curve ﬁtted by Miller to his own results as function of thesidereal time.For the Michelson-Morley experiment, the average observable velocity reported by Milleris about 8.4 km/s. Comparing with the classical prediction for a velocity of 30 km/s, thismeans an experimental 2nd- harmonic amplitude

EXP2

∼

2 (8

430 )

∼

016 (47)which is about twelve times smaller than the expected result.Neither Hicks nor Miller reported an estimate of the error on the 2nd harmonic extractedfrom the Michelson-Morley data. To understand the precision of their readings, we can lookat the original paper [61] where one ﬁnds the following statement : ”The readings are divisionsof the screw-heads. The width of the fringes varied from 40 to 60 divisions, the mean valuebeing near 50, so that one division means 0.02 wavelength”. Now, in their tables Michelsonand Morley reported the readings with an accuracy of 1/10 of a division (example 44.7, 44.0,43.5,..). This means that the nominal accuracy of the readings was

002 wavelengths. Infact, in units of wavelengths, they reported values such as 0.862, 0.832, 0.824,.. Furthermore,this estimate of the error agrees well with Born’s book [67]. In fact, Born, when discussingthe classically expected fractional fringe shift upon rotation of the apparatus by 90

, about15

Figure 2:

The magnitude of the observable velocity measured in various experiments as reported by Miller [ 65 ].

0.37, explicitly says: “Michelson was certain that the one-hundredth part of this displacementwould still be observable” (i.e. 0.0037). Therefore, to be consistent with both the originalMichelson-Morley article and Born’s quotation of Michelson’s thought, we shall adopt

004as an estimate of the error

.With this premise, the Michelson-Morley data were re-analyzed in ref.[51]. To this end,one should ﬁrst follow the well deﬁned procedure adopted in the classical experiments asdescribed in Miller’s paper [65]. Namely, by starting from each set of seventeen entries (oneevery 22

), say

(

), one has ﬁrst to correct the data for the observed linear thermal drift.This is responsible for the diﬀerence

(1)

−

(17) between the 1st entry and the 17th entryobtained after a complete rotation of the apparatus. In this way, by adding 15/16 of thecorrection to the 16th entry, 14/16 to the 15th entry and so on, one obtains a set of 16corrected entries

corr

(

) =

−

116 (

(1)

−

(17)) +

(

) (48)The fringe shifts are then deﬁned by the diﬀerences between each of the corrected entries

To conﬁrm that such estimate should not be considered unrealistically small, we report explicitly Michel-son’s words from ref.[63]:“I must say that every beginner thinks himself lucky if he is able to observe a shiftof 1/20 of a fringe. It should be mentioned however that with some practice shifts of 1/100 of a fringe can bemeasured, and that in very favorable cases even a shift of 1/1000 of a fringe may be observed.”

Table 1:

The fringe shifts

∆

(

)

for all noon (n.) and evening (e.) sessions of the Michelson-Morley experiment.

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 +0.0362 +0.024

−

0.004

−

0.034 +0.008

−

0.015 +0.0443 +0.053

−

0.004

−

0.038

−

0.010 +0.006 +0.0474 +0.015

−

0.003

−

0.066 +0.070 +0.004 +0.0275

−

0.036

−

0.031

−

0.042 +0.041 +0.027

−

0.0026

−

0.007

−

0.020

−

0.014 +0.055 +0.015

−

0.0127 +0.024

−

0.025 +0.000 +0.057

−

0.022 +0.0078 +0.026

−

0.021 +0.028 +0.029

−

0.036

−

0.0119

−

0.021

−

0.049 +0.002

−

0.005

−

0.033

−

0.02810

−

0.022

−

0.032

−

0.010 +0.023 +0.001

−

0.06411

−

0.031 +0.001

−

0.004 +0.005

−

0.008

−

0.09112

−

0.005 +0.012 +0.012

−

0.030

−

0.014

−

0.05713

−

0.024 +0.041 +0.048

−

0.034

−

0.007

−

0.03814

−

0.017 +0.042 +0.054

−

0.052 +0.015 +0.04015

−

0.002 +0.070 +0.038

−

0.084 +0.026 +0.05916 +0.022

−

0.005 +0.006

−

0.062 +0.024 +0.043

corr

(

) and their average value



corr



as∆

(

)

corr

(

)

−

corr



(49)The resulting data are reported in Table 1.With this procedure, the fringe shifts Eq.(49) are given as a periodic function, with van-ishing mean, in the range 0

≤

, with

−

116

, so that they can be reproduced in aFourier expansion. Notice that in the evening observations the apparatus was rotated in theopposite direction to that of noon.One can thus extract the amplitude and the phase of the 2nd-harmonic component byﬁtting the even combination of fringe shifts

(

) = ∆

(

) + ∆

(

(50)(see Fig.3). This is essential to cancel the 1st-harmonic contribution originally pointed outby Hicks [66]. Its theoretical interpretation is in terms of the arrangements of the mirrors17

Table 2:

The amplitude of the ﬁtted second-harmonic component

EXP2

for the six experimental sessions of the Michelson-Morley experiment.

SESSION

EXP2

July 8 (noon) 0

010

005July 9 (noon) 0

015

005July 11 (noon) 0

025

005July 8 (evening) 0

014

005July 9 (evening) 0

011

005July 12 (evening) 0

024

005and, as such, this eﬀect has to show up in the outcome of real experiments. For more details,see the discussion given by Miller, in particular Fig.30 of ref.[65], where it is shown that hisobservations were well consistent with Hicks’ theoretical study. The observed 1st-harmoniceﬀect is sizeable, of comparable magnitude or even larger than the second-harmonic eﬀect.The same conclusion was also obtained by Shankland et

al.

[68] in their re-analysis of Miller’sdata. The 2nd-harmonic amplitudes from the six individual sessions are reported in Table 2.Due to their reasonable statistical consistency, one can compute the mean and variance of the six determinations reported in Table 2 by obtaining

EXP2

∼

016

006. This value isconsistent with an observable velocity

obs

∼

−

s (51)Then, by using Eq.(30), which connects the observable velocity to the projection of the kine-matical velocity in the plane of the interferometer through the refractive index of the mediumwhere light propagation takes place (in our case air where

N ∼

00029), we can deduce theaverage value

∼

349

+62

−

s (52)While the individual values of

show a reasonable consistency, there are substantialchanges in the apparent direction

of the ether-drift eﬀect in the plane of the interferometer.This is the reason for the strong cancelations obtained when ﬁtting together all noon sessionsor all evening sessions [69]. For instance, for the noon sessions, by taking into account that theazimuth is always deﬁned up to

180

, one choice for the experimental azimuths is 357

,285

and 317

respectively for July 8th, 9th and 11th. For this assignment, theindividual velocity vectors

obs

(cos

−

sin

) and their mean are shown in Fig.4. According18

-0.04-0.0200.020.04

Β ( θ )

July 11 noon

π π

Figure 3:

A ﬁt to the even combination

(

)

Eq.( 50 ). The second harmonic amplitude is

EXP2

= 0

025

005

and the fourth harmonic is

EXP4

= 0

004

005

. The ﬁgure is taken from ref.[ 51] . Compare the data with the solid curve of July 11th shown in Fig.1.

to the usual interpretation, the large spread of the azimuths is taken as indication that anynon-zero fringe shift is due to pure instrumental eﬀects. However, as anticipated in Sect.2,this type of discrepancy could also indicate an unconventional form of ether-drift where thereare substantial deviations from Eq.(24) and/or from the smooth trend in Eqs.(31)

−

(34). Forinstance, in agreement with the general structure Eq.(23), and diﬀerently from July 11 noon,which represents a very clean indication, there are sizeable 4th- harmonic contributions (here

EXP4

= 0

019

005 and

EXP4

= 0

008

005 for the noon sessions of July 8 and July 9respectively). In any case, the observed strong variations of

are in qualitative agreementwith the analogous values reported by Miller. To this end, compare with Fig.22 of ref.[65] andin particular with the large scatter of the data taken around August 1st, as this represents theepoch of the year which is closer to the period of July when the Michelson-Morley observationswere actually performed. Thus one could also conclude that individual experimental sessionsindicate a deﬁnite non-zero ether-drift but the azimuth does not exhibit the smooth trendexpected from the conventional picture Eqs.(31)

−

(34).For completeness, we add that the large spread of the

−

values might also reﬂect aparticular systematic eﬀect pointed out by Hicks [66]. As described by Miller [65], “ before beginning observations the end mirror on the telescope arm is very carefully adjusted to secure19

0 1 2 3 4 5 6 7 80123456789

obs

July 8July 11July 9

Mean

(Km/s)(Km/s)

Figure 4:

The observable velocities for the three noon sessions and their mean. The x-axis corresponds to

= 0

≡

360

and the y-axis to

= 270

. Statistical uncertainties of the various determinations are ignored. All individual directions could also be reversed by 180

vertical fringes of suitable width. There are two adjustments of the angle of this mirror whichwill give fringes of the same width but which produce opposite displacements of the fringesfor the same change in one of the light-paths”. Since the relevant shifts are extremely small,“...the adjustments of the mirrors can easily change from one type to the other on consecutivedays. It follows that averaging the results of diﬀerent days in the usual manner is not allowableunless the types are all the same. If this is not attended to, the average displacement maybe expected to come out zero

−

at least if a large number are averaged” [66]. Thereforeaveraging the fringe shifts from various sessions represents a delicate issue and can introduceuncontrolled errors. Clearly, this relative sign does not aﬀect the values of

and this iswhy averaging the 2nd-harmonic amplitudes is a safer procedure. However, it can introducespurious changes in the apparent direction

of the ether-drift. In fact, an overall change of sign of the fringe shifts at all

−

values is equivalent to replacing

→

π/

2. As a matterof fact, Hicks concluded that the fringes of July 8th were of diﬀerent type from those of theremaining days. Thus for his averages (in our Fig.1) “the values of the ordinates are one-thirdof July 9 + July 11

−

July 8 and one-third of July 9 + July 12

−

July 8” [66] for noon andevening sessions respectively. If this were true, one choice for the azimuth of July 8th couldnow be

EXP0

= 267

. This would orient the arrow of July 8th in Fig.4 in the direction20

of the y

−

axis and change the average azimuth from



EXP0

 ∼

317



EXP0

 ∼

290

. We’llreturn to this particular aspect in our Appendix II.Let us ﬁnally compare with the interpretation that Michelson and Morley gave of theirdata. They start from the observation that ”...the displacement to be expected was 0.4 fringe”while ”...the actual displacement was certainly less than the twentieth part of this”. In thisway, since the displacement is proportional to the square of the velocity, ”...the relative velocityof the earth and the ether is... certainly less than one-fourth of the orbital earth’s velocity”.The straightforward translation of this upper bound is

obs

7.5 km/s. However, this estimateis likely aﬀected by a theoretical uncertainty. In fact, in their Fig.6, Michelson and Morleyreported their measured fringe shifts together with the plot of a theoretical second-harmoniccomponent. In doing so, they plotted a wave of amplitude

= 0

05, that they interpretas

one-eight

of the theoretical displacement expected on the base of classical physics, thusimplicitly assuming

class2

=0.4. As discussed above, the amplitude of the classically expectedsecond-harmonic component is

not

0.4 but is just one-half of that,

i.e.

0.2. Therefore, theirexperimental upper bound

exp2

420

=0.02 is actually equivalent to

obs

9.5 km/s. If we now consider that their estimates were obtained after superimposing the fringe shiftsobtained from various sessions (where the overall eﬀect is reduced, see our Fig.1), we deducea substantial agreement with our result Eq. (51).

4. Morley-Miller

After the original 1887 experiment, there was much interest in the Michelson-Morley resultthat, being too small to meet any classical prediction, was apparently contradicting twocornerstones of physics: Galilei’s transformations and/or the existence of the ether. For thisreason, one of the most inﬂuential physicists of the time, Lord Kelvin, after his conference atthe 1900 Paris Expo, induced Morley and his young collaborator Dayton Miller to design anew interferometer (where the eﬀective optical path was increased up to 32 meters) to improvethe accuracy of the measurement over the 1887 result.It must be emphasized that Morley and Miller [70], in their observations of 1905, super-imposed the data of the morning with those of the evening. As explained by Miller [63], thetwo physicists were assuming that the ether drift had to be obtained by combining the motionof the solar system relative to nearby stars, i.e. toward the constellation of Hercules with avelocity of about 19 km/s, with the annual orbital motion (“We now computed the directionand the velocity of the motion of the centre of the apparatus by compounding the annualmotion in the orbit of the earth with the motion of the solar system toward a certain point21

in the heavens...There are two hours in each day when the motion is in the desired planeof the interferometer” [70]). The observations at the two times (about 11:30 a.m. and 9:00p.m.) were, therefore, combined in such a way that the presumed azimuth for the morningobservations coincided with that for the evening (“The direction of the motion with referenceto a ﬁxed line on the ﬂoor of the room being computed for the two hours, we were able tosuperimpose those observations which coincided with the line of drift for the two hours of observation” [70]). However, the observations for the two times of the day gave results havingnearly opposite phases. When these were combined, the result was nearly zero. For thisreason, the value then reported of an observable velocity of 3.5 km/s is incorrect and does

not

correspond to the actual results of the basic observations. The error was later understoodand corrected by Miller who found that the two sets of data were each indicating an eﬀectivevelocity of about 7.5 km/s (see Figure 11 of Miller’s paper [65]). For this reason, the correctaverage observable velocities for the entire period 1902-1905 are those shown in our Figure 2between 7 and 10 km/s or

obs

∼

5) km

s (53)By using Eq.(30), we then deduce the average value

∼

(353

62) km

s (54)

5. Kennedy-Illingworth

An interesting development was proposed by Kennedy in 1926. As summarized in his con-tribution to the previously mentioned Conference on the Michelson-Morley experiment [63],his small optical system was enclosed in an eﬀectively insulated, sealed metal case containinghelium at atmospheric pressure. Because of its small size, ”...circulation and variation indensity of the gas in the light paths were nearly eliminated. Furthermore, since the value of

N −

1 is only about 1/10 that for the air at the same pressure, the disturbing changes indensity of the gas correspond to those in air to only 1/10 of the atmospheric pressure”. Theessential ingredient of Kennedy’s apparatus consisted in the introduction of a small step, 1/20of wavelength thick, in one of the total reﬂecting mirrors of the interferometer allowing, inprinciple, for an ultimate fringe shift accuracy 1

−

. To take full advantage of this possibil-ity, Kennedy should have disposed of perfect mirrors and of a suitable (hotter) source of light.In the original version of the experiment, these reﬁnements were not implemented giving anactual fringe shift accuracy of 2

−

. In these conditions, as Kennedy explicitly says[63],”...the velocity of 10 km/s found by Prof. Miller would produce a fringe shift corresponding22

Table 3:

The infra-session averages





and





obtained from the 10 sets of rotations in each of the 32 sessions of Illingworth’s experiment. These values have been obtained from the weights of Illingworth’s Table III by applying the conversion factor 0.002.

5 A.M. 5 A.M. 11 A.M. 11 A.M. 5 P.M. 5 P.M. 11 P.M. 11 P.M.



 



00024

−

00066 +0

00070

−

00022 +0

00024 +0

00044

−

00010 +0

00024+0

00114 +0

00024

−

00042

−

00036

−

00056

−

00046 +0

00018 +0

00018+0

00000 +0

00000

−

00006

−

00052

−

00144

−

00080

−

00126

−

00006+0

00020

−

00044

−

00030 +0

00012

−

00016 +0

00004

−

00044

−

00026+0

00064 +0

00000

−

00022 +0

00038 +0

00018 +0

00016 +0

00000 +0

00024

−

00002

−

00010 +0

00048 +0

00020 +0

00030 +0

00030

−

00040

−

00004

−

00014

−

00006 +0

00030 +0

00014

−

00006 +0

00004 +0

00036

−

00036

−

00006 +0

00016 +0

00006

−

00006+0

00000 +0

00024

−

00010 +0

00010to 8

−

”, four times larger than the experimental resolution. Since the eﬀect is quadraticin the velocity, Kennedy’s result, fringe shifts

−

, can then be summarized as

obs

5 km

s (55)By using Eq.(30), for helium at atmospheric pressure where

N ∼

000035, this boundamounts to restrict the kinematical value by

v <

600 km/s.Kennedy’s apparatus was further reﬁned by Illingworth in 1927 [71]. Besides improvingthe quality of the mirrors and of the source, Illingworth’s data taking was also designed toreduce the presence of steady thermal drift and of odd harmonics. Looking at Illingworth’spaper, one ﬁnds that his reﬁnements reached indeed the nominal

(10

−

) accuracy mentionedby Kennedy, namely about 1/1500 of wavelength for the individual readings and (1

−

at the level of average values.Let us now analyze Illingworth’s results. He performed four series of observations in theﬁrst ten days of July 1927. These consisted of 32 experimental sessions, conducted dailyat 5 A.M. (6), 11 A.M. (10), 5 P.M. (10) and 11 P.M.(6), in which he was measuring thefringe displacement caused by a rotation through a right angle of the apparatus. To take into23

account 90

rotations let us ﬁrst re-write Eq.(29) as∆

(

)

cos2(

−

) (56)Therefore Illingworth, in his ﬁrst set (set A) of 10 rotations, North, East, South, West andback to North, was actually measuring

≡

cos2

. In a second set (set B), North-East,North-West, South-West, South-East and back to North-East, performed immediately afterthe set A, he was then measuring

≡

sin2

. Notice that both

and

diﬀer fromthe positive-deﬁnite quantity

≡

that should be inserted in Illingworth’s numericalrelation for his apparatus

obs

= 112

√

. Therefore, the reported values for the two velocities

= 112



and

= 112



should only be taken as

lower

bounds for the true

obs

.The mean values





and





obtained from the 10 sets of rotations in the 32 individualsessions can be obtained from Illingworth’s Table III and, for the convenience of the reader,are reported in our Table 3.From Table 3, one ﬁnds that the quantity











has a mean value of about0.00045, which corresponds to

obs

∼

4 km/s. Thus, by using Eq.(30) for helium at atmo-spheric pressure, we would tentatively deduce an average value

∼

284 km/s.However, this is only a very partial view. To go deeper into Illingworth’s experiment wehave to consider his basic measurements, i.e. the individual turns of his interferometer. Inthis case, the only known basic set of data reported by Illingworth is set A of July 9th, 11A.M. This set has been re-analyzed by M´unera [72] and his values for the fringe shifts are reported in our Table 4.As one can see, the fringe shifts are not small and correspond to an observable velocity inthe range 2-5 km/s. However, their sign seems to change randomly. Therefore, if one attemptsto extract the observable velocity from the mean of the 10 determinations,



∼−

00006,the resulting value 0.9 km/s is much smaller than all individual determinations. The basisof M´unera’s analysis was instead to estimate

obs

from

|

|

, from which he obtained anaverage velocity

obs

= 3

04 km/s.Now, the standard interpretation of such apparently random changes of sign is in termsof typical instrumental eﬀects and the standard method for eliminating these is the originalaveraging procedure as employed by Illingworth. But we will now show that they could alsoindicate an unconventional form of stochastic drift, of the type already mentioned in theprevious sections, and in which M´unera’s re-estimate has a deﬁnite signiﬁcance. To this end,we shall ﬁrst use the relations

(

) = 4

(

)

(

) = 4

(

) (57)24

Table 4:

Illingworth’s set A of July 9th, 11 A.M. as re-analyzed by M´ unera [ 72 ].

Rotation D

[km/s]1

−

00100 +0

00100 3.542 +0

00066 +0

00066 2.893

−

00066 +0

00066 2.894

−

00066 +0

00066 2.895

−

00166 +0

00166 4.576 +0

00234 +0

00234 5.417 +0

00100 +0

00100 3.548 +0

00034 +0

00034 2.049 +0

00000 +0

00000 0.0010

−

00100 +0

00100 3.54where the two functions

(

) and

(

) have been introduced in Eqs.(36) and (44). Thus Eqs.(57) can be re-written as

(

) = 8

(

N −

λv

(

)

−

(

) = 8

(

N −

λv

(

)

(

)

(58)where

(

) =

(

)cos

(

) and

(

) =

(

)sin

(

). In this way, by using the numericalrelation for Illingworth’s experiment

Lλ

(30km

∼

035 and the value of the helium refractiveindex, we obtain

(

)

∼

(

)

−

(

(300 km

−

(

)

∼

(

)

(

)(300 km

−

(59)The required random ingredient can then be introduced by characterizing the two velocitycomponents

(

) and

(

) as turbulent ﬂuctuations. To this end, there can be several ways.Here we shall restrict to the simplest choice of a turbulence which, at small scales, appearsstatistically isotropic and homogeneous

. This represents a zeroth-order approximation whichis motivated by the substantial reading error of the Illingworth measurements (it turns out tobe comparable to the eﬀects of turbulence). However, it is a useful example to illustrate basicphenomenological features associated with an underlying stochastic vacuum. To explore theresulting temporal pattern of the data, we have followed refs.[74, 75] where velocity ﬂows, in statistically isotropic and homogeneous 3-dimensional turbulence, are generated by unsteadyrandom Fourier series. The perspective is that of an observer moving in the turbulent ﬂuid

This picture reﬂects the basic Kolmogorov theory [73] of a ﬂuid with vanishingly small viscosity.

who wants to simulate the two components of the velocity in his x-y plane at a given ﬁxedlocation in his laboratory. This leads to the general expressions

(

) =

∞



[

(1)cos

(2)sin

] (60)

(

) =

∞



[

(1)cos

(2)sin

] (61)where

= 2

nπ/T

, T being a time scale which represents a common period of all stochasticcomponents. We have adopted the typical value

day

= 24 hours. However, we havealso checked with a few runs that the statistical distributions of the various quantities do notchange substantially by varying

in the rather wide range 0

day

≤

day

.The coeﬃcients

(

= 1

2) and

(

= 1

2) are random variables with zero mean. Theyhave the physical dimension of a velocity and we shall denote by [

−

] the common intervalfor these four parameters. In terms of ˜

the statistical average of the quadratic values can beexpressed as



(

= 1



stat



(

= 1



stat

= ˜

(62)for the uniform probability model (within the interval [

−

]) which we have chosen forour simulations. Finally, the exponent

controls the power spectrum of the ﬂuctuatingcomponents. For the simulations, between the two values

= 5

6 and

= 1 reported inref.[75], we have chosen

= 1 which corresponds to the point of view of an observer movingin the ﬂuid.Thus, within this simple model for

(

) and

(

), ˜

is the only parameter whosenumerical value could reﬂect the properties of a large-scale motion, for instance of the Earth’smotion with respect to the Cosmic Microwave Background (CMB). For this reason, here, wehave adopted the ﬁxed value ˜

CMB

= 370 km/s. With these premises, our results can beillustrated by ﬁrst considering the basic set of 10 complete rotations of the apparatus duringwhich Illingworth’s fringe shifts (produced by 90

rotations) were recorded every 30 seconds.Therefore, this type of simulations consists in generating 40 values during a total time of 1200seconds. As an illustration, two typical sequences of D

(t) and D

(t), in units 10

−

, areshown in Fig.5.As one can see, the magnitude

(10

−

) and the random nature of the instantaneous valuesis completely consistent with the entries of Table 4. Also the resulting infra-session averages





= 0

00028 and





= 0

00011 are completely consistent with the typical entries of Table 3.26

1400 1600 1800 2000 2200 2400

time (s)

-4-2024

D (t)

200 400 600 800 1000 1200-3-2-10123

D (t)

Figure 5:

A simulation of

(t)

and

(t)

, in units

−

and every 30 seconds, from typical sequences of 1200 seconds. The average values are





= 0

00028

and





= 0

00011

. The velocity parameter is

CMB

370 km/s.

To obtain further insight, we have then performed extensive simulations for large sequencesof measurements. The histograms of a set of 10000 determinations of

(

) and

(

) (againgenerated every 30 seconds) are reported in panels (a) and (b) of Fig.6.Notice that these distributions are clearly “fat-tailed” and very diﬀerent from a Gaussianshape. This kind of behavior is characteristic of probability distributions for instantaneousdata in turbulent ﬂows (see e.g. [76, 77]). To better appreciate the deviation from Gaussianbehavior, in panels (c) and (d) we plot the same data in a log

−

log scale. The resultingdistributions are well ﬁtted by the so-called

−

exponential function [78]

(

) =

−

)

−

)

(63)with entropic index

∼

1. For such large samples of data, the statistical averages





and





are vanishingly small in units of the typical instantaneous values

(10

−

) and any non-zero average has to be considered as statistical ﬂuctuation. On the other hand, the standarddeviations

(

) and

(

) have deﬁnite non-zero values which reﬂect the magnitude of thescale parameter ˜

. By keeping ˜

ﬁxed at 370 km/s, we have found

(

)

∼

05)

−

(

)

∼

06)

−

(64)whose uncertainties reﬂect the observed variations due to the truncation of the Fourier modesin Eqs.(60), (61) and to the dependence on the random sequence. Taking this calculation 27

-1

-2

-1

q-Exponential Fit

-4 -2 0 2 4

0,000,501,001,5010

-1

-2

-1

q-Exponential Fit

-4 -2 0 2 4

0,000,501,001,50

[10 ]

-3

(a)(b)(c)(d)

[10 ]

AA| |BB

[10 ][10 ]

-3

WWWW

-3| |-3

Figure 6:

We show, see (a) and (b), the histograms

obtained from a simulation for

= D

(t)

and

= D

(t)

. The vertical normalization is to a unit area. The mean values are





= 0

−





−

and the standard deviations

) = 0

−

) = 0

−

. We also show, see (c) and (d), the corresponding plots in logarithmic scale and the ﬁts with Eq.( 63 ). The parameters of the ﬁt are q=1.07, a=2 and b=2.2 for

and q=1.12, a=2 and b=2.3 for

. The total statistics correspond to 10.000 values generated at steps of 30 seconds. The velocity parameter is

CMB

370 km/s

.28

Table 5:

Illingworth’s ﬁnal inter-session averages.

Observations



 



5 A.M. +0

00036

00012

−

00016

0000911 A.M.

−

00001

00007

−

00000

000065 P.M.

−

00008

00012

−

00005

0000811 P.M.

−

00034

00014 +0

00005

00006into account gives a mean spread slightly less, about 0

−

, for the eﬀect of stochasticdrift in Illingworth’s measurements. This is comparable to the uncertainty of the individualreadings which, in the best case, was of 1/1500 wavelengths, i.e.

−

. By combiningin quadrature the two uncertainties, one gets a good agreement with our Table 4 where thevariance of the mean is about

−

. Finally, the simulation is also useful to get indicationson the expected value of the observable velocity. In fact, with vanishingly small values of





and





one gets



∼

(

) and



∼

(

). Therefore one obtains the followingtwo average estimates of

obs

∼

112



(

)

∼

05 km

obs

∼

112



(

)

∼

23 km

s (65)with a mean value of 3.14 km/s which is very close to M´unera’s determination

obs

= 3

04km/s.We emphasize that one could further improve the stochastic model by introducing timemodulations and/or slight deviations from isotropy. For instance, ˜

could become a functionof time ˜

= ˜

(

). By still retaining statistical isotropy, this could be used to simulate thepossible modulations of the projection of the Earth’s velocity in the plane of the interferometer.Or, one could ﬁx a range, say [

−

], for the two random parameters

(1) and

(2),which is diﬀerent from the range [

−

] for the other two parameters

(1) and

(2).Finally, ˜

and ˜

could also become given functions of time, for instance ˜

(

)

≡

(

)cos ˜

(

)˜

(

)

≡

(

)sin ˜

(

), ˜

(

) and ˜

(

) being deﬁned in Eqs. (31)

−

(34). We shall discuss thisother alternative later on, in connection with the much more accurate Joos 1930 experiment.In any case, by accepting this type of picture of the ether-drift, it is clear that furtherreduction of the data by performing inter-session averages (



...



) among the various sessions,can wash out completely the physical information contained in the original observations. InTable 5, we report the ﬁnal inter-session averages





and





obtained by Illingworthfor the various observation times.29

Nevertheless, in spite of the strong cancelations expected from the averaging reductionprocess mentioned above, some non-zero value is still surviving. Therefore, regardless of oursimulations, one could draw the following conclusions. Traditionally, from these ﬁnal averagesfor





at 5 A.M. and at 11 P.M. one has been deducing the values

∼

12 km/s and

∼

07 km/s respectively. Therefore, from these two estimates of

that, as anticipated,represent

lower

bounds for

obs

, it follows that there were values of

obs

which clearly had to be

larger

than both. For this reason, this 2.1 km/s velocity value reported by Illingworth, ratherthan being interpreted as an

upper

bound could also be interpreted as a

lower

bound placed byhis experiment. In this way, by combining with the previous Kennedy’s upper bound

obs

5km/s, one would deduce that these two experiments, where light was propagating in heliumat atmospheric pressure, give a range for the observable velocity(Kennedy + Illingworth) 2 km



obs

5 km

s (66)in complete agreement with M´unera’s determination

obs

= 3

0 km

s (67)From this last estimate, by using Eq.(30) and taking into account that for helium at atmo-spheric pressure the refractive index is

N ∼

000035, one obtains a kinematical velocity

∼

(370

120) km

s (68)consistently with the velocity values Eqs.(52) and (54) from the Michelson-Morley and Morley- Miller experiments.

6. Miller

M´unera’s analysis [72] is also interesting because he applied the same method used for Illing- worth’s observations to the only known Miller set of data explicitly reported in the literature.In this case, his value

obs

= 8

4 km/s, after correcting with Eq.(30), conﬁrms theestimate

∼

350 km/s for the average velocity in the plane of the interferometer.This close agreement with the Michelson-Morley value 8.4 km/s is also conﬁrmed by thecritical re-analysis of Shankland et

al.

[68]. Diﬀerently from the original Michelson-Morleyexperiment, Miller’s data were taken over the entire day and in four epochs of the year.However, after the critical re-analysis of the original raw data performed by the Shanklandteam, there is now an independent estimate of the average determinations

EXP2

for the fourepochs. Their values 0.042, 0.049, 0.038 and 0.045, respectively for April 1925, July 1925,30

September 1925 and February 1926 (see page 170 of ref.[68]) are so well statistically consistentthat one can easily average them. The overall determination from Table III of [68]

EXP2

= 0

044

022 (69)when compared with the equivalent classical prediction for Miller’s interferometer

class2

Lλ

(30km

∼

56 corresponds to an average observable velocity

obs

= 8

−

s (70)and, by using Eq.(30), to a true kinematical value

= 349

+79

−

104

s (71)We are aware that our conclusion goes against the widely spread belief, originating preciselyfrom the paper of Shankland et

al.

ref.[68], that Miller’s results might actually have been dueto statistical ﬂuctuation and/or local temperature conditions. To a closer look, however, thearguments of Shankland et

al.

are not so solid as they appear when reading the Abstract of their paper

. In fact, within the paper these authors say that “...there can be little doubtthat statistical ﬂuctuations alone cannot account for the periodic fringe shifts observed byMiller” (see page 171 of ref.[68]). Further, although “...there is obviously considerable scatterin the data at each azimuth position,...the average values...show a marked second harmoniceﬀect” (see page 171 of ref.[68]). In any case, interpreting the observed eﬀects on the basisof the local temperature conditions is certainly not the only explanation since “...we mustadmit that a direct and general quantitative correlation between amplitude and phase of theobserved second harmonic on the one hand and the thermal conditions in the observation huton the other hand could not be established” (see page 175 of ref.[68]).Most surprisingly, however, Shankland et

al.

seem not to realize that Miller’s average value

EXP2

∼

044, obtained after

their own

re-analysis of his observations at Mt.Wilson, whencompared with the reference classical value

class2

= 0

56 for his apparatus, was giving the sameobservable velocity

obs

∼

4 km/s obtained from Miller’s re-analysis of the Michelson-Morleyexperiment in Cleveland. Conceivably, their emphasis on the role of temperature eﬀects wouldhave been re-considered had they realized the perfect identity of two determinations obtainedin completely diﬀerent experimental conditions. In this sense, an interpretation in terms of atemperature gradient is only acceptable provided this gradient represents a

non-local

eﬀect,as in our model of the ether drift from a fundamental vacuum energy-momentum ﬂow.

A detailed rebuttal of the criticism raised by the Shankland team can be found in ref.[79].

Table 6:

The symmetric combination of fringe shifts

(

) =

∆

(

)+∆

(

at the various values of

for the set of 20 turns of the interferometer reported in Fig.8 of ref.[ 65] . For our global ﬁt, following ref.[ 68 ], the nominal accuracy of each entry has been ﬁxed to

050

Turn 0

112

135

157

1 +0.091 +0.159 +0.028 +0.047

−

0.034

−

0.116

−

0.147

−

0.0282

−

0.025 +0.063 +0.050 +0.088

−

0.075

−

0.038 +0.000

−

0.0633 +0.022 +0.103 +0.084 +0.016

−

0.053

−

0.072

−

0.091

−

0.0094 +0.034

−

0.009

−

0.053

−

0.047

−

0.041 +0.016 +0.022 +0.0785 +0.169 +0.081 +0.044

−

0.044

−

0.081

−

0.169

−

0.056 +0.0566

−

0.025 +0.025 +0.025 +0.025 +0.025

−

0.025

−

0.025

−

0.0257 +0.081 +0.094 +0.056 +0.069

−

0.119

−

0.106

−

0.094 +0.0198 +0.066 +0.072

−

0.022

−

0.066

−

0.059

−

0.003 +0.003 +0.0099 +0.041 +0.084 +0.078 +0.022

−

0.134

−

0.141 +0.003 +0.04710 +0.016 +0.072 +0.078

−

0.016

−

0.009

−

0.003

−

0.047

−

0.09111 +0.009 +0.053 +0.097

−

0.009

−

0.116

−

0.072 +0.022 +0.01612 +0.022 +0.016 +0.059 +0.003

−

0.053

−

0.009

−

0.016

−

0.02213 +0.000 +0.063 +0.025 +0.038 +0.050

−

0.038

−

0.075

−

0.06314

−

0.034 +0.047 +0.078 +0.009

−

0.009

−

0.028

−

0.047

−

0.01615 +0.113 +0.125 +0.138 +0.000

−

0.088

−

0.125

−

0.113

−

0.05016 +0.025 +0.050 +0.025 +0.050

−

0.025

−

0.050

−

0.025

−

0.05017 +0.000

−

0.012

−

0.025 +0.063 +0.000

−

0.012

−

0.025 +0.01318 +0.044 +0.050 +0.019

−

0.019

−

0.056

−

0.044

−

0.031 +0.03119 +0.053 +0.059 +0.016

−

0.028

−

0.022

−

0.066

−

0.009

−

0.00320 +0.059 +0.041 +0.122 +0.003

−

0.066

−

0.084

−

0.053

−

0.02232

Another criticism of Miller’s work was recently presented by Roberts [80]. This author,using the set of data reported in Fig.8 of ref.[65], raises several objections to the validityof Miller’s observations. The two main objections concern i) the subtraction of the steadythermal drift, which was approximated by Miller as a pure linear eﬀect, and ii) the statisticalsigniﬁcance of the measurements. Concerning remark i), Roberts reports in his Fig.3 a brokenline that reproduces the expected linear trend. He also reports some chosen points (diﬀeringfrom the corners of the broken line by 180 degrees) that, due to the 2nd-harmonic natureof the ether-drift eﬀect, should lie on the line. However, this expectation ignores that, asalready pointed out for the Michelson-Morley experiment, real measurements contains largeﬁrst-harmonic eﬀects. These only cancel when taking symmetric combinations of data atthe various angles

and

. As a matter of fact, the autocorrelative methods and fur-ther tests applied by the Shankland team over all of Miller’s data conﬁrmed the linear driftapproximation as remarkably good (see their footnote 21 on page 177 of [68]).Concerning remark ii), according to Roberts, the experimental uncertainties are so largethat the observed 2nd-harmonic eﬀect has no statistical signiﬁcance. To check this pointwe have re-computed ourselves the fringe shifts for the set of 20 turns of the interferometers(reported in Fig.8 of ref.[65]) considered by Roberts, by following the same procedureexplainedin Sect.3. The resulting symmetric combinations of fringe shifts

(

) = ∆

(

) + ∆

(

(72)are reported in our Table 6.We have then ﬁtted these data by including both 2nd and 4th harmonic terms. Noticethat, diﬀerently from Roberts’ analysis, we do not perform any averaging of data obtainedfrom diﬀerent turns of the interferometer. For our global ﬁt, to estimate the accuracy of thevarious determinations, we have followed ref.[68] and adopted a nominal uncertainty

050for each entry of Table 6. From the ﬁt, where the 4th harmonic is completely consistent withthe background (

EXP4

= 0

004

012), we have obtained a chi-square of 130 for 157 degreesof freedom and the following values

EXP2

= 0

061

012

EXP0

= 24

(73)Here errors correspond to the overall boundary ∆

= +3

67, as appropriate

for a 70% C.L. in a 3-parameter ﬁt [81]. Notice that, even though the ﬁtted

Eq.(73) is only 20% largerthan the nominal accuracy

050 of each entry, the data are distributed in such a way toproduce a 5

evidence for a non-zero 2nd harmonic.

This probability content assumes a Gaussian distribution as for typical statistical errors.

As for Illingworth’s experiment, we have also analyzed the results obtained from theindividual turns of the interferometer. To this end, we report in Figs. 7 and 8 the plots of the azimuth and of the 2nd harmonic for the 20 rotations.

0 5 10 15 20

Turn

-20020406080

Figure 7:

The azimuth (in degrees) for the 20 individual turns of the interferometer reported in Table 6. The average uncertainty of each determination is about

. The band between the two horizontal lines corresponds to the global ﬁt

= 24

. Each individual value could also be reversed by 180 degrees.

To conclude our analysis of Miller’s experiment, we want to mention that other objectionsto the overall consistency of his solution for the Earth’s cosmic motion [65] were raised byvon Laue [82] and Thirring [83]. Their argument, which concerns the observed displacement of the maximum of the fringe pattern

averaged over all sidereal times

, was also re-proposedby Shankland et al. [68] and amounts to the following.By assuming relations (31)

−

(43) and denoting by



...



the daily average of any givenquantity, one ﬁnds, at any angle

, the daily averaged fringe shift



∆

(

)



= 2



(

)



cos2

(74)since



(

)



= 0 with



(

)



−

(

N −

λV

14(3cos2

−

1)cos

(75)The result can then be cast into the form [68]



∆

(

)



(

γ,φ

)cos2

(76)Therefore, since the latitude

is a constant and the angular declination

is ﬁxed at anyspeciﬁc epoch, the daily averaged fringe shifts should all have a common maximum at the34

0 5 10 15 20

Turn

00,050,10,15

Figure 8:

The 2nd-harmonic amplitude for the 20 individual turns of the interferometer reported in Table 6. The average uncertainty of each determination is about

030

. The band between the two horizontal lines corresponds to the global ﬁt

= 0

061

012

. Within their errors, these individual values correspond to an observable velocity in the range 4

14km/s.

value

= 0. Only the amplitude can be diﬀerent at diﬀerent epochs. Instead, in Miller’sobservations the location of the maximum was diﬀerently displaced from the meridian (seeFigs.25 of ref.[65] and Fig.3 of ref.[68]). The presence of such eﬀect has always represented a problem for the overall consistency of Miller’s solution for the Earth’s cosmic motion [65].However, in this derivation, one assumes that any physical signal should only exhibit thesmooth modulations expected from the Earth’s rotation. As anticipated in Sect.2, and dis-cussed in connections with the Michelson-Morley and Illingworth experiments, one might befaced with the more general scenario where the two velocity components

(

) and

(

) inEq.(44) are not smooth periodic functions but exhibit stochastic behaviour. In this diﬀer-ent perspective, combining observations of diﬀerent days and diﬀerent epochs becomes moredelicate and there might be non-trivial deviations from Eq.(76). We shall therefore concludeour analysis of Miller’s experiments by recalling the remarkable consistency of the velocityvalue

∼

350 km/s (obtained from the 2nd-harmonic amplitude

EXP2

∼

044 computedby the Shankland team) with those from the Michelson-Morley, Morley-Miller and Kennedy-Illingworth experiments. In this sense, this bulk of Miller’s work will remain.35

7. Michelson-Pease-Pearson

Let us further compare with the experiment performed by Michelson, Pease and Pearson[84, 85]. They do not report numbers so that we can only quote from the original article[85] which reports the outcome of the measurements performed in the most reﬁned version of the experiment: “ In the ﬁnal series of experiments, the apparatus was transferred to a well-sheltered basement room of the Mount Wilson Laboratory. The length of the light path wasincreased to eighty-ﬁve feet, and the results showed that the precautions taken to eliminatetemperature and pressure disturbances were eﬀective. The results gave no displacement asgreat as

one-ﬁftieth

of that to be expected on the supposition of an eﬀect due to a motion of the solar system of three hundred kilometers per second”. On the other hand, in ref.[84], aftersimilar comments on the length of the apparatus and on the precautions taken to eliminatethe various disturbances, one ﬁnds this other statement “The results gave no displacement asgreat as

one-ﬁfteenth

of that to be expected on the supposition of an eﬀect due to a motionof the solar system of three hundred kilometers per second. These results are diﬀerencesbetween the displacements observed at maximum and at minimum at sidereal times, thedirections corresponding to Dr. Str¨omberg’s calculations of the supposed velocity of the solarsystem”. In the same paper, the authors report that, according to Str¨omberg’s calculations“ a displacement of 0.017 of the distance between fringes should have been observed at theproper sidereal times”.Clearly, although not explicitly stated, they were assuming that some unknown mechanismwas largely reducing the fringe shifts with respect to the naive non-relativistic value associatedwith a kinematical velocity of 300 km/s. Thus one could try to conclude that their experimentimplies fringe shifts

∆



115

017

∼

001. However this is not what they say (they speakof

diﬀerences

between fringe displacements) and, in any case, this interpretation does not ﬁtwith the result reported by Shankland et al. [68] (see their Table I). According to these otherauthors, the typical observed fringe shifts observed by Michelson, Pease and Pearson were of the order of

005.To try to understand this intricate issue, we have been looking at another article [86]which, surprisingly, was signed by F. G. Pease alone. Here, one discovers that, in the ﬁrststage of the experiment, the fringe shifts had a typical magnitude of about

030. Later on,however, by reducing substantially the rotation speed of the apparatus, the observed eﬀectsbecame considerably smaller.Pease declares that, in their experiment, to test Miller’s claims, they concentrated on a36

purely ‘diﬀerential’ type of measurement. For this reason, he only reports the diﬀerence

(

) =



∆

(

)



−

∆

(

)



(77)between the mean fringe shifts



∆

λλ



, obtained after averaging over a large set of obser-vations performed at sidereal time 5.30, and the mean fringe shifts



∆

λλ



obtained afteraveraging in the same period at sidereal time 17.30. The quantity

(

) has typical magnitudeof

004 or smaller. However, as already anticipated in Sect.2, by averaging observationsperformed at a given sidereal time one is assuming the smooth modulations of the signaldescribed by Eqs.(37)

(38). Otherwise, one will introduce uncontrolled errors. For instanceif, consistently with Illingworth’s and Miller’s data, there were substantial stochastic compo-nents in the signal, the cancelations introduced by a naive averaging process would becomestronger and stronger by increasing the number of observations.Therefore, from these values, nothing can be said about the magnitude of the fringe shifts

∆

(

)

obtained, before any averaging procedure and before any subtraction, in individualmeasurements at various hours of the day. Pease reports a plot of just a single observation,performed when the length of the optical path was still 55 feet, where the even fringe shiftcombinations Eq.(50) vary approximately in the range

007. This is equivalent to fringeshifts of about

011 with a length of 85 feet and could hardly be taken as indicativeof the whole sample of measurements. In this situation, one can only adopt the estimate

∼

010

...

for the value of the 2nd-harmonic amplitude, for optical path L=85 feet,whose uncertainty cannot be estimated in the absence of information on the other individualsessions. Then, for this conﬁguration, where

Lλ

(30km

∼

45, this is equivalent to

obs

= (4

...

) km

s (78)or, by using Eq.(30), to

= (185

...

) km

s (79)We emphasize that Miller’s extensive observations, as reported in Fig.22 of ref.[65] (see alsoour Fig.8), gave ﬂuctuations of the observable velocity lying, within the errors, in the range4

−

14 km/s which has been smoothed in our Fig.2. For this reason, even though Miller’sreconstruction of the Earth’s cosmic motion is not internally consistent, a single observationwhich gives

obs

∼

4.5 km/s does not represent a refutation of the whole Miller experiment.This becomes even more true by noticing that the single session selected by Pease, within aperiod of several months, was chosen to represent an example of extremely small ether-drifteﬀect.37

8. Joos

One more classical experiment, performed by Georg Joos in 1930, has ﬁnally to be considered.For the accuracy of the measurements (data collected at steps of 1 hour to cover the fullsidereal day that were recorded by photocamera), this experiment cannot be compared withthe other experiments (e.g. Michelson-Morley, Illingworth) where only observations at fewselected hours were performed and for which, in view of the strong ﬂuctuations of the azimuth,one can just quote the average magnitude of the observed velocity. Moreover, diﬀerently fromMiller’s, the amplitudes of all basic Joos’ observations can be reconstructed from the publishedarticles [87, 88]. As such, this experiment deserves a more reﬁned analysis and will play a central role in our work.Joos’ optical system was enclosed in a hermetic housing and, traditionally, it was alwaysassumed that the fringe shifts were recorded in a partial vacuum. This is supported by severalelements. For instance, when describing his device for electromagnetic ﬁne movements of themirrors, Joos explicitly refers to the condition of an evacuated apparatus, see p.393 of [87].This aspect is also conﬁrmed by Miller who, quoting Joos’ experiment, explicitly refers to an“evacuated metal housing” in his article [65] of 1933. This is particularly important since lateron, in 1934, Miller and Joos had a public letter exchange [89] and Joos did not correct Miller’sstatement. On the other hand, Swenson [90] explicitly reports that fringe shifts were ﬁnallyrecorded with optical paths placed in a helium bath. In spite of the fact that this importantaspect is never mentioned in Joos’ papers, we shall follow Swenson and assume that duringthe measurements the interferometer was ﬁlled by gaseous helium at atmospheric pressure.The observations were performed in Jena in 1930 starting at 2 P.M. of May 10th andending at 1 P.M. of May 11th. Two measurements, the 1st and the 5th, were ﬁnally deletedby Joos with the motivation that there were spurious disturbances. The data were combinedsymmetrically, in order to eliminate the presence of odd harmonics, and the magnitude of thefringe shifts was typically of the order of a few thousandths of a wavelength. To this end,one can look at Fig.8 of [88] (reported here as our Fig.9) and compare with the shown size of 1/1000 of a wavelength. From this picture, Joos decided to adopt 1/1000 of a wavelength asan upper limit and deduced an observable velocity

obs



5 km/s. To derive this value, heused the fact that, for his apparatus, an observable velocity of 30 km/s would have produceda 2nd-harmonic amplitude of 0.375 wavelengths.Still, since it is apparent from Fig.9 that some fringe displacements were deﬁnitely largerthan 1/1000 of a wavelength, we have decided to extract the values of the 2nd-harmonicamplitude

from the 22 pictures. Diﬀerently from the values of the azimuth, this can be38

Figure 9:

The selected set of data reported by Joos [ 87 , 88 ]. The yardstick corresponds to 1/1000 of a wavelength so that the experimental dots have a size of about

−

. This corresponds to an uncertainty

−

in the extraction of the fringe shifts.

done unambiguously. The point is that, due to the camera eﬀect, it is not clear how to ﬁx thereference angular values in Fig.9 for the fringe shifts. Thus, one could choose for instance theset (k=1, 2, 3, 4)

≡

, 45

, 90

, 135

) or the diﬀerent set

≡

(360

, 315

, 270

, 225

).Or, by noticing that in Fig.9 there is a small misalignment angle

∗

∼

(which actuallyfrom [87] might instead be 22

) between the dots of Joos’ fringe shifts and the N, W, and Smarks, one could also adopt other two set of values, namely

≡

∗

, 45

∗

, 90

∗

,135

∗

) or

≡

(360

−

∗

, 315

−

∗

, 270

−

∗

, 225

−

∗

). By ﬁtting the fringe shifts of Fig.9to the 2nd-harmonic form Eq.(56), these four options for the reference angles

would giveexactly the same amplitude

but four diﬀerent choices for the azimuth, i.e.

−

∗

−

∗

and

. This basic ambiguity should be added to the standard uncertainty in theazimuth that, due to the 2nd-harmonic nature of the measurements, could always be changedby adding

180 degrees

. Therefore, since clearly there is only one correct choice for the

As an example, one can consider the azimuth for Joos’ picture 20. Depending on the choice of the referenceangles

, one ﬁnds

∼

329

, 329

∗

, 31

−

∗

, 31

∼

149

, 149

∗

, 211

−

∗

, 211

angles

, we have preferred not to quote theoretical uncertainties on the azimuth and justconcentrate on the amplitudes. Their values are reported in Table 7 and in Fig.10.

0 5 10 15 20 25

Picture

01234

Figure 10:

Joos’ 2nd-harmonic amplitudes, in units

−

. The vertical band between the twolines corresponds to the range

−

By computing mean and variance of the individual values, we obtain an average 2nd-harmonic amplitude



joos2



= (1

−

(80)and a corresponding observable velocity

obs

∼

−

s (81)By correcting with the helium refractive index, Eqs.(30) and (81) would then imply a true kinematical velocity

∼

217

+66

−

km/s.However, this is only a ﬁrst and very partial view of Joos’ experiment. In fact, we havecompared Joos’ amplitudes with theoretical models of cosmic motion. To this end, one hasﬁrst to transform the civil times of Joos’ measurements into sidereal times. For the longitude11.60 degrees of Jena, one ﬁnds that Joos’ observations correspond to a complete round insidereal time in which the value

= 0

≡

360

is very close to Joos’ picture 20. Then, byusing Eqs.(31) and (34), one can use this input and compare with theoretical predictions for the amplitude which, for the given latitude

= 50

94 degrees of Jena, depend on the rightascension

and the angular declination

. To this end, it is convenient to ﬁrst re-write thetheoretical forms as

(

)cos2

(

) = 2

(

) = 2

(

N −

λv

(

)

−

(

)

∼

−

(

)

−

(

)(300 km

(82)40

Table 7:

The 2nd-harmonic amplitude obtained from the 22 Joos pictures of our Fig.9. The uncertainty in the extraction of these values is about

−

(the size of the dots in Fig.9).The mean amplitude over the 22 determinations is



joos2



= 1

−

Picture

joos2

[10

−

]2 2

053 0

754 1

606 2

007 1

508 1

559 1

1010 0

6011 4

1512 1

2013 2

3514 0

9515 1

1516 1

6517 0

5018 1

0519 1

2520 0

3521 0

4522 1

2523 0

9524 1

6541

and

(

)sin2

(

) = 2

(

) = 2

(

N −

(

)

(

)

∼

−

(

)

(

)(300 km

(83)where we have used the numerical relation for Joos’s experiment

Lλ

(30km

∼

375 and thevalue of the helium refractive index. Then, by approximating

(

)

∼

(

)

∼

(

) andusing Eq.(34) for the scalar combination ˜

(

)

≡



(

) + ˜

(

), we have ﬁtted the amplitudedata of Table 7 to the smooth form

smooth2

(

) = const

sin

(

) (84)where cos

(

) is deﬁned in Eq. (31). The results of the ﬁt

= 168

−

(85)conﬁrm that, as found in connection with the Illingworth experiment, the Earth’s motionwith respect to the CMB (which has

∼

168

and

∼−

) could serve as a useful model todescribe the ether-drift data.Still, in spite of the good agreement with the CMB

−

and

−

values obtained from theﬁt Eq.(85), the nature of the strong ﬂuctuations in Fig.10 remains unclear. Apart from this,there is also a sizeable discrepancy in the absolute normalization of the amplitude. In fact,by assuming the standard picture of smooth time modulations, the mean amplitude over allsidereal times can trivially be obtained from the mean squared velocity Eq.(34)



(

)



−

sin

−

12 cos

cos



(86)For the CMB and Jena, this gives





∼

330 km/s so that one would naively predict fromEqs.(82), (83)



smooth2

(

)

∼

−



(

)



(300 km

∼

−

(87)to be compared with Joos’ mean value



joos2



= (1

−

. In the standard picture, thisexperimental value leads to the previous estimate





∼

217 km/s and

not





∼

330km/s so that it is necessary to change the theoretical model to try to make Joos’ experimentcompletely consistent with the Earth’s motion with respect to the CMB.To try to solve this problem, and understand the origin of the observed strong ﬂuctuations,we have used the same model Eqs.(60), (61) of Sect.5, to simulate stochastic variations of the

Actually, there is another degenerate minimum at

= 348

and

= 13

because sin

(

)remains invariant under the simultaneous replacements

→

+180

and

→−

. However, due to the closeagreement with the CMB parameters we have concentrated on solution (85).

velocity ﬁeld. As anticipated however, due to the higher accuracy of the Joos experiment, wehave modiﬁed the theoretical framework. Namely, we have allowed the two random parameters

(1) and

(2) to vary in the range [

−

(

)

(

)] and the other two parameters

(1)and

(2) to vary in the diﬀerent range [

−

(

)

(

)], where ˜

(

) and ˜

(

) are deﬁned inEqs.(31)

−

(33). In this way, for each time

, Eqs.(62) now become



(

= 1



stat

= ˜

(



(

= 1



stat

=˜

(

(88)It is understood that the latitude corresponds to Joos’ experiment while

and

describethe Earth’s motion with respect to the CMB. Notice that, in this model, there will be asubstantial reduction of the amplitude with respect to its smooth prediction. To estimatethe order of magnitude of the reduction, one can perform a full statistical average (as for aninﬁnite number of measurements) and use Eqs.(88) in Eqs.(82), (83) for our case

= 1. Thisgives



(

)



stat

∼

−

(

)(300 km

∞



smooth2

(

) (89)By also averaging over all sidereal times, for the CMB and Jena, one would now predict amean amplitude of about 1

−

and not of 3

−

.After having ﬁxed all theoretical inputs, we have analyzed the dependence of the numericalresults on the remaining parameters of the simulation, namely the number

of Fourier modes(in the available range



) and the integer number

(the ‘seed’) which determines therandom sequence. In particular, the dependence on the latter is usually quoted as theoreticaluncertainty. For this reason, for Illingworth’s experiment in Sect.5 we had produced severalcopies of the high-statistics simulation in Fig.6 by quoting values for the standard deviationsEq.(64) which take into account the observed

−

dependence of the results.Here, we have started by doing something similar. However, since it is not possible toconsider at once all characteristics of a given conﬁguration, we have ﬁrst concentrated onthe simplest statistical indicator, namely the mean amplitude



simul2



obtained by averagingover all sidereal times. Quite in general, this can be evaluated for a variety of conﬁgurationswhich depend on the number

of measurements that one wants to simulate and the interval∆

between two consecutive measurements. For instance, Joos’ experiment corresponds to

= 24 (actually

= 22 since Joos ﬁnally deleted two observations) and ∆

∼

3600 seconds.At the same time, the simulations become quite lengthy for large

, large

and small ∆

.Therefore, we have ﬁrst performed a scan of

−

values for

= 10

and then studied a few

by increasing

. To give an idea of the spread of the central values, due to changes of thepair (

N,s

), we report below the approximate results of this analysis for some choices of the43

Table 8:

The 2nd-harmonic amplitude obtained from a single simulation of 22 instantaneous measurements performed at Joos’ times. The stochastic velocity components are controlled by the kinematical parameters

(

V,α,γ

)

CMB

as explained in the text. The mean amplitude over the 22 determinations is



simul2



= 1

−

Picture

simul2

[10

−

]2 1

263 3

504 0

466 0

347 2

718 0

359 2

1910 0

5211 5

2412 0

2413 1

1914 1

9315 0

0816 1

5217 2

2918 0

2419 1

0220 0

0721 0

0922 2

1823 1

5024 1

5244

pair (

∆

)



simul2

(

= 24

∆

= 3600

)

∼

−

(90)



simul2

(

= 1440

∆

= 60

)

∼

−

(91)



simul2

(

= 240

∆

= 3600

)

∼

−

(92)As it might be expected, the average



simul2



becomes more stable by increasing the numberof observations. Concerning the individual values

simul2

(

), with

= 1

,..,n

, they have alarge spread, about (1

−

. This is in agreement with the ‘fat-tailed’ distributions of instantaneous values expected in turbulent ﬂows [76, 77] (compare with Fig. 6 in Sect.5).However this other spread can be reduced by starting to average the data in some intervalof time

. In this case, the spread of the resulting average values



simul2

(

)



decreases as

√

. We emphasize that, by performing extensive simulations, there are occasionally verylarge spikes of the amplitude at some sidereal times, of the order (10

20)

−

. The eﬀectof these spikes gets smoothed when averaging over many conﬁgurations but their presence ischaracteristic of a stochastic-ether model. With a standard attitude, where the ether driftis only expected to exhibit smooth time modulations, the observation of such eﬀects wouldnaturally be interpreted as a spurious disturbance (Joos’ omitted observations 1 and 5?).

0 5 10 15 20 25

Picture

0123456Joos DataPoly FitSimulationPoly Fit

Figure 11:

Joos’ experimental amplitudes in Table 7 are compared with the single simulation of 22 measurements for ﬁxed

(

N,s

)

in Table 8. By changing the pair

(

N,s

)

, the typical variation of each simulated entry is

−

depending on the sidereal time. We alsoshow two 5th-order polynomial ﬁts to the two diﬀerent sets of values.

After this preliminary study, we have then concentrated on the real goal of our simulation,i.e. to compare with the

single

Joos conﬁguration of 22 entries in Table 7. To this end, one45

could ﬁrst try to look for the ‘best seed’, or subset of seeds, which can minimize the diﬀerencebetween the generated conﬁgurations and Joos’ data. This standard task, usually accom-plished by minimizing a chi-square, is diﬃcult to implement here. In fact, it is problematic toconstruct a function

(

) and look for its minima because a seed

and the closest seeds

1give often vastly diﬀerent conﬁgurations and chi-square. For this reason, we have followedan empirical procedure by forming a grid and selecting a set of seeds whose mean amplitude(for

= 24 and ∆

= 3600 s) gets close to Joos’s mean amplitude



joos2



= 1

−

for alarge number

of Fourier modes. One of such seeds gave a sequence



simul2



=1.66, 1.40,1.08, 1.21 and 1.38 (in units 10

−

), for

= 10

, 10

and 5

respectively, andthe conﬁguration with

= 5

was ﬁnally chosen to give an idea of the agreement onecan achieve between data and a single numerical simulation for ﬁxed (

N,s

). The simulatedvalues are reported in Table 8 and a graphical comparison with Joos’ data is shown in Fig.11. We emphasize that one should not compare each individual entry with the correspondingdata since, by changing (

N,s

), the simulated instantaneous values vary typically of about(1

−

depending on the sidereal time. Instead, one should compare the overall trend of data and simulation. To this end, we show two 5th-order polynomial ﬁts to the two diﬀerentsets of values.A more conventional comparison with the data consists in quoting for the various 22entries simulated average values and uncertainties. To this end, we have considered the meanamplitudes



simul2

(

)



deﬁned by averaging, for each Joos’ time

, over 10 hypotheticalmeasurements performed on 10 consecutive days. For each

, the observed eﬀect of varying(

N,s

) has been summarized into a central value and a symmetric error. The values arereported in Table 9 and the comparison with Joos’ amplitudes is shown in Fig.12.The spread of the various entries is larger at the sidereal times where the projection at Jenaof the cosmic Earth’s velocity becomes larger. The tendency of Joos’ data to lie in the lowerpart of the theoretical predictions in Table 9 mostly depends on our use of symmetric errors.In fact, by comparing in some case with the histograms of the basic generated conﬁgurations

simul2

(

), we have seen that our sampling method of



simul2

(

)



, based on a grid of (

N,s

)values, typically underestimates the weight of the low-amplitude region in a prediction at the70% C.L. . This can also be checked by considering the single simulation of Table 8 andcounting the sizeable fraction of amplitudes

simul2

(

)



−

. For this reason, one couldimprove the evaluation of the probability content. However, in view of the good agreementalready found in Fig.12 (

= 13

22), we did not attempt to carry out this more reﬁnedanalysis.In conclusion, after the ﬁrst indication obtained from the ﬁt Eq.(85), we believe that the46

Table 9:

The 2nd-harmonic amplitudes obtained by simulating the averaging process over 10 hypothetical measurements performed, at each Joos’ time, on 10 consecutive days. The stochastic velocity components are controlled by the kinematical parameters

(

V,α,γ

)

CMB

as explained in the text. The eﬀect of varying the pair

(

N,s

)

has been approximated into a central value and a symmetric error. The mean amplitude over the 22 determinations is



simul2



= 1

−

Picture

simul2

[10

−

]2 2

03 1

854 1

856 1

857 1

908 2

09 2

010 2

211 2

412 2

613 2

514 2

415 1

8516 1

7517 1

7518 1

7019 1

7020 1

7021 1

6022 1

6023 1

8024 1

047

0 5 10 15 20 25

Picture

0123456Joos DataSimulation

Figure 12:

Joos’ experimental amplitudes in Table 7 are compared with our simulation in Table 9.

link between Joos’ data and the Earth’s motion with respect to the CMB gets reinforced byour simulations. In fact, by inspection of Figs.11 and 12, the values of the amplitudes and thecharacteristic scatter of the data are correctly reproduced. In principle, there could be spacefor further reﬁnements by taking into account the Earth’s orbital motion in the input valuesfor

and

.From this agreement with the data, we then deduce that the previous value for the kine-matical velocity

∼

217

+66

−

km/s, obtained by simply correcting with the helium refractiveindex the average observable velocity (81), has to be considerably increased if one allows forstochastic variations of the velocity ﬁeld. In fact, the magnitude of the ﬂuctuations in

and

is controlled by the same scalar parameter ˜

(

)

≡



(

) + ˜

(

) of Eq.(34). In view of the good agreement between data and our numerical simulations, we conclude that Joos’ dataare consistent with a range of kinematical velocity

= 330

+40

−

km/s which corresponds toEq.(34) for

= 50

= 370 km/s,

= 168

and

−

9. Summary and conclusions

The condensation of elementary quanta and their macroscopic occupation of the same quan-tum state is the essential ingredient of the degenerate vacuum of present-day elementaryparticle physics. In this description, one introduces implicitly a reference frame Σ, where thecondensing quanta have

= 0, which characterizes the physically realized form of relativ-ity and could play the role of preferred reference frame in a modern re-formulation of the48

Lorentzian approach. To this end, we have given in the Introduction some general theoreticalarguments related to the problematic notions of a non-zero vacuum energy and of an exactLorentz-invariant vacuum state. These arguments suggest the possibility of a tiny vacuumenergy-momentum ﬂux, associated with an Earth’s absolute velocity

, which could aﬀect

diﬀerently

the various forms of matter. Namely, it could produce small convective currentsin a loosely bound system such as a gas or dissipate mainly by heat conduction with no ap-preciable particle ﬂow in strongly bound systems as liquid or solid transparent media. In theformer case, by introducing the refractive index

of the gas, convective currents of the gasmolecules would produce a small anisotropy, proportional to (

N −

1)(

v/c

)

, of the two-wayvelocity of light in agreement with the general structure Eq.(23) or with its particular limitEq.(24). Notice that this tiny anisotropy refers to the system

′

where the container of thegas is at rest. In this sense, contrary to standard Special Relativity,

′

might not deﬁne atrue frame of rest. This conceptual possibility can be objectively tested with a new seriesof dedicated ether-drift experiments where two orthogonal optical resonators are ﬁlled withvarious gaseous media by measuring the fractional frequency shift ∆

ν/ν

between the tworesonators. By assuming the typical value

∼

300 km/s of most cosmic motions, one expectsfrequency shifts



−

for gaseous helium and



−

for air, which are well within thepresent technology.Given the heuristic nature of our approach, and to further motivate the new series of dedicated experiments, we have tried to get a ﬁrst consistency check. In fact, by adoptingEq.(24), the frequency shift between the optical resonators is governed by the same classicalformula for the

fringe shifts

in the old ether-drift experiments with the only replacement

→

N −

≡

2obs

(93)In this way, where one re-obtains the same classical formulas (with the only replacement

→

obs

), testing the present scheme is very simple: one should just check the consistency of the true kinematical

′

s obtained in diﬀerent experiments.In the old times, experiments were performed with interferometers where light was propa-gating in gaseous media, air or helium at atmospheric pressure, where (

N −

1) is a very smallnumber. In this regime, the theoretical fringe shifts expected on the basis of Eqs.(23) and(24) are much smaller than the classical prediction (

v/c

)

. Another important aspect of theseclassical experiments is that one was always expecting smooth sinusoidal modulations of thedata due to the Earth’s rotation, see Eqs. (35), (37) and (38). As emphasized in Sect.2, we now understand the logical gap missed so far. The relation between the macroscopic Earth’smotions (daily rotation, annual orbital revolution,...) and the ether-drift experiments depends49

on the physical nature of the vacuum. Assuming Eqs.(37) and (38), to describe the eﬀect of the Earth’s daily rotation, amounts to considering the vacuum as some kind of ﬂuid in astate of regular, laminar motion for which global and local properties of the ﬂow coincide.Instead, several theoretical arguments (see e.g. refs.[14, 15, 16, 17, 56, 57, 58, 59, 60]) suggest that the physical vacuum might behave as a stochastic medium similar to a turbulent ﬂuidwhere large-scale and small-scale motions are only

indirectly

related. In this case, there mightbe non-trivial implications. For instance, due to the irregular behaviour of turbulent ﬂows,vectorial observables collected at the same sidereal time might average to zero. However, thisdoes not mean that there is no ether-drift. More generally, the relevant Earth’s motion withrespect to Σ might well correspond to that indicated by the anisotropy of the CMB, but it be-comes non trivial to reconstruct the kinematical parameters from microscopic measurementsof the velocity of light in a laboratory. These arguments make more and more plausible thata genuine physical phenomenon, much smaller than expected and characterized by stochasticvariations, might have been erroneously interpreted as an instrumental artifact thus leadingto the standard ‘null interpretation’ of the experiments reported in all textbooks.Now, our analysis of Sects.3

−

8 shows that this traditional interpretation is far from obvi-ous. In fact, by using Eqs.(24), (29) and (30), the small residuals point to an average velocity of about 300 km/s, as in most cosmic motions. In this alternative interpretation, the indica-tions of the various experiments are summarized in our Table 10

. As a summary of ourwork, we emphasize the following points:i) an analysis of the individual sessions of the original Michelson-Morley experiment, inagreement with Hicks [66] and Miller [65] (see our Figs. 1 and 2), gives no justiﬁcation to its standard null interpretation. As discussed in Sect.3, this type of analysis is more reliable. Infact, averaging directly the fringe displacements of diﬀerent sessions requires two additionalassumptions, on the nature of the ether-drift as a smooth periodic eﬀect and on the absenceof systematic errors introduced by the re-adjustment of the mirrors on consecutive days, that

Other determinations of less accuracy could also be included, as for the 1881 Michelson experiment inPotsdam [91] or Tomaschek’s starlight experiment [92] or the Piccard and Stahel experiment which was ﬁrst performed in a ballon [93] and later [94] on the summit of Mt. Rigi in Switzerland. These results were summarized in Table I of ref.[68] and by Miller [65]. In the 1881 Potsdam experiment the fringe shifts were in the range 0

002

007 to be compared with an expected 2nd-harmonic of 0.02 for 30 km/s. This meansobservable velocities (9

18) km/s which are comparable and even larger than those of the 1887 experiment.In Tomaschek’s starlight experiment, fringe shifts were about 15 times smaller than those classically expectedfor an Earth’s velocity of 30 km/s. This gives

obs



7.7 km/s or



320 km/s. From Piccard and Stahel,in the most reﬁned version of Mt. Rigi, one gets an observable velocity

obs



1.5 km/s. Since their opticalpaths were enclosed in an evacuated enclosure, this very low value can easily be reconciled with the typicalkinematical velocity

∼

300 km/s of the most accurate experiments in Table 10.

Table 10:

The average velocity observed (or the limits placed) by the classical ether-drift experiments in the alternative interpretation of Eqs.( 24), ( 29 ), ( 30 ).

Experiment gas in the interferometer

obs

(km

(km/s)Michelson-Morley(1887) air 8

−

349

+62

−

Morley-Miller(1902-1905) air 8

5 353

62Kennedy(1926) helium

600Illingworth(1927) helium 3

0 370

120Miller(1925-1926) air 8

−

349

+79

−

104

Michelson-Pease-Pearson(1929) air 4

...

185

...

Joos(1930) helium 1

−

330

+40

−

in the end may turn out to be wrong.ii) one gets consistent indications from the Michelson-Morley, Morley-Miller, Miller andIllingworth-Kennedy experiments. In view of this consistency, an interpretation of Miller’sobservations in terms of a temperature gradient [68] is only acceptable provided this gradientrepresents a non-local eﬀect as in our picture where the ether-drift is the consequence of afundamental vacuum energy-momentum ﬂow. We have also produced numerical simulationsof the Illingworth experiment in a simple statistically isotropic and homogeneous turbulent-ether model. This represents a zeroth-order approximation and is useful to illustrate basicphenomenological features associated with the picture of the vacuum as an underlying stochas-tic medium. In this scheme, Illingworth’s data are consistent with ﬂuctuations of the velocityﬁeld whose absolute scale is controlled by ˜

CMB

∼

370 km/s, the velocity of the Earth’smotion with respect to the CMB.iii) on the other hand, there is some discrepancy with the experiment performed byMichelson, Pease and Pearson (MPP). However, as discussed in Sect.7, the uncertainty cannotbe easily estimated since only a single basic MPP observation is explicitly reported in theliterature. Therefore, since Miller’s extensive observations (see Fig.22 of ref.[65] and ourFig.8), within their errors, gave ﬂuctuations of the observable velocity in the wide range 4

−

14km/s, a single observation giving

obs

∼

4.5 km/s cannot be interpreted as a refutation. Thisbecomes even more true by noticing that the single session selected by Pease, within a periodof several months, was chosen to represent an example of extremely small ether-drift eﬀect.iv) some more details are needed to account for the Joos observations. This experiment isparticularly important since the data were collected at steps of 1 hour to cover the full sidereal51

day and were recorded by photocamera. For this reason, Joos’ experiment is not comparablewith other experiments (e.g. Michelson-Morley, Illingworth) where only observations at fewselected hours were performed and for which, in view of the strong ﬂuctuations of the azimuth,one can just quote the average magnitude of the observed velocity. Moreover, diﬀerently fromMiller’s, the amplitudes of all Joos’s observations can be reconstructed from the publishedarticles [87, 88]. For these reasons, this experiment has deserved a more reﬁned analysisand is central for our work. As discussed in Sect.8, due to uncertainties in the originaldata analysis, the standard 1.5 km/s velocity value quoted for this experiment should beunderstood as an order of magnitude estimate and not as a true upper limit. Instead, ourreported observable velocity

obs

∼

−

km/s has been obtained from a direct analysisof Joos’ fringe shifts. From this value, to deduce a kinematical velocity, one still needs therefractive index. The traditional view, motivated by Miller’s review article [65] and Joos’sown statements in ref.[87], is that the experiment was performed in an evacuated housing. Inthese conditions, it would be easy to reconcile a large kinematical velocity

∼

350 km/s withthe very small values of the observable velocity. On the other hand, Swenson [90] explicitlyreports that fringe shifts were ﬁnally recorded with optical paths placed in a helium bath.Since Joos’ papers do not provide any deﬁnite clue on this aspect, we have decided to followSwenson’s indications. In this case, by simply correcting with the helium refractive index theresult

obs

∼

−

km/s, one would get a kinematical velocity

∼

217

+66

−

km/s. However,as discussed in detail in Sect.8, this is only a ﬁrst partial view of Joos’ experiment. In fact,by ﬁtting the experimental amplitudes in Table 7 to various forms of cosmic motion (seeEq.(85)) we have obtained angular parameters which are very close to those that describethe CMB anisotropy (right ascension

CMB

∼

168

and angular declination

CMB

∼ −

).Still, to get a complete agreement, one should explain the absolute normalization of theamplitudes and the strong ﬂuctuations of the data. Thus we have improved our analysisby performing various numerical simulations where the velocity components in the plane of the interferometer

(

) and

(

), which determine the basic functions

(

) and

(

) throughEqs.(44) and the fringe shifts through Eq.(35), are not smooth functions but are represented as turbulent ﬂuctuations. Their Fourier components in Eqs.(60) and (61) now vary within time- dependent ranges Eqs.(32)

−

(33), [

−

(

)

(

)] and [

−

(

)

(

)] respectively, controlled bythe macroscopic parameters (

V,α,γ

)

CMB

. Taking into account these stochastic ﬂuctuationsof the velocity ﬁeld tends to increase the ﬁtted average Earth’s velocity, see Eq.(89), and canreproduce correctly Joos’ 2nd-harmonic amplitudes and the characteristic scatter of the data,see Figs. 11 and 12. In view of this consistency, we conclude that the range

= 330

+40

−

km/s(corresponding to Eq.(34) for CMB and Joos’ laboratory) is actually the most appropriate52

one.The more reﬁned analysis adopted for the Joos experiment provides an explicit exampleof the previously mentioned non-trivial ingredients that might be required to reconstruct theglobal Earth’s motion from microscopic measurements performed in a laboratory. For thisreason, the results reported in Table 10, besides providing an impressive evidence for a lightanisotropy proportional to (

N −

1)(

v/c

)

, with the realistic velocity values

∼

300 km/s of most cosmic motions, could also represent the ﬁrst experimental indication for the Earth’smotion with respect to the CMB. Due to the importance of this result, and to provide thereader with all elements of the analysis, we present in a second Appendix a brief numericalsimulation of one noon session of the Michelson-Morley experiment. This has been performedin the same framework adopted for the Joos experiment where the velocity components

(

)and

(

) in the plane of the interferometer are represented as turbulent ﬂuctuations varyingwithin time-dependent ranges controlled by the macroscopic parameters (

V,α,γ

)

CMB

. Wepostpone to a future publication the non-trivial task of performing a complete numericalsimulation of the whole Michelson-Morley experiment and of the Illingworth experiment (withits 32 sessions and the associated sets of 20 rotations for each session) where we’ll also comparethe various theoretical schemes mentioned in Sect.5 to handle the stochastic components of the velocity ﬁeld.We emphasize that the simulation reported in our second Appendix corresponds to asingle conﬁguration whereas taking into account more and more conﬁgurations is essential toproperly estimate theoretical uncertainties (as for the Joos experiment with the results in ourTable 9 and Fig.12). Nevertheless, even this very small sample can provide interesting clueson the real data. For instance, the strong scatter of the fringe shifts at the same

−

values inconsecutive rotations and the good agreement with the experimental azimuths obtained byaccepting Hicks’ interpretation of the observations of July 8th (see Sect.3). In this sense, thisbrief numerical analysis reinforces the picture of the classical experiments emerging from ourTable 10. According to the usual view, the theoretical predictions, with a very low velocity

∼

30 km/s, were much larger than the observed values. Instead, in a modern view of thevacuum as a stochastic medium, theoretical predictions, for the realistic velocities

∼

300km/s of most cosmic motions, are now well compatible and, sometimes, even smaller thanthe actual outcome of the observations. This latter case simply means that the experimentaldata were also aﬀected by spurious eﬀects such as deformations induced by the rotation of theapparatus or local thermal conditions. This gives a strong motivation to repeat these crucialmeasurements with today’s much greater accuracy.To this end, let us now brieﬂy consider the modern ether-drift experiments. As anticipated,53

in the modern experiments, the test of the isotropy of the velocity of light consists in measuringthe relative frequency shift ∆

of two orthogonal optical resonators [35, 36]. Here, the analog of Eq.(29), for a hypothetical physical part of the frequency shift (after subtraction of allspurious eﬀects), is∆

phys

(

)

= ¯

(

π/

2 +

)

−

(

)

medium

cos2(

−

) (94)where

is the direction of the ether-drift. This can be interpreted within Eq.(109) where

medium

|∼N

medium

−

1 (95)

medium

being the refractive index of the gaseous medium ﬁlling the optical resonators. Test-ing this prediction, requires replacing the high vacuum usually adopted within the opticalresonators with a gaseous medium and studying the substantially larger frequency shift in-troduced with respect to the vacuum experiments.As a rough check, a comparison was made [7, 51] with the results obtained by Jaseja et. al [95] in 1963 when looking at the frequency shift of two orthogonal He-Ne masers placed on arotating platform. To this end, one has to preliminarily subtract a large systematic eﬀect thatwas present in the data and interpreted by the authors as probably due to magnetostrictionin the Invar spacers induced by the Earth’s magnetic ﬁeld. As suggested by the same authors,this spurious eﬀect, which was only aﬀecting the normalization of the experimental ∆

, can besubtracted by looking at the variations of the data. As discussed in refs.[7, 51], the measured variations of a few kHz are roughly consistent with the refractive index

−

∼

00004and the typical variations of an Earth’s velocity as in Eq.(52).More recent experiments [44]

−

[50] have always been performed in a very high vacuumwhere, as emphasized in the Introduction, the diﬀerences between Special Relativity and theLorentzian interpretation are at the limit of visibility. In fact, in a perfect vacuum by deﬁnition

vacuum

= 1 so that

vacuum

will vanish

. Thus one should switch to the new generationof dedicated ether-drift experiments in gaseous systems. Our conclusion is that these newexperiments should just conﬁrm Joos’ remarkable observations of eighty years ago.

Throughout this paper we have assumed the limit of a zero light anisotropy for experiments performed invacuum. However, as discussed at the end of Appendix I, one could also consider the more general scenariowhere a metric of the form [97]

µν

+ ∆

µν

is introduced from the very beginning. In this case, thevacuum behaves as a medium and light can spread with diﬀerent velocities for diﬀerent directions. As anexample, by adopting various parameterizations for ∆

µν

, the non-zero one-way light anisotropy reported bythe GRAAL experiment [98] requires typical values of the matrix elements

∆

µν

= 10

−

[99].In anycase, as anticipated in Sect.2, these genuine vacuum eﬀects are much smaller than those discussed in the presentpaper in connection with a gas refractive index.

Acknowledgments

We thank Angelo Pagano for useful discussions.55

References

[1] A. Einstein, Ann. der Physik,

(1905) 891.[2] H. A. Lorentz, Proceedings of the Academy of Sciences of Amsterdam,

, 1904.[3] H. Poincar´e, La Science et l’Hypothese, Flammarion, Paris 1902; C. R. Acad. Sci. Paris

140

(1905) 1504.[4] H. A. Lorentz, The Theory of Electrons, Leipzig 1909, B. G. Teubner Ed.[5] J. S. Bell, How to teach special relativity, in Speakable and unspeakable in quantummechanics, Cambridge University Press 1987, pag. 67.[6] H. R. Brown and O. Pooley, The origin of the space-time metric: Bell’s Lorentzianpedagogy and its signiﬁcance in general relativity, in ‘Physics meets Philosophy atthe Planck Scale’, C. Callender and N. Hugget Eds., Cambridge University Press 2000(arXiv:gr-qc/9908048).[7] M. Consoli and E. Costanzo, Phys. Lett.

A333

(2004) 355.[8] S. Liberati, S. Sonego and M. Visser, Ann. Phys.

298

(2002) 167.[9] V. Scarani et al., Phys. Lett.

A276

(2000) 1.[10] G. E. Volovik, Phys. Rep.

351

(2001) 195.[11] G. E. Volovik, Found.Phys.

(2003) 349.[12] S. Chadha and H. B. Nielsen, Nucl. Phys.

217

(1983) 125.[13] C. D. Froggatt and H. B. Nielsen, Origin of Symmetries, World Scientiﬁc, 1991.[14] J. A. Wheeler, in Relativity, Groups and Topology, B. S. DeWitt and C. M. DeWitt Eds.,Gordon and Breach New York 1963, p. 315.[15] A. A. Migdal, Int. J. Mod. Phys.

(1994) 1197.[16] V. Jejjala, D. Minic, Y. J. Ng and C. H. Tze, Int. J. Mod. Phys.

D19

(2010) 2311.[17] Y. J. Ng, Various Facets of Spacetime Foam, in Proceedings of the Third Conference onTime and Matter, Budva, Montenegro 2010, arXiv:1102.4109 [gr-qc].[18] G. Amelino-Camelia, Nature

418

(2002) 34.56

[19] G. Amelino-Camelia, Int.J.Mod.Phys.

D11

(2002) 35.[20] G. Amelino-Camelia, Symmetry

(2010) 230.[21] P. Jizba and H. Kleinert, Phys. Rev.

D82

(2010) 085016.[22] G. ’t Hooft, In Search of the Ultimate Building Blocks, Cambridge Univ. Press, 1997.[23] M. Consoli and P.M. Stevenson, Int. J. Mod. Phys.

A15

(2000) 133, hep-ph/9905427.[24] M. Consoli, A. Pagano and L. Pappalardo, Phys. Lett.

A318

, (2003) 292.[25] M. Consoli and E. Costanzo, Eur. Phys. Journ.

C54

(2008) 285.[26] M. Consoli and E. Costanzo, Eur. Phys. Journ.

C55

(2008) 469.[27] See, for instance, R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and allthat, W. A. Benjamin, New York 1964.[28] Y. B. Zeldovich , Sov. Phys. Usp.

, 381 (1968).[29] S. Weinberg , Rev. Mod. Phys.

, 1 (1989).[30] C. Barcelo, S. Liberati and M. Visser, Class. Quantum Grav.

, 3595 (2001).[31] M. Visser, C. Barcelo and S. Liberati, Gen. Rel. Grav.

, 1719 (2002).[32] M. Consoli, Class. Quantum Grav.

, 225008 (2009).[33] R. P. Feynman, in Superstrings: A Theory of Everything ?, P. C. W. Davies and J.Brown Eds., Cambridge University Press, 1997, pag. 201.[34] C. Eckart, Phys. Rev.

, 919 (1940).[35] For a comprehensive review, see H. M¨uller et al., Appl. Phys. B

, 719 (2003).[36] Special Relativity, J. Ehlers and C. L¨ammerzahl Eds., Lectures Notes in Physics,Springer, New York 2006.[37] V. W. Hughes, H. G. Robinson, and V. Beltran-Lopez, Phys. Rev. Lett.

, 342 (1960).[38] R. W. P. Drever, Phil. Mag.

, 683 (1961).[39] C. M. Will, The Confrontation between General Relativity and Experiment,arXiv:gr-qc/0510072.57

[40] M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge 1999.[41] V. C. Ballenegger and T. A. Weber, Am. J. Phys.

, 599 (1999).[42] H. P. Robertson, Rev. Mod. Phys.

, 378 (1949).[43] R. M. Mansouri and R. U. Sexl, Gen. Rel. Grav.

, 497 (1977).[44] A. Brillet and J. L. Hall, Phys. Rev. Lett.

(1979) 549.[45] H. M¨uller et al., Phys. Rev. Lett.

(2003) 020401.[46] S. Herrmann, et al., Phys. Rev. Lett.

(2005) 150401.[47] P. Antonini et al., Phys. Rev.

A71

(2005) 050101(R).[48] Ch. Eisele et al., Opt. Comm.

281

, 1189 (2008).[49] S. Herrmann, et al., Phys.Rev. D

, 105011 (2009).[50] Ch. Eisele, A. Newski and S. Schiller, Phys. Rev. Lett.

103

, 090401 (2009).[51] M. Consoli and E. Costanzo, N. Cim.

119B

(2004) 393.[52] J. Shamir and R. Fox, N. Cim.

62B

(1969) 258.[53] R. De Abreu and V. Guerra, Relativity-Einstein’s Lost Frame, 2005, Extra]muros[ Publ.[54] H. M¨uller, Phys. Rev.

D71

, 045004 (2005).[55] J. J. Nassau and P. M. Morse, Astrophys. Journ.

(1927) 73.[56] O. V. Troshkin, Physica

A168

(1990) 881.[57] H. E. Puthoﬀ, Linearized turbulent ﬂow as an analog model for linearized General Rela-tivity, arXiv:0808.3404 [physics.gen-ph].[58] T. D. Tsankov, Classical Electrodynamics and the Turbulent Aether Hypothesis, PreprintFebruary 2009.[59] M. Consoli, A. Pluchino and A. Rapisarda, Chaos, Solitons and Fractals

, 1089 (2011).[60] M. Consoli, Phys. Lett.

A 376

, 3377 (2012).[61] A. A. Michelson and E. W. Morley, Am. J. Sci.

(1887) 333.58

[62] An incomplete list of references includes: W. Sutherland, Phil. Mag.

(1898) 23; A.Righi, N. Cimento

XVI

(1918) 213;

ibidem

XIX

(1920) 141;

ibidem

XXI

(1921) 187; G.Dalla Noce, N. Cimento

XXIV

(1922) 17. Righi’s theory was also re-analyzed by P. DiMauro, S. Notarrigo and A. Pagano, Quaderni di Storia della Fisica,

(1997) 101.[63] A. A. Michelson, et al., Astrophys. Journ.

(1928) pag. 341-402.[64] R. J. Kennedy, Phys. Rev.

(1935) 965.[65] D. C. Miller, Rev. Mod. Phys.

(1933) 203.[66] W. M. Hicks, Phil. Mag.

(1902) 9.[67] M. Born, Einstein’s Theory of Relativity, Dover Publ., New York, 1962.[68] R. S. Shankland et al., Rev. Mod. Phys.,

, (1955) 167.[69] M. A. Handshy, Am. J. of Phys.

(1982) 987.[70] E. W. Morley and D. C. Miller, Phil. Mag.

(1905) 680.[71] K. K. Illingworth, Phys. Rev.

(1927) 692.[72] H. A. M´unera, APEIRON

(1998) 37.[73] A. N. Kolmogorov, Dokl. Akad. Nauk SSSR

(1940) 4; English translation in Proc. R.Soc.

A 434

(1991) 9.[74] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press 1959, Chapt. III.[75] J. C. H. Fung et al., J. Fluid Mech.

236

, 281 (1992).[76] K. R. Sreenivasan, Rev. Mod. Phys.

, Centenary Volume 1999, S383.[77] C. Beck, Phys. Rev. Lett.

, 064502 (2007).[78] C. Tsallis, Introduction to Nonextensive Statistical Mechanics, Springer, 2009.[79] J. DeMeo, Dayton C. Miller Revisited, in Should the Laws of Gravitation be Reconsid-ered?, H. A. M´unera Ed., APEIRON Montreal 2011, pp. 285-315.[80] T. Roberts, An Explanation of Dayton Miller’s Anomalous ”Ether Drift” Result,arXiv:physics/0608238.59

[81] F. James, MINUIT: Function minimization and error analysis, CERN Computing andNetworks Division, Long Writeup D506, Geneva 1994.[82] M. von Laue, Handbuch der Experimentalphysik, Vol. XVIII (1926) 95.[83] H. Thirring, Zeit. Physik

(1926) 723; Nature

118

(1926) 81.[84] A. A. Michelson, F. G. Pease and F. Pearson, Nature,

123

, (1929) 88[85] A. A. Michelson, F. G. Pease and F. Pearson, J. Opt. Soc. Am.

(1929) 181.[86] F. G. Pease, Publ. of the Astr. Soc. of the Paciﬁc,

XLII

, 197 (1930).[87] G. Joos, Ann. d. Physik

(1930) 385.[88] G. Joos, Naturwiss.

(1931) 784.[89] G. Joos, D. Miller, Letters to the Editor, Phys. Rev.

, 114 (1934).[90] Loyd S. Swenson Jr., Journ. for the History of Astronomy,

, 56 (1970).[91] A. A. Michelson, Am. J. Sci.

, 120 (1881).[92] R. Tomaschek, Ann. d. Physik,

, 105 (1924).[93] A. Piccard and E. Stahel, Compt. Rend.

183

, 420 (1926); Naturwiss.

, 935 (1926).[94] A. Piccard and E. Stahel, Compt. Rend.

185

, 1198 (1927); Naturwiss.

, 25 (1928).[95] T. S. Jaseja, et al., Phys. Rev.

133

(1964) A1221.[96] J. M. Jauch and K. M. Watson, Phys. Rev.

, 950 (1948).[97] Zhou L. L. and B. -Q. Ma, Mod. Phys. Lett.

A25

, 2489 (2010).[98] V. G. Gurzadyan et al., arXiv:1004.2867 [physics.acc-ph], and references quoted therein,in Proc. 12th M. Grossmann Meeting on General Relativity, p.1495, World Sci. (2012).[99] Zhou L. L. and B. -Q. Ma, Astropart. Phys.

, 37 (2012), arXiv:1009.1675 [hep-ph].60

Appendix I

To derive Eq.(24), one should start from Eq.(23) which describes light propagation in a gaseous system in the presence of convective currents of the gas molecules originating froma fundamental vacuum energy-momentum ﬂow. Due to these convective currents, isotropyof the velocity of light would only hold if the solid container of the gas and the observerwere both at rest in the particular reference frame Σ where the macroscopic condensation of quanta correspond to the state

= 0. This introduces obvious diﬀerences with respect tothe standard analysis. For instance, let us compare with Jauch and Watson [96] who workedout the quantization of the electromagnetic ﬁeld in a moving medium of refractive index

.They noticed that the procedure introduces unavoidably a preferred frame, the one where thephoton energy does not depend on the direction of propagation. Their conclusion, that thisframe is “usually taken as the system for which the medium is at rest”, reﬂects however thepoint of view of Special Relativity with

preferred frame. Instead, one could consider adiﬀerent scenario where, at least in some limit, the angle-independence of the photon energymight only hold for some special frame Σ.To discuss this diﬀerent case, let us ﬁrst consider a dielectric medium of refractive index

whose container is at rest in Σ. For an observer at rest in this reference frame, lightpropagation within the medium is isotropic and described by

µν

= 0 (96)where

µν

= diag(

−

1) (97)and

denotes the light 4-momentum vector for the Σ observer. Let us now consider thatthe container of the medium is moving with some velocity

with respect to Σ and

is at rest in some other frame

′

. By analogy, light propagation within the medium for the observer in

′

will be described by

µν

= 0

(98)where

≡

(

E/c,

) and

µν

denote respectively the light 4-momentum and the eﬀectivemetric for

′

. On this basis, by introducing the

′

dimensionless velocity 4-vector

≡

(

) (with

= 1), one can deﬁne a transformation matrix

µν

(

) andexpress

µν

µσ

ν ρ

σρ

(99)61

In this context, requiring the consistency of vacuum condensation with Special Relativitycorresponds to place all reference frames on the same footing and assume

µν

(

) =

µν

(100)This identiﬁcation is independent of the physical nature of the medium, being valid for gaseoussystems as well as for liquid or solid transparent media. In this sense Special Relativity, byconstruction, cannot describe light propagation in the presence of a vacuum energy-momentumﬂow which could aﬀect the various forms of matter diﬀerently.Instead, consistently with the basic ambiguity in the interpretation of relativity mentionedin the Introduction, and with Lorentz’ point of view [4] (“it seems natural not to assume atstarting that it can never make any diﬀerence whether a body moves through the ether ornot”), one could adopt diﬀerent choices without pretending to determine

a priori

the outcomeof any ether-drift experiment. Thus, by noticing that we have at our disposal two matrices,namely

µν

and the Lorentz transformation matrix Λ

µν

associated with

, one could stillmaintain Eq.(100) for strongly bound systems, such as solid or liquid transparent media,where the small energy ﬂux generated by the motion with respect to Σ should mainly dissipateby heat conduction with no appreciable particle ﬂow and no light anisotropy in the rest frameof the medium. One could instead identify

µν

(

= 1) = Λ

µν

(101)to solve non-trivially the equation

µν

when

= 1, i.e. when light propagatesin vacuum and

µν

reduces to the Minkowski tensor

µν

. But then, by continuity, it isconceivable that Eq.(101), up to higher-order terms, can also describe the case

= 1 +

of gaseous media. This choice provides a simple interpretative model and a particular formof the more general structure Eq.(23) which corresponds to the Robertson-Mansouri-Sexl(RMS) [42, 43] parametrization for the two-way velocity of light. Moreover, when comparingwith experiments with optical resonators, the resulting frequency shift is governed by thesame classical formula for the fringe shifts in the old ether-drift experiments with the onlyreplacement

→

N −

. To see this, let us compute

µν

through the relation

µν

= Λ

µσ

ν ρ

σρ

(102)with

µν

as in Eq.(97). This gives the eﬀective metric for

′

µν

κu

(103)with

−

1 (104)62

In this way, Eq.(98) gives a photon energy (

= 1 +

)

(

,θ

) =

−

κu



(1 +

κu

)

−

κζ

1 +

κu

(105)with

cos

θ,

(106)where

and

≡

lab

indicates the angle deﬁned, in the laboratory

′

frame, betweenthe photon momentum and

. By using the above relation, one gets the one-way velocity of light

(

,θ

)

(

) =

−

κβ



1 +

cos



1 +

κβ

sin

1 +

(1 +

)

(107)or to

(

) and

(

)

(

)

∼



−

κβ

cos

−

(1 + cos

)



(108)From this one can compute the two-way velocity¯

(

) = 2

(

)

(

)

(

) +

(

)

∼



−



−

2 sin



(109)which, as anticipated, is a special form of the more general Eq.(23). We can then deﬁne theRMS anisotropy parameter

(

π/

2 +

)

−

(

)



 ∼B

cos2(

−

) (110)where the pair (

v,θ

) describes the projection of

onto the relevant plane and

|B|∼

∼

(

N −

1) (111)

There is a subtle diﬀerence between our Eqs.(108) and(109) and the corresponding Eqs. (6) and (10) of ref. [7] that has to do with the relativistic aberration of the angles. Namely, in ref.[7], with the (wrong) motivation that the anisotropy is

(

), no attention was paid to the precise deﬁnition of the angle betweenthe Earth’s velocity and the direction of the photon momentum. Thus the two-way velocity of light in the

′

frame was parameterized in terms of the angle

≡

as seen in the Σ frame. This can be explicitlychecked by replacing in our Eqs. (108) and(109) the aberration relation cos

lab

= (

−

+cos

)

−

cos

)or equivalently by replacing cos

= (

+ cos

lab

)

(1 +

cos

lab

) in Eqs. (6) and (10) of ref. [7]. However,the apparatus is at rest in the laboratory frame, so that the correct orthogonality condition of two opticalcavities at angles

and

π/

2 +

is expressed in terms of

lab

and not in terms of

. This trivialremark produces however a non-trivial diﬀerence in the value of the anisotropy parameter. In fact, the correctresulting

|B|

Eq. (111) is now smaller by a factor of 3 than the one computed in ref.[7] by adopting the wrong deﬁnition of orthogonality in terms of

From the previous analysis, by replacing the two-way velocity in Eq.(28), one ﬁnally obtainsthe observable velocity

2obs

∼

|B|

∼

N −

(112)to be used in Eq.(29). In this way, where one re-obtains the classical formulas with the onlyreplacement

→

obs

, testing the present scheme requires to check the consistency of thekinematical

′

s obtained in diﬀerent experiments.Before concluding this Appendix, we emphasize that throughout this paper we have as-sumed the limit of a zero light anisotropy for experiments performed in vacuum. In fact, theeﬀective metric Eq.(103) reduces to the Minkowski tensor

µν

in the limit

N →

1. Admit-tedly, this might represent a restrictive scenario and one could also consider the more generalcase where a metric of the form [97]

µν

+ ∆

µν

(113)is introduced from the very beginning in extensions of the Standard Model. In this sense, onceEq.(113) is adopted, the vacuum behaves as a medium and the dispersion relations that de-scribe light and particle propagation can have several solutions. For instance, light will spreadwith diﬀerent velocities in diﬀerent directions as with anisotropic media in optics. Therefore,by adopting various parameterizations for ∆

µν

, one can restrict its size by comparing withmeasurements of the one- and two-way velocity of light. As an example, the one-way lightanisotropy reported by the GRAAL experiment [98] requires typical values of the matrix ele-ments

∆

µν

= 10

−

[99]. In any case, as anticipated in Sect.2, these genuine vacuumeﬀects are much smaller than those discussed in the present paper in connection with a gasrefractive index.64

Appendix II

In this second Appendix we’ll report the results of a single simulation of an individualnoon session of the Michelson-Morley experiment. This will be performed within the samestochastic-ether model described in Sect.8 for the Joos experiment. For sake of clarity, werecapitulate the various steps so that an interested reader can also run his own simulations.One should ﬁrst express the functions

(

) and

(

) as in Eqs.(44) and model the twovelocity components

(

) and

(

) as in Eqs.(60) and (61). A basic input value is the sidereal time of the observation. This has to be inserted, together with the CMB kinematicalparameters

CMB

∼

370 km/s,

CMB

∼

168

CMB

∼ −

, in Eqs.(31)

−

(33) to ﬁx theboundaries [

−

(

)

(

)] and [

−

(

)

(

)] respectively for the random parameters

(

2) and

(

= 1

2) entering Eqs.(60) and (61). In the end, with the simulated

(

) and

(

), one should form the fringe shift combination∆

(

)

≡

(

)cos2

+ 2

(

)sin2

(

)cos2(

−

(

)) (114)As recalled in Sect.3, an individual session of the Michelson-Morley experiment consisted of 6 rotations. Each complete rotation of the interferometer took 6 minutes and the consecutivereadings of the fringe shifts were performed every 22.5 degrees. Therefore, two consecutivereadings diﬀered by 22.5 seconds. In these conditions, a numerical simulation of a singlerotation consists in generating 16 pairs [

(

)

(

)] at steps of 22.5 seconds.As the central time of the observations, we have chosen 12 A. M. of July 10, 1887 which, forCleveland, corresponds to a sidereal time

∼

102

. To select the parameters of the simulation,we have compared with the traditional analysis of the experiment where one performs a ﬁtto the fringe shifts obtained by averaging the results of the various experimental sessions. Inour case, averaging the data of the three noon sessions in our Table 1 gives a 2nd-harmonicamplitude

ﬁt2

(average data

−

noon)

∼

012 (115)We have thus considered the exact amplitude

exact2

(

) = 2



(

) +

(

) (116)and selected a particular conﬁguration whose global average over the 6 turns gives



exact2

(

)

∼

0.012. Of course, this condition can be realized by a very large number of conﬁgurations.These can produce very diﬀerent fringe shifts at the same

−

values and sizeable variations of the ﬁtted amplitude and azimuth. Taking into account these variations is essential to perform65

a full numerical simulation and estimate theoretical uncertainties (as done for the Joos ex-periment with Table 9 and Fig.12). However, our intention here is just to give an idea of theagreement one can achieve between data and a single numerical simulation. We thus postponeto a future publication a complete analysis of the whole Michelson-Morley experiment andof the Illingworth experiment (with its 32 sessions and the associated sets of 20 rotations foreach session) where we’ll also compare the various theoretical schemes mentioned in Sect.5 tohandle the stochastic components of the velocity ﬁeld.The results of our single simulation for [2

(

)

(

)] are reported in Tables 11 and 12while the combinations

∆

(

)

Eq.(114) are reported in Table 13, for

−

116

togetherwith the results of 2-parameter ﬁts to the simulated data. Notice the strong scatter of thesimulated data at the same

−

values. Of course, to compare with the

real

data, one shouldﬁrst take the even combination Eq.(50) of the entries in Table 1 which otherwise also containodd-harmonic terms.We conclude this brief analysis by emphasizing the importance of Hicks’ observation (seeSect.3) concerning the fringe shifts from the session of July 8th. By accepting his interpre-tation, the experimental azimuths from the three noon sessions of July 8th, 9th and 11th,respectively

EXP0

∼

357, 285 and 317 degrees, would become

EXP0

∼

267, 285 and 317 degreesand thus be in rather good agreement with the simulated azimuths reported in Table 13.66

Table 11:

The coeﬃcients 2C(t) Eqs.( 44) from a single simulation of 6 rotations in one noon session of the Michelson-Morley experiment. The stochastic components of

(

)

and

(

)

in Eqs.( 60 ) and ( 61) are controlled by the kinematical parameters

(

V,α,γ

)

CMB

as explained in the text.

i 1 2 3 4 5 61

−

023 +0

002

−

024 +0

001

−

004

−

0032 +0

003

−

011

−

000

−

006

−

021

−

0343

−

001

−

001 +0

000

−

009 +0

002 +0

0074

−

002 +0

003

−

008 +0

002

−

060

−

0305 +0

002

−

017 +0

002

−

001 +0

003

−

0086

−

007

−

006

−

059

−

013

−

008

−

0477

−

020

−

001

−

019

−

000

−

003

−

0038

−

011

−

001

−

011

−

002

−

026 +0

0019

−

015

−

000

−

008

−

001

−

008

−

02210

−

037

−

005 +0

000

−

002

−

003 +0

00311 +0

003

−

022

−

015

−

005

−

003 +0

00212

−

002

−

049

−

023

−

016

−

009

−

00613

−

001 +0

002 +0

000 +0

001 +0

003

−

00114 +0

003

−

003 +0

003

−

023

−

001

−

01915

−

012

−

034

−

013

−

001

−

001

−

01116

−

004

−

017

−

004 +0

002

−

010

−

01067

Table 12:

The coeﬃcients 2S(t) Eqs.( 44) from a single simulation of 6 rotations in one noon session of the Michelson-Morley experiment. The stochastic components of

(

)

and

(

)

in Eqs.( 60 ) and ( 61) are controlled by the kinematical parameters

(

V,α,γ

)

CMB

as explained in the text.

i 1 2 3 4 5 61 +0

011 +0

001 +0

011 +0

000 +0

003 +0

0012 +0

004

−

003

−

001

−

008 +0

007

−

0093 +0

003

−

005 +0

000

−

007 +0

0004 +0

001

−

007

−

003

−

001 +0

0165 +0

002

−

007 +0

000 +0

001

−

000

−

0036

−

004 +0

010 +0

001

−

001 +0

006 +0

0097

−

017 +0

002 +0

005

−

000

−

003 +0

0028 +0

011

−

002

−

012

−

001 +0

010

−

0009 +0

011 +0

001 +0

008 +0

011 +0

005

−

01110 +0

016 +0

005

−

001 +0

001

−

002 +0

00111 +0

000

−

013 +0

015 +0

005

−

002

−

00112

−

001

−

022 +0

004

−

003

−

000 +0

00213

−

001

−

001 +0

000

−

004 +0

00114

−

001 +0

002 +0

018 +0

001 +0

00915

−

012 +0

018

−

002 +0

001

−

004 +0

00716 +0

002 +0

008 +0

001 +0

002

−

005

−

00668

Table 13:

The fringe shifts

∆

(

)

Eq.( 114) for the single simulation of one noon session of the Michelson-Morley experiment reported in Tables 11 and 12. The angular values are deﬁned as

−

116

. The variance of the averages is about

004

for the amplitude and about

for the azimuth.

i 1 2 3 4 5 6 average1

−

023 +0

002

−

024 +0

001

−

004

−

003

−

0092 +0

005

−

010

−

001

−

010

−

010

−

031

−

0103 +0

003

−

005 +0

000

−

007 +0

000

−

0004 +0

002

−

007 +0

004

−

002 +0

043 +0

033 +0

0125

−

002 +0

017

−

002 +0

001

−

003 +0

008 +0

0036 +0

008

−

003 +0

041 +0

010 +0

002 +0

027 +0

0147 +0

017

−

002

−

005 +0

000 +0

003

−

002 +0

0028

−

016 +0

001 +0

001

−

001

−

025 +0

001

−

0079

−

015

−

000

−

008

−

001

−

008

−

022

−

00910

−

015

−

000

−

000

−

001

−

003 +0

003

−

00311 +0

000

−

013 +0

015 +0

005

−

002

−

001 +0

00112 +0

000 +0

019 +0

020 +0

009 +0

006 +0

01013 +0

001

−

002

−

000

−

001

−

003 +0

001

−

00114

−

001 +0

001

−

004 +0

004

−

000 +0

007 +0

00115 +0

012

−

018 +0

002

−

001 +0

004

−

007

−

00116

−

005

−

018

−

003

−

000

−

004

−

003

−

005

ﬁt2

0.009 0.005 0.009 0.003 0.011 0.013 0.008

ﬁt0

279

259

266

285

255

272

269