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The classical ether-drift experiments: A modern re-interpretation
Maurizio Consoli
2013, European Physical Journal Plus
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  a   r    X    i   v   :    1    3    0    2  .    3    5    0    8   v    2    [   p    h   y   s    i   c   s  .   g   e   n   -   p    h    ]    6    J   u   n    2    0    1    3
The classical ether-drift experiments: a modern re-interpretation
M. Consoli
(
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 quan- tum 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 pre- ferred reference frame in a modern re-formulation of the Lorentzian approach. In spite of this, the so calle d ‘nul l resul ts’ of the class ical ether -drif t experim ent s, tradit ionall y int erpre ted as conﬁrmations of Special Relativity, have so deeply inﬂuenced scientiﬁc thought as to prevent a critical discussio n on the real reasons underlyi ng its allege d supre macy . In this paper, we argue that this traditiona l null int erpre tatio n is far from obv ious. In fact, by using Loren tz transformations to connect the Earth’s frame to Σ, the small observed eﬀects point to an av erag e Earth’ s velocit y of about 300 km/s, as in most cosmic motion s. A common feature is the irregular behaviour of the data. While this has motivated, so far, their standard inter- pretation as instrumental artifacts, our new re-analysis of the very accurate Joos experiment gives clear indications for the type of Earth’s motion associated with the CMB anisotropy and lea ve s little space for this traditio nal interpr etatio n. The new expla natio n requ ires 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 asp ects of both quantum physics and relativit y . The over- all 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. In tr odu ct io n
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 ]. Th er e is no dou bt tha t the forme r in ter pre tat ion is toda y wid ely acce pte d. Ho we ve r, in spi te 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 Lorent z. It is, in fact, because of this circumstan ce, that according to Einstein’s theory an observer must observe exactly the same contractions, changes of rate, etc . in the measu rin g rods, clock s, etc . mo vin g wit h respec t to him as in the Lore nt zia n the ory . And let it be sai d her e rig ht aw ay and in all gene ral it y . As a mat ter of pri nci ple , there is no experime ntu m crucis b et wee n the tw o theories”. This can b e understood since, independently of all interpretative aspects, the basic quantitative ingredients, namely Lorentz tra nsf orm ati ons , are the sam e in both form ula tio ns. The ir v ali dit y will be ass ume d 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 thi s alt ern ati ve approa ch , diﬀ ere nt ly fro m the usual deriv ati ons , one start s fro m 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
β
rel
 ﬁxed by the relativistic composition rule
β
rel
 =
  β
′
−
β
′′
1
−
β
′
β
′′
  (1) (for simplic ity we restric t to the case of one-dimens ional motion) . This produces a substan- tial quantitative equivalence with Einstein’s formulation for most standard experimental tests where one just compares the relativ e measu remen ts of a pair of obser ve rs. Henc e 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
  c
γ
 propagating in the various interferometers coinc ides with the basic parame ter
  c
 ente ring Loren tz trans format ions, relat ivisti c eﬀec ts 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 unobserv able, 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 induc e one to re-a nalyz e the class ical ether-d rift experime nts . Let us ﬁrst give some general theoretical arguments that could motivate this apparently startling idea. A poss ibl e obs erv ati on is tha t Lor en tz sym met ry mig ht not be an exact symme try . In this case, one could conce iv ably detect the eﬀec ts of absolute motion. F or insta nce Loren tz 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 ph ysi cs and the cos mol ogi cal ter m nee ded in Ein ste in’ s equ ati ons to ﬁt the obs erv ati ons ) ii) the idea of Lorentz symmetry as associated with an infrared ﬁxed point  [12, 13 ] in non- symmetric quantum ﬁeld theories iii) the quantum-gravity literature which, by starting 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 ]. Her e, ho we ve r, we sha ll ado pt a diﬀe ren t pers pect iv e and con ce nt rat e our ana lys is on a p eculi ar aspect of today ’s quantum ﬁeld theorie s: the represe nta tion of the vac uum as a ‘cond ensat e’ of element ary quanta. These condens e b ecaus e their triviall y empt y v acuum is a meta-stabl e state and not the true ground state of the theory . In the phy sical ly relev ant case of the Standard Model of elec tro wea k int eract ions, this situation can b e summarized by saying [ 22]  that “What we experience as empty space is nothing but the conﬁguration of  the Higgs ﬁeld that has the low est 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 conde nsed”. The explicit translat ion from ﬁeld jargon to partic le jargon, with the substa ntia l equivalence between the eﬀective potential of quantum ﬁeld theory and the energy density of  2

a dilute particle condensate, can be found for instance in ref.[ 23] . The trivial empty vacuum will eventually be re-established by heating the system above a critical temperature
  T
  =
  T
c
 where the conde nsate ‘ev aporate s’. This temperatu re in the Standard Model is so high that one can safely approximate the ordinary vacuum as a zero- temperature system (think of
  4
He at a temperature 10
−
12
  o
K). This observation allows one to view the physical vacuum as a superﬂuid medium [ 10]  where bodies can ﬂow without any appar ent friction , consi sten tly with the experimen tal resul ts. Clear ly , this form of quan tum 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 Einste in had in mind in 1905. There fore , one migh t ask [ 24]  if Bose conden sation , i.e. the macroscopic occupation of the same quantum state, say
 k
 = 0 in some reference frame Σ, can repre sen t the operativ e const ructi on of a ‘quantum ether’. This charac teriz es 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
. Namel y , Lore ntz transform ation operators   ˆ
U
′
,   ˆ
U
′′
,..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 Lorentz- invariant operator   ˆ
G
, one would ﬁnd

ˆ
G

Ψ
(0)
  =

ˆ
G

Ψ
′
  =

ˆ
G

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
  P
α
,
  M
α,β
  (
  α
  ,
β
=0, 1, 2, 3) where
P
α
 are the 4 generators of the space-time translations and
 M
αβ
  =
−
M
βα
 are the 6 generators of the Lorentzian rotations with commutation relations [
P
α
, P
β
] = 0 ( 3 ) [
M
αβ
, P
γ
] =
 η
βγ
P
α
−
η
αγ
P
β
  (4) [
M
αβ
, M
γδ
] =
 η
αγ
M
βδ
 +
η
βδ
M
αγ
 −
η
βγ
M
αδ
−
η
αδ
M
βγ
  (5) where
 η
αβ
 = 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) ˆ
P
i
|
Ψ
(0)

=   ˆ
M
ij
|
Ψ
(0)

= 0 ( 6 ) but, at the same time, be characterized by a non-vanishing energy ˆ
P
0
|
Ψ
(0)

=
 E
0
|
Ψ
(0)

  (7) This vacuum energ y might have diﬀeren t explanations. Here, we shall limit ourselves to explor- ing 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
  E
0
 = 0 without imposing an unbr ok en supersymmetr y , whic h is not pheno menolo gical ly acce ptable . In this framework, by using the Poincar´ e algebra of the b oost and energy-moment um op erators, 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
 
ˆ
P
i

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

ˆ
W
0
i

Ψ
′
 through the relation (i=1,2,3)

ˆ
P
i

Ψ
′
 ≡

  d
3
x
 
ˆ
W
0
i

Ψ
′
 
= 0 ( 8 ) On the other hand, there is an alternative approach where one tends to consider the vacuum energy
 E
0
 as a spurious concept and only concentrate on an energy-momentum tensor of the following form [ 28, 29 ]

ˆ
W
µν

Ψ
(0)
  =
 ρ
v
  η
µν
  (9) (
ρ
v
 being a spa ce- tim e ind epen den t con sta nt ). In thi s cas e, one is dri ve n to com ple tel y diﬀer ent conclusio ns since , by introducin g the Lorentz transforma tion matric es Λ
µ ν
 to any moving frame
 S
′
, deﬁning
 
ˆ
W
µν

Ψ
′
 through the relation

ˆ
W
µν

Ψ
′
  = Λ
σ µ
Λ
ρ ν
 
ˆ
W
σρ

Ψ
(0)
 (10) and using Eq.( 9 ), it follows that the expectation value of    ˆ
W
0
i
 in any boosted vacuum state
|
Ψ
′

 vanishes, just as it vanishes in
 |
Ψ
(0)

, i.e.

  d
3
x
 
ˆ
W
0
i

Ψ
′
 ≡ 
ˆ
P
i

Ψ
′
  = 0 ( 1 1 )
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
  E
1
(
p
) are deﬁned as the energies of  the corresponding one-particle states
 |
p

 minus the energy of the zero-particle, vacuum state. If
  E
0
 is considered a spurious concept,
  E
1
(
p
) will also b ecome an ill-de ﬁned quan tit y . 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   ˆ
M
01
. Once
|
Ψ
(0)

 is considered an eigenstate of the energy-momentum operator, one can deﬁnitely show [ 25]  that, for
  E
0
 
= 0,
 |
Ψ
′

  and
 |
Ψ
(0)

 diﬀer non-tr ivial ly . On the other hand, if
  E
0
 = 0 there are only two alternatives: either   ˆ
M
01
|
Ψ
(0)

= 0, so that
 |
Ψ
′

=
|
Ψ
(0)

, or   ˆ
M
01
|
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  ˆ
M
0
i
 =
−
i

  d
3
x
 (
x
i
  ˆ
W
00
−
x
0
  ˆ
W
0
i
) ( 1 2 ) wi th zero eig en v al ue . As far as we can see , th er e is no wa y to ob ta in su c h a re su lt by  just starting from Eq.( 9 ) (this only amounts to the weaker condition
 
  ˆ
M
0
i

Ψ
(0)
 = 0) . Th us, independently of the ﬁniteness of
  E
0
, 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 ). F or these reasons, it is not obvious that the local relations ( 9 ) represent a more fundamental approach to the vacuum. Alter nativ ely , one could argue that a satisf actor y soluti on of the v acuum -ener gy prob- lem lies deﬁni tel y bey ond ﬂat spa ce. A non-z ero
  ρ
v
, 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 obser v ati ons . F or ins tan ce , as an tic ipa ted , the pictur e of the vac uum as a supe rﬂu id 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 an y liqui d, in fact, curv a- ture requires
 deviations
 from the equil ibrium state . The same happens for a crys tal 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 under- stand why curvature eﬀects can be orders of magnitude smaller than those naively expected by solving Einstein’s equations with the full
 
ˆ
W
µν

Ψ
(0)
 as a cos mol ogi cal term . In this per- spective, ‘emergent-gravity’ approaches   [30 ,  31, 32] , where gravity somehow arises from the same physical ﬂat-space vacuum, may become natural
  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 phy sical picture of a superﬂuid medium . T o decid e betw een 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. V acu um ene rgy -mo men tum ﬂo w as an eth er dr ift
T o ex plo re the idea of a non -ze ro v acu um ene rgy -mo men tum ﬂow, one can ado pt a phe - nomenological model [ 25]  where the physical vacuum is described as a relativistic ﬂuid  [34] . In this representation, a non-zero
 
ˆ
W
0
i

Ψ
′
 gives rise to a tiny heat ﬂow and an eﬀective ther- mal 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 calor imetr ic devic es. Ho wev er, it could even tually be detec ted throug h ve ry accu rate ether- drift 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
2
L
(
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] .
6

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 cav ities. These can produc e an anisotrop y of the tw o-w ay velocit y of light. In this sense , the referen ce 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 recen t ve rsion s of the origi nal Hughes-D rev er experim ent   [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. There fore, mech anism a), if presen t, is completel y negli gible and, from now on, we shall assume
  L
(
θ
) =
 L
 =cons tan t. In this wa y , one re-obtain s the standard relation adopted in the analysis of the experiments ∆
ν
phys
(
θ
)
ν
0
=   ¯
c
γ
(
π/
2 +
θ
)
−
¯
c
γ
(
θ
)
c
  ≡
  ∆¯
c
θ
c
 (14) where
 ν
0
 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 wav eleng th. In the opposite case of an isotro pic, dilute random medium  [41] as a gas, it is relatively easy to compute the scattered wave in the forward direction and obtai n the refractiv e index . How eve r, the presence of con ve ctiv e curre nts 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 isotr opy , the time
  t
 spent by refracted light to cover some given distance
L
 within the medium is
  t
  =
  N
L/c
. Th is ca n be exp re ss ed as the sum of
  t
0
  =
  L/c
  and
t
1
 = (
 N −
1)
L/c
 where
 t
0
 is the same time as in the vacuum and
 t
1
 represents the additional, av erag e time by which refra cted light is slowe d dow n by the presenc e of matte r. If there are convective currents, due to the motion of the laboratory with respect to a preferred reference frame Σ, then
  t
1
 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 consi der lig ht prop aga ting in a 2-dimensional plane and express
  t
1
  as
t
1
 =
  L c  f
(
 N
,θ,β
) ( 1 5 ) 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 ( 1 6 ) is assumed . By expand ing 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
) ( 1 7 ) 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 c
γ
(
 N
,θ,β
)
 ∼
  L c
  +
  L c
 (
 N −
1)
  F
(
θ, β
) ( 1 9 ) one gets
c
γ
(
 N
,θ,β
)
∼
  c
 N
  [1
−
(
 N −
1) (
F
(
θ, β
)
−
1 ) ] ( 2 0 ) Analogous relations hold for the two-way velocity ¯
c
γ
(
 N
,θ,β
) ¯
c
γ
(
 N
,θ,β
) =   2
  c
γ
(
 N
,θ,β
)
c
γ
(
 N
, π
 +
θ, β
)
c
γ
(
 N
,θ,β
) +
c
γ
(
 N
, π
 +
θ, β
)
 ∼
  c
 N

1
−
(
 N −
1)

F
(
θ, β
) +
F
(
π
 +
θ, β
) 2
  −
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 =
 β F
1
(
θ
) +
β
2
F
2
(
θ
) +
...
 (22) and tak ing in to acc oun t tha t, by the very deﬁ nit ion of tw o-w ay ve loci ty , ¯
c
γ
(
 N
,θ,β
) = ¯
c
γ
(
 N
, θ,
−
β
), it follows that
  F
1
(
θ
) =
 −
F
1
(
π
 +
 θ
). The ref ore , to
 O
(
β
2
), we get the general structure [ 26 ] ¯
c
γ
(
 N
,θ,β
)
∼
  c
 N

1
−
(
 N −
1)
 β
2
∞

n
=0
ζ
2
n
P
2
n
(cos
θ
)

 (23) in which we have expressed the combination
  F
2
(
θ
) +
 F
2
(
π
 +
 θ
) as an inﬁnite expansion of  even-order Legendre polynomials with unknown coeﬃcients
  ζ
2
n
  =
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 compa red 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
−
β
2
(
 N −
1)(
A
 +
B
sin
2
θ
) ] ( 2 4 ) with
  A
 = 2 and
  B
  =
 −
1 which corresponds to setting
  ζ
0
  = 4
/
3,
  ζ
2
  = 2
/
3 and all
  ζ
2
n
  = 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
. Summ ari zin g: in thi s sc hem e, the theor eti cal est ima te for a possi ble aniso tro py of the two-way velocity of light is

∆
ν  ν
0

Theor gas
=

∆¯
c
θ
c

Theor gas
∼
(
 N
gas
−
1)
  V
2
c
2
 (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
.
00003 5. There fore these potent ial eﬀects are much larger than those p ossibl y associated with vacuum cavities. In fact, from experiments one ﬁnds  [44]
−
[ 50 ]

∆
ν  ν
0

EXP vacuum
=

∆¯
c
θ
c

EXP vacuum
∼
10
−
15
(27)
3
One 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
 N
air
 = 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/
 N
air
 as an angular average of Eq.( 24 ), i.e.
c
 N
air
≡ 
¯
c
γ
(  ¯
 N
air
,θ,β
)

θ
  =
  c
¯
 N
air
[1
−
  3 2 (  ¯
 N
air
−
1)
β
2
] ( 2 5 ) 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
  N
air
  ≡ N
(
Earth
) and of
  V
 . In prac tic e, 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 complet ely negligi ble. The same holds true for the other gaseous syst ems at STP (say nitroge n, carbon dioxi de, helium,. .) for which the pres ent experi men tal 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 give n 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 mediu m. This conclu sion is in agree men t with the experimen ts [ 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 th e diﬀerenc e 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). Altho ugh it woul d b e diﬃcult to describe in a fully quan titati ve 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 mean time, due to the heuristi c nature of our approac h, and to further motiv ate 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 propa gatin g in air or helium at atmos pheric pres sure. 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
(
β
2
) 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 ). The n, an aniso tro py of the tw o-w ay ve locit y of ligh t could be measu red by rotat ing a Mic helson inter ferom eter. As anticipa ted, 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
) =   2
L
¯
c
γ
(
 N
,θ,β
)
 −
  2
L
¯
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
  v
2
c
2
 the fringe shift ∆
λ
(
θ
)
λ
  ∼
  c
∆
T
(
θ
)
 N
λ
  ∼
  L λ v
2 obs
c
2
 cos2(
θ
−
θ
0
) ( 2 9 ) 10

In the above equation the angle
  θ
0
  =
 θ
0
(
t
) indicates the apparent direction of the ether-drift in the plane of the interferometer (the ‘azimuth’) and the square of the
 observable
 velocity
v
2 obs
(
t
)
∼
2(
 N −
1)
v
2
(
t
) ( 3 0 ) is re-scaled by the tiny factor 2(
 N −
1) with respect to the true
 kinematical
 velocity
  v
2
(
t
). We emphasize that
 v
obs
 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 avo id its intr oduction altog ether . Ho wev er, 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
 →
v
obs
. F or this reason , as we shall see in the following sections, it is in terms of
 v
obs
, 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 conclusio n, in this sc heme, the int erpre tation of the experim ent s is transp aren t. Ac- cordi ng to Special Relativ ity , there can b e no fring e shift upon rotat ion of the inte rfero mete r. 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
  v
obs
  =
  v
 = 0. On the other hand , if there we re fring e shifts , one cou ld try to deduce the existence of a preferred frame Σ

=
 S
′
 provided the following minimal requirements are fulﬁlle d : i) the fringe shifts exhibi t an angula r 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
 v
obs
 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 exis tence of an ether-dri ft. Ho wev er inte rpret ativ e models are requi red to compar e result s obtai ned at diﬀere nt times and in diﬀer ent place s. In the scheme of Eqs.( 29 ) and ( 30 ), the crucial information is contained in the two time-dependent functions
 v
 =
 v
(
t
) and
 θ
0
 =
 θ
0
(
t
), respectively the magnitude of the velocity and the apparent direction of the azimuth in the plane of the interf erome ter. F or their determ inatio n, the standard assumpt ion is to consider a cos mic Ear th’ s ve loci ty with we ll deﬁ ned mag nitu de
  V
, rig ht asc ens ion
  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
  θ
0
(
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
) ( 3 1 ) ˜
v
x
(
t
)
V
  ≡
sin
z
(
t
)cos  ˜
θ
0
(
t
) = sin
γ
 cos
φ
−
cos
γ
sin
φ
cos(
τ
 −
α
) ( 3 2 ) ˜
v
y
(
t
)
V
  ≡
sin
z
(
t
)sin  ˜
θ
0
(
t
) = cos
γ
 sin(
τ
 −
α
) ( 3 3 ) ˜
v
(
t
)
≡

˜
v
2
x
(
t
) + ˜
v
2
y
(
t
) =
 V
  sin
z
(
t
)
,
 (34) Here
  z
  =
  z
(
t
) is the zenithal distance of
  V
,
  φ
 is the latitude of the observatory,
  τ
  =
  ω
sid
t
is the sidereal time of the observation in degrees (
ω
sid
 ∼
  2
π
23
h
56
′
) and the angle
  θ
0
 is counted conventionally from North through East so that North is
  θ
0
 = 0 and East is
 θ
0
 = 90
o
. To explore the observable implications, let us ﬁrst re-write the basic Eq.( 29 ) as ∆
λ
(
θ
)
λ
  ∼
  2
L
(
 N −
1)
λ v
2
(
t
)
c
2
 co s 2(
θ
−
θ
0
(
t
))
≡
2
C
(
t
)cos2
θ
 + 2
S
(
t
)sin2
θ
 (35) where
C
(
t
) =
  L
(
 N −
1)
λ v
2
(
t
)
c
2
 cos2
θ
0
(
t
)
  S
(
t
) =
  L
(
 N −
1)
λ v
2
(
t
)
c
2
 sin2
θ
0
(
t
) ( 3 6 ) Then Eqs. ( 31 )
−
( 34 ) amount to the structure
S
(
t
)
≡
  ˜
S
(
t
) =
 S
s
1
sin
τ
 +
S
c
1
cos
τ
 +
S
s
2
sin(2
τ
) +
S
c
2
cos(2
τ
) ( 3 7 )
C
(
t
)
≡
  ˜
C
(
t
) =
 C
0
 +
C
s
1
sin
τ
 +
C
c
1
cos
τ
  +
C
s
2
sin(2
τ
) +
C
c
2
cos(2
τ
) ( 3 8 ) with Fourier coeﬃcients (
R ≡
  L
(
 N −
1)
λ V
 2
c
2
 )
C
0
 =
−
1 4
R
(3cos2
γ
−
1)cos
2
φ
 (39)
C
s
1
 =
−
1 2
R
sin
α
sin2
γ
 s in 2
φ C
c
1
 =
−
1 2
R
cos
α
sin2
γ
sin2
φ
 (40)
C
s
2
 =   1 2
R
sin2
α
cos
2
γ
(1 + sin
2
φ
)
  C
c
2
 =   1 2
R
cos2
α
cos
2
γ
(1 + sin
2
φ
) ( 4 1 ) and
S
s
1
 =
−
  C
c
1
sin
φ   S
c
1
 =
  C
s
1
sin
φ
 (42)
S
s
2
 =
−
 2sin
φ
1 + sin
2
φ C
c
2
  S
c
2
 =  2sin
φ
1 + sin
2
φ C
s
2
 (43) These standard forms are nowadays adopted in the analysis of the data of the ether-drift experiments [ 46] . Ho we ve r, one shou ld not forge t tha t 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 regul ar motion. Inste ad, the phy sical va cuum 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 eth er- dri ft mig ht exh ibi t for ms of tim e modu lat ion s tha t do
  not
 ﬁt in the scheme of  Eqs.( 37 )
,
( 38 ). T o ev alu ate the poten tia l eﬀe cts , and by sti ll ret ain ing the fun cti ona l for m Eq.( 35 ), one could ﬁrst re-write Eqs.( 36 ) as
C
(
t
) =
  L
(
 N −
1)
λ v
2
x
(
t
)
−
v
2
y
(
t
)
c
2
  S
(
t
) =
  L
(
 N −
1)
λ
2
v
x
(
t
)
v
y
(
t
)
c
2
 (44) where
  v
x
(
t
) =
  v
(
t
)cos
θ
0
(
t
) and
  v
y
(
t
) =
  v
(
t
)sin
θ
0
(
t
). The n, by exp loi tin g the turb ule nce scenario, one could model the two velocity components
  v
x
(
t
) and
  v
y
(
t
) as stochastic ﬂuctua- tions . In this diﬀeren t sch eme, where now
  v
(
t
)

= ˜
v
(
t
) and
  θ
0
(
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
 ne cessa rily mean that there is no ether-dr ift. This partic ular aspect will be discussed at length in the rest of the paper. After this importan t premise, we shall now proceed in Sects . 3-8 with our re-a nalysi s of  the classical experiments. In the end, Sect.9 will contain a summary, a brief discussion of the modern experimen ts and our concl usions .
3. The o rig ina l Mic hel son -Mo rle y exper ime n t
The Michelson-Morley experiment   [61 ] is probably the most celebrated experiment in the his tor y of ph ysi cs. Its resul t and its in ter pre tat ion ha v e 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 large ly reduc ed and become unobserv able. These points of view are summar ized in Hedr ick ’s contribution to the ‘Conference on the Michelson-Morley experiment’ [ 63]  (Pasadena, Febru- ary 1927) which was attended by the greatest experts of the time, in particular Lorentz and 13

Michelson. The arguments presented by Hedrick were, how ever, refuted by Kennedy [ 64]  in a paper of 1935 where, by using Huygens principle, he re-obtained to order
  v
2
/c
2
the classical result of Eq.( 29 ) (with the identiﬁcation
  v
obs
  =
 v
). Figure 1:
 The Michelson-Morley fringe shifts as reported by Hicks [  66   ]. Solid and dashe d lines  refer respectively to noon and evening observations.
In this framewo rk, the fringe shift is a seco nd-har monic eﬀe ct, i.e. periodic in the range [0
, π
], whose amplitude
  A
2
 is predicted diﬀerently by using the classical formulas or Lorentz transformations ( 29 )
A
class 2
  =
  L λ v
2
c
2
  A
rel 2
  =
  L λ v
obs 2
c
2
  ∼
2(
 N −
1)
A
class 2
 (45) Notice also that upon rotation of
 π /
2 with respect to
 θ
 =
 θ
0
 the predicted fringe shift is 2
A
2
. No w, for the Mic helso n-Mor ley inte rfero meter the whole eﬀectiv e optic al path was about
L
 = 11 meters, or about 2
·
10
7
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
 A
class 2
  ∼
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
  A
2
 through the relation
v
obs
 ∼
30

A
2
0
.
2   km
/
s ( 4 6 ) 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 min ute s. As well summar ize d by Mille r in 193 3   [65 ], “The brief series of observations was suﬃci ent to sho w clea rly that the eﬀect did not have the antici pated magnitude . Ho wev er, 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 dat a publishe d by Michelson and Morley, instead of giving a null result, show distinct evidence for an eﬀect of the kind to be expected” . Name ly , there was a seco nd-har monic eﬀec t. 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. Com par ing with the cla ssi cal predi cti on for a ve loci ty of 30 km/s, this means an experimental 2nd- harmonic amplitude
A
EXP 2
  ∼
0
.
2 ( 8
.
4 30  )
2
∼
0
.
0 1 6 ( 4 7 ) 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 Michels on-Mo rley data. T o unders tand the precision of their reading s, we can look at the original paper [ 61]  where one ﬁnds the following statement : ”The readings are divisions of the screw-hea ds. The width of the fringe s varie d from 40 to 60 divisions, the mean valu e being near 50, so that one divisi on means 0.02 wav eleng th”. No w, in their tables Mich elson 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 accur acy of the readings was
 ±
0
.
002 wav eleng ths. 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 , Bor n, when discu ssi ng the classically expected fractional fringe shift upon rotation of the apparatus by 90
o
, 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 wo uld still b e obs erv abl e” (i. e. 0.0 037 ). The ref ore , to be con sis ten t wit h both the ori gin al 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 ]. T o this en d, one should ﬁrst follow the well deﬁned procedure adopted in the classical experiments as described in Miller’s paper [ 65] . Name ly , by starti ng from each set of seven teen entrie s (one every 22
.
5
o
), 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 obt ain ed after a com ple te rot ati on of the app ara tus . In this wa y , by addi ng 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
E
corr
(
i
) =
  i
−
1 16   (
E
(1)
−
E
(17)) +
E
(
i
) ( 4 8 ) The fringe shifts are then deﬁned by the diﬀerences between each of the corrected entries
4
To conﬁrm that such estimate should not be considered unrealistically small, we report explicitly Michel- son’s words from ref. [63] :“I must say that every beginner thinks himself lucky if he is able to observe a 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 no on (n.) and even ing (e. ) sessi ons of th e Miche lson- Morley experiment.
i J u l y 8 ( n . ) J u l y 9 ( n . ) J u l y 1 1 ( n . ) J u l y 8 ( e . ) J u l y 9 ( e .) J ul y 1 2 ( e . ) 1
  −
0 . 0 0 1 + 0 . 0 1 8 + 0 . 0 1 6
  −
0 . 0 1 6 + 0 . 0 0 7 + 0 . 0 3 6 2 + 0 . 0 2 4
  −
0.004
  −
0 . 0 3 4 + 0 . 0 0 8
  −
0 . 0 1 5 + 0 . 0 4 4 3 + 0 . 0 5 3
  −
0.004
  −
0.038
  −
0 . 0 1 0 + 0 . 0 0 6 + 0 . 0 4 7 4 + 0 . 0 1 5
  −
0.003
  −
0 . 0 6 6 + 0 . 0 7 0 + 0 . 0 0 4 + 0 . 0 2 7 5
  −
0.036
  −
0.031
  −
0 . 0 4 2 + 0 . 0 4 1 + 0 . 0 2 7
  −
0.002 6
  −
0.007
  −
0.020
  −
0 . 0 1 4 + 0 . 0 5 5 + 0 . 0 1 5
  −
0.012 7 + 0 . 0 2 4
  −
0 . 0 2 5 + 0 . 0 0 0 + 0 . 0 5 7
  −
0 . 0 2 2 + 0 . 0 0 7 8 + 0 . 0 2 6
  −
0 . 0 2 1 + 0 . 0 2 8 + 0 . 0 2 9
  −
0.036
  −
0.011 9
  −
0.021
  −
0 . 0 4 9 + 0 . 0 0 2
  −
0.005
  −
0.033
  −
0.028 10
  −
0.022
  −
0.032
  −
0 . 0 1 0 + 0 . 0 2 3 + 0 . 0 0 1
  −
0.064 11
  −
0 . 0 3 1 + 0 . 0 0 1
  −
0 . 0 0 4 + 0 . 0 0 5
  −
0.008
  −
0.091 12
  −
0 . 0 0 5 + 0 . 0 1 2 + 0 . 0 1 2
  −
0.030
  −
0.014
  −
0.057 13
  −
0 . 0 2 4 + 0 . 0 4 1 + 0 . 0 4 8
  −
0.034
  −
0.007
  −
0.038 14
  −
0 . 0 1 7 + 0 . 0 4 2 + 0 . 0 5 4
  −
0 . 0 5 2 + 0 . 0 1 5 + 0 . 0 4 0 15
  −
0 . 0 0 2 + 0 . 0 7 0 + 0 . 0 3 8
  −
0 . 0 8 4 + 0 . 0 2 6 + 0 . 0 5 9 1 6 + 0 . 0 2 2
  −
0 . 0 0 5 + 0 . 0 0 6
  −
0 . 0 6 2 + 0 . 0 2 4 + 0 . 0 4 3
E
corr
(
i
) and their average value
 
E
corr

  as ∆
λ
(
i
)
λ
  =
 E
corr
(
i
)
− 
E
corr

 (49) The resulting data are reported in Table 1. With this procedure, the fringe shifts Eq.( 49 ) are given as a periodic function, with van- ishing mean, in the range 0
≤
θ
 ≤
2
π
, with
  θ
 =
  i
−
1 16
  2
π
, 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
 +
θ
) 2
λ
 (50) (see Fig.3). This is essen tial to cance l the 1st-harm onic contr ibutio n origin ally pointed out by Hicks   [66 ]. Its theor eti cal in ter pre tat ion is in ter ms of the arran gem en ts of the mirro rs 17

Table 2:
 The amplitude of the ﬁtted second-harmonic component
 A
EXP 2
  for the six experimenta l  sessions of the Michelson-Morley experiment.
SESSION
  A
EXP 2
J u l y 8 ( n oo n ) 0
.
010
±
0
.
005 J u l y 9 ( n oo n ) 0
.
015
±
0
.
005 J u l y 1 1 ( n oo n ) 0
.
025
±
0
.
005 Ju ly 8 (e v en in g) 0
.
014
±
0
.
005 Ju ly 9 (e v en in g) 0
.
011
±
0
.
005 Ju ly 12 (e v en in g) 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 obser v ation s wer e wel l consi sten t with Hic ks’ theor etica l study . The obser ve d 1st-h armoni c 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
  A
EXP 2
  ∼
0
.
016
±
0
.
006. This v alue is consistent with an observable velocity
v
obs
 ∼
8
.
4
+1
.
5
−
1
.
7
  km
/
s ( 5 1 ) 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
+62
−
70
  km
/
s ( 5 2 ) While the individual values of
  A
2
 show a reaso nable consisten cy , there are substa ntia l changes in the apparent direction
 θ
0
 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
±
180
o
, one choice for the experimental azimuths is 357
o
±
14
o
, 285
o
±
10
o
and 317
o
±
8
o
respec tiv ely for July 8th, 9th and 11th. F or this assignmen t, the individual velocity vectors
 v
obs
(cos
θ
0
,
−
sin
θ
0
) and their mean are shown in Fig.4. According 18

θ
-0.04 -0.02 0 0.02 0.04
       Β        (        θ        )
July 11 noon
0
  π   π
 /2
Figure 3:
 A ﬁt to the even combination
  B
(
θ
)
 Eq.(  50  ). The sec ond harmo nic ampli tud e is
A
EXP 2
  = 0
.
025
±
0
.
005
 and the fourth harmonic is
  A
EXP 4
  = 0
.
004
±
0
.
005
. The ﬁgure is taken   fro m 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-z ero fringe shift is due to pure instrum ent al eﬀects. Ho wev er, as anticipa ted 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 insta nce, in agree men t with the gene ral struc ture Eq.( 23 ), and diﬀerently from July 11 noon, which represents a very clean indication, there are sizeable 4th- harmonic contributions (here
A
EXP 4
  = 0
.
019
±
0
.
005 and
  A
EXP 4
  = 0
.
008
±
0
.
005 for the noon sessions of July 8 and July 9 respec tiv ely). In any case, the observ ed strong variat ions of
  θ
0
 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 wer e actua lly perform ed. Thus one could also concl ude that indiv idual experimen tal sess ions indicate a deﬁnite non-zero ether-drift but the azimuth does not exhibit the smooth trend expected from the conventional picture Eqs.( 31 )
−
( 34 ). F or com ple ten ess , we add tha t the large spr ead of the
  θ
0
−
values might also reﬂect a particular systematic eﬀect pointed out by Hicks   [66 ]. As desc ribe d by Mille r  [65] , “ before beginning observations the end mirror on the telescope arm is very carefully adjusted to secure 19

0 1 2 3 4   5 6   7   8 0 1 2 3 4 5 6 7 8 9
V
x
obs
V
obs
y
July 8 July 11 July 9
Mean
(Km/s) (Km/s)
Figure 4:
 The obser vable velo citie s for the thre e no on sessions and their me an. The x-axis  corr espond s to
  θ
0
  = 0
o
≡
  360
o
and the y-axis to
  θ
0
 = 270
o
. Stati stic al uncert ainti es of the  various determinations are ignored. All individual directions could also be reversed by 180
o
.
ve rtica l fring es of suitab le width. There are two adjustme nts of the angle of this mirror whic h 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 unl ess the types are all the sam e. If this is not att end ed to, the av era ge dis pla cem en t ma y be expected to come out zero
 −
 at least if a large number are averaged” [ 66] . The ref ore averaging the fringe shifts from various sessions represents a delicate issue and can introduce uncon trolle d errors. Clearl y , this relativ e sign does not aﬀect the va lues of
  A
2
 and this is why av eragi ng the 2nd-harmon ic amplitudes is a safer procedur e. How eve r, it can introduce spurious changes in the apparent direction
  θ
0
 of the ether-drift. In fact, an overall change of  sign of the fringe shifts at all
  θ
−
values is equivalent to replacing
 θ
0
 →
θ
0
±
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 ev ening session s respect ive ly . If this wer e true, one cho ice for the azimut h of July 8th could now be
 θ
EXP 0
  = 267
o
±
14
o
. 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
 
θ
EXP 0
   ∼
  317
o
to
 
θ
EXP 0
   ∼
 290
o
. W e’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 displace men t was certa inly less than the twe ntie th 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... certa inly less than one-f ourth of the orbit al earth’s velocit y”. The straightforward translation of this upper bound is
 v
obs
 <
 7.5 km/s. However, this estimate is likely aﬀecte d by a theoretic al uncer tain ty . In fact, in their Fig.6, Mic helson and Morley reported their measured fringe shifts together with the plot of a theoretical second-harmonic com pone nt . In doing so, the y plo tte d a wa ve of amp lit ude
  A
2
  = 0
.
05, that they interpret as
 one-eight
 of the theoretical displacement expected on the base of classical physics, thus implicitly assuming
 A
class 2
  =0.4. As discus sed abov e, the amplitu de of the class ically expec ted second-harmonic component is
  not
 0.4 but is just one-half of that,
 i.e.
 0.2. There fore, the ir experimental upper bound
  A
exp 2
  <
  0
.
4 20
 =0.02 is actually equivalent to
  v
obs
  <
 9 .5 km /s . If  we no w con sid er tha t the ir est ima tes we re obt ain ed aft er supe rim posin g the fri nge shi fts 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 . Mo rl e y- Mi ll er
After the original 1887 experiment, there was much interest in the Michelson-Morley result tha t, bein g too sma ll to mee t an y cla ssi cal pre dic tio n, wa s app are nt ly con tra dic tin g tw o corne rstone s of physics : Galile i’s trans format ions and/or the exist ence of the ether. F or 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, super- imposed the data of the mornin g with those of the eve ning. As explaine d by Miller  [63] , the two physicists were assuming that the ether drift had to be obtained by combining the motion of the solar syste m relativ e to nearby stars , i.e. tow ard the const ellat ion of Herc ules 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 obse rv ati ons at the tw o tim es (about 11:3 0 a.m . and 9:00 p.m.) wer e, therefore , com bined in such a wa y that the presume d 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 nea rly oppos ite phas es. Whe n these wer e com bin ed, the res ult was near ly zero . F or this reason, the value then reported of an observable velocity of 3.5 km/s is incorrect and does
not
 corres pond to the actual results of the basic observ ations . The error was later underst ood 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
v
obs
 ∼
(8
.
5
±
1
.
5) km
/
s ( 5 3 ) By using Eq.( 30 ), we then deduce the average value
v
 ∼
(353
±
62) km
/
s ( 5 4 )
5. Ken nedy -Il lin gw ort h
An inter esting dev elopm ent was proposed by Kenne dy in 1926. As summariz ed in his con- tribution to the previously mentioned Conference on the Michelson-Morley experiment  [63] , his small optical system was enclosed in an eﬀectively insulated, sealed metal case containing hel ium at atm osp her ic pre ssu re. Bec aus e of its small siz e, ”.. .ci rcu lat ion and var iat ion in densi ty of the gas in the ligh t paths were nearl y eliminated . F urthe rmore , since the valu e 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 princ iple, for an ultima te fringe shift accur acy 1
·
10
−
4
. To take full advantage of this possibil- ity, Kennedy should have disposed of perfect mirrors and of a suitable (hotter) source of light. In the original version of the experiment, these reﬁnements were not implemented giving an actual fringe shift accuracy of 2
·
10
−
3
. In these condit ions, as Kennedy expli citly say s[ 63] , ”...th e ve locity of 10 km/s found by Prof. Mille r woul d produce a fringe shift corre sponding 22

Table 3:
 The infra-session averages
 
D
A

  and
 
D
B

 obtained from the 10 sets of rotations in  each of the 32 sessions of Illingworth’s experiment. These values have been obtained from the  weights of Illingworth’s Table III by applying the conversion factor 0.002.
5 A . M . 5 A . M . 1 1 A . M . 1 1 A . M . 5 P . M . 5 P . M . 1 1 P . M . 1 1 P . M .

D
A
 
D
B
 
D
A
 
D
B
 
D
A
 
D
B
 
D
A
 
D
B

+0
.
00024
  −
0
.
0 0 0 6 6 + 0
.
00070
  −
0
.
0 0 0 2 2 + 0
.
0 0 0 2 4 + 0
.
00044
  −
0
.
0 0 0 1 0 + 0
.
00024 +0
.
0 0 1 1 4 + 0
.
00024
  −
0
.
00042
  −
0
.
00036
  −
0
.
00056
  −
0
.
0 0 0 4 6 + 0
.
0 0 0 1 8 + 0
.
00018 +0
.
0 0 0 0 0 + 0
.
00000
  −
0
.
00006
  −
0
.
00052
  −
0
.
00144
  −
0
.
00080
  −
0
.
00126
  −
0
.
00006 +0
.
00020
  −
0
.
00044
  −
0
.
0 0 0 3 0 + 0
.
00012
  −
0
.
0 0 0 1 6 + 0
.
00004
  −
0
.
00044
  −
0
.
00026 +0
.
0 0 0 6 4 + 0
.
00000
  −
0
.
0 0 0 2 2 + 0
.
0 0 0 3 8 + 0
.
0 0 0 1 8 + 0
.
0 0 0 1 6 + 0
.
0 0 0 0 0 + 0
.
00024
−
0
.
00002
  −
0
.
0 0 0 1 0 + 0
.
0 0 0 4 8 + 0
.
0 0 0 2 0 + 0
.
0 0 0 3 0 + 0
.
00030
  −
0
.
00040
  −
0
.
00004
−
0
.
00014
  −
0
.
0 0 0 0 6 + 0
.
0 0 0 3 0 + 0
.
00014
−
0
.
0 0 0 0 6 + 0
.
0 0 0 0 4 + 0
.
00036
  −
0
.
00036
−
0
.
0 0 0 0 6 + 0
.
0 0 0 1 6 + 0
.
00006
  −
0
.
00006 +0
.
0 0 0 0 0 + 0
.
00024
  −
0
.
0 0 0 1 0 + 0
.
00010 to 8
·
10
−
3
”, four times large r than the experimen tal resoluti on. Since the eﬀect is quadr atic in the velocity, Kennedy’s result, fringe shifts
 <
 2
·
10
−
3
, can then be summarized as
v
obs
 <
 5 km
/
s ( 5 5 ) By usi ng Eq. ( 30 ), for helium at atm osp her ic pre ssu re whe re
  N ∼
  1
.
000 035 , thi s bound amounts to restrict the kinematical value by
  v <
 600 km/s. Kennedy’s apparatus was further reﬁned by Illingworth in 1927   [71 ]. Besid es impro ving the quality of the mirrors and of the source, Illingworth’s data taking was also designed to reduc e the prese nce of stead y thermal drift and of odd harmo nics. Looking at Illingwor th’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 analyz e Illingwor th’s results . He performed four serie s of obser v ation s in the ﬁrs t ten days of Jul y 192 7. The se con sis ted of 32 expe rim en tal ses sio ns, con duc ted 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 fring e displaceme nt caused by a rotat ion through a righ t angle of the apparatu s. T o tak e into 23

account 90
o
rotations let us ﬁrst re-write Eq.( 29 ) as ∆
λ
(
θ
)
λ
  =
 A
2
 co s 2(
θ
−
θ
0
) ( 5 6 ) Therefore Illingworth, in his ﬁrst set (set A) of 10 rotations, North, East, South, West and back to North, was actually measuring
 D
A
 ≡
2
A
2
 c os 2
θ
0
. 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
 D
B
 ≡
2
A
2
 si n 2
θ
0
. Notice that both
 D
A
  and
 D
B
 diﬀer from the positiv e-de ﬁnite quan tity
  D
  ≡
  2
A
2
 that should be inserted in Illingworth’s numerical relation for his apparatus
 v
obs
 = 112
√
D
. Therefore, the reported values for the two velocities
v
A
 = 112

|
D
A
|
 and
 v
B
 = 112

|
D
B
|
 should only be taken as
 lower
 bounds for the true
 v
obs
. The mean values
 
D
A

  and
 
D
B

 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


D
A

2
+

D
B

2
has a mean value of about 0.00045, which corresponds to
  v
obs
 ∼
 2
.
4 km/ s. Th us, by usi ng Eq.( 30 ) for helium at atmo- spheric pressure, we would tentatively deduce an average value
  v
 ∼
284 km/s. How ev er, this is only a ve ry partial view. T o go deeper into Illingw orth’s experimen t we hav e to consid er his basic measure men ts, i.e. the individ ual turns of his interf erome ter. 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 M´ unera [ 72 ] and his values for the fringe shifts are reported in our Table 4. As one can see, the fringe shifts are not small and correspond to an observable velocity 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,
 
D
A
 ∼ −
0
.
00006, the resultin g value 0.9 km/s is muc h smaller than all indivi dual determi nation s. The basis of M´ unera’s analysis was instead to estimate
  v
obs
 from
 |
D
A
|
, from which he obtained an av erage velocity
  v
obs
  = 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 av erag ing procedur e as emplo ye d by Illingw orth. 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 M´ unera’s re-estimate has a deﬁnite signiﬁcance. To this end, we shall ﬁrst use the relations
D
A
(
t
) = 4
C
(
t
)
  D
B
(
t
) = 4
S
(
t
) ( 5 7 ) 24

Table 4:
 Illingworth’s set A of July 9th, 11 A.M. as re-analyzed by M´  unera [  72   ].
R o t a t i o n D
A
  |
D
A
|
  v
A
[km/s] 1
  −
0
.
0 0 1 0 0 + 0
.
00 1 00 3. 54 2 +0
.
0 0 0 6 6 + 0
.
00 0 66 2. 89 3
  −
0
.
0 0 0 6 6 + 0
.
00 0 66 2. 89 4
  −
0
.
0 0 0 6 6 + 0
.
00 0 66 2. 89 5
  −
0
.
0 0 1 6 6 + 0
.
00 1 66 4. 57 6 +0
.
0 0 2 3 4 + 0
.
00 2 34 5. 41 7 +0
.
0 0 1 0 0 + 0
.
00 1 00 3. 54 8 +0
.
0 0 0 3 4 + 0
.
00 0 34 2. 04 9 +0
.
0 0 0 0 0 + 0
.
00 0 00 0. 00 10
  −
0
.
0 0 1 0 0 + 0
.
00 1 00 3. 54 where the two functions
  C
(
t
) and
  S
(
t
) have been introduced in Eqs.( 36 ) and ( 44 ). Th us Eqs.( 57 ) can be re-written as
D
A
(
t
) =   8
L
(
 N −
1)
λ v
2
x
(
t
)
−
v
2
y
(
t
) 2
c
2
  D
B
(
t
) =   8
L
(
 N −
1)
λ v
x
(
t
)
 v
y
(
t
)
c
2
 (58) where
  v
x
(
t
) =
  v
(
t
)cos
θ
0
(
t
) and
  v
y
(
t
) =
  v
(
t
)sin
θ
0
(
t
). In this way , by usin g the num eri cal relation for Illingworth’s experiment
  L λ
(30km
/
s)
2
c
2
  ∼
0
.
035 and the value of the helium refractive index, we obtain
D
A
(
t
)
∼
  v
2
x
(
t
)
−
v
2
y
(
t
) 2
·
(300 km
/
s)
2
 ·
10
−
3
D
B
(
t
)
∼
  v
x
(
t
)
  v
y
(
t
) (300 km
/
s)
2
 ·
10
−
3
(59) The required random ingredient can then be introduced by characterizing the two velocity components
 v
x
(
t
) and
 v
y
(
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 eﬀec ts of turbul ence ). Ho wev er, it is a usefu l example to illust rate basic pheno menolo gical feat ures associated with an underl ying stochast ic v acuum. T o explo re 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 rando m Fou rier series . The perspectiv e is that of an observ er moving in the turbul ent ﬂuid
5
This 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
v
x
(
t
) =
∞

n
=1
[
x
n
(1)cos
ω
n
t
+
x
n
(2)sin
ω
n
t
] ( 6 0 )
v
y
(
t
) =
∞

n
=1
[
y
n
(1)cos
ω
n
t
+
y
n
(2)sin
ω
n
t
] ( 6 1 ) where
 ω
n
 = 2
nπ/T
, T being a time scale which represents a common period of all stochastic compone nts . W e have adopte d the typi cal value
  T
  =
  T
day
= 24 hou rs. Ho we v er, we hav e 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
  T
day
 ≤
T
 ≤
10
 T
day
. The coeﬃcients
  x
n
(
i
 = 1
,
2) and
  y
n
(
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

x
2
n
(
i
 = 1
,
2)

stat
  =

y
2
n
(
i
 = 1
,
2)

stat
  =   ˜
v
2
3
 n
2
η
 (62) for the uni for m pro bab ili ty mode l (wi thi n the in ter v al [
−
˜
v,
˜
v
]) which we have chosen for our simu lat ion s. Fin all y , the expone nt
  η
 con tro ls the pow er spec tru m of the ﬂuc tua ting compone nts . F or the simulat ions, b etw een the tw o v alues
  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. Th us, wit hin this sim ple model for
  D
A
(
t
) and
  D
B
(
t
), ˜
v
 is the only par ame ter 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
 =
 V
CMB
 = 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 90
o
rotations) were recorded every 30 seconds. Therefore, this type of simulations consists in generating 40 values during a total time of 1200 sec ond s. As an ill ust rat ion , tw o ty pic al seq uen ces of D
A
(t) and D
B
(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

D
A

  = 0
.
00028 and
 
D
B

  = 0
.
00011 are completely consistent with the typical entries of  Table 3. 26

1400  1600  1 8 0 0 2 0 0 0 2 2 0 0 2 4 0 0
time (s)
-4 -2 0 2 4
D (t)
2 0 0 4 0 0   600  8 0 0 1 0 0 0 1 2 0 0 -3 -2 -1 0 1 2 3
D (t)
A B
Figure 5:
  A sim ulati on of
 D
A
(t)
 and
 D
B
(t)
, in units
 10
−
3
and every 30 seconds, from typical  sequenc es of 1200 sec onds. The average values are
 
D
A

= 0
.
00028
  and
 
D
B

= 0
.
00011
. The  velocity parameter is
  ˜
v
 =
 V
CMB
 =
 370 km/s.
To obtain further insight, we have then performed extensive simulations for large sequences of measurem ent s. The histogr ams of a set of 10000 deter minatio ns of
 D
A
(
t
) and
 D
B
(
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 behav ior is cha racte ristic of proba bilit y distributio ns for instan taneo us data in turbulen t ﬂows (see e.g. [ 76, 77] ). T o b ette r appreciate the deviatio n from Gaussi an behavior, in panels (c) and (d) we plot the same data in a log
−
log sc ale . The re sul ting distributions are well ﬁtted by the so-called
  q
−
exponential function   [78 ]
f
q
(
x
) =
 a
(1
−
(1
−
q
)
xb
)
1
/
(1
−
q
)
(63) with entropic index
 q
 ∼
1
.
1. F or such large samples of data, the statistical averages
 
D
A

and

D
B

are vanishingly small in units of the typical instantaneous values
O
(10
−
3
) and any non- zero average has to be considered as statistical ﬂuctuation. On the other hand, the standard deviations
 σ
(
D
A
) and
 σ
(
D
B
) 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
(
D
A
)
∼
(0
.
74
±
0
.
05)
·
10
−
3
σ
(
D
B
)
∼
(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 depend enc e on the rando m seq uen ce. T aki ng this calc ula tio n 27

10
-1
10
0
D
10
-2
10
-1
10
0
q-Exponential Fit
- 4 - 2 0 2 4
D
0,00 0,50 1,00 1,50 10
-1
10
0
D
10
-2
10
-1
10
0
q-Exponential Fit
- 4 - 2 0 2 4
D
0,00 0,50 1,00 1,50
[10 ]
-3
(a) (b) (c) (d)
[10 ]
A A | | B B
[10 ] [10 ]
-3
W W W W
-3 | | -3
Figure 6:
  We show , se e (a) and (b), the hist o gr ams
  W
 obtaine d fr om a simul ation for
D
A
 = D
A
(t)
  and
  D
B
  = D
B
(t)
. The vertical normalization is to a unit area. The mean values  ar e
 
D
A

= 0
.
75
·
10
−
5
,
 
D
B

=
−
1
.
1
·
10
−
5
and the standard deviations
  σ
(D
A
) = 0
.
75
·
10
−
3
,
σ
(D
B
) = 0
.
83
·
10
−
3
. We also show, se e (c) and (d), the co rr esp onding plots in lo garit hmic  scale and the ﬁts with Eq.(  63  ). The parameters of the ﬁt are q=1.07, a=2 and b=2.2 for
  D
A
and q=1.12, a=2 and b=2.3 for
 D
B
. The total statistics correspond to 10.000 values generated  at steps of 30 se co nds. The veloci ty p ar amete r is
  ˜
v
 =
 V
CMB
 =
 370 km/s
. 28

Table 5:
 Illingworth’s ﬁnal inter-session averages.
Observations
  
D
A
   
D
B

5 A . M . + 0
.
00036
±
0
.
00012
  −
0
.
00016
±
0
.
00009 11 A.M.
  −
0
.
00001
±
0
.
00007
  −
0
.
00000
±
0
.
00006 5 P.M.
  −
0
.
00008
±
0
.
00012
  −
0
.
00005
±
0
.
00008 11 P.M.
  −
0
.
00034
±
0
.
0 0 0 1 4 + 0
.
00005
±
0
.
00006 into account gives a mean spread slightly less, about 0
.
65
·
10
−
3
, for the eﬀect of stochastic drift in Illing wort h’s measureme nts . This is compa rable to the uncertain ty of the individua l readings which, in the best case, was of 1/1500 wavelengths, i.e.
  ±
0
.
7
·
10
−
3
. By com bin ing 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 va nishingly small values of

D
A

and
 
D
B

 one gets
 
D
2
A
 ∼
σ
2
(
D
A
) and
 
D
2
B
 ∼
σ
2
(
D
B
). Therefore one obtains the following two average estimates of
  v
obs
v
obs
 ∼
112

σ
(
D
A
)
∼
3
.
05 km
/
s
  v
obs
 ∼
112

σ
(
D
B
)
∼
3
.
23 km
/
s ( 6 5 ) with a mean value of 3.14 km/s which is very close to M´ unera’s determination
 v
obs
 = 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 ret ain ing stati sti cal isotro py , thi s cou ld be use d to sim ula te 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
  x
n
(1) and
  x
n
(2), which is diﬀerent from the range [
−
˜
v
y
,
˜
v
y
] for the other two parameters
  y
n
(1) and
  y
n
(2). Finally, ˜
v
x
 and ˜
v
y
 could also become give n functions of time, for instanc e ˜
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 de ﬁn ed in Eqs . ( 31 )
−
( 34 ). W e sha ll discu ss 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 compl etely the phy sical informat ion contain ed in the original observ ations . In Table 5, we report the ﬁnal inter-session averages
 
D
A

 and

D
B

 obtained by Illingworth for the various observation times. 29

Nev erthe less, in spite of the strong cance lation s expect ed from the av erag ing reduc tion process ment ioned above , some non-zer o v alue is still surviving . There fore, regardl ess of our simulations, one could draw the following conclusions. Traditionally, from these ﬁnal averages for
 
D
A

 at 5 A.M. and at 11 P.M. one has been deducing the values
  v
A
 ∼
 2
.
12 km/s and
v
A
 ∼
2
.
07 km/s respectiv ely . There fore, from these tw o estim ates of
  v
A
 that, as anticipated, represent
 lower
 bounds for
 v
obs
, it follows that there were values of
 v
obs
 which clearly had to be
larger
 than b oth. F or this reas on, this 2.1 km/s velocit y v alue reported by Illingw orth, 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
 v
obs
  <
 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 ( K e n n e d y + I l l i n g w o r t h ) 2 k m
/
s

v
obs
 <
 5 km
/
s ( 6 6 ) in complete agreement with M´ unera’s determination
v
obs
 = 3
.
1
±
1
.
0 km
/
s ( 6 7 ) From this last estimate, by using Eq.( 30 ) and taking into account that for helium at atmo- spheric pressure the refractive index is
 N ∼
1
.
000035, one obtains a kinematical velocity
v
 ∼
(370
±
120) km
/
s ( 6 8 ) consistently with the velocity values Eqs.( 52 ) and ( 54 ) from the Michelson-Morley and Morley- Miller experiments.
6 . M i l l e r
M´ unera’s analysis  [72 ] is also interesting because he applied the same method used for Illing- worth’s observations to the only known Miller set of data explicitly reported in the literature. In thi s cas e, his va lue
  v
obs
  = 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 ]. Diﬀe ren tly from the original Miche lson-M orley ex peri men t, Mil ler ’s dat a we re tak en ov er the enti re da y and in fou r epoc hs of the yea r. 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
  A
EXP 2
  for the four epoc hs. The ir v alu es 0.0 42, 0.0 49, 0.0 38 and 0.045 , res pect iv ely for Apr il 192 5, Jul y 192 5, 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]
A
EXP 2
  = 0
.
044
±
0
.
0 2 2 ( 6 9 ) when compared with the equivalent classical prediction for Miller’s interferometer
  A
class 2
  =
L λ
(30km
/
s)
2
c
2
  ∼
0
.
56 corresponds to an average observable velocity
v
obs
  = 8
.
4
+1
.
9
−
2
.
5
  km
/
s ( 7 0 ) and, by using Eq.( 30 ), to a true kinematical value
v
 = 349
+79
−
104
  km
/
s ( 7 1 ) 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 actual ly hav e 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 autho rs say that “.. .th ere can be lit tle 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, inter pretin g the obser ve d 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, howev er, Shankland et
 al.
 seem not to realize that Miller’s average value
A
EXP 2
  ∼
  0
.
044, obtained after
 their own
 re-analysis of his observations at Mt.Wilson, when compared with the reference classical value
 A
class 2
  = 0
.
56 for his apparatus, was giving the same observ able velocity
 v
obs
 ∼
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.
6
A 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
) 2
λ
  at the var iou s  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 accur acy of each entry has be en ﬁxed to
 ±
0
.
050
.
T u r n 0
o
22
.
5
o
45
o
67
.
5
o
90
o
112
.
5
o
135
o
157
.
5
o
1 + 0 . 0 9 1 + 0 . 1 5 9 + 0 . 0 2 8 + 0 . 0 4 7
  −
0.034
  −
0.116
  −
0.147
  −
0.028 2
  −
0. 02 5 + 0. 06 3 +0 .0 50 +0 .0 88
  −
0.075
  −
0. 03 8 +0 .0 00
  −
0.063 3 + 0 . 0 2 2 + 0 . 1 0 3 + 0 . 0 8 4 + 0 . 0 1 6
  −
0.053
  −
0.072
  −
0.091
  −
0.009 4 + 0 . 0 3 4
  −
0.009
  −
0.053
  −
0.047
  −
0. 04 1 +0 .0 16 +0 .0 22 + 0. 07 8 5 + 0 . 1 6 9 + 0 . 0 8 1 + 0 . 0 4 4
  −
0.044
  −
0.081
  −
0.169
  −
0. 05 6 +0 .0 56 6
  −
0. 02 5 + 0. 02 5 +0 .0 25 +0 .0 25 + 0. 02 5
  −
0.025
  −
0.025
  −
0.025 7 + 0 . 0 8 1 + 0 . 0 9 4 + 0 . 0 5 6 + 0 . 0 6 9
  −
0.119
  −
0.106
  −
0. 09 4 +0 .0 19 8 + 0 . 0 6 6 + 0 . 0 7 2
  −
0.022
  −
0.066
  −
0.059
  −
0. 0 03 +0 .0 03 + 0. 00 9 9 + 0 . 0 4 1 + 0 . 0 8 4 + 0 . 0 7 8 + 0 . 0 2 2
  −
0.134
  −
0. 1 41 +0 .0 03 + 0. 04 7 1 0 + 0 . 0 1 6 + 0 . 0 7 2 + 0 . 0 7 8
  −
0.016
  −
0.009
  −
0.003
  −
0.047
  −
0.091 1 1 + 0 . 0 0 9 + 0 . 0 5 3 + 0 . 0 9 7
  −
0.009
  −
0.116
  −
0. 0 72 +0 .0 22 + 0. 01 6 1 2 + 0 . 0 2 2 + 0 . 0 1 6 + 0 . 0 5 9 + 0 . 0 0 3
  −
0.053
  −
0.009
  −
0.016
  −
0.022 1 3 + 0 . 0 0 0 + 0 . 0 6 3 + 0 . 0 2 5 + 0 . 0 3 8 + 0 . 0 5 0
  −
0.038
  −
0.075
  −
0.063 14
  −
0. 03 4 + 0. 04 7 +0 .0 78 +0 .0 09
  −
0.009
  −
0.028
  −
0.047
  −
0.016 1 5 + 0 . 1 1 3 + 0 . 1 2 5 + 0 . 1 3 8 + 0 . 0 0 0
  −
0.088
  −
0.125
  −
0.113
  −
0.050 1 6 + 0 . 0 2 5 + 0 . 0 5 0 + 0 . 0 2 5 + 0 . 0 5 0
  −
0.025
  −
0.050
  −
0.025
  −
0.050 1 7 + 0 . 0 0 0
  −
0.012
  −
0. 0 25 +0 .0 63 + 0. 00 0
  −
0.012
  −
0. 02 5 +0 .0 13 1 8 + 0 . 0 4 4 + 0 . 0 5 0 + 0 . 0 1 9
  −
0.019
  −
0.056
  −
0.044
  −
0. 03 1 +0 .0 31 1 9 + 0 . 0 5 3 + 0 . 0 5 9 + 0 . 0 1 6
  −
0.028
  −
0.022
  −
0.066
  −
0.009
  −
0.003 2 0 + 0 . 0 5 9 + 0 . 0 4 1 + 0 . 1 2 2 + 0 . 0 0 3
  −
0.066
  −
0.084
  −
0.053
  −
0.022 32

Another criticism of Miller’s work was recently presented by Roberts   [80 ]. This a utho r, using the set of data reported in Fig.8 of ref. [65 ], raise s sev eral objections to the vali dity of Miller’s obse rv ations . The tw o main objection s conc ern i) the subtra ction 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 ethe r-d rif t eﬀe ct, sho uld lie on the line. Ho we ve r, thi s expe cta tio n ign ore s tha t, as already pointed out for the Michelson-Morley experiment, real measurements contains large ﬁrs t-h arm oni c eﬀe cts . The se onl y can ce l whe n tak ing sym met ric comb ina tio ns of dat a at the various angles
  θ
  and
  π
 +
 θ
. As a mat ter of fac t, the autoc orr ela tiv e met hods and fur- ther 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 tha t the obser ve d 2nd -ha rmo nic eﬀect has no sta tis tic al sig niﬁ can ce. T o c hec k thi s poin t 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 b y Roberts, by follo wing the same procedure explained in Sect.3. The resulting symmetric combinations of fringe shifts
B
(
θ
) =   ∆
λ
(
θ
) + ∆
λ
(
π
 +
θ
) 2
λ
 (72) are reported in our Table 6. W e have then ﬁtted these data by includin g both 2nd and 4th harmonic terms. Notic e that, diﬀerently from Roberts’ analysis, we do not perform any averaging of data obtained from diﬀere nt turns of the interf erome ter. F or our globa l ﬁt, to estima te the accurac y 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 (
A
EXP 4
  = 0
.
004
±
0
.
012), we have obtained a chi-square of 130 for 157 degrees of freedom and the following values
A
EXP 2
  = 0
.
061
±
0
.
012
  θ
EXP 0
  = 24
o
±
7
o
(73) Here errors correspond to the overall boundary ∆
χ
2
= +3
.
67, as appropriate
  7
for a 70% C. L. in a 3-parameter ﬁt [ 81 ]. Notice that, even though the ﬁtted
 A
2
 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 5
σ
 evidence for a non-zero 2nd harmonic.
7
This probability content assumes a Gaussian distribution as for typical statistical errors.
33

As for Illi ngw ort h’s ex peri men t, we ha ve als o ana lyz ed the res ult s obt ain ed fro m the ind ivi dua l turns of the in ter fer ome ter . T o thi s end, we repor t in Figs. 7 and 8 the plot s of  the azimuth and of the 2nd harmonic for the 20 rotations.
0   5   10   15   20
Turn
-20 0 20 40 60 80
θ
0
Figure 7:
  The azimuth (in degre es) for the 20 individual turns of the interferometer r eport ed  in T able 6. The avera ge unc ertai nty of ea ch deter minat ion is ab out
 ±
20
o
. The band betw ee n  the two horizontal lines corresponds to the global ﬁt
 θ
0
 = 24
o
±
7
o
. 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 argumen t, whic h conce rns the observe d displa ceme nt 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 dai ly av era ge of an y giv en quantity, one ﬁnds, at any angle
  θ
, the daily averaged fringe shift

∆
λ
(
θ
)
λ
  
= 2

˜
C
(
t
)

c os2
θ
 (74) since
 
˜
S
(
t
)

= 0 with

˜
C
(
t
)

=
−
L
(
 N −
1)
λ V
2
c
2
1 4 (3cos2
γ
−
1)cos
2
φ
 (75) The result can then be cast into the form  [68 ]

∆
λ
(
θ
)
λ
  
=
 V
2
F
(
γ, φ
)cos2
θ
 (76) 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   5   10   15   20
Turn
0 0,05 0,1 0,15
A
2
Figure 8:
  The 2nd-har monic ampli tude for the 20 indivi dual turns of the interfer omete r  r ep orte d in T able 6. The averag e unc ertai nty of each determina tion is ab out
 ±
0
.
030
. T h e  band between the two horizontal lines corresponds to the global ﬁt
 A
2
 = 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. Onl y the amp lit ude can be diﬀ ere nt at diﬀe ren t epoc hs. Ins tea d, in Mille r’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 modulation s expected from the Earth’s rotat ion. As antic ipate d in Sect.2, and dis- cussed in connections with the Michelson-Morley and Illingworth experiments, one might be faced with the more general scenario where the two velocity components
  v
x
(
t
) and
  v
y
(
t
) in Eq.( 44 ) are not smoo th peri odic funct ion s but exhib it stoc has tic b eha vio ur. In thi s diﬀe r- ent 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
  A
EXP 2
  ∼
  0
.
044 computed by the Shankland team) with those from the Michelson-Morley, Morley-Miller and Kennedy- Illingworth experiments. In this sense, this bulk of Miller’s work will remain. 35

7. Mic helso n-P ease- Pea rson
Le t us fur the r com par e with the expe rim en t perf orm ed by Mic hel son , Pe ase and Pe ars on [ 84, 85 ]. The y do not repor t num bers so that we can onl y quote from the orig ina l art icl e [ 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 tempera ture and press ure disturba nces were eﬀect ive . The results gav e no displacem ent as great as
 one-ﬁftieth
  of that to be expected on the supposit ion of an eﬀec t due to a motion of  the solar system of three hundre d kilomete rs per seco nd”. 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 syste m of thr ee hu ndr ed kilome ter s per sec ond . The se res ult s are diﬀer enc es bet we en the dis pla cem en ts obs erv ed at max im um and at mini mu m at sid ere al tim es, the direc tions corres ponding to Dr. Str¨ omberg ’s calculati ons of the supposed ve locit y of the solar syste m”. In the same paper, the authors repor t that, accord ing 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. How eve r 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] whic h, surprising ly , was signe d by F. G. Pease alone. Here , one disco ver s 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 typica l magnit ude of
 ±
0
.
004 or smalle r. Ho wev er, as already anticipa ted in Sect. 2, by ave raging observ ation s performed at a given sidereal time one is assuming the smooth modulations of the signal described by Eqs.( 37 )
,
( 38 ). Other wise, one will introduce uncon trolle d errors. F or 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
  obtain ed, b efore an y ave raging procedur e and before any subtrac tion, in indivi dual measu remen ts at v ariou s hours of the day . Pe ase reports a plot of just a single observ ation , 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 eq uiv ale nt to fri nge shifts of about
 ±
0
.
011 with a length of 85 feet and could hardly be taken as indicative of the who le sample of mea sur eme nt s. In thi s sit uat ion , one can onl y ado pt the est ima te
A
2
  ∼
  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
c
2
  ∼
0
.
45, this is equivalent to
v
obs
 = (4
.
5
±
...
) km
/
s ( 7 8 ) or, by using Eq.( 30 ), to
v
 = (185
±
...
) km
/
s ( 7 9 ) 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 whic h has been smoothe d in our Fig .2. F or this rea son , ev en though Mill er’ s reconstruction of the Earth’s cosmic motion is not internally consistent, a single observation which gives
  v
obs
 ∼
 4.5 km/s does not represent a refutation of the whole Miller experiment. This becomes even more true by noticing that the single session selected by Pease, within a period of several months, was chosen to represent an example of extremely small ether-drift eﬀect. 37

8 . J o o s
One more classica l experime nt, performe d by Georg Joos in 1930, has ﬁnally to b e consider ed. 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 experim ent s (e.g. Mic helso n-Mor ley , Illingw orth) where only observ ations 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 suc h, thi s expe rim en t des erv es a mor e reﬁ ned anal ysi s 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 particula rly importan t since later on, in 1934, Miller and Joos had a public letter exchange  [89]  and Joos did not correct Miller’s state men t. On the other hand, Swenso n [ 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 endin g at 1 P.M . of May 11th. Two measur emen ts, the 1st and the 5th, were ﬁnally deleted by Joos with the motivation that there were spurious disturbances. The data were combined symme trica lly , in order to elimina te the presence of o dd harmonics , and the magnit ude of the fri nge shift s wa s ty pic all y of the ord er of a few thous and ths of a wa ve len gth . T o thi s end , one can look at Fig.8 of [ 88]  (reported here as our Fig.9) and compare with the shown size of  1/100 0 of a wa vel ength . F rom this pictu re, Joos decid ed to adopt 1/1000 of a wa ve lengt h as an upper limit and deduced an observable velocity
  v
obs
  
 1
.
5 km/s. T o derive this v alue, 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 tha n 1/1 000 of a wa v ele ngt h, we hav e dec ide d to ext rac t the v alu es of the 2nd-ha rmo nic amplitude
  A
2
 from the 22 pictures. Diﬀe ren tly from the value s of the azim uth, this can b e 38

Figure 9:
 The selected set of data reported by Joos [  87  ,   88   ]. The yard stick corr esponds to 1/1000 of a wavelength so that the experimental dots have a size of about
  0
.
4
·
10
−
3
. T hi s  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)
  θ
k
 ≡
(0
o
, 45
o
, 90
o
, 135
o
) or the diﬀerent set
  θ
k
 ≡
(360
o
, 315
o
, 270
o
, 225
o
). Or, by noticing that in Fig.9 there is a small misalignment angle
  θ
∗
 ∼
  17
o
(which actually from  [87 ] might instead be 22
.
5
o
) between the dots of Joos’ fringe shifts and the N, W, and S marks, one could also adopt other two set of values, namely
 θ
k
 ≡
(0
o
+
θ
∗
, 45
o
+
θ
∗
, 90
o
+
θ
∗
, 135
o
+
θ
∗
) or
 θ
k
 ≡
(360
o
−
θ
∗
, 315
o
−
θ
∗
, 270
o
−
θ
∗
, 225
o
−
θ
∗
). By ﬁtting the fr inge shift s of Fig.9 to the 2nd-harmonic form Eq.( 56 ), these four options for the reference angles
  θ
k
 would give exactly the same amplitude
 A
2
 but four diﬀerent choices for the azimuth, i.e.
 −
θ
0
,
 −
θ
0
 +
θ
∗
,
θ
0
 −
θ
∗
  and
  θ
0
. This basi c am big uit y sho uld be add ed to the stand ard unce rta in ty in the azimuth that, due to the 2nd-harmonic nature of the measurements, could always be changed by adding
 ±
180 degrees
  8
. The ref ore , sin ce clear ly there is onl y one corr ec t ch oic e for the
8
As an example, one can consider the azimuth for Joos’ picture 20. Depending on the choice of the reference angles
  θ
k
, one ﬁnds
 θ
0
 ∼
 329
o
, 329
o
+
 θ
∗
, 31
o
−
θ
∗
, 31
o
or
 θ
0
 ∼
 149
o
, 149
o
+
 θ
∗
, 211
o
−
θ
∗
, 211
o
.
39

angles
  θ
k
, 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.
0   5   10   15   20   25
Picture
0 1 2 3 4
2
A
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

A
 joos 2
  
= (1
.
4
±
0
.
8)
·
10
−
3
(80) and a corresponding observable velocity
v
obs
 ∼
1
.
8
+0
.
5
−
0
.
6
  km
/
s ( 8 1 ) By correcting with the helium refractive index, Eqs.( 30 ) and ( 81 ) would then imply a true kinematical velocity
  v
 ∼
217
+66
−
79
 km/s. How ev er, this is only a ﬁrst and very part ial view of Joos’ experime nt. In fact, we have compa red Joos’ amplitudes with theore tical models of cosmic motion . T o this end, one has ﬁrst to trans form the civil times of Joos’ measureme nts into siderea l times. F or the longit ude 11.60 degrees of Jena, one ﬁnds that Joos’ observations correspond to a complete round in sidereal time in which the value
  τ
  = 0
o
≡
  360
o
is ver y close to Joos’ pictu re 20. The n, 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
. T o this end, it is con ve nien t to ﬁrst re-write the theoretical forms as
A
2
(
t
)cos2
θ
0
(
t
) = 2
C
(
t
) =   2
L
(
 N −
1)
λ v
2
x
(
t
)
−
v
2
y
(
t
)
c
2
  ∼
2
.
6
·
10
−
3
  v
2
x
(
t
)
−
v
2
y
(
t
) (300 km
/
s)
2
 (82) 40

Table 7:
 The 2nd-h armon ic ampli tude obtaine d fr om the 22 Joos pictur es 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
 
A
 joos 2
  
= 1
.
4
·
10
−
3
.
Picture
  A
 joos 2
  [10
−
3
] 2 2
.
05 3 0
.
75 4 1
.
60 6 2
.
00 7 1
.
50 8 1
.
55 9 1
.
10 10 0
.
60 11 4
.
15 12 1
.
20 13 2
.
35 14 0
.
95 15 1
.
15 16 1
.
65 17 0
.
50 18 1
.
05 19 1
.
25 20 0
.
35 21 0
.
45 22 1
.
25 23 0
.
95 24 1
.
65 41

and
A
2
(
t
)sin2
θ
0
(
t
) = 2
S
(
t
) =   2
L
(
 N −
1)
λ
2
v
x
(
t
)
  v
y
(
t
)
c
2
  ∼
2
.
6
·
10
−
3
  2
v
x
(
t
)
  v
y
(
t
) (300 km
/
s)
2
 (83) where we have used the numerical relation for Joos’s experiment
  L λ
(30km
/
s)
2
c
2
  ∼
0
.
375 and the value of the helium refractive index. Then, by approximating
 v
x
(
t
)
∼
 ˜
v
x
(
t
),
 v
y
(
t
)
∼
 ˜
v
y
(
t
) and using Eq.( 34 ) for the scalar combination ˜
v
(
t
)
≡

˜
v
2
x
(
t
) + ˜
v
2
y
(
t
), we have ﬁtted the amplitude data of Table 7 to the smooth form
A
smooth 2
  (
t
) = const
·
sin
2
z
(
t
) ( 8 4 ) where cos
z
(
t
) is deﬁned in Eq. ( 31 ). The results of the ﬁt
  9
α
 = 168
o
±
30
o
γ
  =
−
13
o
±
14
o
(85) conﬁrm that, as found in connection with the Illingworth experiment, the Earth’s motion with respect to the CMB (which has
 α
∼
168
o
and
 γ
 ∼ −
6
o
) could serve as a useful model to desc ribe 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 sizea ble discrepa ncy in the absolu te normaliza tion of the amplitud e. 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
−
sin
2
γ
sin
2
φ
−
  1 2  cos
2
γ
 cos
2
φ

 (86) For the CMB and Jena, this gives


˜
v
2
 ∼
330 km/s so that one would naively predict from Eqs.( 82 ), ( 83 )

A
smooth 2
  (
t
)
 ∼
2
.
6
·
10
−
3
  
˜
v
2
(
t
)

(300 km
/
s)
2
 ∼
3
.
2
·
10
−
3
(87) to be compared with Joos’ mean value

A
 joos 2
  
= (1
.
4
±
0
.
8)
·
10
−
3
. In the standa rd pictur e, 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
9
Actually, there is another degenerate minimum at
  α
 = 348
o
±
30
o
and
  γ
  = 13
o
±
14
o
because sin
2
z
(
t
) remains invariant under the simultaneous replacements
 α
→
α
+ 18 0
o
and
 γ
 → −
γ
. However, due to the close agreement with the CMB parameters we have concentrated on solution ( 85 ).
42

ve locit y ﬁeld. As anticip ated howe ve r, due to the higher accurac y of the Joos experi ment , we hav e modiﬁed the theoretical framework. Namely , we hav e allowed the two random parameters
x
n
(1) and
  x
n
(2) to vary in the range [
−
˜
v
x
(
t
)
,
˜
v
x
(
t
)] and the oth er tw o par ame ter s
  y
n
(1) and
  y
n
(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

x
2
n
(
i
 = 1
,
2)

stat
  =   ˜
v
2
x
(
t
) 3
 n
2
η
  
y
2
n
(
i
 = 1
,
2)

stat
  = ˜
v
2
y
(
t
) 3
 n
2
η
 (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 subst anti al reduction of the amplitude with respect to its smooth predict ion. T o 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

A
2
(
t
)

stat
 ∼
2
.
6
·
10
−
3
  ˜
v
2
(
t
) (300 km
/
s)
2
1 3
∞

n
=1
1
n
2
  =
  π
2
18
  A
smooth 2
  (
t
) ( 8 9 ) 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
  
10
7
) and the integer number
  s
 (the ‘seed’) which determines the rando m sequence . In particula r, the dependence on the latter is usuall y quoted as theor etica l uncer tain ty . F or this reas on, for Illingwo rth’s experimen t in Sect. 5 we had produce d sev eral 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. Her e, we hav e sta rte d by doing some thin g sim ila r. Ho we ve r, sin ce it is not possib le to consider at once all characteristics of a given conﬁguration, we have ﬁrst concentrated on the simplest statistical indicator, namely the mean amplitude
 
A
simul 2
  
 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
 betw een two consecuti ve measureme nts . F or instan ce, Joos’ experim ent corres ponds 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
  = 10
4
and then studied a few
  s
by increasing
  N
. T o giv e an idea of the spre ad of the cen tra l v alu es, due to ch ang es 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 exp lai ne d in the tex t. The me an amp lit ude over  the 22 determinations is
 
A
simul 2
  
= 1
.
38
·
10
−
3
.
Picture
  A
simul 2
  [10
−
3
] 2 1
.
26 3 3
.
50 4 0
.
46 6 0
.
34 7 2
.
71 8 0
.
35 9 2
.
19 10 0
.
52 11 5
.
24 12 0
.
24 13 1
.
19 14 1
.
93 15 0
.
08 16 1
.
52 17 2
.
29 18 0
.
24 19 1
.
02 20 0
.
07 21 0
.
09 22 2
.
18 23 1
.
50 24 1
.
52 44

pair (
n,
∆
t
)

A
simul 2
  (
n
 = 24
,
∆
t
 = 3600
 s
)
 ∼
(1
.
7
±
0
.
8)
·
10
−
3
(90)

A
simul 2
  (
n
 = 1440
,
∆
t
 = 60
 s
)
 ∼
(1
.
7
±
0
.
3)
·
10
−
3
(91)

A
simul 2
  (
n
 = 240
,
∆
t
 = 3600
  s
)
 ∼
(1
.
8
±
0
.
5)
·
10
−
3
(92) As it might be expected, the average
 
A
simul 2
  
 becomes more stable by increasing the number of observ ation s. Conce rning the individual va lues
  A
simul 2
  (
t
i
), with
  i
  = 1
,..,n
, they have a large spread, about (1
÷
4)
·
10
−
3
. This is in agre emen t with the ‘fat-tai led’ distributi ons of  instantaneous values expected in turbulent ﬂows [ 76, 77 ] (comp are wit h Fig. 6 in Sect. 5). However this other spread can be reduced by starting to average the data in some interval of time
  t
0
. In this case, the sprea d of the resulti ng ave rage val ues
 
A
simul 2
  (
t
i
)

t
0
 decreases as
1
√
t
0
. W e empha size that, by performing exte nsiv e sim ulatio ns, there are occasional ly ve ry 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 ch aract erist ic of a stoch astic -ethe r model. With a standa rd 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?).
0   5   10   15   20   25
Picture
0 1 2 3 4 5 6 Joos Data Poly Fit Simulation Poly Fit
2
A
Figure 11:
  Joo s’ experimental amplitudes in T able 7 are co mpar ed with the single simulation  of 22 me asur ement s for ﬁxe d
  (
N, s
)
 in T ab le 8. By cha ngi ng the p ai r
  (
N, s
)
, the typ ic al  variation of each simulated entry is
  (1
÷
4)
·
10
−
3
dep endin g on the sider e al 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 compa re with the
 single
 Joos conﬁgur ation of 22 ent ries in Tab le 7. T o this end, one 45

could ﬁrst try to look for the ‘best seed’, or subset of seeds, which can minimize the diﬀerence betw een the generate d conﬁg uratio ns and Joos’ data. This standar d task, usually acc om- plishe d by minimizing a chi -squa re, is diﬃcul t to implem ent here. In fact, it is problemat ic to construct a function
 χ
2
(
s
) and look for its minima because a seed
 s
 and the closest seeds
 s
±
1 giv e often vas tly diﬀere nt conﬁgurat ions and chi -squa re. F or this reason, we have follo wed 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
 
A
 joos 2
  
 = 1
.
4
·
10
−
3
for a large number
  N
 of Fou rier modes. One of such seeds gav e a sequenc e
 
A
simul 2
  
 =1.66, 1.40, 1.08, 1.21 and 1.38 (in units 10
−
3
), for
  N
  = 10
3
, 10
4
, 10
5
, 10
6
and 5
·
10
6
respectively, and the conﬁguration with
  N
  = 5
·
10
6
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 sim ula ted values are reported in Table 8 and a graphical comparison with Joos’ data is shown in Fig. 11. W e empha size that one should not compa re each individual ent ry with the correspond ing data since, by changing (
N, s
), the simulated instantaneous values vary typically of about (1
÷
4)
·
10
−
3
dependin g on the siderea l time. Inste ad, one should compa re the overa ll trend of  data and simulation. To this end, we show two 5th-order polynomial ﬁts to the two diﬀerent sets of values. A mor e con ve nt ion al com par iso n with the dat a con sis ts in quo tin g for the v ari ous 22 entries simulated average values and uncertainties. To this end, we have considered the mean amplitudes
 
A
simul 2
  (
t
i
)

 deﬁn ed by ave rag ing , for eac h Joos ’ tim e
  t
i
, ov er 10 hy pothe tic al measu remen ts performe d on 10 conse cutiv e day s. F or eac h
  t
i
, the observed eﬀect of varying (
N, s
) has been summa riz ed into a cen tra l v alu e and a sym met ric erro r. The va lue s 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
A
simul 2
  (
t
i
), we have seen that our sampling method of
 
A
simul 2
  (
t
i
)

, 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 ch ec ke d by cons ide rin g the sing le simu lat ion of T abl e 8 and counting the sizeable fraction of amplitudes
 A
simul 2
  (
t
i
)

0
.
5
·
10
−
3
. For this reason, one could impro ve the evalua tion of the probabili ty conte nt. Ho wev er, in view of the good agree men t already found in Fig.12 (
χ
2
= 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

T able 9:
  The 2nd-ha rmoni c ampl itude s obtai ne d by simul ating the aver aging pr o ce ss over  10 hyp othet ic al me asur ement s p erfor me d, at ea ch Jo os’ time, on 10 c onse cutive days. The  stochastic velocity components are controlled by the kinematical parameters
  (
V,α,γ
)
CMB
  as  exp lai ne d in the text. The eﬀe ct of var yin g the pa ir
  (
N, s
)
 has be en appr oxima te d into a  c ent r al val ue and a sym met ric err or. The mea n amp lit ude ove r the 22 det erm ina tio ns is

A
simul 2
  
= 1
.
8
·
10
−
3
.
Picture
  A
simul 2
  [10
−
3
] 2 2
.
5
±
1
.
0 3 1
.
80
±
0
.
85 4 1
.
95
±
0
.
85 6 1
.
90
±
0
.
85 7 1
.
65
±
0
.
90 8 2
.
1
±
1
.
0 9 2
.
0
±
1
.
0 10 2
.
2
±
1
.
2 11 2
.
4
±
1
.
4 12 2
.
7
±
1
.
6 13 2
.
3
±
1
.
5 14 2
.
4
±
1
.
4 15 1
.
85
±
0
.
85 16 1
.
70
±
0
.
75 17 1
.
20
±
0
.
75 18 1
.
20
±
0
.
70 19 1
.
15
±
0
.
70 20 1
.
05
±
0
.
70 21 1
.
25
±
0
.
60 22 1
.
55
±
0
.
60 23 1
.
60
±
0
.
80 24 1
.
7
±
1
.
0 47

0   5   10   15   20   25
Picture
0 1 2 3 4 5 6 Joos Data Simulation
2
A
Figure 12:
  Jo os’ experi menta l ampl itudes in T able 7 ar e co mp ar ed with our simulati on in  Table 9.
link between Joos’ data and the Earth’s motion with respect to the CMB gets reinforced by our simul ations . In fact, by inspection of Figs.11 and 12, the va lues 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 kine- matical velocity
  v
 ∼
  217
+66
−
79
 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
 v
x
  and
v
y
 is controlled by the same scalar parameter ˜
v
(
t
)
 ≡

˜
v
2
x
(
t
) + ˜
v
2
y
(
t
) of Eq.( 34 ). In v ie w 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
+40
−
70
 km/s which corresponds to Eq.( 34 ) for
  φ
 = 50
.
94
o
,
  V
 = 370 km/s,
  α
 = 168
o
and
  γ
  =
−
6
o
.
9. Sum ma ry an d co nc lu si on s
The condensation of elementary quanta and their macroscopic occupation of the same quan- tum state is the esse ntia l ingre dien t of the dege nerat e v acuum of prese nt- day elemen tary 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 relativ- ity 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 Lore ntz -in v arian t v acuum state. These argumen ts sugge st the p ossibi lit y of a tiny va cuum energy-momentum ﬂux, associated with an Earth’s absolute velocity
  v
, which could aﬀect
diﬀerently
 the vario us forms of matter. Name ly , it could produce small con ve ctiv e curre nts in a loosely bound system such as a gas or dissipate mainly by heat conduction with no ap- preciable particle ﬂow in strongly bound systems as liquid or solid transparent media. In 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 ). Notic e that this tiny aniso trop y refers to the system
  S
′
 where the container of the gas is at rest. In this sense, contra ry to standa rd Special Relativ ity ,
  S
′
 might not deﬁne a tru e fra me of res t. Thi s con cep tua l possi bil it y can be objec tiv ely teste d wit h a new seri es 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  dedic ated experime nts , we hav e tried to get a ﬁrst consis tency che ck . In fact, by adopt ing Eq.( 24 ), the frequency shift between the optical resonators is governed by the same classical formula for the
 fringe shifts
  in the old ether-dr ift experimen ts with the only replac emen t
v
2
→
2(
 N −
1)
v
2
≡
v
2 obs
 (93) In thi s wa y , whe re one re- obt ain s the sam e cla ssi cal for mu las (wi th the onl y rep lac eme nt
v
 →
v
obs
), 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 propa- gating in gaseous media, air or helium at atmospheric pressure, where (
 N −
1) is a very small nu mbe r. In this reg ime , the theo ret ica l fri nge shift s expe cte d on the bas is of Eqs.( 23 ) and ( 24 ) are muc h smalle r than the class ical predictio n (
v/c
)
2
. Another important aspect of these classical experiments is that one was always expecting smooth sinusoidal modulations of the dat a due to the Ear th’ s rot ati on, see Eqs. ( 35 ), ( 37 ) and ( 38 ). As empha sized in Sec t.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-t rivia l implic ations . F or instance, due to the irregular behavi our of turbul ent 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 be- comes 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 obvi- ous. In fact, by using Eqs.( 24 ), ( 29 ) and ( 30 ), the small residuals point to an average velocity of about 300 km/s, as in most cosmic motions. In this alternative interpretation, the indica- tions of the various experiments are summarized in our Table 10
  10
. As a sum ma ry of ou r 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
10
Other 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 sum mit of Mt. Rig i in Swi tze rla nd. The se res ult s wer e summarized in Table I of ref. [68 ] and by Miller  [65] . In the 1881 Pot sdam experim ent the fring e shifts were in the range 0
.
002
÷
0
.
007 to be compare d with an expecte d 2nd-harmo nic 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 vel ocity of 30 km/s. This giv es
  v
obs
  
 7.7 km/s or
  v
  
 320 km/s. F rom Piccar d and Stahel , in the most reﬁne d versio n of Mt. Rigi, one gets an obse rv able veloci ty
  v
obs
  
 1.5 km/s. Since the ir 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

T able 10:
  The aver age velo city obser ve d (or the limi ts plac ed ) by the class ic al ether -drif t  experiments in the alternative interpretation of Eqs.(  24 ), (  29  ), (  30  ).
E x pe r i m e n t g a s i n t h e i n t e r f e r o m e t e r
  v
obs
(km
/
s)
  v
(km/s) M i c h e l s o n - M o r l e y ( 1 8 8 7 ) a i r 8
.
4
+1
.
5
−
1
.
7
  349
+62
−
70
M o r l e y - M i l l e r ( 1 9 0 2 - 1 9 0 5 ) a i r 8
.
5
±
1
.
5 3 5 3
±
62 K e n n e d y ( 1 9 2 6 ) h e l i u m
  <
 5
  <
 600 I l l i n g w o r t h ( 1 9 2 7 ) h e l i u m 3
.
1
±
1
.
0 3 7 0
±
120 M i l l e r ( 1 9 2 5 - 1 9 2 6 ) a i r 8
.
4
+1
.
9
−
2
.
5
  349
+79
−
104
M i c h e l s o n - P e a s e - P e a r s o n ( 1 9 2 9 ) a i r 4
.
5
±
...
  185
±
...
J o o s ( 1 9 3 0 ) h e l i u m 1
.
8
+0
.
5
−
0
.
6
  330
+40
−
70
in the end may turn out to be wrong. ii) one gets consistent indications from the Michelson-Morley, Morley-Miller, Miller and Illing wor th-Ke nnedy experimen ts. In view of this consiste ncy , an inte rpret ation of Mille r’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 v acuum energy-momentum ﬂow. W e have also pro duced numerical simulations of the Illingworth experiment in a simple statistically isotropic and homogeneous turbulent- ethe r model. This represe nts a zero th-or der approxi mation and is usefu l to illustr ate basic phenomenological features associated with the picture of the vacuum as an underlying stochas- tic medium. In this sch eme, Illingw orth’s data are consis ten t with ﬂuctuat ions of the velocit y ﬁeld whose absolute scale is controlled by ˜
v
  =
 V
CMB
 ∼
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 liter ature . There fore , since Mille r’s extensi ve observ ations (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
 v
obs
 ∼
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 experim ent s (e.g. Mic helso n-Mor ley , Illing wort h) where only obser v ations 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] . F or the se reaso ns, thi s expe rime nt has des erv ed a mor e reﬁ ned analy sis and is cen tra l for our wor k. As discu sse d in Sec t.8 , due to unc ert ain tie s in the ori gin al data analysis, the standard 1.5 km/s velocity value quoted for this experiment should be und ers tood as an order of magnit ude esti mat e and not as a tru e upper limit . Ins tea d, our reported observable velocity
  v
obs
  ∼
  1
.
8
+0
.
5
−
0
.
6
 km/s has been obtained from a direct analysis of Joos’ fring e shifts. F rom this value , to deduc e a kinem atica l ve locit y , one still needs the refr activ e index . The tradition al view, motiv ated by Miller’s review articl e [ 65]  and Joos’s own statements in ref.[ 87] , is that the experiment was performed in an evacuated housing. In these condit ions, it wou ld be easy to recon cile a large kinemati cal velocit y
 v
∼
350 km/s with the very small value s of the obser v able velocit y . On the other hand, Swenso n   [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 Swe nson’s indica tions. In this case , by simply correc ting with the helium refractiv e index the result
  v
obs
 ∼
1
.
8
+0
.
5
−
0
.
6
 km/s, one would get a kinematical velocity
  v
 ∼
217
+66
−
79
 km/s. How ever, as discuss ed in detail in Sect.8, this is only a ﬁrst partial view of Joos’ experimen t. 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
  α
CMB
 ∼
  168
o
and angular declination
  γ
CMB
 ∼ −
6
o
). Sti ll, to get a com ple te agr ee men t, one should expl ain the absolu te nor mal iza tio n of the amp lit ude s and the str ong ﬂuctu ati ons of the data. Th us we ha ve impro ve d our analy sis by performing various numerical simulations where the velocity components in the plane of  the interferometer
 v
x
(
t
) and
 v
y
(
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 time- dependent 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
. T aking into accoun t these stoch astic ﬂuctuatio ns 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
+40
−
70
 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 globa l Earth ’s motion from microsco pic measureme nts performed in a laborato ry . F or 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 sim ulatio n of one noon session of the Miche lson-M orley experimen t. This has been p erfor med in the same framework adopted for the Joos experiment where the velocity components
 v
x
(
t
) and
 v
y
(
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
. W e 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. W e emp has ize tha t the simul ati on repo rte d in our secon d Appe ndix corr espo nds to a single conﬁguration whereas taking into account more and more conﬁgurations is essential to properl y estimate theoretic al uncer tain ties (as for the Joos experimen t with the result s in our T able 9 and Fig.12). Nev erthe less, even this ve ry small sample can pro vide inter esting 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 acce pting Hicks ’ inte rpret ation of the observ ations of July 8th (see Sect. 3). In this sense , this brief numerical analysis reinforces the picture of the classical experiments emerging from our T able 10. Acco rding to the usual view, the theor etica l predic tions, with a ve ry low velocit y
v
 ∼
 30 km/s, were mu ch large r than the observ ed valu es. Instea d, 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
)
ν
0
=   ¯
c
γ
(
π/
2 +
θ
)
−
¯
c
γ
(
θ
)
c
  =
B
medium
v
2
c
2
 co s 2(
θ
−
θ
0
) ( 9 4 ) where
 θ
0
 is the direction of the ether-drift. This can be interpreted within Eq.( 109 ) where
|B
medium
| ∼ N
medium
−
1 ( 9 5 )
 N
medium
 b eing the refracti ve index of the gaseous medium ﬁlling the optical resonator s. T est- ing this prediction, requires replacing the high vacuum usually adopted within the optical resonators with a gaseous medium and studying the substantially larger frequency shift in- troduced with respect to the vacuum experiments. As a rough check, a comparison was made [ 7 ,  51 ] with the resul ts obtained by Jase ja 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
 N
He
−
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
 N
vacuum
 = 1 so that
 B
vacuum
 will vanish
  11
. Th us one sho uld swit ch to the new gener ati on of dedicate d ether-drif t experim ent s in gaseo us systems. Our conclu sion is that these new experiments should just conﬁrm Joos’ remarkable observations of eighty years ago.
11
Throughout this paper we have assumed the limit of a zero light anisotropy for experiments performed in va cuum. How eve r, as discu ssed at the end of Appendi x I, one could also consi der the more gene ral scenario where a metric of the form   [97 ]
  g
µν
=
  η
µν
+ ∆
µν
is introdu ced from the ve ry beginnin g. In this case, the va cuum behav es as a medi um and ligh t can spread with diﬀere nt velocit ies for diﬀe ren t dire ctio ns. 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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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 fundam ent al v acuum energy- momen tum ﬂow. Due to these conv ecti ve curre nts, isotrop y 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. Thi s in trodu ces obv iou s diﬀ ere nce s with respe ct 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 photo n energy does not depend on the direction of propagat ion. Their conc lusion , 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
 pr efer red frame. Inste ad, 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 con tai ner is at rest in Σ. F or an obs erv er at res t in this refer enc e frame , lig ht propagation within the medium is isotropic and described by
π
µ
π
ν
γ
µν
= 0 ( 9 6 ) where
γ
µν
= diag(
 N
2
,
−
1
,
−
1
,
−
1 ) ( 9 7 ) and
  π
µ
 denot es the ligh t 4-mome ntu m vec tor for the Σ obser ve r. Let us now consi der 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
 p
µ
 p
ν
g
µν
= 0
,
 (98) where
  p
µ
  ≡
  (
E/c,
p
) and
  g
µν
denote respectively the light 4-momentum and the eﬀective metric for
  S
′
. On this bas is , b y in tr odu ci ng the
  S
′
 dimensionless velocity 4-vector
  u
µ
≡
(
u
0
,
V
/c
) (with
  u
µ
u
µ
= 1), one can deﬁne a transformation matrix
  A
µ ν
  =
  A
µ ν
(
u
µ
,
 N
) and express
g
µν
=
 A
µ σ
A
ν  ρ
γ
σρ
(99) 61

In this context , requi ring the consi stenc y of v acuum condensa tion with Special Relativi ty corresponds to place all reference frames on the same footing and assume
 g
µν
=
 γ
µν
or
A
µ ν
(
u
µ
,
 N
) =
 δ
µ ν
 (100) This identiﬁcation is independent of the physical nature of the medium, being valid for gaseous syste ms as wel l as for liquid or solid transpa ren t media. In this sense Specia l Relativit y , 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-dr ift experimen t. Thus , by notici ng that we hav e at our disposal tw o matric es, namely
  δ
µ ν
 and the Lorentz transformation matrix Λ
µ ν
 associated with
  V
, one could still main tain 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
A
µ ν
(
u
µ
,
 N
 = 1) = Λ
µ ν
 (101) to solv e non-t rivia lly the equa tion
  g
µν
=
  γ
µν
when
  N
 = 1, i. e. whe n lig ht pr op ag at es in vacuum and
  γ
µν
reduc es to the Mink owsk i tenso r
  η
µν
. Bu t th en , by co nt in uit y , it is conceivable that Eq.( 101 ), up to higher-order terms, can also describe the case
 N
  = 1 +
 ǫ
of gaseous media. This choic e pro vides a simple interpr etativ e model and a particular form of the more general structure Eq.( 23 ) whic h corre sponds to the Roberts on-Ma nsouri -Sex l (RMS) [ 42, 43 ] parametrization for the two-w ay velocity of light. Moreov er, when comparing with experimen ts with optical reso nators , the resulting frequenc y shift is gov erne d 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
  g
µν
throug h the relat ion
g
µν
= Λ
µ σ
Λ
ν  ρ
γ
σρ
(102) with
  γ
µν
as in Eq.( 97 ). This gives the eﬀective metric for
  S
′
g
µν
=
 η
µν
+
κu
µ
u
ν
(103) with
κ
 =
 N
2
−
1 ( 1 0 4 ) 62

In this way, Eq.( 98 ) gives a photon energy (
u
2 0
 = 1 +
V
2
/c
2
)
E
(
|
p
|
, θ
) =
 c
 −
κu
0
ζ
 +

|
p
|
2
(1 +
κu
2 0
)
−
κζ
2
1 +
κu
2 0
(105) with
ζ
  =
 p
·
  V
c
  =
|
p
|
β
cos
θ,
 (106) where
  β
  =
  |
V
|
c
  and
  θ
 ≡
 θ
lab
 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
|
  =
 c
γ
(
θ
) =
 c
 −
κβ

1 +
β
2
cos
θ
 +

1 +
κ
+
κβ
2
sin
2
θ
1 +
κ
(1 +
β
2
)
  .
 (107) or to
 O
(
κ
) and
 O
(
β
2
)
c
γ
(
θ
)
∼
  c
 N

1
−
κβ
cos
θ
−
 κ
2
β
2
(1 + cos
2
θ
)

 (108) From this one can compute the two-way velocity ¯
c
γ
(
θ
) =   2
c
γ
(
θ
)
c
γ
(
π
 +
θ
)
c
γ
(
θ
) +
c
γ
(
π
 +
θ
)
∼
  c
 N

1
−
β
2

κ
−
 κ
2  sin
2
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
(
θ
)

¯
c
γ
   ∼ B
v
2
c
2
 co s 2(
θ
−
θ
0
) ( 1 1 0 ) where the pair (
v, θ
0
) describes the projection of
  V
 onto the relevant plane and
|B|∼
  κ
2
 ∼
(
 N −
1 ) ( 1 1 1 )
12
There is a subtle diﬀerence between our Eqs.( 108 ) and( 109 ) and the cor re spon din g Eqs . (6) and (10 ) of ref.   [7]  that has to do with the rela tivis tic aberratio n of the angle s. Namel y , in ref. [7] , with the (wrong) motivation that the anisotropy is
 O
(
β
2
), no attention was paid to the precise deﬁnition of the angle between the Earth ’s vel ocity and the dire ction of the photon momen tum. Thu s the tw o-w ay veloci ty of ligh t in the
S
′
frame was parameterized in terms of the angle
  θ
  ≡
  θ
Σ
 as seen in the Σ fra me. Th is can be exp lic itl y checked by replacing in our Eqs. ( 108)  and( 109 ) the aberration relation cos
θ
lab
 = (
−
β
 + co s
θ
Σ
)
/
(1
−
β
 cos
θ
Σ
) or equivalently by replacing cos
θ
Σ
  = (
β
 + cos
θ
lab
)
/
(1 +
 β
 cos
θ
lab
) in Eqs . (6) and (1 0) of ref.   [7] . How eve r, 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
  =
  θ
lab
 and not in terms of
  θ
  =
  θ
Σ
. Th is tri via l 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
 =
 θ
Σ
.
63

From the previous analysis, by replacing the two-way velocity in Eq.( 28 ), one ﬁnally obtains the observable velocity
v
2 obs
 ∼
2
|B|
v
2
∼
2(
 N −
1)
v
2
(112) to be used in Eq.( 29 ). In this wa y , where one re-obta ins the class ical formul as with the only replacement
  v
 →
  v
obs
, testing the present scheme requires to check the consistency of the kinematical
  v
′
s obtain ed in diﬀer ent experimen ts. Before concluding this Appendix, we emphasize that throughout this paper we have as- sumed the limit of a zero light anisotropy for experiments performed in vacuum. In fact, the eﬀective metric Eq.( 103 ) reduces to the Minkowski tensor
  η
µν
in the limit
 N →
  1. Adm it- tedly, this might represent a restrictive scenario and one could also consider the more general case where a metric of the form  [ 97 ]
g
µν
=
 η
µν
+ ∆
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 de- scribe light and particle propagation can have several solutions. For instance, light will spread with diﬀeren t vel ocities in diﬀere nt directio ns as with anisotrop ic media in optic s. There fore, by adopting various parameterizations for ∆
µν
, one can restrict its size by comparing with measu remen ts of the one- and two- wa y ve locit y of ligh t. As an examp le, the one-w ay light anisotropy reported by the GRAAL experiment  [98 ] requires typical values of the matrix ele- ments
|
∆
µν
|
= 10
−
13
÷
10
−
14
[ 99] . In any case, as anti cipat ed in Sect. 2, these gen uine va cuum 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 Michels on-M orley experime nt. This will be performe d within the same stoch astic -ethe r model described in Sect.8 for the Joos experim ent . F or sake of clari ty , 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
  v
x
(
t
) and
  v
y
(
t
) as in Eqs.( 60 ) and ( 61 ). A ba si c in pu t va lu e is the sidereal time of the observation. This has to be inserted, together with the CMB kinematical parameters
  V
CMB
  ∼
 370 km/s,
  α
CMB
  ∼
  168
o
,
  γ
CMB
  ∼ −
6
o
, 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
  x
n
(
i
  = 1
,
2) and
  y
n
(
i
  = 1
,
2) entering Eqs.( 60 ) and ( 61 ). In the end, with the sim ula ted
  C
(
t
) and
S
(
t
), one should form the fringe shift combination ∆
λ
(
θ
)
λ
  ≡
2
C
(
t
)cos2
θ
 + 2
S
(
t
)sin2
θ
 =
 A
2
(
t
)cos2(
θ
−
θ
0
(
t
) ) ( 1 1 4 ) As recalled in Sect.3, an individual session of the Michelson-Morley experiment consisted of  6 rotat ions. Eac h compl ete rotation of the interf erome ter took 6 min utes and the consecu tiv e readi ngs of the fringe shifts wer e performe d every 22.5 degre es. There fore, two consecu tiv e rea din gs diﬀ ere d by 22.5 sec ond s. In the se con dit ion s, a nu mer ica l sim ula tio n of a sin gle 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
 ∼
102
o
. 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
A
ﬁt 2
 (average data
−
noon)
∼
0
.
0 1 2 ( 1 1 5 ) We have thus considered the exact amplitude
A
exact 2
  (
t
) = 2

C
2
(
t
) +
S
2
(
t
) ( 1 1 6 ) and selected a particular conﬁguration whose global average over the 6 turns gives

A
exact 2
  (
t
)
 ∼
0.0 12. Of cours e, thi s con dit ion can be rea liz ed by a v ery large num ber of con ﬁgu rat ion s. 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- perimen t with Table 9 and Fig.12). Ho wev er, our intent ion 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 [2
C
(
t
)
,
2
S
(
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
  2
π
 together wit h the resu lts of 2-p ara met er ﬁts to the simu lat ed dat a. Not ice the str ong scatt er of the simulated data at the same
  θ
−
v alues. Of course , to compar e 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) concernin g the fringe shift s from the sessi on of July 8th. By accepti ng his int erpre - tation, the experimental azimuths from the three noon sessions of July 8th, 9th and 11th, respectively
 θ
EXP 0
  ∼
357, 285 and 317 degrees, would become
 θ
EXP 0
  ∼
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 com ponents of
  v
x
(
t
)
  and
  v
y
(
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
.
0 2 3 + 0
.
002
  −
0
.
0 2 4 + 0
.
001
  −
0
.
004
  −
0
.
003 2 +0
.
003
  −
0
.
011
  −
0
.
000
  −
0
.
006
  −
0
.
021
  −
0
.
034 3
  −
0
.
001
  −
0
.
0 0 1 + 0
.
000
  −
0
.
0 0 9 + 0
.
0 0 2 + 0
.
007 4
  −
0
.
0 0 2 + 0
.
003
  −
0
.
0 0 8 + 0
.
002
  −
0
.
060
  −
0
.
030 5 +0
.
002
  −
0
.
0 1 7 + 0
.
002
  −
0
.
0 0 1 + 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
.
0 2 6 + 0
.
001 9
  −
0
.
015
  −
0
.
000
  −
0
.
008
  −
0
.
001
  −
0
.
008
  −
0
.
022 10
  −
0
.
037
  −
0
.
0 0 5 + 0
.
000
  −
0
.
002
  −
0
.
0 0 3 + 0
.
003 1 1 + 0
.
003
  −
0
.
022
  −
0
.
015
  −
0
.
005
  −
0
.
0 0 3 + 0
.
002 12
  −
0
.
002
  −
0
.
049
  −
0
.
023
  −
0
.
016
  −
0
.
009
  −
0
.
006 13
  −
0
.
0 0 1 + 0
.
0 0 2 + 0
.
0 0 0 + 0
.
0 0 1 + 0
.
003
  −
0
.
001 1 4 + 0
.
003
  −
0
.
0 0 3 + 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
.
0 0 4 + 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 com ponents of
  v
x
(
t
)
  and
  v
y
(
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
.
0 1 1 + 0
.
0 0 1 + 0
.
0 1 1 + 0
.
0 0 0 + 0
.
0 0 3 + 0
.
001 2 +0
.
004
  −
0
.
003
  −
0
.
001
  −
0
.
0 0 8 + 0
.
007
  −
0
.
009 3 +0
.
003
  −
0
.
0 0 5 + 0
.
000
  −
0
.
0 0 7 + 0
.
0 0 7 + 0
.
000 4 +0
.
001
  −
0
.
007
  −
0
.
003
  −
0
.
0 0 1 + 0
.
0 0 1 + 0
.
016 5 +0
.
002
  −
0
.
0 0 7 + 0
.
0 0 0 + 0
.
001
  −
0
.
000
  −
0
.
003 6
  −
0
.
0 0 4 + 0
.
0 1 0 + 0
.
001
  −
0
.
0 0 1 + 0
.
0 0 6 + 0
.
009 7
  −
0
.
0 1 7 + 0
.
0 0 2 + 0
.
005
  −
0
.
000
  −
0
.
0 0 3 + 0
.
002 8 +0
.
011
  −
0
.
002
  −
0
.
012
  −
0
.
0 0 1 + 0
.
010
  −
0
.
000 9 +0
.
0 1 1 + 0
.
0 0 1 + 0
.
0 0 8 + 0
.
0 1 1 + 0
.
005
  −
0
.
011 1 0 + 0
.
0 1 6 + 0
.
005
  −
0
.
0 0 1 + 0
.
001
  −
0
.
0 0 2 + 0
.
001 1 1 + 0
.
000
  −
0
.
0 1 3 + 0
.
0 1 5 + 0
.
005
  −
0
.
002
  −
0
.
001 12
  −
0
.
001
  −
0
.
0 2 2 + 0
.
004
  −
0
.
003
  −
0
.
0 0 0 + 0
.
002 13
  −
0
.
001
  −
0
.
0 0 1 + 0
.
0 0 1 + 0
.
000
  −
0
.
0 0 4 + 0
.
001 14
  −
0
.
0 0 1 + 0
.
0 0 2 + 0
.
0 0 2 + 0
.
0 1 8 + 0
.
0 0 1 + 0
.
009 15
  −
0
.
0 1 2 + 0
.
018
  −
0
.
0 0 2 + 0
.
001
  −
0
.
0 0 4 + 0
.
007 1 6 + 0
.
0 0 2 + 0
.
0 0 8 + 0
.
0 0 1 + 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
  2
π
. The varian ce of the averag es is ab out
 ±
0
.
004
 for the amplitude and about
 ±
11
o
 for the azimuth.
i 1 2 3 4 5 6 a v e r a g e 1
  −
0
.
0 2 3 + 0
.
002
  −
0
.
0 2 4 + 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
.
0 0 5 + 0
.
000
  −
0
.
0 0 7 + 0
.
0 0 7 + 0
.
000
  −
0
.
000 4 +0
.
002
  −
0
.
0 0 7 + 0
.
004
  −
0
.
0 0 2 + 0
.
0 4 3 + 0
.
0 3 3 + 0
.
012 5
  −
0
.
0 0 2 + 0
.
017
  −
0
.
0 0 2 + 0
.
001
  −
0
.
0 0 3 + 0
.
0 0 8 + 0
.
003 6 +0
.
008
  −
0
.
0 0 3 + 0
.
0 4 1 + 0
.
0 1 0 + 0
.
0 0 2 + 0
.
0 2 7 + 0
.
014 7 +0
.
017
  −
0
.
002
  −
0
.
0 0 5 + 0
.
0 0 0 + 0
.
003
  −
0
.
0 0 2 + 0
.
002 8
  −
0
.
0 1 6 + 0
.
0 0 1 + 0
.
001
  −
0
.
001
  −
0
.
0 2 5 + 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
.
0 0 3 + 0
.
003
  −
0
.
003 1 1 + 0
.
000
  −
0
.
0 1 3 + 0
.
0 1 5 + 0
.
005
  −
0
.
002
  −
0
.
0 0 1 + 0
.
001 1 2 + 0
.
0 0 0 + 0
.
0 1 9 + 0
.
0 2 0 + 0
.
0 0 9 + 0
.
0 0 6 + 0
.
0 0 6 + 0
.
010 1 3 + 0
.
001
  −
0
.
002
  −
0
.
000
  −
0
.
001
  −
0
.
0 0 3 + 0
.
001
  −
0
.
001 14
  −
0
.
0 0 1 + 0
.
001
  −
0
.
0 0 4 + 0
.
004
  −
0
.
0 0 0 + 0
.
0 0 7 + 0
.
001 1 5 + 0
.
012
  −
0
.
0 1 8 + 0
.
002
  −
0
.
0 0 1 + 0
.
004
  −
0
.
007
  −
0
.
001 16
  −
0
.
005
  −
0
.
018
  −
0
.
003
  −
0
.
000
  −
0
.
004
  −
0
.
003
  −
0
.
005
A
ﬁt 2
  0 . 0 0 9 0 . 0 0 5 0 . 0 0 9 0 . 0 0 3 0 . 0 1 1 0 . 0 1 3 0 . 0 0 8
θ
ﬁt 0
  279
o
259
o
266
o
285
o
255
o
272
o
269
o
69
About Author
Maurizio Consoli

Istituto Nazionale di Fisica Nucleare

Faculty Member
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34 more by Maurizio Consoli View All ▸
Relativistic analysis of Michelson-Morley experiments and Miller's cosmic solution for the Earth's motion
Maurizio Consoli
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