Photon-Phonon Analogy in a Superfluid Vacuum. Marco Fedi

Photon-phonon analogy in a superfluid vacuum. Marco Fedi

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Marco Fedi. Photon-phonon analogy in a superfluid vacuum. . 2017. hal-01532718v1

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Distributed under a Creative Commons Attribution| 4.0 International License Photon-phonon analogy in a superuid vacuum.

Marco Fedi* Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR), Rome, Italy

June 2, 2017

Abstract for uids and superuids, between phonons and photons and (3) referring to Bolmatov [14], we discuss transver- We discuss clues to consider quantum vacuum as a su- sal wave propagation and heat transmission in uids on peruid, probably superuid dark energy, in which we acoustic basis (phonons). From Gremaud (4) [25] we re- analyze a photon-phonon analogy. The discussion is sort to a complete analogy between Maxwell's equations structured in four parts: (a) shared features and behav- for electromagnetism and non-divergent deformations of ior photon-phonon; (b) phonons in uids and their role an isotropic lattice in Euler's coordinates, translated in in expressing energy, along with the transient solid-like our case into the quasi-lattice structure of uids which (quasi-lattice) structure arising in uids and superuids manifests within their relaxation time, a structural prop- during Frenkel relaxation time; (c) Gremaud's analogy erty which has been discussed also for superuid 4He [38]. of Maxwell's equations in a lattice; (d) Lorentz factor as Finally (5), arguing a possible dilatant behavior of some the rheogram of dark energy and a possible basis for a superuids under shear stress in a relativistic regime, we quantum interpretation of special relativity. present Lorentz factor as the rheogram of dark energy, opening a perspective onto the explanation of special rel- PACS numbers: 42.25.Bs, 95.36.+x, 47.35.Rs, 12.20.-m, ativity on a quantum hydrodynamic basis and reinforcing 95.35.+d, 03.30.+p transverse propagation of phonons in superuid dark energy due its dilatancy within a relativistic regime. As a consequence, we understand that dark energy would be undetectable only as long as it remains unperturbed, be- Introduction ing light its most evident manifestation, along with its (at the moment more known) repulsive action which avoids We know that light propagates through a quantum vac- the gravitational collapse of the universe, probably due uum but also through dark energy, since according to to its internal pressure as a superuid. recent measurings it constitutes 69.1% of the universe mass-energy, which, along with dark matter, comes to 95%. We show that quantum vacuum uctuations pos- 1 Light propagates through a sess a hydrodynamic nature appearing as quantized vortices, so we can interpret quantum vacuum as the sponta- quantum vacuum, correspond- neous hydrodynamic perturbation of an ubiquitous cos- ing to superuid dark energy. mic superuid, which may correspond to dark energy with superuid characteristics, as a cosmic Bose-Einstein The existence of a false vacuum with non-zero energy con- condensate [1, 2, 3, 4, 5, 6, 7, 13]. Thus, dark energy tent is denitively accepted and proven in a lot of phys- would confer on space the features of a superuid quan- ical phenomena: the Lamb shift, the Casimir eect, the tum space (SQS). According to our considerations, here a Unruh eect, the anomalous magnetic moment, vacuum photon might propagate as a transverse phonon (arguing birefringence [12]etc. From the point of view of quantum that light is the sound of dark energy) and we support physics, light travels through such a quantum vacuum. this thesis in four steps. We reect (2) on an interest- This environment is known for the continuous appearance ingly wide set of currently known analogies, also valid and annihilation of virtual particle-antiparticle pairs, as *[email protected] initially formulated by Dirac. The relationship for these

1 uctuations is with the structure of a vortex web arising in a familiar h 4E4t ≥ = . (1) superuid such as 4He. In this analogy all the space 2π ~ among the laments is occupied by liquid helium in (a) The Bohr-Sommerfeld quantization condition, expressing and by superuid dark energy (SDE) in (b). mass circulation in a quantized vortex

p · dx = nh (2) ˛C for n = 1 tells us that the quantum of action, h, actually refers to a complete turn along a circular path of a quantum whose momentum is p. In (1) 2π also refers to a 360° turn and we can therefore interpret vacuum uctuations as quantized vortices. The sign≥ states that we consider n ≥ 1 complete rotations of the vortex during a time 4t as the vacuum uctuation. Quantum vortices are known to manifest in superuids, as in 4He [8, 37]. We also observe vortex-antivortex pairs, which form and annihilate [39, 8, 37], exactly as particles-antiparticles pairs in quantum vacuum. In our opinion, these are therefore clues for considering quantum vacuum as a superuid. The analogy particles-quantized vortices is reinforced by the fact that fermions spin-½ may be described in hydrodynamic terms as the circulation of quanta in a torus vortex (see [4], 3.1). Thus, if vacuum Figure 1: Left [10]: (a) Metal atoms trapped in superuid helium vortices highlight a structure of vortex laments; (b) galactic uctuations are superuid vortices, what is the underly- laments of dark matter which galaxies aggregate on [11]. Here ing superuid in which they arise? the relationship between dark energy and dark matter is the same The possible answers are the Higgs eld or dark energy, existing between superuid helium and the vortex laments which manifest in it, i.e. dark matter is a hydrodynamic manifestation of both observed as dark scalar elds. Being the Higgs dark energy [1]. boson the fundamental excitation of the Higgs eld and very massive, it is probably a vortex itself, so we opt for dark energy, as a cosmic fundamental superuid. After The internal pressure of SDE would be responsible [9] all, we know it constitutes of the mass-energy of ∼ 69% for the repulsive force traditionally attributed to dark the universe, also expressed in the cosmological constant, energy in cosmology. Moreover, the equation of state , where ( 00 as regards the stress-energy Λ = kρ0 ρ0 T , of cosmology for a single-uid model can be referred to tensor) indicates the density of dark energy. Along with SDE, where P and ρ are respectively the pressure and dark matter, which can be interpreted as condensed dark d d the density of dark energy [13] energy [1, 2, 3, 4] and whose existence is for instance evident in the dark halos of spiral galaxies which the at P w = d (3) proles of orbital velocities are believed to be due to, we ρd arrive at . ∼ 95% As far as the propagation of light through this ubiquitous As for any form of energy, also dark energy has to superuid is concerned, we can still believe that photon be quantized. We speak of dark energy quanta (DEQ). is a real particle whose energy is not aected by any min- The hydrodynamical perturbation of these quanta would imal friction while traveling through this superuid or we produce the known picture of quantum vacuum as well can analyze, as below, the possibility that a photon is ac- as the Higgs boson itself. tually a transverse phonon (a quasi-particle) propagating The temperature of the cosmic microwaves background in superuid dark energy. According to this view, dark radiation (CMB), ∼ 2.72 K, would be in agreement with energy does not interact with baryon matter unless it is the temperature of other superuids such as 4He. Fig. 1 hydrodynamically perturbed and the most evident pertur- shows dark matter distribution in the universe in analogy bation might coincide with light itself.

2 2 Current photon-phonon analo- of light is given as [4, 9]

gies. 1 c = √ . (4) βdρd Let us start with listing all current analogies be- Indeed, starting from the equation which denes the tween photons and phonons (which can also manifest speed of sound in a uid, a = pK/ρ, where K is the in superuids [16, 26]). Both are bosons [17]; have bulk modulus, and putting β = 1 as isentropic com- wave-particle duality [18, 19]; obey the doppler eect, S K pressibility (in the specic case of SDE we say βd), we z = (f − f ) /f ; are symmetric under exchange, emit obs obs obtain (4). This acoustic analogy of the speed of light is |α, βi = |β, αi; can be created by repeatedly applying the also conrmed possible in [25], as discussed in 4. creation operator, a†; possess a momentum, where that Amendola and Tsujikawa [13], by introducing the of a phonon1 is with (hence pph ≡ ~k = h/λ, k = 2π/λ speed of sound through a ultra-light scalar eld φ, state the parallelism: radiation pressure ⇔ sound pressure); that it is the key parameter to understand the (back- are involved in photoelectric eect and Compton scat- ground) dynamics of such a eld. Using the ratio tering thanks to their momentum; they can spin [20, 21] pressure/density under root as the the speed of sound (rotating phonons has been also described as regards the through this cosmic uid, they dene such a speed as physics of nanotubes [22]); can form squeezed coherent states [23]; can interact via parametric down conversion s s [24]. Both for photons and phonons, ~ω/2 is vacuum's 2 0 0 02 δPφ H (φ ϕ − φ Ψ) − V,φϕ (5) (we say dark energy's) contribution, where the harmonic cs,φ = = 2 0 0 02 . δρφ H (φ ϕ − φ Ψ) + V,φϕ oscillator eigenvalues for the mode ωk (k is the wave number) are and to con- where we see that, when the potential of the eld becomes En = (n + 1/2) ~ωk n = 1, 2, 3, ... rm the presence of a false vacuum we see that also at, V,φ → 0, we may have the speed of sound through for n = 0 the energy is not zero. This means that the eld coinciding with that of light (natural units are what we think to be the vacuum actually contains en- used by the authors, where the speed of light is c = 1). ergy and according to E = mc2, energy implies a certain mass density ( in the cosmological constant), where ρ0 3 Phonons-driven thermal energy 2 −1 c = (βdρd) would be precisely linked to dark energy's intrinsic parameters, as from Eq. (4). There is a medium transmission in uids. throughout the universe owning density ρ 6= 0 which light propagates through. In other words and according to We have remarked that phonons, which are typically as- quantum physics, light does not propagate in a vacuum sociated to a solid state, also manifest in uids and super- but in a quantum vacuum, which, from a cosmologi- uids [14, 15, 16]. Particularly relevant for our case is the cal point of view, may be identied as dark energy with paper of Bolmatov, Brazhkin and Trachenko [14], about superuid features. a phonon theory of heat diusion in classical and quantum uids where longitudinal and transversal phonons are described considering Frenkel's work [15], who rst 2.1 The formula for the speed of light. noticed that the density of liquids is much dierent from It is worth recalling that Maxwell derived the dielectric that of gases but only slightly dierent from the density constant (ε0) and the magnetic permeability (µ0) of vac- of solids and who also dened the existence of transversal √ uum, from which the formula c = 1/ ε0µ0 follows, in waves in liquids, as previously observed in solids, for fre- terms of density and transverse elasticity of the ether (see quencies larger than 1/τ, where τ is the relaxation time 4, [25]). We excluded the existence of the ether but we of the uid, i.e. the time during which the structure of still need quantum vacuum. Thus, if we now consid- the liquid remains unaltered, similar to a solid lattice. ered the old ether to be the modern dark energy, whose After many years this has been observed and also for existence is on the contrary accepted, and we equate low-viscosity uids [16]. The importance of what investi-

ε0µ0 = βdρd, where ρd is the density of SDE and βd its gated in [14] is linked to the fact that we need to describe isentropic compressibility, we may state that the speed phonons through SDE as transverse waves (as light is) and to the evidence that photons too transmit heat, en- 1it is said that a phonon possesses a pseudo-momentum but following our reasoning this can be true also for a photon. ergy. Indeed, any body whose temperature is not at ab-

3 solute zero (i.e. any object, according to Nernst theorem) Planck constant coming from phonon free energy emits photons, whose frequency is in the infrared range X ωi for common objects around us, except higher frequencies F = E + T ln 1 − exp −~ , (9) ph 0 T of visible light sources. The relationship photon-phonon i as far as heat/energy transmission is concerned is inter- where E0 is the temperature-dependent zero-point energy esting and we believe it may end up into the coincidence (that we can assume as the cosmic microwave background photon-phonon if the propagation occurs in the scalar temperature in our case, i.e. as the intrinsic superuidity eld of SDE. It is interesting to also reect that the exis- temperature of dark energy, ∼ 2.72K), and N the number tence of optical phonons, i.e. phonons created via photon of modes. In (7) the zero-point energy has been omitted. scattering, could represent, from our point of view, the The authors conclude that as we have a good understand- passage of a phonon from SDE to a baryon lattice. We ing of thermodynamics in solids based on phonons, de- understand now that dark energy could be actually in- spite their structural complexity, the same can apply for teracting with our baryon world in most common ways liquids. For the present investigation, this means that but still we do not interpret its interactions in the right phonon-based transversal heat transmission through su- way, exclusively thinking of its mere repulsive action far peruid dark energy is possible. The thermal signicance into the cosmos. of a photon would then be comprised in phonon-based In superuids, energy is dissipated as heat at small quasi-lattice vibrations of dark energy quanta. Not only. scales by phonon radiation [27]. So let us analyze the is- Below, we discuss how Maxwell equations describing pho- sue of phonons carrying heat in uids, useful to describe ton's electromagnetic eld can equally express the lattice photons as transversal phonons in a superuid. Brazhkin dynamics of SDE, theoretically completing the analogy and collegues come to the result that there are two kinds phonon-photon. of atomic motion in uids: phonon motion, consisting in one longitudinal mode and two transverse modes with fre- 4 Maxwell's equations express quency ω > ωF , where ωF = 2π/τ is Frenkel frequency, and diusive motion. Both kinds of motion possess a ki- SDE quasi-lattice dynamics. netic (K) and a potential (P ) component, so the energy of the uid is expressed as Important for our photon-phonon analogy in SDE is Gremaud's work at the Institute of Condensed Matter Physics of the Swiss Federal Institute of Technology in Lausanne, who discusses a complete analogy between the E = Kl +Pl +Ks (ω > ωF )+Ps (ω > ωF )+Kd +Pd (6) equations of a non-divergent deformation of an isotropic solid lattice in Euler's coordinates and Maxwell's equa- where the subscripts l and s refers to longitudinal and tions of electromagnetism [25, 29, 30]. In his work he con- shear waves (transversal phonons) and d to diusion. cludes that Maxwell's equations can be seen as a model By applying several steps including the virial theorem, for describing also dierent physical systems, not only phonon free energy, Grüneisen approximation and Debye electromagnetism. Simmetrically speaking, we go further vibrational density of states, for the details of which we and state that electromagnetism is the dynamic eect of refer to [14], and neglecting the diusive potential compo- a dierent physical system. This is fundamental if we nent since Pd Ps (ω > ωF ), a nal equation expressing want to dene photon's electromagnetic eld as acoustic a phonon theory of liquids is obtained in the form perturbations of SDE, which, for excitation frequencies greater than the reciprocal of Frenkel's relaxation time 3 ! (1/τ), as discussed above, may behave as a solid-like lat- αT ~ωD ωF ~ωF E = NT 1 + 3D − D tice (quasi-lattice transient structure), despite possess- 2 T ωD T (7) ing very low viscosity [38, 16]. For radio waves of about 250MHz relaxation time should be for instance greater where x 3 than −9 while −15 for visible light. 3 z dz (8) 4 · 10 s >∼ 1.7 · 10 s D(x) = 3 x ˆ0 exp(z) − 1 Gremaud's analogy is complete since, along with is Debye function [28], ωD is Debye frequency, α is the Maxwell equations, it describes the dielectric polariza- coecient of thermal expansion of the uid, ~ the reduced tion and magnetization of matter, as well as electrical

4 charges and currents. The author introduces the concept of dislocation charges in the lattice [29], in analogy with  −→ ˜rot ~ −→ − ∂~ω + rot φ = J˜ ⇔ − ∂D + rotH~ = ~j the electrical charges, associated to plastic distortions. It ∂t 2 ∂t (11) is shown that the transversal waves of rotation and shear div ~ω = λ ⇔ div D~ = ρ strain are associated with a propagation velocity given by r where ~ω is the rotation eld corresponding to the elec- K2 + K3 ct = (10) ~ ~ nm tric eld of displacement, D; J is the vector ow of rotation charges [29], equivalent to the density of electric where the subscript on the left means transversal and current ~j; λ is the density of rotation charges analogous

K2 and K3 respectively represent shear stress modulus to the density of electric charges ρ and H~ is the magnetic and rotation modulus. The dielectric permittivity of vac- eld. uum, ε0, is given as 1/ (2 (K2 + K3)), and this corresponds to 1/K = βd, the isentropic compressibility of  rot −→ 0 ~ −→ SDE used in (4), while corresponds to the mass  ∂n~p = −rot ˜m ⇔ ∂B = rotE~ 2nm ∂t 2 ∂t (12) rot density of the lattice and in our case to ρd. div n~p = 0 ⇔ div B~ = 0 In this uid approach to light, photon's electromagnetic eld is produced as transversal acoustic lattice os- where B~ is the magnetic induction eld, E~ the electric cillations (Fig. 2), probably due to angular momentum eld, n~prot the volume linear momentum of lattice (mass transfer from the particle which emits the photon. ow of lattice) and ~m0 the generalized torque momentum.

 0 ~ω = 1 ~m + ~ωan  2(K2+K3) 2 rot rot ~rot ~rot rot φ~ φ~ (JI −JL ) n~p = 2nm 2 + C 2 + 2n

m (13)

 D~ = ε0E~ + P~ Figure 2: Density and compressibility of dark energy (ρd) (βd) ~ h ~ ~ ~ i at the origin of photon's transverse EM eld (b), whose oscilla- B = µ0 H + χH + M tions are due to harmonic, orthogonal compressions of dark energy quanta occurring within the relaxation time, while the main pulse propagates along the z−axis. On the left (a), the probable quan- being 1/ (2 (K2 + K3)) ⇔ ε0 and analogous to 1/K = tum mechanism at the origin of B ⊥ E, due to compression and an βd in (4); ~ω the vector of anelastic shear and local ro- intrinsic angular momentum of DEQ (here and are two ex- ¯d1 ¯d2 tation, analogous to the dielectric polarization of matter emplifying quanta) arising from that of the emitting particle (e.g. ~ of an electron). P ; C = (CI − CL) as the atomic concentrations of interstitials and vacancies (for which we take instead into consideration DEQ), analogous to the paramagnetic and Below we summarize the analogy between alterations diamagnetic susceptibility of matter χ = χpara + χdia; of lattice geometry presenting homogeneous expansion J~rot − J~rot is the surface ux of interstitials and va- in a mobile frame O0x0y0z0 and Maxwell's equations as I L argued in detail by Gremaud in [25]. cancies; n the density of lattice sites and M~ the magnetization of matter. To do that, according to Frenkel [15] and Bolmatov [14], we treat the uid medium as momentarily solid-like, And nally assuming that electromagnetic waves have a frequency ν > 1/τ, being τ Frenkel relaxation time. This allows ∂λ ∂ρ = −div J~ ⇐⇒ = −div~j (14) transversal waves propagation. Besides that concerning ∂t ∂t the speed of light, just discussed above, we also obtain the following analogies and

5 ! ~m0 φ~rot ∂n~prot ~m0 ∂~ω φ~rot ~m0 − J~ = + − div ∧ 2 2 ∂t 2 ∂t 2 2

m (15)

∂B~ ∂D~ −E~~j = H~ + E~ − div H~ ∧ E~ ∂t ∂t Superuid behavior of light has been also observed Figure 3: Lorentz factor as the rheogram of dark energy. Because in polaritons condensates from 2007 [31] up to recent of its quantum, granular nature, superuid dark energy should be- experiences [32]. Another approach to the many body have as a dilatant uid when shear stress enters into a relativistic physics in uids of light has been resorting to a bulk non- regime, that is when the body velocity approaches the speed of linear medium with intensity-dependent refractive index, sound in dark energy. This would imply that the so-called relativistic mass increase is actually the eect of apparent viscosity, as showed by Carusotto [33], where, under the parax- which acts as a force in the opposite direction to acceleration) . ial approximation, photon propagation can be described Assumed that the speed of sound in a dilatant uid can't be ex- through a Gross-Pitaevskii equation for the order param- ceeded without cracking it, this would explain the upper limit to eter, as the electric eld amplitude of a monochromatic acceleration experienced in synchrotrons. laser beam. Another noteworthy investigation on superuid propagation of light is that of Leboeuf and Moulieras sound in dark energy) [34], although it has to be pointed out that these studies do not treat light itself as a hydrodynamic, acoustic 0 v 1 1 (16) γ ≡ arcsin = r = p , phenomenon, as we do, but only analyze the superuid vsd 2 1 − v2β ρ 1 − v d d behavior of light under the right circumstances. vsd

2 where βdρd = 1/c from (4) and the derivative of the arc- sine species that it is not possible to exceed the speed 5 Insurmountability of the speed of sound through a dilatant uid unless to crack it, as ex- perimentally veriable, generating an asymptote to shear of light: Lorentz factor as the stress (and to acceleration). Our reasoning would imply rheogram of SDE. that also some among familiar superuids could manifest a dilatant behavior under relativistic shear stress, a Dark energy's quanta which ll up the universe as a sus- phenomenon for the verication of which we invoke spe- pension in space, would cause a non-Newtonian, dilatant cic tests. Dark energy would therefore show a double behavior of dark energy. However, the dilatancy of this side: superuid within a non-relativistic regime and di- granular, dark substance would be detectable only under latant inside a relativistic regime, helping to explain the relativistic shear stress, i.e. for accelerations occurring in microscopic, quantum basis of special relativity. relativistic regime, while for non-relativistic speeds, the As far as the hypothesized dilatancy of space (lled cosmic scalar eld with positive, near-zero viscosity called of quantized dark energy) is concerned, we can cite the dark energy behaves as a superuid. The dierence be- words of the Nobel laureate for the quantum Hall eect tween a relativistic and non-relativistic regime (see Fig. R. B. Laughlin [35]: Studies with large particle acceler- 3) would be then reduced to the apparent viscosity of ators have now led us to understand that space is more SDE coming into play with the increasing of accelera- like a piece of window glass than ideal Newtonian empti- tion, as observed in synchrotrons, a phenomenon which ness. It is lled with 'stu' that is normally transparent is currently interpreted as relativistic mass increase. In but can be made visible by hitting it suciently hard Lorentz factor, we can consider β = v/c as the ratio to knock out a part. The modern concept of the vac- v/vsd of the velocity of a body through SDE to the speed uum of space, conrmed every day by experiments, is a of sound in dark energy (we write vsd instead of c to relativistic ether. But we do not call it this because it remark that we describe the speed of light as speed of is taboo. Indeed, we call it quantum vacuum. Or, in

6 our case, SDE, using a term closer to cosmology and to ing apparent viscosity of dark energy, paving the way the need of general relativity itself of having a huge, in- for a possible explanation of special relativity at a quan- visible mass-energy present throughout the universe to tum level. From this investigation, light appears as the impede a gravitational collapse. Finally, it is important sound2 of dark energy and a photon as a quasi-particle to point out that dark energy dilatancy at relativistic (just like a phonon) propagating through dark energy. regime (then we can also include light propagation in- This framework would also explain why light possesses terpreted as sound through SDE traveling at a speed a precise propagation speed in vacuum exactly as me- c = 299792458 m/s) would play a fundamental role in chanical waves possess one for each dierent substance phonon propagation as a transverse wave, compared to they propagate through. As regards light, this substance standard, longitudinal sound propagation in other medi- may be called dark energy in cosmology or quantum vac- ums, by producing a local, transient solid-like environ- uum in QFT. Accordingly, if the density of dark energy ment for the wave, making us remember Stoke's theory should be dierent in a distant part of the universe, light of light propagation, and this in addition to the Frenkel's would travel at a dierent speed. To conclude, it should solid-like behavior at frequency ω > ωF . be noted that if the gravitational eld corresponded to an ether wind (or better SDE wind), as theorized in the theory of Superuid Quantum Gravity ([4], 4), then a 6 Conclusion Michelson-Morley interferometric test [36] could not de- tect any variation in light propagation due to the relative The propagation of light through a quantum vacuum for motion Earth-ether, as indeed happened, but light would which there are strong hints of superuid features, let us be only inuenced by the gravitational eld, as general wonder whether a photon could be a transversal pulse relativity conrms. Cosmological implications of acoustic through this dark medium, identied as superuid dark propagation of light through the cosmic superuid have energy [4, 2, 13, 1]. The fact that photons and phonons been discussed in [9]. share virtually all their features, including bosonic nature, wave-particle duality, doppler eect, symmetry un- Acknowledgements der exchange, application of creation-annihilation opera- The author thanks Valeriy Sbitnev for collaboration and tors, momentum, squeezed coherent states, photoelectric for the exchange of views concerning the issue of a super- eect, interaction via parametric down conversion and uid quantum space. the harmonic oscillator along with the vacuum contribution it expresses, has driven us to write a formula for the speed of light as that of sound through a uid medium References (using in our case density an isentropic compressibility of superuid dark energy) as originally done by Maxwell, [1] V.I. Sbitnev, Dark matter is a manifestation of √ the vacuum Bose-Einstein condensate. 2016, URL: in c = 1/ ε0µ0, who considered vacuum's permittivity and magnetic permeability as, respectively, elasticity and http://arxiv.org/abs/1601.04536 density of the ether (see also [25]). [2] K. Huang, A Superuid Universe. World Scientic, Transversal phonon propagation and heat transporta- Singapore (2016) tion, as well as transient solid-like behavior of uids according to specic frequencies, were necessary and [3] K. Huang, Dark energy and dark matter in a su- have been analyzed through the work of Bolmatov and peruid universe. https://arxiv.org/abs/1309.5707 collegues [14], while a complete analogy between lat- (2013) tice deformations, useful for phonon propagation, and [4] M. Fedi, A Superuid Theory of Everything?, v.4, Maxwell's equations of electromagnetism has been re- 2017, URL: hal.archives-ouvertes.fr/hal-01312579 ported from Gremaud [25, 29, 30]. Finally, by hypothe- sizing a dilatant behavior of SDE (and perhaps of some [5] V.I. Sbitnev, Hydrodynamics of the physical vac- familiar superuids) if exposed to relativistic shear stress, uum: II. Vorticity dynamics. Found. of Physics. we have justied the universally insurmountable limit of 2016; URL: http://rdcu.be/kdon. the speed of light (which in Einstein's theory of relativ- 2What? Is it the light I hear?, R.Wagner, Tristan und Isolde, ity is used as a matter of fact) as due to the increas- Act 3, Scene 2.

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