zotero-db/storage/MYLNTTLH/.zotero-ft-cache

arXiv:1302.3508v2 [physics.gen-ph] 6 Jun 2013

The classical ether-drift experiments: a modern re-interpretation
M. Consoli(a), C. Matheson(b) and A. Pluchino (a,c)
a) Istituto Nazionale di Fisica Nucleare, Sezione di Catania, Italy b) Selwyn College, Cambridge, United Kingdom c) Dipartimento di Fisica e Astronomia dell’Universit`a di Catania, Italy
Abstract
The condensation of elementary quanta and their macroscopic occupation of the same quantum state, say k = 0 in some reference frame Σ, is the essential ingredient of the degenerate vacuum 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 preferred 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 as conﬁrmations of Special Relativity, have so deeply inﬂuenced scientiﬁc thought as to prevent a critical discussion on the real reasons underlying its alleged supremacy. In this paper, we argue that this traditional null interpretation is far from obvious. In fact, by using Lorentz transformations to connect the Earth’s frame to Σ, the small observed eﬀects point to an average Earth’s velocity of about 300 km/s, as in most cosmic motions. A common feature is the irregular behaviour of the data. While this has motivated, so far, their standard interpretation as instrumental artifacts, our new re-analysis of the very accurate Joos experiment gives clear indications for the type of Earth’s motion associated with the CMB anisotropy and leaves little space for this traditional interpretation. The new explanation requires instead a view of the vacuum as a stochastic medium, similar to a ﬂuid in a turbulent state of motion, in agreement with basic foundational aspects of both quantum physics and relativity. The overall consistency of this picture with the present experiments with vacuum optical resonators and 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 diﬀerences

between 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 between

the two formulations. This type of conclusion was, for instance, already clearly expressed by

Ehrenfest in his lecture ‘On the crisis of the light ether hypothesis’ (Leyden, December 1912)

as follows: “So, we see that the ether-less theory of Einstein demands exactly the same here

as the ether theory of Lorentz. It is, in fact, because of this circumstance, that according to

Einstein’s theory an observer must observe exactly the same contractions, changes of rate,

etc. in the measuring rods, clocks, etc. moving with respect to him as in the Lorentzian

theory. And let it be said here right away and in all generality. As a matter of principle,

there is no experimentum crucis between the two theories”. This can be understood since,

independently of all interpretative aspects, the basic quantitative ingredients, namely Lorentz

transformations, are the same in both formulations. Their validity will be assumed in the

following 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 length

contraction in the direction of motion) to deduce Lorentz transformations. In this way, due to the fundamental group properties, the relation between two observers S′ and S′′, individually

related to the preferred frame Σ by Lorentz transformations with dimensionless parameters

β′ = v′/c and β′′ = v′′/c, is also a Lorentz transformation with relative velocity parameter

βrel ﬁxed by the relativistic composition rule

βrel

=

β′ − β′′ 1 − β′β′′

(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 tests

where 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 cγ propagating in the various interferometers coincides with the basic parameter c entering Lorentz transformations, relativistic eﬀects

conspire to make undetectable the individual β′, β′′,...This means that a null result of the

ether-drift experiments should not be automatically interpreted as a conﬁrmation of Special

1

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 reconcile faster-than-light signals with causality [8] and thus provide a diﬀerent view of the apparent non-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 assume at starting that it can never make any diﬀerence whether a body moves through the ether or not..”, may induce one to re-analyze the classical ether-drift experiments. Let us ﬁrst give some general theoretical arguments that could motivate this apparently startling idea.
A possible observation is that Lorentz symmetry might not be an exact symmetry. In this case, one could conceivably detect the eﬀects of absolute motion. For instance Lorentz symmetry could represent an ‘emergent’ phenomenon and thus reﬂect the existence of some underlying form of ether. This is an interesting conceptual possibility which, in many diﬀerent forms, objectively reﬂects the fast growing interest of part of the physics community, a partial list 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 particle physics 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 nonsymmetric quantum ﬁeld theories iii) the quantum-gravity literature which, by starting from the original concept [14] of ‘space-time foam’, explicitly models the vacuum as a turbulent ﬂuid [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 particle propagation from the superposition, at very short time scales, of non-relativistic particle paths with diﬀerent Newtonian mass [21].
Here, however, we shall adopt a diﬀerent perspective and concentrate our analysis on a 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 is a meta-stable state and not the true ground state of the theory. In the physically relevant case of the Standard Model of electroweak interactions, this situation can be summarized by 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 Bose condensed”. The explicit translation from ﬁeld jargon to particle jargon, with the substantial equivalence between the eﬀective potential of quantum ﬁeld theory and the energy density of
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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 above
a critical temperature T = Tc where the condensate ‘evaporates’. This temperature in the Standard Model is so high that one can safely approximate the ordinary vacuum as a zerotemperature system (think of 4He at a temperature 10−12 oK). This observation allows one to view the physical vacuum as a superﬂuid medium [10] where bodies can ﬂow without any apparent friction, consistently with the experimental results. Clearly, this form of quantum vacuum is not the kind of ether imagined by Lorentz. However, if possible, this modern view of the vacuum state is even more diﬀerent from the empty space-time of Special Relativity that Einstein had in mind in 1905. Therefore, one might ask [24] if Bose condensation, i.e. the macroscopic occupation of the same quantum state, say k = 0 in some reference frame Σ, can represent 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 modern Lorentzian approach.
Usually this possibility is not considered with the motivation, perhaps, that the average properties of the condensed phase are summarized into a single quantity which transforms as a world scalar under the Lorentz group, for instance, in the Standard Model, the vacuum expectation value Φ of the Higgs ﬁeld. However, this does not imply that the vacuum state itself has to be Lorentz invariant. Namely, Lorentz transformation operators Uˆ ′, Uˆ ′′,..might transform non trivially the reference vacuum state |Ψ(0) (appropriate to an observer at rest in Σ) into |Ψ′ , |Ψ′′ ,.. (appropriate to moving observers S′, S′′,..) and still, for any Lorentzinvariant operator Gˆ, one would ﬁnd

Gˆ Ψ(0) = Gˆ Ψ′ = Gˆ Ψ′′ = ..

(2)

Here, we are assuming the existence of a suitable operatorial representation of the Poincar´e algebra for the quantum theory in terms of 10 generators Pα, Mα,β ( α ,β=0, 1, 2, 3) where Pα are the 4 generators of the space-time translations and Mαβ = −Mβα are the 6 generators of the Lorentzian rotations with commutation relations

[Pα, Pβ] = 0

(3)

[Mαβ , Pγ ] = ηβγPα − ηαγ Pβ

(4)

[Mαβ, Mγδ ] = ηαγ Mβδ + ηβδMαγ − ηβγMαδ − ηαδMβγ

(5)

where ηαβ = diag(1, −1, −1, −1). With these premises, the possibility of a Lorentz-non-invariant vacuum state was addressed
in 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 vacuum as an eigenstate of the energy-momentum vector. This physical vacuum state |Ψ(0) 1 would maintain both zero momentum and zero angular momentum, i.e. (i,j=1,2,3)

Pˆi|Ψ(0) = Mˆ ij |Ψ(0) = 0

(6)

but, at the same time, be characterized by a non-vanishing energy

Pˆ0|Ψ(0) = E0|Ψ(0)

(7)

This vacuum energy might have diﬀerent explanations. Here, we shall limit ourselves to exploring the physical implications of its existence by just observing that, in interacting quantum ﬁeld theories, there is no known way to ensure consistently the condition E0 = 0 without imposing an unbroken supersymmetry, which is not phenomenologically acceptable. In this framework, by using the Poincar´e algebra of the boost and energy-momentum operators, one then deduces that the physical vacuum cannot be a Lorentz-invariant state and that, in any moving frame, there should be a non-zero vacuum spatial momentum Pˆi Ψ′ = 0 along the direction of motion. In this way, for a moving observer S′ the physical vacuum would look like some kind of ethereal medium for which, in general, one can introduce a momentum density Wˆ 0i Ψ′ through the relation (i=1,2,3)

Pˆi Ψ′ ≡ d3x Wˆ 0i Ψ′ = 0

(8)

On the other hand, there is an alternative approach where one tends to consider the vacuum

energy E0 as a spurious concept and only concentrate on an energy-momentum tensor of the

following form [28, 29]

Wˆ µν Ψ(0) = ρv ηµν

(9)

(ρv being a space-time independent constant). In this case, one is driven to completely diﬀerent conclusions since, by introducing the Lorentz transformation matrices Λµν to any moving frame S′, deﬁning Wˆ µν Ψ′ through the relation

Wˆ µν Ψ′ = ΛσµΛρν Wˆ σρ Ψ(0)

(10)

and using Eq.(9), it follows that the expectation value of Wˆ 0i in any boosted vacuum state |Ψ′ vanishes, just as it vanishes in |Ψ(0) , i.e.

d3x Wˆ 0i Ψ′ ≡ Pˆi Ψ′ = 0

(11)

1 We ignore here the problem of vacuum degeneracy by assuming that any overlapping among equivalent vacua 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).

4

As discussed in ref.[25], both alternatives have their own good motivations and it is not so
obvious how to decide between Eq.(8) and Eq.(11) on purely theoretical grounds. For instance,
in a second-quantized formalism, single-particle energies E1(p) are deﬁned as the energies of the corresponding one-particle states |p minus the energy of the zero-particle, vacuum state.
If E0 is considered a spurious concept, E1(p) will also become an ill-deﬁned quantity. At a deeper level, one should also realize that in an approach based solely on Eq.(9) the properties of |Ψ(0) under a Lorentz transformation are not well deﬁned. In fact, a transformed vacuum state |Ψ′ is obtained, for instance, by acting on |Ψ(0) with the boost generator Mˆ01. Once |Ψ(0) is considered an eigenstate of the energy-momentum operator, one can deﬁnitely show [25] that, for E0 = 0, |Ψ′ and |Ψ(0) diﬀer non-trivially. On the other hand, if E0 = 0 there are only two alternatives: either Mˆ01|Ψ(0) = 0, so that |Ψ′ = |Ψ(0) , or Mˆ01|Ψ(0) is a state vector proportional to |Ψ(0) , so that |Ψ′ and |Ψ(0) diﬀer by a phase factor.
Therefore, if the structure in Eq.(9) were really equivalent to the exact Lorentz invariance of the vacuum, it should be possible to show similar results, for instance that such a |Ψ(0)
state can remain invariant under a boost, i.e. be an eigenstate of

Mˆ 0i = −i d3x (xiWˆ 00 − x0Wˆ 0i)

(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 Mˆ0i Ψ(0) = 0). Thus, independently of the ﬁniteness of E0, it should not come as a surprise that one can run into contradictory statements once |Ψ(0) is instead characterized by means of Eqs.(6)−(7). For these reasons, it is not obvious that the local relations (9) represent a more fundamental approach to the vacuum.
Alternatively, one could argue that a satisfactory solution of the vacuum-energy problem lies deﬁnitely beyond ﬂat space. A non-zero ρv, in fact, should induce a cosmological term in Einstein’s ﬁeld equations and a non-vanishing space-time curvature which anyhow dynamically 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 be useful to clarify a crucial point that, so far, remains obscure: the huge renormalization eﬀect which is seen when comparing the typical vacuum-energy scales of modern particle physics with the experimental value of the cosmological term needed in Einstein’s equations to ﬁt the observations. For instance, as anticipated, the picture of the vacuum as a superﬂuid can explain in a natural way why there might be no non-trivial macroscopic curvature in the equilibrium state where any liquid is self-sustaining [10]. In any liquid, in fact, curvature requires deviations from the equilibrium state. The same happens for a crystal at zero

5

temperature where all lattice distortions vanish and electrons can propagate freely as in a perfect vacuum. In such representations of the lowest energy state, where large condensation energies (of the liquid and of the crystal) play no observable role, one can intuitively understand why curvature eﬀects can be orders of magnitude smaller than those naively expected by solving Einstein’s equations with the full Wˆ µν Ψ(0) as a cosmological term. In this perspective, ‘emergent-gravity’ approaches [30, 31, 32], where gravity somehow arises from the same physical ﬂat-space vacuum, may become natural 2 and, to ﬁnd the eﬀective form for the cosmological term to be inserted in Einstein’s ﬁeld equations, we are lead to sharpen our understanding of the vacuum structure and of its excitation mechanisms by starting from the physical 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 a fundamental 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 phenomenological model [25] where the physical vacuum is described as a relativistic ﬂuid [34]. In this representation, a non-zero Wˆ 0i Ψ′ gives rise to a tiny heat ﬂow and an eﬀective thermal gradient in a moving frame S′. This would represent a fundamental perturbation which, if present, is likely too small to be detectable in most experimental conditions by standard calorimetric devices. However, it could eventually be detected through very accurate etherdrift experiments performed in forms of matter that react by producing convective currents in 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 these experiments one looks for a possible anisotropy of the two-way velocity of light through the relative frequency shift ∆ν(θ) of two orthogonal optical cavities [35, 36]. Their frequency

ν(θ)

=

c¯γ (θ)m 2L(θ)

(13)

is proportional to the two-way velocity of light c¯γ(θ) within the cavity through an integer number m, which ﬁxes the cavity mode, and the length of the cavity L(θ) as measured in the laboratory. In principle, by ﬁlling the resonating cavities with some gaseous medium, the existence of a vacuum energy-momentum ﬂow could produce two basically diﬀerent eﬀects:
2 In this sense, by exploring emergent-gravity approaches based on an underlying superﬂuid medium, one is taking seriously Feynman’s indication : ”...the ﬁrst thing we should understand is how to formulate gravity so that it doesn’t interact with the energy in the vacuum” [33].

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a) modiﬁcations of the solid parts of the apparatus. These can change the cavity length

upon active rotations of the apparatus or under the Earth’s rotation.

b) convective currents of the gas molecules inside the optical cavities. These can produce an anisotropy of the two-way velocity of light. In this sense, the reference frame S′ where the

solid 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 an

anisotropy 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 to

the ﬁxed stars, possible deviations from isotropy have been found below the 10−20 level [39].

This is incomparably smaller than any other eﬀect on the velocity of light that we are going

to discuss. Therefore, mechanism a), if present, is completely negligible and, from now on, we

shall assume L(θ) = L =constant. In this way, one re-obtains the standard relation adopted

in the analysis of the experiments

∆ν phys (θ ) ν0

=

c¯γ (π/2

+ θ) c

−

c¯γ (θ)

≡

∆c¯θ c

(14)

where ν0 is the reference frequency of the two optical resonators and the suﬃx “phys” indicates

a 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 rigorous

treatments 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 smaller

than 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 and

obtain the refractive index. However, the presence of convective currents would produce an

anisotropy of the velocity of refracted light.

To derive the relevant relations, let us introduce from scratch the refractive index N of the

gas. By assuming isotropy, the time t spent by refracted light to cover some given distance

L within the medium is t = N L/c. This can be expressed as the sum of t0 = L/c and t1 = (N − 1)L/c where t0 is the same time as in the vacuum and t1 represents the additional, average time by which refracted light is slowed down by the presence of matter. If there are

convective currents, due to the motion of the laboratory with respect to a preferred reference

frame Σ, then t1 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 a

2-dimensional plane and express t1 as

t1

=

L c

f

(N

,

θ,

β)

(15)

7

with β = V /c, V being (the projection on the considered plane of) the relevant velocity with respect to Σ where the isotropic form

f (N , θ, 0) = N − 1

(16)

is assumed. By expanding around N = 1 where, whatever β, f vanishes by deﬁnition, one ﬁnds for gaseous systems (where N − 1 ≪ 1) the universal trend

f (N , θ, β) ∼ (N − 1)F (θ, β)

(17)

with

F (θ, β) ≡ (∂f /∂N )|N =1

(18)

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

t(N , θ, β)

=

L cγ(N , θ, β)

∼

L c

+

L c

(N

−

1)

F (θ, β)

(19)

one gets

cγ(N ,

θ,

β)

∼

c N

[1 − (N − 1) (F (θ, β) − 1)]

(20)

Analogous relations hold for the two-way velocity c¯γ(N , θ, β)

c¯γ(N , θ, β)

=

2 cγ(N , θ, β)cγ (N , π + θ, β) cγ(N , θ, β) + cγ(N , π + θ, β)

∼

c N

1 − (N − 1)

F

(θ,

β)

+

F 2

(π

+

θ,

β)

−

1

(21)

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

tion F (θ, β). By expanding in powers of β

F (θ, β) − 1 = βF1(θ) + β2F2(θ) + ...

(22)

and taking into account that, by the very deﬁnition of two-way velocity, c¯γ(N , θ, β) =

c¯γ(N , θ, −β), it follows that F1(θ) = −F1(π + θ). Therefore, to O(β2), we get the general

structure [26]

c¯γ(N , θ, β)

∼

c N

∞
1 − (N − 1) β2 ζ2nP2n(cos θ)

(23)

n=0

in which we have expressed the combination F2(θ) + F2(π + θ) as an inﬁnite expansion of even-order Legendre polynomials with unknown coeﬃcients ζ2n = O(1) which depend on the characteristics 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 wants

to compare with real experiments. In fact, it would require the complete control of all possible

mechanisms that can produce the gas convective currents by starting from scratch with the

8

macroscopic Earth’s motion in the physical vacuum. This general structure can, however, be compared with the particular form (see Eq.(109) of the Appendix) obtained by using Lorentz transformations to connect S′ to the preferred frame

c¯γ(N , θ, β)

∼

c N

[1 − β2

(N

− 1)(A + B sin2 θ)]

(24)

with A = 2 and B = −1 which corresponds to setting ζ0 = 4/3, ζ2 = 2/3 and all ζ2n = 0 for 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 the problems posed by our limited knowledge of the electromagnetic properties of gaseous systems and will be adopted in the following as a tentative model for the two-way velocity of light 3.
Summarizing: in this scheme, the theoretical estimate for a possible anisotropy of the two-way velocity of light is

∆ν ν0

Theor
=
gas

∆c¯θ c

Theor gas

∼ (Ngas

− 1)

V2 c2

(26)

Then, by assuming the typical velocity of most Earth’s cosmic motions V ∼ 300 km/s, one

would

expect

∆c¯θ c

10−9 for experiments performed in air at atmospheric pressure, where

N

∼

1.00029,

or

∆c¯θ c

10−10 for experiments performed in helium at atmospheric pressure,

where N ∼ 1.000035. Therefore these potential eﬀects are much larger than those possibly

associated with vacuum cavities. In fact, from experiments one ﬁnds [44]−[50]

∆ν EXP = ∆c¯θ EXP ∼ 10−15

ν0 vacuum

c vacuum

(27)

3One conceptual detail concerns the gas refractive index whose reported values are experimentally measured on the Earth by two-way measurements. For instance for air, the most precise determinations are at the level 10−7, say Nair = 1.0002926.. at STP (Standard Temperature and Pressure). By assuming a non-zero anisotropy in the Earth’s frame, one should interpret the isotropic value c/Nair as an angular average of Eq.(24), i.e.

c Nair

≡

c¯γ (N¯air, θ, β)

θ

=

c N¯air

[1

−

3 2

(N¯air

−

1)β2]

(25)

From this relation, one can determine in principle the unknown value N¯air ≡ N (Σ) (as if the gas were at

rest in Σ), in terms of the experimentally known quantity Nair ≡ N (Earth) and of V . In practice, for the

standard velocity values involved in most cosmic motions, say V ∼ 300 km/s, the diﬀerence between N (Σ)

and N (Earth) is at the level 10−9 and thus completely negligible. The same holds true for the other gaseous

systems at STP (say nitrogen, carbon dioxide, helium,..) for which the present experimental accuracy in the

refractive index is, at best, at the level 10−6. Finally, the isotropic two-way speed of light is better determined

in the low-pressure limit where (N − 1) → 0. In the same limit, for any given value of V , the approximation

N (Σ) = N (Earth) becomes better and better.

9

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 motion

with respect to the vacuum condensate should mainly dissipate by heat conduction with no

appreciable particle ﬂow and no light anisotropy in the rest frame of the container of the

medium. 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 are

small residuals which are roughly consistent with Eq.(26). A latter region where the diﬀerence

of N from unity is substantial, (e.g. N ∼ 1.5 as with perspex in the experiment by Shamir

and 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 way

the transition between the two regimes, some simple arguments can be given along the lines

suggested 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 be

along the lines of ref.[54] where just the use of optical cavities ﬁlled with diﬀerent materials

was 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 motivate

this new series of experiments, one could try to obtain quantitative checks by applying the

same interpretative scheme to the classical ether-drift experiments (Michelson-Morley, Miller,

Illingworth, Joos,...). These old experiments were performed with interferometers where light

was propagating in air or helium at atmospheric pressure. In this regime, where (N − 1) is

a very small number, the theoretical fringe shifts expected on the basis of Eqs.(23) and (24)

are much smaller than the classical prediction O(β2) and it becomes conceivable that tiny

non-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-way

velocity of light could be measured by rotating a Michelson interferometer. As anticipated,

in the rest frame S′ of the apparatus, the length L of its arms does not depend on their

orientation so that the interference pattern between two orthogonal beams of light depends

on the time diﬀerence

∆T (θ)

=

2L c¯γ(N , θ, β)

−

2L c¯γ(N , π/2

+

θ, β)

(28)

In this way, by introducing the wavelength λ of the light source and the projection v of the

relative

velocity

in

the

plane

of

the

interferometer,

one

ﬁnds

to

order

v2 c2

the

fringe

shift

∆λ(θ) λ

∼

c∆T (θ) Nλ

∼

L λ

vo2bs c2

cos 2(θ

−

θ0)

(29)

10

In the above equation the angle θ0 = θ0(t) indicates the apparent direction of the ether-drift in the plane of the interferometer (the ‘azimuth’) and the square of the observable velocity

vo2bs(t) ∼ 2(N − 1)v2(t)

(30)

is re-scaled by the tiny factor 2(N − 1) with respect to the true kinematical velocity v2(t). We emphasize that vobs is just a short-hand notation to summarize into a single quantity the combined eﬀects of a given kinematical v and of the gas refractive index N . In this sense, one could also avoid its introduction altogether. However, in our opinion, it is a useful, compact parametrization since, in this way, relation (29) is formally identical to the classical prediction of a second-harmonic eﬀect with the only replacement v → vobs. For this reason, as we shall see in the following sections, it is in terms of vobs, rather than in terms of the true kinematical v, that one can more easily compare with the original analysis of the classical ether-drift experiments.
In conclusion, in this scheme, the interpretation of the experiments is transparent. According 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 assumed to coincide with the laboratory frame S′, where the container of the medium is at rest, and thus one has vobs = v = 0. On the other hand, if there were fringe shifts, one could try to deduce the existence of a preferred frame Σ = S′ provided the following minimal requirements are fulﬁlled : i) the fringe shifts exhibit an angular dependence of the type in Eq.(29) ii) by using gaseous media with diﬀerent refractive index one gets consistency with Eq.(30) in such a way that diﬀerent vobs correspond to the same kinematical v.
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 the existence of an ether-drift. However interpretative models are required to compare results obtained 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 v = v(t) and θ0 = θ0(t), respectively the magnitude of the velocity and the apparent direction of the azimuth in the plane of the interferometer. For their determination, the standard assumption is to consider a cosmic Earth’s velocity with well deﬁned magnitude V , right ascension α and angular declination γ that can be considered constant for short-time observations of a few days where there are no appreciable changes due to the Earth’s orbital velocity around the Sun. In this framework, where the only time dependence is due to the Earth’s rotation, one identiﬁes v(t) ≡ v˜(t) and θ0(t) ≡ θ˜0(t) where v˜(t) and θ˜0(t) derive from the simple application of

11

spherical trigonometry [55]

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

(31)

v˜x(t) V

≡

sin z(t) cos θ˜0(t)

=

sin γ

cos φ

−

cos γ

sin φ cos(τ

−

α)

(32)

v˜y (t) V

≡

sin z(t) sin θ˜0(t)

=

cos γ

sin(τ

−

α)

(33)

v˜(t) ≡ v˜x2(t) + v˜y2(t) = V sin z(t),

(34)

Here z = z(t) is the zenithal distance of V, φ is the latitude of the observatory, τ = ωsidt

is

the

sidereal

time

of

the

observation

in

degrees

(ωsid

∼

2π 23h 56′

)

and

the

angle

θ0

is

counted

conventionally from North through East so that North is θ0 = 0 and East is θ0 = 90o.

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

∆λ(θ) λ

∼

2L(N − λ

1)

v2(t) c2

cos 2(θ

−

θ0(t))

≡

2C(t) cos 2θ

+

2S(t) sin 2θ

(35)

where

C (t)

=

L(N − λ

1)

v2(t) c2

cos 2θ0(t)

S(t)

=

L(N − λ

1)

v2(t) c2

sin

2θ0(t)

(36)

Then Eqs. (31)−(34) amount to the structure

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

(37)

C(t) ≡ C˜(t) = C0 + Cs1 sin τ + Cc1 cos τ + Cs2 sin(2τ ) + Cc2 cos(2τ )

(38)

with

Fourier

coeﬃcients

(R

≡

L(N −1) λ

V2 c2

)

C0

=

−

1 4

R(3

cos

2γ

−

1)

cos2

φ

(39)

Cs1

=

−

1 2

R

sin

α

sin

2γ

sin

2φ

Cs2

=

1 2

R

sin

2α

cos2

γ(1

+

sin2

φ)

Cc1

=

−

1 2

R

cos

α

sin

2γ

sin

2φ

(40)

Cc2

=

1 2

R

cos

2α

cos2

γ(1

+

sin2

φ)

(41)

and

Ss1

=

−

Cc1 sin φ

Sc1

=

Cs1 sin φ

(42)

Ss2

=

−

1

2 sin φ + sin2

φ

Cc2

Sc2

=

1

2 sin φ + sin2

φ

Cs2

(43)

These standard forms are nowadays adopted in the analysis of the data of the ether-drift

experiments [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 might

12

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 in the physical vacuum as the motion of a body in a ﬂuid, the scheme Eqs.(37),(38) of smooth sinusoidal 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 vacuum might 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 aspects of 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 form Eq.(35), one could ﬁrst re-write Eqs.(36) as

C(t) =

L(N − 1) λ

vx2(t) − vy2(t) c2

S(t) =

L(N − 1) λ

2vx (t)vy (t) c2

(44)

where vx(t) = v(t) cos θ0(t) and vy(t) = v(t) sin θ0(t). Then, by exploiting the turbulence scenario, one could model the two velocity components vx(t) and vy(t) as stochastic ﬂuctuations. In this diﬀerent scheme, where now v(t) = v˜(t) and θ0(t) = θ˜0(t), experimental results which, 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 time average to zero this does not necessarily mean that there is no ether-drift. This particular aspect 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 the modern experiments and our conclusions.

3. The original Michelson-Morley experiment
The Michelson-Morley experiment [61] is probably the most celebrated experiment in the history 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 into account the reﬂection from a moving mirror and other eﬀects, the predicted shifts would be largely reduced and become unobservable. These points of view are summarized in Hedrick’s contribution to the ‘Conference on the Michelson-Morley experiment’ [63] (Pasadena, February 1927) which was attended by the greatest experts of the time, in particular Lorentz and
13

Michelson. The arguments presented by Hedrick were, however, refuted by Kennedy [64] in a paper of 1935 where, by using Huygens principle, he re-obtained to order v2/c2 the classical result of Eq.(29) (with the identiﬁcation vobs = v).

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 A2 is predicted diﬀerently by using the classical formulas or Lorentz transformations (29)

Ac2lass

=

L v2 λ c2

Ar2el

=

L vobs2 λ c2

∼

2(N

− 1)Ac2lass

(45)

Notice also that upon rotation of π/2 with respect to θ = θ0 the predicted fringe shift is 2A2. Now, for the Michelson-Morley interferometer the whole eﬀective optical path was about L = 11 meters, or about 2 · 107 in units of light wavelengths, so for a velocity v ∼ 30 km/s (the Earth’s orbital velocity about the Sun, and consequently the minimum anticipated drift

14

velocity) the expected classical 2nd-harmonic amplitude was Ac2lass ∼ 0.2. This value can thus be used as a reference point to obtain an observable velocity, in the plane of the interferometer, from the actual measured value of A2 through the relation

vobs ∼ 30

A2 0.2

km/s

(46)

Michelson and Morley performed their six observations in 1887, on July 8th, 9th, 11th and 12th, 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 36 minutes. As well summarized by Miller in 1933 [65], “The brief series of observations was suﬃcient to show clearly that the eﬀect did not have the anticipated magnitude. However, and this fact must be emphasized, the indicated eﬀect was not zero”.
The same conclusion had already been obtained by Hicks in 1902 [66]: ”..the data published by Michelson and Morley, instead of giving a null result, show distinct evidence for an eﬀect of the kind to be expected”. Namely, there was a second-harmonic eﬀect. But its amplitude was substantially smaller than the classical expectation (see Fig.1).
Quantitatively, the situation can be summarized in Figure 2, taken from Miller [65], where the values of the eﬀective velocity measured in various ether-drift experiments are reported and compared with a smooth curve ﬁtted by Miller to his own results as function of the sidereal time.
For the Michelson-Morley experiment, the average observable velocity reported by Miller is about 8.4 km/s. Comparing with the classical prediction for a velocity of 30 km/s, this means an experimental 2nd- harmonic amplitude

AE2 XP

∼

0.2

(

8.4 30

)2

∼

0.016

(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 extracted
from the Michelson-Morley data. To understand the precision of their readings, we can look at the original paper [61] where one ﬁnds the following statement : ”The readings are divisions of the screw-heads. The width of the fringes varied from 40 to 60 divisions, the mean value being near 50, so that one division means 0.02 wavelength”. Now, in their tables Michelson and 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 ±0.002 wavelengths. In fact, 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 discussing the classically expected fractional fringe shift upon rotation of the apparatus by 90o, about

15

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 displacement

would still be observable” (i.e. 0.0037). Therefore, to be consistent with both the original

Michelson-Morley article and Born’s quotation of Michelson’s thought, we shall adopt ±0.004

as an estimate of the error 4.

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 as

described in Miller’s paper [65]. Namely, by starting from each set of seventeen entries (one

every 22.5o), say E(i), one has ﬁrst to correct the data for the observed linear thermal drift.

This is responsible for the diﬀerence E(1) − E(17) between the 1st entry and the 17th entry

obtained after a complete rotation of the apparatus. In this way, by adding 15/16 of the

correction to the 16th entry, 14/16 to the 15th entry and so on, one obtains a set of 16

corrected entries

Ecorr(i)

=

i

−1 16

(E(1)

−

E(17))

+

E(i)

(48)

The fringe shifts are then deﬁned by the diﬀerences between each of the corrected entries
4To conﬁrm that such estimate should not be considered unrealistically small, we report explicitly Michelson’s words from ref.[63]:“I must say that every beginner thinks himself lucky if he is able to observe a shift of 1/20 of a fringe. It should be mentioned however that with some practice shifts of 1/100 of a fringe can be measured, and that in very favorable cases even a shift of 1/1000 of a fringe may be observed.”

16

Table

1:

The

fringe

shifts

∆λ(i) λ

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

+0.036

2 +0.024

−0.004

−0.034

+0.008

−0.015

+0.044

3 +0.053

−0.004

−0.038

−0.010

+0.006

+0.047

4 +0.015

−0.003

−0.066

+0.070

+0.004

+0.027

5 −0.036

−0.031

−0.042

+0.041

+0.027

−0.002

6 −0.007

−0.020

−0.014

+0.055

+0.015

−0.012

7 +0.024

−0.025

+0.000

+0.057

−0.022

+0.007

8 +0.026

−0.021

+0.028

+0.029

−0.036

−0.011

9 −0.021

−0.049

+0.002

−0.005

−0.033

−0.028

10 −0.022

−0.032

−0.010

+0.023

+0.001

−0.064

11 −0.031

+0.001

−0.004

+0.005

−0.008

−0.091

12 −0.005

+0.012

+0.012

−0.030

−0.014

−0.057

13 −0.024

+0.041

+0.048

−0.034

−0.007

−0.038

14 −0.017

+0.042

+0.054

−0.052

+0.015

+0.040

15 −0.002

+0.070

+0.038

−0.084

+0.026

+0.059

16 +0.022

−0.005

+0.006

−0.062

+0.024

+0.043

Ecorr(i) and their average value Ecorr as

∆λ(i) λ

=

Ecorr(i)

−

Ecorr

(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

≤

θ

≤

2π,

with

θ

=

i−1 16

2π,

so

that

they

can

be

reproduced

in

a

Fourier expansion. Notice that in the evening observations the apparatus was rotated in the

opposite direction to that of noon.

One can thus extract the amplitude and the phase of the 2nd-harmonic component by

ﬁtting the even combination of fringe shifts

B(θ)

=

∆λ(θ)

+ ∆λ(π 2λ

+

θ)

(50)

(see Fig.3). This is essential to cancel the 1st-harmonic contribution originally pointed out

by Hicks [66]. Its theoretical interpretation is in terms of the arrangements of the mirrors

17

Table 2: The amplitude of the ﬁtted second-harmonic component AE2 XP for the six experimental sessions of the Michelson-Morley experiment.

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

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

and, 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 his observations were well consistent with Hicks’ theoretical study. The observed 1st-harmonic eﬀect 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’s data. 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 AE2 XP ∼ 0.016 ± 0.006. This value is consistent with an observable velocity

vobs ∼ 8.4+−11..57 km/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 medium

where light propagation takes place (in our case air where N ∼ 1.00029), we can deduce the

average value

v ∼ 349+−7602 km/s

(52)

While the individual values of A2 show a reasonable consistency, there are substantial changes in the apparent direction θ0 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 sessions or all evening sessions [69]. For instance, for the noon sessions, by taking into account that the azimuth is always deﬁned up to ±180o, one choice for the experimental azimuths is 357o ± 14o, 285o ± 10o and 317o ± 8o respectively for July 8th, 9th and 11th. For this assignment, the individual velocity vectors vobs(cos θ0, − sin θ0) and their mean are shown in Fig.4. According

18

July 11 noon
0.04

0.02

Β(θ)

0

-0.02

-0.04

0

π /2

π

θ

Figure 3: A ﬁt to the even combination B(θ) Eq.(50). The second harmonic amplitude is AE2 XP = 0.025 ± 0.005 and the fourth harmonic is AE4 XP = 0.004 ± 0.005. The ﬁgure is taken from ref.[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 any non-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 there are substantial deviations from Eq.(24) and/or from the smooth trend in Eqs.(31)−(34). For instance, 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 AE4 XP = 0.019 ± 0.005 and AE4 XP = 0.008 ± 0.005 for the noon sessions of July 8 and July 9 respectively). In any case, the observed strong variations of θ0 are in qualitative agreement with the analogous values reported by Miller. To this end, compare with Fig.22 of ref.[65] and in particular with the large scatter of the data taken around August 1st, as this represents the epoch of the year which is closer to the period of July when the Michelson-Morley observations were actually performed. Thus one could also conclude that individual experimental sessions indicate a deﬁnite non-zero ether-drift but the azimuth does not exhibit the smooth trend expected from the conventional picture Eqs.(31)−(34).
For completeness, we add that the large spread of the θ0−values might also reﬂect a particular 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 secure

19

9

8

7

6

V obs y

5

(Km/s) 4

3

2

1

0 0

July 9

1

2

Mean

3

4

5

V obs x (Km/s)

July 11

July 8

6

7

8

Figure 4: The observable velocities for the three noon sessions and their mean. The x-axis corresponds to θ0 = 0o ≡ 360o and the y-axis to θ0 = 270o. Statistical uncertainties of the various determinations are ignored. All individual directions could also be reversed by 180o.

vertical fringes of suitable width. There are two adjustments of the angle of this mirror which will give fringes of the same width but which produce opposite displacements of the fringes for 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 consecutive days. It follows that averaging the results of diﬀerent days in the usual manner is not allowable unless the types are all the same. If this is not attended to, the average displacement may be expected to come out zero − at least if a large number are averaged” [66]. Therefore averaging the fringe shifts from various sessions represents a delicate issue and can introduce uncontrolled errors. Clearly, this relative sign does not aﬀect the values of A2 and this is why averaging the 2nd-harmonic amplitudes is a safer procedure. However, it can introduce spurious changes in the apparent direction θ0 of the ether-drift. In fact, an overall change of sign of the fringe shifts at all θ−values is equivalent to replacing θ0 → θ0 ± π/2. As a matter of fact, Hicks concluded that the fringes of July 8th were of diﬀerent type from those of the remaining days. Thus for his averages (in our Fig.1) “the values of the ordinates are one-third of July 9 + July 11 − July 8 and one-third of July 9 + July 12 − July 8” [66] for noon and evening sessions respectively. If this were true, one choice for the azimuth of July 8th could now be θ0EXP = 267o ± 14o. This would orient the arrow of July 8th in Fig.4 in the direction

20

of the y−axis and change the average azimuth from θ0EXP ∼ 317o to θ0EXP ∼ 290o. We’ll return to this particular aspect in our Appendix II.

Let us ﬁnally compare with the interpretation that Michelson and Morley gave of their

data. 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 this

way, since the displacement is proportional to the square of the velocity, ”...the relative velocity

of 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 vobs < 7.5 km/s. However, this estimate is likely aﬀected by a theoretical uncertainty. In fact, in their Fig.6, Michelson and Morley

reported their measured fringe shifts together with the plot of a theoretical second-harmonic

component. In doing so, they plotted a wave of amplitude A2 = 0.05, that they interpret

as one-eight of the theoretical displacement expected on the base of classical physics, thus

implicitly assuming Ac2lass=0.4. As discussed above, the amplitude of the classically expected

second-harmonic component is not 0.4 but is just one-half of that, i.e. 0.2. Therefore, their

experimental upper bound Ae2xp <

0.4 20

=0.02 is actually equivalent to vobs < 9.5 km/s.

If

we now consider that their estimates were obtained after superimposing the fringe shifts

obtained from various sessions (where the overall eﬀect is reduced, see our Fig.1), we deduce

a substantial agreement with our result Eq. (51).

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

21

in the heavens...There are two hours in each day when the motion is in the desired plane

of the interferometer” [70]). The observations at the two times (about 11:30 a.m. and 9:00

p.m.) were, therefore, combined in such a way that the presumed azimuth for the morning

observations coincided with that for the evening (“The direction of the motion with reference

to a ﬁxed line on the ﬂoor of the room being computed for the two hours, we were able to

superimpose 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 having

nearly opposite phases. When these were combined, the result was nearly zero. For this

reason, 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 understood

and corrected by Miller who found that the two sets of data were each indicating an eﬀective

velocity of about 7.5 km/s (see Figure 11 of Miller’s paper [65]). For this reason, the correct

average observable velocities for the entire period 1902-1905 are those shown in our Figure 2

between 7 and 10 km/s or

vobs ∼ (8.5 ± 1.5) km/s

(53)

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

v ∼ (353 ± 62) km/s

(54)

5. Kennedy-Illingworth
An interesting development was proposed by Kennedy in 1926. As summarized in his contribution to the previously mentioned Conference on the Michelson-Morley experiment [63], his small optical system was enclosed in an eﬀectively insulated, sealed metal case containing helium at atmospheric pressure. Because of its small size, ”...circulation and variation in density 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 in density of the gas correspond to those in air to only 1/10 of the atmospheric pressure”. The essential ingredient of Kennedy’s apparatus consisted in the introduction of a small step, 1/20 of wavelength thick, in one of the total reﬂecting mirrors of the interferometer allowing, in principle, for an ultimate fringe shift accuracy 1 · 10−4. To take full advantage of this possibility, 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 an actual fringe shift accuracy of 2 · 10−3. In these conditions, as Kennedy explicitly says[63], ”...the velocity of 10 km/s found by Prof. Miller would produce a fringe shift corresponding

22

Table 3: The infra-session averages DA and DB 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. DA
+0.00024 +0.00114 +0.00000 +0.00020 +0.00064 −0.00002

5 A.M. DB
−0.00066 +0.00024 +0.00000 −0.00044 +0.00000 −0.00010

11 A.M. DA
+0.00070 −0.00042 −0.00006 −0.00030 −0.00022 +0.00048 −0.00014 −0.00006 −0.00006 +0.00000

11 A.M. DB
−0.00022 −0.00036 −0.00052 +0.00012 +0.00038 +0.00020 −0.00006 +0.00004 +0.00016 +0.00024

5 P.M. DA
+0.00024 −0.00056 −0.00144 −0.00016 +0.00018 +0.00030 +0.00030 +0.00036 +0.00006 −0.00010

5 P.M. DB
+0.00044 −0.00046 −0.00080 +0.00004 +0.00016 +0.00030 +0.00014 −0.00036 −0.00006 +0.00010

11 P.M. DA
−0.00010 +0.00018 −0.00126 −0.00044 +0.00000 −0.00040

11 P.M. DB
+0.00024 +0.00018 −0.00006 −0.00026 +0.00024 −0.00004

to 8 · 10−3”, four times larger than the experimental resolution. Since the eﬀect is quadratic in the velocity, Kennedy’s result, fringe shifts < 2 · 10−3, can then be summarized as

vobs < 5 km/s

(55)

By using Eq.(30), for helium at atmospheric pressure where N ∼ 1.000035, this bound amounts to restrict the kinematical value by v < 600 km/s.
Kennedy’s apparatus was further reﬁned by Illingworth in 1927 [71]. Besides improving the quality of the mirrors and of the source, Illingworth’s data taking was also designed to reduce the presence of steady thermal drift and of odd harmonics. Looking at Illingworth’s paper, one ﬁnds that his reﬁnements reached indeed the nominal O(10−4) accuracy mentioned by Kennedy, namely about 1/1500 of wavelength for the individual readings and (1 ÷ 2) · 10−4 at the level of average values.
Let us now analyze Illingworth’s results. He performed four series of observations in the ﬁrst ten days of July 1927. These consisted of 32 experimental sessions, conducted daily at 5 A.M. (6), 11 A.M. (10), 5 P.M. (10) and 11 P.M.(6), in which he was measuring the fringe displacement caused by a rotation through a right angle of the apparatus. To take into

23

account 90o rotations let us ﬁrst re-write Eq.(29) as

∆λ(θ) λ

=

A2

cos

2(θ

−

θ0)

(56)

Therefore Illingworth, in his ﬁrst set (set A) of 10 rotations, North, East, South, West and back to North, was actually measuring DA ≡ 2A2 cos 2θ0. In a second set (set B), North-East, North-West, South-West, South-East and back to North-East, performed immediately after the set A, he was then measuring DB ≡ 2A2 sin 2θ0. Notice that both DA and DB diﬀer from the positive-deﬁnite quantity D ≡ √2A2 that should be inserted in Illingworth’s numerical relation for his apparatus vobs = 112 D. Therefore, the reported values for the two velocities vA = 112 |DA| and vB = 112 |DB| should only be taken as lower bounds for the true vobs. The mean values DA and DB obtained from the 10 sets of rotations in the 32 individual sessions 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 DA 2 + DB 2 has a mean value of about 0.00045, which corresponds to vobs ∼ 2.4 km/s. Thus, by using Eq.(30) for helium at atmospheric pressure, we would tentatively deduce an average value v ∼ 284 km/s.
However, this is only a very partial view. To go deeper into Illingworth’s experiment we have to consider his basic measurements, i.e. the individual turns of his interferometer. In this case, the only known basic set of data reported by Illingworth is set A of July 9th, 11 A.M. This set has been re-analyzed by Mu´nera [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 in the range 2-5 km/s. However, their sign seems to change randomly. Therefore, if one attempts to extract the observable velocity from the mean of the 10 determinations, DA ∼ −0.00006, the resulting value 0.9 km/s is much smaller than all individual determinations. The basis of Mu´nera’s analysis was instead to estimate vobs from |DA| , from which he obtained an average velocity vobs = 3.13 ± 1.04 km/s.
Now, the standard interpretation of such apparently random changes of sign is in terms of typical instrumental eﬀects and the standard method for eliminating these is the original averaging procedure as employed by Illingworth. But we will now show that they could also indicate an unconventional form of stochastic drift, of the type already mentioned in the previous sections, and in which Mu´nera’s re-estimate has a deﬁnite signiﬁcance. To this end, we shall ﬁrst use the relations

DA(t) = 4C(t)

DB(t) = 4S(t)

(57)

24

Table 4: Illingworth’s set A of July 9th, 11 A.M. as re-analyzed by Mu´nera [72].

Rotation 1 2 3 4 5 6 7 8 9 10

DA −0.00100 +0.00066 −0.00066 −0.00066 −0.00166 +0.00234 +0.00100 +0.00034 +0.00000 −0.00100

|DA| +0.00100 +0.00066 +0.00066 +0.00066 +0.00166 +0.00234 +0.00100 +0.00034 +0.00000 +0.00100

vA[km/s] 3.54 2.89 2.89 2.89 4.57 5.41 3.54 2.04 0.00 3.54

where the two functions C(t) and S(t) have been introduced in Eqs.(36) and (44). Thus Eqs.(57) can be re-written as

DA(t) =

8L(N − 1) λ

vx2(t) − vy2(t) 2c2

DB(t) =

8L(N − 1) λ

vx(t) vy(t) c2

(58)

where vx(t) = v(t) cos θ0(t) and vy(t) = v(t) sin θ0(t). In this way, by using the numerical

relation for Illingworth’s experiment

L λ

(30km/s)2 c2

∼ 0.035 and the value of the helium refractive

index, we obtain

DA(t)

∼

vx2(t) − vy2(t) 2 · (300 km/s)2

·

10−3

DB (t)

∼

vx(t) vy(t) (300 km/s)2

·

10−3

(59)

The required random ingredient can then be introduced by characterizing the two velocity components vx(t) and vy(t) 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, appears statistically isotropic and homogeneous 5. This represents a zeroth-order approximation which is motivated by the substantial reading error of the Illingworth measurements (it turns out to be comparable to the eﬀects of turbulence). However, it is a useful example to illustrate basic phenomenological features associated with an underlying stochastic vacuum. To explore the resulting 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 unsteady random Fourier series. The perspective is that of an observer moving in the turbulent ﬂuid

5This picture reﬂects the basic Kolmogorov theory [73] of a ﬂuid with vanishingly small viscosity.

25

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

∞

vx(t) = [xn(1) cos ωnt + xn(2) sin ωnt]

(60)

n=1

∞

vy(t) = [yn(1) cos ωnt + yn(2) sin ωnt]

(61)

n=1

where ωn = 2nπ/T , T being a time scale which represents a common period of all stochastic components. We have adopted the typical value T = Tday= 24 hours. However, we have

also checked with a few runs that the statistical distributions of the various quantities do not

change substantially by varying T in the rather wide range 0.1 Tday ≤ T ≤ 10 Tday. The coeﬃcients xn(i = 1, 2) and yn(i = 1, 2) are random variables with zero mean. They
have the physical dimension of a velocity and we shall denote by [−v˜, v˜] the common interval

for these four parameters. In terms of v˜ the statistical average of the quadratic values can be

expressed as

x2n(i = 1, 2) stat =

yn2 (i

=

1, 2)

stat

=

v˜2 3 n2η

(62)

for the uniform probability model (within the interval [−v˜, v˜]) which we have chosen for

our simulations. Finally, the exponent η controls the power spectrum of the ﬂuctuating

components. For the simulations, between the two values η = 5/6 and η = 1 reported in

ref.[75], we have chosen η = 1 which corresponds to the point of view of an observer moving

in the ﬂuid.

Thus, within this simple model for DA(t) and DB(t), v˜ is the only parameter whose numerical value could reﬂect the properties of a large-scale motion, for instance of the Earth’s

motion with respect to the Cosmic Microwave Background (CMB). For this reason, here, we

have adopted the ﬁxed value v˜ = VCMB = 370 km/s. With these premises, our results can be

illustrated by ﬁrst considering the basic set of 10 complete rotations of the apparatus during

which Illingworth’s fringe shifts (produced by 90o rotations) were recorded every 30 seconds.

Therefore, this type of simulations consists in generating 40 values during a total time of 1200 seconds. As an illustration, two typical sequences of DA(t) and DB(t), in units 10−3, are

shown in Fig.5. As one can see, the magnitude O(10−3) and the random nature of the instantaneous values

is completely consistent with the entries of Table 4. Also the resulting infra-session averages

DA = 0.00028 and DB = 0.00011 are completely consistent with the typical entries of

Table 3.

26

3 2 1
DA(t) 0
-1 -2 -3 4
2
DB(t) 0
-2
-4

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

time (s)

Figure 5: A simulation of DA(t) and DB(t), in units 10−3 and every 30 seconds, from typical sequences of 1200 seconds. The average values are DA = 0.00028 and DB = 0.00011. The velocity parameter is v˜ = VCMB = 370 km/s.

To obtain further insight, we have then performed extensive simulations for large sequences of measurements. The histograms of a set of 10000 determinations of DA(t) and DB(t) (again generated 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 Gaussian shape. This kind of behavior is characteristic of probability distributions for instantaneous data in turbulent ﬂows (see e.g. [76, 77]). To better appreciate the deviation from Gaussian behavior, in panels (c) and (d) we plot the same data in a log−log scale. The resulting distributions are well ﬁtted by the so-called q−exponential function [78]

fq(x) = a(1 − (1 − q)xb)1/(1−q)

(63)

with entropic index q ∼ 1.1. For such large samples of data, the statistical averages DA and DB are vanishingly small in units of the typical instantaneous values O(10−3) and any nonzero average has to be considered as statistical ﬂuctuation. On the other hand, the standard deviations σ(DA) and σ(DB) have deﬁnite non-zero values which reﬂect the magnitude of the scale parameter v˜. By keeping v˜ ﬁxed at 370 km/s, we have found

σ(DA) ∼ (0.74 ± 0.05) · 10−3

σ(DB) ∼ (0.83 ± 0.06) · 10−3

(64)

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

27

1,50
W 1,00

0,50

0,00

-4

-2

1,50

W 1,00

0,50

0,00

-4

-2

(a)

100

W

10-1

10-2

0

2

DA [10 -3]

4

10-1

(b)
100 W
10-1

10-2

0

2

4

DB [10 -3]

10-1

(c)
q-Exponential Fit 100
|DA| [10-3 ]
(d)
q-Exponential Fit 100
|DB| [10-3 ]

Figure 6: We show, see (a) and (b), the histograms W obtained from a simulation for
DA = DA(t) and DB = DB(t). The vertical normalization is to a unit area. The mean values are DA = 0.75 · 10−5, DB = −1.1 · 10−5 and the standard deviations σ(DA) = 0.75 · 10−3, σ(DB) = 0.83 · 10−3. 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 DA and q=1.12, a=2 and b=2.3 for DB. The total statistics correspond to 10.000 values generated at steps of 30 seconds. The velocity parameter is v˜ = VCMB = 370 km/s.

28

Table 5: Illingworth’s ﬁnal inter-session averages.

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

DA +0.00036 ± 0.00012 −0.00001 ± 0.00007 −0.00008 ± 0.00012 −0.00034 ± 0.00014

DB −0.00016 ± 0.00009 −0.00000 ± 0.00006 −0.00005 ± 0.00008 +0.00005 ± 0.00006

into account gives a mean spread slightly less, about 0.65 · 10−3, for the eﬀect of stochastic drift in Illingworth’s measurements. This is comparable to the uncertainty of the individual readings which, in the best case, was of 1/1500 wavelengths, i.e. ±0.7 · 10−3. By combining in quadrature the two uncertainties, one gets a good agreement with our Table 4 where the variance of the mean is about ±1·10−3. Finally, the simulation is also useful to get indications on the expected value of the observable velocity. In fact, with vanishingly small values of DA and DB one gets DA2 ∼ σ2(DA) and DB2 ∼ σ2(DB). Therefore one obtains the following two average estimates of vobs

vobs ∼ 112 σ(DA) ∼ 3.05 km/s

vobs ∼ 112 σ(DB) ∼ 3.23 km/s (65)

with a mean value of 3.14 km/s which is very close to Mu´nera’s determination vobs = 3.13±1.04 km/s.
We emphasize that one could further improve the stochastic model by introducing time modulations and/or slight deviations from isotropy. For instance, v˜ could become a function of time v˜ = v˜(t). By still retaining statistical isotropy, this could be used to simulate the possible modulations of the projection of the Earth’s velocity in the plane of the interferometer. Or, one could ﬁx a range, say [−v˜x, v˜x], for the two random parameters xn(1) and xn(2), which is diﬀerent from the range [−v˜y, v˜y] for the other two parameters yn(1) and yn(2). Finally, v˜x and v˜y could also become given functions of time, for instance v˜x(t) ≡ v˜(t) cos θ˜0(t) v˜y(t) ≡ v˜(t) sin θ˜0(t), v˜(t) and θ˜0(t) being deﬁned in Eqs. (31)−(34). We shall discuss this other 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 further reduction of the data by performing inter-session averages ( ... ) among the various sessions, can wash out completely the physical information contained in the original observations. In Table 5, we report the ﬁnal inter-session averages DA and DB obtained by Illingworth for the various observation times.

29

Nevertheless, in spite of the strong cancelations expected from the averaging reduction process mentioned above, some non-zero value is still surviving. Therefore, regardless of our simulations, one could draw the following conclusions. Traditionally, from these ﬁnal averages for DA at 5 A.M. and at 11 P.M. one has been deducing the values vA ∼ 2.12 km/s and vA ∼ 2.07 km/s respectively. Therefore, from these two estimates of vA that, as anticipated, represent lower bounds for vobs, it follows that there were values of vobs which clearly had to be larger than both. For this reason, this 2.1 km/s velocity value reported by Illingworth, rather than being interpreted as an upper bound could also be interpreted as a lower bound placed by his experiment. In this way, by combining with the previous Kennedy’s upper bound vobs < 5 km/s, one would deduce that these two experiments, where light was propagating in helium at atmospheric pressure, give a range for the observable velocity

(Kennedy + Illingworth)

2 km/s vobs < 5 km/s

(66)

in complete agreement with Mu´nera’s determination

vobs = 3.1 ± 1.0 km/s

(67)

From this last estimate, by using Eq.(30) and taking into account that for helium at atmospheric pressure the refractive index is N ∼ 1.000035, one obtains a kinematical velocity

v ∼ (370 ± 120) km/s

(68)

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

6. Miller
Mu´nera’s analysis [72] is also interesting because he applied the same method used for Illingworth’s observations to the only known Miller set of data explicitly reported in the literature. In this case, his value vobs = 8.2 ± 1.4 km/s, after correcting with Eq.(30), conﬁrms the estimate v ∼ 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 the critical re-analysis of Shankland et al. [68]. Diﬀerently from the original Michelson-Morley experiment, 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 Shankland team, there is now an independent estimate of the average determinations AE2 XP for the four epochs. 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 consistent that one can easily average them. The overall determination from Table III of [68]

AE2 XP = 0.044 ± 0.022

(69)

when compared with the equivalent classical prediction for Miller’s interferometer Ac2lass =

L (30km/s)2

λ

c2

∼

0.56

corresponds

to

an

average

observable

velocity

vobs = 8.4+−21..95 km/s

(70)

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

v = 349+−17094 km/s

(71)

We are aware that our conclusion goes against the widely spread belief, originating precisely from the paper of Shankland et al. ref.[68], that Miller’s results might actually have been due to statistical ﬂuctuation and/or local temperature conditions. To a closer look, however, the arguments of Shankland et al. are not so solid as they appear when reading the Abstract of their paper 6. In fact, within the paper these authors say that “...there can be little doubt that statistical ﬂuctuations alone cannot account for the periodic fringe shifts observed by Miller” (see page 171 of ref.[68]). Further, although “...there is obviously considerable scatter in the data at each azimuth position,...the average values...show a marked second harmonic eﬀect” (see page 171 of ref.[68]). In any case, interpreting the observed eﬀects on the basis of the local temperature conditions is certainly not the only explanation since “...we must admit that a direct and general quantitative correlation between amplitude and phase of the observed second harmonic on the one hand and the thermal conditions in the observation hut on 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 AE2 XP ∼ 0.044, obtained after their own re-analysis of his observations at Mt.Wilson, when compared with the reference classical value Ac2lass = 0.56 for his apparatus, was giving the same observable velocity vobs ∼ 8.4 km/s obtained from Miller’s re-analysis of the Michelson-Morley experiment in Cleveland. Conceivably, their emphasis on the role of temperature eﬀects would have been re-considered had they realized the perfect identity of two determinations obtained in completely diﬀerent experimental conditions. In this sense, an interpretation in terms of a temperature 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.
6A detailed rebuttal of the criticism raised by the Shankland team can be found in ref.[79].

31

Table 6:

The

symmetric

combination

of

fringe

shifts

B(θ) =

∆λ(θ)+∆λ(π+θ) 2λ

at

the

various

values of θ 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 ±0.050.

Turn 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0o +0.091 −0.025 +0.022 +0.034 +0.169 −0.025 +0.081 +0.066 +0.041 +0.016 +0.009 +0.022 +0.000 −0.034 +0.113 +0.025 +0.000 +0.044 +0.053 +0.059

22.5o +0.159 +0.063 +0.103 −0.009 +0.081 +0.025 +0.094 +0.072 +0.084 +0.072 +0.053 +0.016 +0.063 +0.047 +0.125 +0.050 −0.012 +0.050 +0.059 +0.041

45o +0.028 +0.050 +0.084 −0.053 +0.044 +0.025 +0.056 −0.022 +0.078 +0.078 +0.097 +0.059 +0.025 +0.078 +0.138 +0.025 −0.025 +0.019 +0.016 +0.122

67.5o +0.047 +0.088 +0.016 −0.047 −0.044 +0.025 +0.069 −0.066 +0.022 −0.016 −0.009 +0.003 +0.038 +0.009 +0.000 +0.050 +0.063 −0.019 −0.028 +0.003

90o −0.034 −0.075 −0.053 −0.041 −0.081 +0.025 −0.119 −0.059 −0.134 −0.009 −0.116 −0.053 +0.050 −0.009 −0.088 −0.025 +0.000 −0.056 −0.022 −0.066

112.5o −0.116 −0.038 −0.072 +0.016 −0.169 −0.025 −0.106 −0.003 −0.141 −0.003 −0.072 −0.009 −0.038 −0.028 −0.125 −0.050 −0.012 −0.044 −0.066 −0.084

135o −0.147 +0.000 −0.091 +0.022 −0.056 −0.025 −0.094 +0.003 +0.003 −0.047 +0.022 −0.016 −0.075 −0.047 −0.113 −0.025 −0.025 −0.031 −0.009 −0.053

157.5o −0.028 −0.063 −0.009 +0.078 +0.056 −0.025 +0.019 +0.009 +0.047 −0.091 +0.016 −0.022 −0.063 −0.016 −0.050 −0.050 +0.013 +0.031 −0.003 −0.022

32

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 validity of Miller’s observations. The two main objections concern i) the subtraction of the steady thermal drift, which was approximated by Miller as a pure linear eﬀect, and ii) the statistical signiﬁcance of the measurements. Concerning remark i), Roberts reports in his Fig.3 a broken line that reproduces the expected linear trend. He also reports some chosen points (diﬀering from the corners of the broken line by 180 degrees) that, due to the 2nd-harmonic nature of the ether-drift eﬀect, should lie on the line. However, this expectation ignores that, as already pointed out for the Michelson-Morley experiment, real measurements contains large ﬁrst-harmonic eﬀects. These only cancel when taking symmetric combinations of data at the various angles θ and π + θ. As a matter of fact, the autocorrelative methods and further tests applied by the Shankland team over all of Miller’s data conﬁrmed the linear drift approximation as remarkably good (see their footnote 21 on page 177 of [68]).
Concerning remark ii), according to Roberts, the experimental uncertainties are so large that the observed 2nd-harmonic eﬀect has no statistical signiﬁcance. To check this point we 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 procedure explained in Sect.3. The resulting symmetric combinations of fringe shifts

B(θ)

=

∆λ(θ)

+ ∆λ(π 2λ

+

θ)

(72)

are reported in our Table 6. We have then ﬁtted these data by including both 2nd and 4th harmonic terms. Notice
that, diﬀerently from Roberts’ analysis, we do not perform any averaging of data obtained from diﬀerent turns of the interferometer. For our global ﬁt, to estimate the accuracy of the various determinations, we have followed ref.[68] and adopted a nominal uncertainty ±0.050 for each entry of Table 6. From the ﬁt, where the 4th harmonic is completely consistent with the background (AE4 XP = 0.004 ± 0.012), we have obtained a chi-square of 130 for 157 degrees of freedom and the following values

AE2 XP = 0.061 ± 0.012

θ0EXP = 24o ± 7o

(73)

Here errors correspond to the overall boundary ∆χ2 = +3.67, as appropriate 7 for a 70% C. L. in a 3-parameter ﬁt [81]. Notice that, even though the ﬁtted A2 Eq.(73) is only 20% larger than the nominal accuracy ±0.050 of each entry, the data are distributed in such a way to produce a 5σ evidence for a non-zero 2nd harmonic.
7This probability content assumes a Gaussian distribution as for typical statistical errors.

33

As for Illingworth’s experiment, we have also analyzed the results obtained from the individual 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.
80

60

40
θ0 20

0

-20

0

5

10

15

20

Turn

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 ±20o. The band between the two horizontal lines corresponds to the global ﬁt θ0 = 24o ± 7o. Each individual value could also be reversed by 180 degrees.

To conclude our analysis of Miller’s experiment, we want to mention that other objections

to the overall consistency of his solution for the Earth’s cosmic motion [65] were raised by

von 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-proposed

by Shankland et al. [68] and amounts to the following.

By assuming relations (31)−(43) and denoting by ... the daily average of any given

quantity, one ﬁnds, at any angle θ, the daily averaged fringe shift

∆λ(θ) λ

= 2 C˜(t)

cos 2θ

(74)

since S˜(t) = 0 with

C˜(t)

=

−

L(N − λ

1)

V2 c2

1 4

(3

cos

2γ

−

1)

cos2

φ

(75)

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

∆λ(θ) = V 2F (γ, φ) cos 2θ

(76)

λ

Therefore, since the latitude φ is a constant and the angular declination γ is ﬁxed at any

speciﬁc epoch, the daily averaged fringe shifts should all have a common maximum at the

34

0,15

0,1
A2
0,05

0

0

5

10

15

20

Turn

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 ±0.030. The band between the two horizontal lines corresponds to the global ﬁt A2 = 0.061 ± 0.012. Within their errors, these individual values correspond to an observable velocity in the range 4÷14 km/s.
value θ = 0. Only the amplitude can be diﬀerent at diﬀerent epochs. Instead, in Miller’s observations the location of the maximum was diﬀerently displaced from the meridian (see Figs.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 the smooth modulations expected from the Earth’s rotation. As anticipated in Sect.2, and discussed in connections with the Michelson-Morley and Illingworth experiments, one might be faced with the more general scenario where the two velocity components vx(t) and vy(t) in Eq.(44) are not smooth periodic functions but exhibit stochastic behaviour. In this diﬀerent perspective, combining observations of diﬀerent days and diﬀerent epochs becomes more delicate and there might be non-trivial deviations from Eq.(76). We shall therefore conclude our analysis of Miller’s experiments by recalling the remarkable consistency of the velocity value v ∼ 350 km/s (obtained from the 2nd-harmonic amplitude AE2 XP ∼ 0.044 computed by the Shankland team) with those from the Michelson-Morley, Morley-Miller and KennedyIllingworth 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 was

increased to eighty-ﬁve feet, and the results showed that the precautions taken to eliminate

temperature and pressure disturbances were eﬀective. The results gave no displacement as

great 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], after

similar comments on the length of the apparatus and on the precautions taken to eliminate

the various disturbances, one ﬁnds this other statement “The results gave no displacement as

great as one-ﬁfteenth 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. These results are diﬀerences

between the displacements observed at maximum and at minimum at sidereal times, the

directions corresponding to Dr. Str¨omberg’s calculations of the supposed velocity of the solar

system”. 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 the

proper sidereal times”.

Clearly, although not explicitly stated, they were assuming that some unknown mechanism

was largely reducing the fringe shifts with respect to the naive non-relativistic value associated

with a kinematical velocity of 300 km/s. Thus one could try to conclude that their experiment

implies

fringe

shifts

|∆λ| λ

1 15

0.017 ∼ 0.001.

However

this

is

not

what

they

say

(they

speak

of diﬀerences between fringe displacements) and, in any case, this interpretation does not ﬁt

with the result reported by Shankland et al. [68] (see their Table I). According to these other

authors, the typical observed fringe shifts observed by Michelson, Pease and Pearson were of

the order of ±0.005.

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 ﬁrst

stage of the experiment, the fringe shifts had a typical magnitude of about ±0.030. Later on,

however, by reducing substantially the rotation speed of the apparatus, the observed eﬀects

became considerably smaller.

Pease declares that, in their experiment, to test Miller’s claims, they concentrated on a

36

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

ǫ(θ) =

∆λ(θ) λ

5.30 −

∆λ(θ) λ 17.30

(77)

between the mean fringe shifts

∆λ λ

5.30,

obtained

after

averaging

over

a

large

set

of

obser-

vations performed at sidereal time 5.30, and the mean fringe shifts

∆λ λ

17.30

obtained

after

averaging in the same period at sidereal time 17.30. The quantity ǫ(θ) has typical magnitude

of ±0.004 or smaller. However, as already anticipated in Sect.2, by averaging observations

performed at a given sidereal time one is assuming the smooth modulations of the signal

described by Eqs.(37),(38). Otherwise, one will introduce uncontrolled errors. For instance

if, 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 become

stronger and stronger by increasing the number of observations.

Therefore, from these values, nothing can be said about the magnitude of the fringe shifts

∆λ(θ) λ

obtained,

before

any

averaging

procedure

and

before

any

subtraction,

in

individual

measurements 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 shift

combinations Eq.(50) vary approximately in the range ±0.007. This is equivalent to fringe

shifts of about ±0.011 with a length of 85 feet and could hardly be taken as indicative

of the whole sample of measurements. In this situation, one can only adopt the estimate

A2 ∼ 0.010 ± ... 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 individual

sessions.

Then, for this conﬁguration, where

L λ

(30km/s)2 c2

∼ 0.45, this is equivalent to

vobs = (4.5 ± ...) km/s

(78)

or, by using Eq.(30), to

v = (185 ± ...) km/s

(79)

We emphasize that Miller’s extensive observations, as reported in Fig.22 of ref.[65] (see also our Fig.8), gave ﬂuctuations of the observable velocity lying, within the errors, in the range 4−14 km/s which has been smoothed in our Fig.2. For this reason, even though Miller’s reconstruction of the Earth’s cosmic motion is not internally consistent, a single observation which gives vobs ∼ 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 a period of several months, was chosen to represent an example of extremely small ether-drift eﬀect.

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 full sidereal day that were recorded by photocamera), this experiment cannot be compared with the other experiments (e.g. Michelson-Morley, Illingworth) where only observations at few selected 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 from Miller’s, the amplitudes of all basic Joos’ observations can be reconstructed from the published articles [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 always assumed that the fringe shifts were recorded in a partial vacuum. This is supported by several elements. For instance, when describing his device for electromagnetic ﬁne movements of the mirrors, 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 later on, in 1934, Miller and Joos had a public letter exchange [89] and Joos did not correct Miller’s statement. On the other hand, Swenson [90] explicitly reports that fringe shifts were ﬁnally recorded with optical paths placed in a helium bath. In spite of the fact that this important aspect is never mentioned in Joos’ papers, we shall follow Swenson and assume that during the 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 and ending at 1 P.M. of May 11th. Two measurements, the 1st and the 5th, were ﬁnally deleted by Joos with the motivation that there were spurious disturbances. The data were combined symmetrically, in order to eliminate the presence of odd harmonics, and the magnitude of the fringe 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 as an upper limit and deduced an observable velocity vobs 1.5 km/s. To derive this value, he used the fact that, for his apparatus, an observable velocity of 30 km/s would have produced a 2nd-harmonic amplitude of 0.375 wavelengths.
Still, since it is apparent from Fig.9 that some fringe displacements were deﬁnitely larger than 1/1000 of a wavelength, we have decided to extract the values of the 2nd-harmonic amplitude A2 from the 22 pictures. Diﬀerently from the values of the azimuth, this can be
38

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 0.4 · 10−3. This corresponds to an uncertainty ±0.2 · 10−3 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 the reference angular values in Fig.9 for the fringe shifts. Thus, one could choose for instance the set (k=1, 2, 3, 4) θk ≡(0o, 45o, 90o, 135o) or the diﬀerent set θk ≡(360o, 315o, 270o, 225o). Or, by noticing that in Fig.9 there is a small misalignment angle θ∗ ∼ 17o (which actually from [87] might instead be 22.5o) between the dots of Joos’ fringe shifts and the N, W, and S marks, one could also adopt other two set of values, namely θk ≡(0o + θ∗, 45o + θ∗, 90o + θ∗, 135o +θ∗) or θk ≡(360o −θ∗, 315o −θ∗, 270o −θ∗, 225o −θ∗). By ﬁtting the fringe shifts of Fig.9 to the 2nd-harmonic form Eq.(56), these four options for the reference angles θk would give exactly the same amplitude A2 but four diﬀerent choices for the azimuth, i.e. −θ0, −θ0 + θ∗, θ0 − θ∗ and θ0. This basic ambiguity should be added to the standard uncertainty in the azimuth that, due to the 2nd-harmonic nature of the measurements, could always be changed by adding ±180 degrees 8. Therefore, since clearly there is only one correct choice for the
8As an example, one can consider the azimuth for Joos’ picture 20. Depending on the choice of the reference angles θk, one ﬁnds θ0 ∼ 329o, 329o + θ∗, 31o − θ∗, 31o or θ0 ∼ 149o, 149o + θ∗, 211o − θ∗, 211o.
39

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

4

3
A2 2

1

0

0

5

10

15

20

25

Picture

Figure 10: Joos’ 2nd-harmonic amplitudes, in units 10−3. The vertical band between the two lines corresponds to the range (1.4 ± 0.8) · 10−3.

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

harmonic amplitude

Aj2oos = (1.4 ± 0.8) · 10−3

(80)

and a corresponding observable velocity

vobs ∼ 1.8+−00..56 km/s

(81)

By correcting with the helium refractive index, Eqs.(30) and (81) would then imply a true kinematical velocity v ∼ 217+−7696 km/s.
However, this is only a ﬁrst and very partial view of Joos’ experiment. In fact, we have

compared Joos’ amplitudes with theoretical models of cosmic motion. To this end, one has

ﬁrst to transform the civil times of Joos’ measurements into sidereal times. For the longitude

11.60 degrees of Jena, one ﬁnds that Joos’ observations correspond to a complete round in

sidereal time in which the value τ = 0o ≡ 360o is very close to Joos’ picture 20. Then, by

using Eqs.(31) and (34), one can use this input and compare with theoretical predictions for

the amplitude which, for the given latitude φ = 50.94 degrees of Jena, depend on the right

ascension α and the angular declination γ. To this end, it is convenient to ﬁrst re-write the

theoretical forms as

A2(t) cos 2θ0(t) = 2C(t) =

2L(N − 1) λ

vx2(t) − vy2(t) c2

∼ 2.6 · 10−3

vx2(t) − vy2(t) (300 km/s)2

(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 ±0.2 · 10−3 (the size of the dots in Fig.9). The mean amplitude over the 22 determinations is Aj2oos = 1.4 · 10−3.

Picture 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Aj2oos [10−3 ] 2.05 0.75 1.60 2.00 1.50 1.55 1.10 0.60 4.15 1.20 2.35 0.95 1.15 1.65 0.50 1.05 1.25 0.35 0.45 1.25 0.95 1.65

41

and

A2(t) sin 2θ0(t) = 2S(t) =

2L(N − 1) λ

2vx(t) vy(t) c2

∼ 2.6 · 10−3

2vx(t) vy(t) (300 km/s)2

(83)

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

L λ

(30km/s)2 c2

∼ 0.375 and the

value of the helium refractive index. Then, by approximating vx(t) ∼ v˜x(t), vy(t) ∼ v˜y(t) and

using Eq.(34) for the scalar combination v˜(t) ≡ v˜x2(t) + v˜y2(t), we have ﬁtted the amplitude

data of Table 7 to the smooth form

As2mooth(t) = const · sin2 z(t)

(84)

where cos z(t) is deﬁned in Eq. (31). The results of the ﬁt 9

α = 168o ± 30o

γ = −13o ± 14o

(85)

conﬁrm that, as found in connection with the Illingworth experiment, the Earth’s motion with respect to the CMB (which has α ∼ 168o and γ ∼ −6o) could serve as a useful model to describe the ether-drift data.
Still, in spite of the good agreement with the CMB α− and γ−values obtained from the ﬁt 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 all sidereal times can trivially be obtained from the mean squared velocity Eq.(34)

v˜2(t) = V 2

1

−

sin2

γ

sin2

φ

−

1 2

cos2

γ

cos2

φ

(86)

For the CMB and Jena, this gives v˜2 ∼ 330 km/s so that one would naively predict from

Eqs.(82), (83)

As2mooth (t)

∼ 2.6 · 10−3

v˜2(t) (300 km/s)2

∼

3.2 · 10−3

(87)

to be compared with Joos’ mean value Aj2oos = (1.4±0.8)·10−3. In the standard picture, this experimental value leads to the previous estimate v˜2 ∼ 217 km/s and not to v˜2 ∼ 330

km/s so that it is necessary to change the theoretical model to try to make Joos’ experiment

completely 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

9Actually, there is another degenerate minimum at α = 348o ± 30o and γ = 13o ± 14o because sin2 z(t) remains invariant under the simultaneous replacements α → α + 180o and γ → −γ. However, due to the close agreement with the CMB parameters we have concentrated on solution (85).

42

velocity ﬁeld. As anticipated however, due to the higher accuracy of the Joos experiment, we have modiﬁed the theoretical framework. Namely, we have allowed the two random parameters xn(1) and xn(2) to vary in the range [−v˜x(t), v˜x(t)] and the other two parameters yn(1) and yn(2) to vary in the diﬀerent range [−v˜y(t), v˜y(t)], where v˜x(t) and v˜y(t) are deﬁned in Eqs.(31)−(33). In this way, for each time t, Eqs.(62) now become

x2n(i

=

1, 2)

stat

=

v˜x2(t) 3 n2η

yn2 (i

=

1, 2)

stat

=

v˜y2(t) 3 n2η

(88)

It is understood that the latitude corresponds to Joos’ experiment while V , α and γ describe

the Earth’s motion with respect to the CMB. Notice that, in this model, there will be a

substantial reduction of the amplitude with respect to its smooth prediction. To estimate

the order of magnitude of the reduction, one can perform a full statistical average (as for an

inﬁnite number of measurements) and use Eqs.(88) in Eqs.(82), (83) for our case η = 1. This

gives

A2(t) stat ∼ 2.6 · 10−3

v˜2(t) (300 km/s)2

1 3

∞

1 n2

=

π2 18

As2mooth (t)

(89)

n=1

By also averaging over all sidereal times, for the CMB and Jena, one would now predict a

mean amplitude of about 1.7 · 10−3 and not of 3.2 · 10−3.

After having ﬁxed all theoretical inputs, we have analyzed the dependence of the numerical

results on the remaining parameters of the simulation, namely the number N of Fourier modes

(in the available range N 107) and the integer number s (the ‘seed’) which determines the

random sequence. In particular, the dependence on the latter is usually quoted as theoretical

uncertainty. For this reason, for Illingworth’s experiment in Sect.5 we had produced several

copies of the high-statistics simulation in Fig.6 by quoting values for the standard deviations

Eq.(64) which take into account the observed s−dependence of the results.

Here, we have started by doing something similar. However, since it is not possible to

consider at once all characteristics of a given conﬁguration, we have ﬁrst concentrated on

the simplest statistical indicator, namely the mean amplitude As2imul obtained by averaging over all sidereal times. Quite in general, this can be evaluated for a variety of conﬁgurations

which depend on the number n of measurements that one wants to simulate and the interval

∆t between two consecutive measurements. For instance, Joos’ experiment corresponds to

n = 24 (actually n = 22 since Joos ﬁnally deleted two observations) and ∆t ∼ 3600 seconds.

At the same time, the simulations become quite lengthy for large N , large n and small ∆t.

Therefore, we have ﬁrst performed a scan of s−values for N = 104 and then studied a few s

by increasing N . To give an idea of the spread of the central values, due to changes of the

pair (N, s), we report below the approximate results of this analysis for some choices of the

43

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 As2imul = 1.38 · 10−3.

Picture 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

As2imul[10−3] 1.26 3.50 0.46 0.34 2.71 0.35 2.19 0.52 5.24 0.24 1.19 1.93 0.08 1.52 2.29 0.24 1.02 0.07 0.09 2.18 1.50 1.52

44

pair (n, ∆t)

As2imul(n = 24, ∆t = 3600 s) ∼ (1.7 ± 0.8) · 10−3

(90)

As2imul(n = 1440, ∆t = 60 s) ∼ (1.7 ± 0.3) · 10−3

(91)

As2imul(n = 240, ∆t = 3600 s) ∼ (1.8 ± 0.5) · 10−3

(92)

As it might be expected, the average As2imul becomes more stable by increasing the number of observations. Concerning the individual values As2imul(ti), with i = 1, .., n, they have a large spread, about (1 ÷ 4) · 10−3. 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 interval

of time t0. In this case, the spread of the resulting average values As2imul(ti) t0 decreases as

√1 t0

.

We emphasize that, by performing extensive simulations, there are occasionally very

large spikes of the amplitude at some sidereal times, of the order (10 ÷ 20) · 10−3. The eﬀect

of these spikes gets smoothed when averaging over many conﬁgurations but their presence is

characteristic of a stochastic-ether model. With a standard attitude, where the ether drift

is only expected to exhibit smooth time modulations, the observation of such eﬀects would

naturally be interpreted as a spurious disturbance (Joos’ omitted observations 1 and 5?).

6

Joos Data

5

Poly Fit

Simulation

Poly Fit

4

A2 3

2

1

0

0

5

10

15

20

25

Picture

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 (1 ÷ 4) · 10−3 depending on the sidereal time. We also show 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, one
45

could ﬁrst try to look for the ‘best seed’, or subset of seeds, which can minimize the diﬀerence between the generated conﬁgurations and Joos’ data. This standard task, usually accomplished by minimizing a chi-square, is diﬃcult to implement here. In fact, it is problematic to construct a function χ2(s) and look for its minima because a seed s and the closest seeds s ± 1 give often vastly diﬀerent conﬁgurations and chi-square. For this reason, we have followed an empirical procedure by forming a grid and selecting a set of seeds whose mean amplitude (for n = 24 and ∆t = 3600 s) gets close to Joos’s mean amplitude Aj2oos = 1.4 · 10−3 for a large number N of Fourier modes. One of such seeds gave a sequence As2imul =1.66, 1.40, 1.08, 1.21 and 1.38 (in units 10−3), for N = 103, 104, 105, 106 and 5 · 106 respectively, and the conﬁguration with N = 5 · 106 was ﬁnally chosen to give an idea of the agreement one can achieve between data and a single numerical simulation for ﬁxed (N, s). The simulated values 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 corresponding data since, by changing (N, s), the simulated instantaneous values vary typically of about (1 ÷ 4) · 10−3 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 diﬀerent sets of values.
A more conventional comparison with the data consists in quoting for the various 22 entries simulated average values and uncertainties. To this end, we have considered the mean amplitudes As2imul(ti) deﬁned by averaging, for each Joos’ time ti, over 10 hypothetical measurements performed on 10 consecutive days. For each ti, the observed eﬀect of varying (N, s) has been summarized into a central value and a symmetric error. The values are reported 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 Jena of the cosmic Earth’s velocity becomes larger. The tendency of Joos’ data to lie in the lower part 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 As2imul(ti), we have seen that our sampling method of As2imul(ti) , based on a grid of (N, s) values, typically underestimates the weight of the low-amplitude region in a prediction at the 70% C.L. . This can also be checked by considering the single simulation of Table 8 and counting the sizeable fraction of amplitudes As2imul(ti) 0.5 · 10−3. For this reason, one could improve the evaluation of the probability content. However, in view of the good agreement already found in Fig.12 (χ2 = 13/22), we did not attempt to carry out this more reﬁned analysis.
In conclusion, after the ﬁrst indication obtained from the ﬁt Eq.(85), we believe that the
46

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 As2imul = 1.8 · 10−3.

Picture 2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

As2imul[10−3] 2.5 ± 1.0 1.80 ± 0.85 1.95 ± 0.85 1.90 ± 0.85 1.65 ± 0.90 2.1 ± 1.0 2.0 ± 1.0 2.2 ± 1.2 2.4 ± 1.4 2.7 ± 1.6 2.3 ± 1.5 2.4 ± 1.4 1.85 ± 0.85 1.70 ± 0.75 1.20 ± 0.75 1.20 ± 0.70 1.15 ± 0.70 1.05 ± 0.70 1.25 ± 0.60 1.55 ± 0.60 1.60 ± 0.80 1.7 ± 1.0

47

6 5 4
A2 3
2 1 0
0

Joos Data Simulation

5

10

15

20

25

Picture

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 by our simulations. In fact, by inspection of Figs.11 and 12, the values of the amplitudes and the characteristic scatter of the data are correctly reproduced. In principle, there could be space for further reﬁnements by taking into account the Earth’s orbital motion in the input values for V , α and γ.
From this agreement with the data, we then deduce that the previous value for the kinematical velocity v ∼ 217+−7696 km/s, obtained by simply correcting with the helium refractive index the average observable velocity (81), has to be considerably increased if one allows for stochastic variations of the velocity ﬁeld. In fact, the magnitude of the ﬂuctuations in vx and vy is controlled by the same scalar parameter v˜(t) ≡ v˜x2(t) + v˜y2(t) of Eq.(34). In view of the good agreement between data and our numerical simulations, we conclude that Joos’ data are consistent with a range of kinematical velocity v = 330+−7400 km/s which corresponds to Eq.(34) for φ = 50.94o, V = 370 km/s, α = 168o and γ = −6o.
9. Summary and conclusions
The condensation of elementary quanta and their macroscopic occupation of the same quantum state is the essential ingredient of the degenerate vacuum of present-day elementary particle physics. In this description, one introduces implicitly a reference frame Σ, where the condensing quanta have k = 0, which characterizes the physically realized form of relativity and could play the role of preferred reference frame in a modern re-formulation of the

48

Lorentzian approach. To this end, we have given in the Introduction some general theoretical arguments related to the problematic notions of a non-zero vacuum energy and of an exact Lorentz-invariant vacuum state. These arguments suggest the possibility of a tiny vacuum energy-momentum ﬂux, associated with an Earth’s absolute velocity v, which could aﬀect diﬀerently the various forms of matter. Namely, it could produce small convective currents in a loosely bound system such as a gas or dissipate mainly by heat conduction with no appreciable particle ﬂow in strongly bound systems as liquid or solid transparent media. In the former case, by introducing the refractive index N of the gas, convective currents of the gas molecules would produce a small anisotropy, proportional to (N − 1)(v/c)2, of the two-way velocity of light in agreement with the general structure Eq.(23) or with its particular limit Eq.(24). Notice that this tiny anisotropy refers to the system S′ where the container of the gas is at rest. In this sense, contrary to standard Special Relativity, S′ might not deﬁne a true frame of rest. This conceptual possibility can be objectively tested with a new series of dedicated ether-drift experiments where two orthogonal optical resonators are ﬁlled with various gaseous media by measuring the fractional frequency shift ∆ν/ν between the two resonators. By assuming the typical value v ∼ 300 km/s of most cosmic motions, one expects frequency shifts 10−10 for gaseous helium and 10−9 for air, which are well within the present 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 adopting Eq.(24), the frequency shift between the optical resonators is governed by the same classical formula for the fringe shifts in the old ether-drift experiments with the only replacement

v2 → 2(N − 1)v2 ≡ vo2bs

(93)

In this way, where one re-obtains the same classical formulas (with the only replacement v → vobs), testing the present scheme is very simple: one should just check the consistency of the true kinematical v′s obtained in diﬀerent experiments.
In the old times, experiments were performed with interferometers where light was propagating in gaseous media, air or helium at atmospheric pressure, where (N − 1) is a very small number. 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)2. Another important aspect of these classical experiments is that one was always expecting smooth sinusoidal modulations of the data 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’s motions (daily rotation, annual orbital revolution,...) and the ether-drift experiments depends

49

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 a state 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 ﬂuid where large-scale and small-scale motions are only indirectly related. In this case, there might be 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, this does not mean that there is no ether-drift. More generally, the relevant Earth’s motion with respect to Σ might well correspond to that indicated by the anisotropy of the CMB, but it becomes non trivial to reconstruct the kinematical parameters from microscopic measurements of the velocity of light in a laboratory. These arguments make more and more plausible that a genuine physical phenomenon, much smaller than expected and characterized by stochastic variations, might have been erroneously interpreted as an instrumental artifact thus leading to 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 obvious. 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 indications of the various experiments are summarized in our Table 10 10. As a summary of our work, we emphasize the following points:
i) an analysis of the individual sessions of the original Michelson-Morley experiment, in agreement 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. In fact, averaging directly the fringe displacements of diﬀerent sessions requires two additional assumptions, on the nature of the ether-drift as a smooth periodic eﬀect and on the absence of systematic errors introduced by the re-adjustment of the mirrors on consecutive days, that
10Other determinations of less accuracy could also be included, as for the 1881 Michelson experiment in Potsdam [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 ÷ 0.007 to be compared with an expected 2nd-harmonic of 0.02 for 30 km/s. This means observable 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 expected for an Earth’s velocity of 30 km/s. This gives vobs 7.7 km/s or v 320 km/s. From Piccard and Stahel, in the most reﬁned version of Mt. Rigi, one gets an observable velocity vobs 1.5 km/s. Since their optical paths were enclosed in an evacuated enclosure, this very low value can easily be reconciled with the typical kinematical velocity v ∼ 300 km/s of the most accurate experiments in Table 10.
50

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 Michelson-Morley(1887) Morley-Miller(1902-1905)
Kennedy(1926) Illingworth(1927) Miller(1925-1926) Michelson-Pease-Pearson(1929)
Joos(1930)

gas in the interferometer air air helium helium air air helium

vobs(km/s) 8.4+−11..57 8.5 ± 1.5
<5
3.1 ± 1.0 8.4+−21..95 4.5 ± ... 1.8+−00..56

v(km/s) 349+−7602 353 ± 62
< 600
370 ± 120 349+−17094 185 ± ... 330+−7400

in the end may turn out to be wrong. ii) one gets consistent indications from the Michelson-Morley, Morley-Miller, Miller and
Illingworth-Kennedy experiments. In view of this consistency, an interpretation of Miller’s observations in terms of a temperature gradient [68] is only acceptable provided this gradient represents a non-local eﬀect as in our picture where the ether-drift is the consequence of a fundamental vacuum energy-momentum ﬂow. We have also produced numerical simulations of the Illingworth experiment in a simple statistically isotropic and homogeneous turbulentether model. This represents a zeroth-order approximation and is useful to illustrate basic phenomenological features associated with the picture of the vacuum as an underlying stochastic medium. In this scheme, Illingworth’s data are consistent with ﬂuctuations of the velocity ﬁeld whose absolute scale is controlled by v˜ = VCMB ∼370 km/s, the velocity of the Earth’s motion with respect to the CMB.
iii) on the other hand, there is some discrepancy with the experiment performed by Michelson, Pease and Pearson (MPP). However, as discussed in Sect.7, the uncertainty cannot be easily estimated since only a single basic MPP observation is explicitly reported in the literature. Therefore, since Miller’s extensive observations (see Fig.22 of ref.[65] and our Fig.8), within their errors, gave ﬂuctuations of the observable velocity in the wide range 4−14 km/s, a single observation giving vobs ∼ 4.5 km/s cannot be interpreted as a refutation. This becomes even more true by noticing that the single session selected by Pease, within a period of 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 is particularly important since the data were collected at steps of 1 hour to cover the full sidereal
51

day and were recorded by photocamera. For this reason, Joos’ experiment is not comparable with other experiments (e.g. Michelson-Morley, Illingworth) where only observations at few selected 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 from Miller’s, the amplitudes of all Joos’s observations can be reconstructed from the published articles [87, 88]. For these reasons, this experiment has deserved a more reﬁned analysis and is central for our work. As discussed in Sect.8, due to uncertainties in the original data analysis, the standard 1.5 km/s velocity value quoted for this experiment should be understood as an order of magnitude estimate and not as a true upper limit. Instead, our reported observable velocity vobs ∼ 1.8+−00..56 km/s has been obtained from a direct analysis of Joos’ fringe shifts. From this value, to deduce a kinematical velocity, one still needs the refractive index. The traditional view, motivated by Miller’s review article [65] and Joos’s own statements in ref.[87], is that the experiment was performed in an evacuated housing. In these conditions, it would be easy to reconcile a large kinematical velocity v ∼ 350 km/s with the very small values of the observable velocity. On the other hand, Swenson [90] explicitly reports 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 follow Swenson’s indications. In this case, by simply correcting with the helium refractive index the result vobs ∼ 1.8+−00..56 km/s, one would get a kinematical velocity v ∼ 217+−7696 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 (see Eq.(85)) we have obtained angular parameters which are very close to those that describe the CMB anisotropy (right ascension αCMB ∼ 168o and angular declination γCMB ∼ −6o). Still, to get a complete agreement, one should explain the absolute normalization of the amplitudes and the strong ﬂuctuations of the data. Thus we have improved our analysis by performing various numerical simulations where the velocity components in the plane of the interferometer vx(t) and vy(t), which determine the basic functions C(t) and S(t) through Eqs.(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 timedependent ranges Eqs.(32)−(33), [−v˜x(t), v˜x(t)] and [−v˜y(t), v˜y(t)] respectively, controlled by the macroscopic parameters (V, α, γ)CMB. Taking into account these stochastic ﬂuctuations of the velocity ﬁeld tends to increase the ﬁtted average Earth’s velocity, see Eq.(89), and can reproduce 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 v = 330+−7400 km/s (corresponding to Eq.(34) for CMB and Joos’ laboratory) is actually the most appropriate
52

one. The more reﬁned analysis adopted for the Joos experiment provides an explicit example
of the previously mentioned non-trivial ingredients that might be required to reconstruct the global Earth’s motion from microscopic measurements performed in a laboratory. For this reason, the results reported in Table 10, besides providing an impressive evidence for a light anisotropy proportional to (N − 1)(v/c)2, with the realistic velocity values v ∼ 300 km/s of most cosmic motions, could also represent the ﬁrst experimental indication for the Earth’s motion with respect to the CMB. Due to the importance of this result, and to provide the reader with all elements of the analysis, we present in a second Appendix a brief numerical simulation of one noon session of the Michelson-Morley experiment. This has been performed in the same framework adopted for the Joos experiment where the velocity components vx(t) and vy(t) in the plane of the interferometer are represented as turbulent ﬂuctuations varying within time-dependent ranges controlled by the macroscopic parameters (V, α, γ)CMB. We postpone to a future publication the non-trivial task of performing a complete numerical simulation of the whole Michelson-Morley experiment and of the Illingworth experiment (with its 32 sessions and the associated sets of 20 rotations for each session) where we’ll also compare the 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 a single conﬁguration whereas taking into account more and more conﬁgurations is essential to properly estimate theoretical uncertainties (as for the Joos experiment with the results in our Table 9 and Fig.12). Nevertheless, even this very small sample can provide interesting clues on the real data. For instance, the strong scatter of the fringe shifts at the same θ−values in consecutive rotations and the good agreement with the experimental azimuths obtained by accepting Hicks’ interpretation of the observations of July 8th (see Sect.3). In this sense, this brief numerical analysis reinforces the picture of the classical experiments emerging from our Table 10. According to the usual view, the theoretical predictions, with a very low velocity v ∼ 30 km/s, were much larger than the observed values. Instead, in a modern view of the vacuum as a stochastic medium, theoretical predictions, for the realistic velocities v ∼ 300 km/s of most cosmic motions, are now well compatible and, sometimes, even smaller than the actual outcome of the observations. This latter case simply means that the experimental data were also aﬀected by spurious eﬀects such as deformations induced by the rotation of the apparatus or local thermal conditions. This gives a strong motivation to repeat these crucial measurements 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 measuring the relative frequency shift ∆ν 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 all spurious eﬀects), is

∆ν phys (θ ) ν0

=

c¯γ (π/2

+ θ) c

−

c¯γ (θ)

=

Bmedium

v2 c2

cos 2(θ

−

θ0)

(94)

where θ0 is the direction of the ether-drift. This can be interpreted within Eq.(109) where

|Bmedium| ∼ Nmedium − 1

(95)

Nmedium being the refractive index of the gaseous medium ﬁlling the optical resonators. Testing this prediction, requires replacing the high vacuum usually adopted within the optical resonators with a gaseous medium and studying the substantially larger frequency shift introduced 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 a rotating platform. To this end, one has to preliminarily subtract a large systematic eﬀect that was present in the data and interpreted by the authors as probably due to magnetostriction in 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 ∆ν, can be subtracted 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 NHe−Ne ∼ 1.00004 and 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 vacuum where, as emphasized in the Introduction, the diﬀerences between Special Relativity and the Lorentzian interpretation are at the limit of visibility. In fact, in a perfect vacuum by deﬁnition Nvacuum = 1 so that Bvacuum will vanish 11. Thus one should switch to the new generation of dedicated ether-drift experiments in gaseous systems. Our conclusion is that these new experiments should just conﬁrm Joos’ remarkable observations of eighty years ago.
11Throughout this paper we have assumed the limit of a zero light anisotropy for experiments performed in vacuum. However, as discussed at the end of Appendix I, one could also consider the more general scenario where a metric of the form [97] gµν = ηµν + ∆µν is introduced from the very beginning. In this case, the vacuum behaves as a medium and light can spread with diﬀerent velocities for diﬀerent directions. As an example, by adopting various parameterizations for ∆µν , the non-zero one-way light anisotropy reported by the GRAAL experiment [98] requires typical values of the matrix elements |∆µν | = 10−13 ÷ 10−14 [99].In any case, as anticipated in Sect.2, these genuine vacuum eﬀects are much smaller than those discussed in the present paper in connection with a gas refractive index.

54

Acknowledgments We thank Angelo Pagano for useful discussions.
55

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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 from a fundamental vacuum energy-momentum ﬂow. Due to these convective currents, isotropy of the velocity of light would only hold if the solid container of the gas and the observer were both at rest in the particular reference frame Σ where the macroscopic condensation of quanta correspond to the state k = 0. This introduces obvious diﬀerences with respect to the standard analysis. For instance, let us compare with Jauch and Watson [96] who worked out the quantization of the electromagnetic ﬁeld in a moving medium of refractive index N . They noticed that the procedure introduces unavoidably a preferred frame, the one where the photon energy does not depend on the direction of propagation. Their conclusion, that this frame is “usually taken as the system for which the medium is at rest”, reﬂects however the point of view of Special Relativity with no preferred frame. Instead, one could consider a diﬀerent scenario where, at least in some limit, the angle-independence of the photon energy might only hold for some special frame Σ.
To discuss this diﬀerent case, let us ﬁrst consider a dielectric medium of refractive index N whose container is at rest in Σ. For an observer at rest in this reference frame, light propagation within the medium is isotropic and described by

πµπν γµν = 0

(96)

where

γµν = diag(N 2, −1, −1, −1)

(97)

and πµ denotes the light 4-momentum vector for the Σ observer. Let us now consider that

the container of the medium is moving with some velocity V with respect to Σ and is at rest

in some other frame S′. By analogy, light propagation within the medium for the observer in

S′ will be described by

pµpν gµν = 0,

(98)

where pµ ≡ (E/c, p) and gµν denote respectively the light 4-momentum and the eﬀective

metric for S′. On this basis, by introducing the S′ dimensionless velocity 4-vector uµ ≡

(u0, V/c) (with uµuµ = 1), one can deﬁne a transformation matrix Aµν = Aµν(uµ, N ) and

express

gµν = AµσAν ργσρ

(99)

61

In this context, requiring the consistency of vacuum condensation with Special Relativity corresponds to place all reference frames on the same footing and assume gµν = γµν or

Aµν (uµ, N ) = δνµ

(100)

This identiﬁcation is independent of the physical nature of the medium, being valid for gaseous systems as well as for liquid or solid transparent media. In this sense Special Relativity, by construction, cannot describe light propagation in the presence of a vacuum energy-momentum ﬂow which could aﬀect the various forms of matter diﬀerently.
Instead, consistently with the basic ambiguity in the interpretation of relativity mentioned in the Introduction, and with Lorentz’ point of view [4] (“it seems natural not to assume at starting that it can never make any diﬀerence whether a body moves through the ether or not”), one could adopt diﬀerent choices without pretending to determine a priori the outcome of any ether-drift experiment. Thus, by noticing that we have at our disposal two matrices, namely δνµ and the Lorentz transformation matrix Λµν associated with V, one could still maintain 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 dissipate by heat conduction with no appreciable particle ﬂow and no light anisotropy in the rest frame of the medium. One could instead identify

Aµν (uµ, N = 1) = Λµν

(101)

to solve non-trivially the equation gµν = γµν when N = 1, i.e. when light propagates in vacuum and γµν reduces to the Minkowski tensor ηµν . But then, by continuity, it is conceivable that Eq.(101), up to higher-order terms, can also describe the case N = 1 + ǫ of gaseous media. This choice provides a simple interpretative model and a particular form of 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 comparing with experiments with optical resonators, the resulting frequency shift is governed by the same classical formula for the fringe shifts in the old ether-drift experiments with the only replacement V 2 → 2(N − 1)V 2. To see this, let us compute gµν through the relation

gµν = ΛµσΛν ργσρ

(102)

with γµν as in Eq.(97). This gives the eﬀective metric for S′

gµν = ηµν + κuµuν

(103)

with

κ = N2 −1

(104)

62

In this way, Eq.(98) gives a photon energy (u20 = 1 + V2/c2)

E(|p|, θ) = c −κu0ζ +

|p|2(1 + κu20) − κζ2 1 + κu20

(105)

with

ζ

=

p

·

V c

=

|p|β

cos

θ,

(106)

where

β

=

|V| c

and

θ

≡

θlab

indicates

the

angle

deﬁned,

in

the

laboratory

S′

frame,

between

the photon momentum and V. By using the above relation, one gets the one-way velocity of

light

E(|p|, θ) |p|

=

cγ (θ)

=

c

−κβ

1

+

β2

cos θ + 1 + κ(1

1+ + β2)

κ

+

κβ2

sin2

θ

.

(107)

or to O(κ) and O(β2)

cγ (θ)

∼

c N

1

−

κβ

cos

θ

−

κ 2

β

2

(1

+

cos2

θ)

From this one can compute the two-way velocity

(108)

c¯γ (θ)

=

2cγ(θ)cγ(π + θ) cγ(θ) + cγ(π + θ)

∼

c N

1 − β2

κ

−

κ 2

sin2

θ

(109)

which, as anticipated, is a special form of the more general Eq.(23). We can then deﬁne the RMS anisotropy parameter B 12

c¯γ (π/2

+ θ) c¯γ

−

c¯γ (θ)

∼

B

v2 c2

cos

2(θ

−

θ0)

(110)

where the pair (v, θ0) describes the projection of V onto the relevant plane and

|B|

∼

κ 2

∼

(N

−

1)

(111)

12There 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 O(β2), no attention was paid to the precise deﬁnition of the angle between

the Earth’s velocity and the direction of the photon momentum. Thus the two-way velocity of light in the

S′ frame was parameterized in terms of the angle θ ≡ θΣ as seen in the Σ frame. This can be explicitly checked by replacing in our Eqs. (108) and(109) the aberration relation cos θlab = (−β + cos θΣ)/(1 − β cos θΣ) or equivalently by replacing cos θΣ = (β + cos θlab)/(1 + β cos θlab) 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 optical

cavities at angles θ and π/2 + θ is expressed in terms of θ = θlab and not in terms of θ = θΣ. This trivial remark produces however a non-trivial diﬀerence in the value of the anisotropy parameter. In fact, the correct

resulting |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 θ = θΣ.

63

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

the observable velocity

vo2bs ∼ 2|B|v2 ∼ 2(N − 1)v2

(112)

to be used in Eq.(29). In this way, where one re-obtains the classical formulas with the only replacement v → vobs, testing the present scheme requires to check the consistency of the kinematical v′s obtained in diﬀerent experiments.
Before concluding this Appendix, we emphasize that throughout this paper we have assumed the limit of a zero light anisotropy for experiments performed in vacuum. In fact, the eﬀective metric Eq.(103) reduces to the Minkowski tensor ηµν in the limit N → 1. Admittedly, this might represent a restrictive scenario and one could also consider the more general case where a metric of the form [97]

gµν = ηµν + ∆µν

(113)

is introduced from the very beginning in extensions of the Standard Model. In this sense, once Eq.(113) is adopted, the vacuum behaves as a medium and the dispersion relations that describe light and particle propagation can have several solutions. For instance, light will spread with 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 with measurements of the one- and two-way velocity of light. As an example, the one-way light anisotropy reported by the GRAAL experiment [98] requires typical values of the matrix elements |∆µν | = 10−13 ÷ 10−14 [99]. In any case, as anticipated in Sect.2, these genuine vacuum eﬀects are much smaller than those discussed in the present paper in connection with a gas refractive index.

64

Appendix II

In this second Appendix we’ll report the results of a single simulation of an individual noon session of the Michelson-Morley experiment. This will be performed within the same stochastic-ether model described in Sect.8 for the Joos experiment. For sake of clarity, we recapitulate the various steps so that an interested reader can also run his own simulations.
One should ﬁrst express the functions C(t) and S(t) as in Eqs.(44) and model the two velocity components vx(t) and vy(t) 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 kinematical parameters VCMB ∼ 370 km/s, αCMB ∼ 168o, γCMB ∼ −6o, in Eqs.(31)−(33) to ﬁx the boundaries [−v˜x(t), v˜x(t)] and [−v˜y(t), v˜y(t)] respectively for the random parameters xn(i = 1, 2) and yn(i = 1, 2) entering Eqs.(60) and (61). In the end, with the simulated C(t) and S(t), one should form the fringe shift combination

∆λ(θ) λ

≡

2C(t) cos 2θ

+

2S(t) sin 2θ

=

A2(t) cos 2(θ

−

θ0(t))

(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 consecutive

readings of the fringe shifts were performed every 22.5 degrees. Therefore, two consecutive

readings diﬀered by 22.5 seconds. In these conditions, a numerical simulation of a single

rotation consists in generating 16 pairs [C(t), S(t)] at steps of 22.5 seconds.

As the central time of the observations, we have chosen 12 A. M. of July 10, 1887 which, for

Cleveland, corresponds to a sidereal time τ ∼ 102o. To select the parameters of the simulation,

we have compared with the traditional analysis of the experiment where one performs a ﬁt

to the fringe shifts obtained by averaging the results of the various experimental sessions. In

our case, averaging the data of the three noon sessions in our Table 1 gives a 2nd-harmonic

amplitude

Aﬁ2t(average data − noon) ∼ 0.012

(115)

We have thus considered the exact amplitude

Ae2xact(t) = 2 C2(t) + S2(t)

(116)

and selected a particular conﬁguration whose global average over the 6 turns gives Ae2xact(t) ∼ 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 θ−values and sizeable variations of the ﬁtted amplitude and azimuth. Taking into account these variations is essential to perform

65

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 the

agreement one can achieve between data and a single numerical simulation. We thus postpone

to a future publication a complete analysis of the whole Michelson-Morley experiment and

of the Illingworth experiment (with its 32 sessions and the associated sets of 20 rotations for

each session) where we’ll also compare the various theoretical schemes mentioned in Sect.5 to

handle the stochastic components of the velocity ﬁeld.

The results of our single simulation for [2C(t), 2S(t)] are reported in Tables 11 and 12

while

the

combinations

∆λ(θ) λ

Eq.(114)

are

reported

in

Table

13,

for

θ

=

i−1 16

2π

together

with the results of 2-parameter ﬁts to the simulated data. Notice the strong scatter of the

simulated data at the same θ−values. Of course, to compare with the real data, one should

ﬁrst take the even combination Eq.(50) of the entries in Table 1 which otherwise also contain

odd-harmonic terms.

We conclude this brief analysis by emphasizing the importance of Hicks’ observation (see

Sect.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 θ0EXP ∼ 357, 285 and 317 degrees, would become θ0EXP ∼ 267, 285 and 317 degrees and 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 vx(t) and vy(t) 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

6

1 −0.023 +0.002 −0.024 +0.001 −0.004 −0.003

2 +0.003 −0.011 −0.000 −0.006 −0.021 −0.034

3 −0.001 −0.001 +0.000 −0.009 +0.002 +0.007

4 −0.002 +0.003 −0.008 +0.002 −0.060 −0.030

5 +0.002 −0.017 +0.002 −0.001 +0.003 −0.008

6 −0.007 −0.006 −0.059 −0.013 −0.008 −0.047

7 −0.020 −0.001 −0.019 −0.000 −0.003 −0.003

8 −0.011 −0.001 −0.011 −0.002 −0.026 +0.001

9 −0.015 −0.000 −0.008 −0.001 −0.008 −0.022

10 −0.037 −0.005 +0.000 −0.002 −0.003 +0.003

11 +0.003 −0.022 −0.015 −0.005 −0.003 +0.002

12 −0.002 −0.049 −0.023 −0.016 −0.009 −0.006

13 −0.001 +0.002 +0.000 +0.001 +0.003 −0.001

14 +0.003 −0.003 +0.003 −0.023 −0.001 −0.019

15 −0.012 −0.034 −0.013 −0.001 −0.001 −0.011

16 −0.004 −0.017 −0.004 +0.002 −0.010 −0.010

67

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 vx(t) and vy(t) 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

6

1 +0.011 +0.001 +0.011 +0.000 +0.003 +0.001

2 +0.004 −0.003 −0.001 −0.008 +0.007 −0.009

3 +0.003 −0.005 +0.000 −0.007 +0.007 +0.000

4 +0.001 −0.007 −0.003 −0.001 +0.001 +0.016

5 +0.002 −0.007 +0.000 +0.001 −0.000 −0.003

6 −0.004 +0.010 +0.001 −0.001 +0.006 +0.009

7 −0.017 +0.002 +0.005 −0.000 −0.003 +0.002

8 +0.011 −0.002 −0.012 −0.001 +0.010 −0.000

9 +0.011 +0.001 +0.008 +0.011 +0.005 −0.011

10 +0.016 +0.005 −0.001 +0.001 −0.002 +0.001

11 +0.000 −0.013 +0.015 +0.005 −0.002 −0.001

12 −0.001 −0.022 +0.004 −0.003 −0.000 +0.002

13 −0.001 −0.001 +0.001 +0.000 −0.004 +0.001

14 −0.001 +0.002 +0.002 +0.018 +0.001 +0.009

15 −0.012 +0.018 −0.002 +0.001 −0.004 +0.007

16 +0.002 +0.008 +0.001 +0.002 −0.005 −0.006

68

Table

13:

The

fringe

shifts

∆λ(θ) λ

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

θ

=

i−1 16

2π.

The

variance

of

the

averages

is

about

±0.004

for

the

amplitude

and

about

±11o

for the azimuth.

i

1

2

3

4

5

6 average

1 −0.023 +0.002 −0.024 +0.001 −0.004 −0.003 −0.009

2 +0.005 −0.010 −0.001 −0.010 −0.010 −0.031 −0.010

3 +0.003 −0.005 +0.000 −0.007 +0.007 +0.000 −0.000

4 +0.002 −0.007 +0.004 −0.002 +0.043 +0.033 +0.012

5 −0.002 +0.017 −0.002 +0.001 −0.003 +0.008 +0.003

6 +0.008 −0.003 +0.041 +0.010 +0.002 +0.027 +0.014

7 +0.017 −0.002 −0.005 +0.000 +0.003 −0.002 +0.002

8 −0.016 +0.001 +0.001 −0.001 −0.025 +0.001 −0.007

9 −0.015 −0.000 −0.008 −0.001 −0.008 −0.022 −0.009

10 −0.015 −0.000 −0.000 −0.001 −0.003 +0.003 −0.003

11 +0.000 −0.013 +0.015 +0.005 −0.002 −0.001 +0.001

12 +0.000 +0.019 +0.020 +0.009 +0.006 +0.006 +0.010

13 +0.001 −0.002 −0.000 −0.001 −0.003 +0.001 −0.001

14 −0.001 +0.001 −0.004 +0.004 −0.000 +0.007 +0.001

15 +0.012 −0.018 +0.002 −0.001 +0.004 −0.007 −0.001

16 −0.005 −0.018 −0.003 −0.000 −0.004 −0.003 −0.005

Aﬁ2t 0.009 0.005 0.009 0.003 0.011 0.013 0.008

θ0ﬁt

279o

259o

266o

285o

255o

272o

269o

69