Theoretical and CFD analysis of gravity and galaxy formation and rotation in a dilatant vacuum Marco Fedi, Mario R.L. Artigiani

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Marco Fedi, Mario R.L. Artigiani. Theoretical and CFD analysis of gravity and galaxy formation and rotation in a dilatant vacuum. 2019. ￿hal-02343559v3￿

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Marco Fedi∗ Ministero dell’Istruzione, dell’Universita` e della Ricerca (MIUR), Rome, Italy

Mario R. L. Artigiani† Engineering3D.it, Desenzano del Garda, Italy (Dated: 22 December 2019) Following some successful computations in which the vacuum has been treated as a non-Newtonian dilatant fluid to derive Einstein formula for the precession of perihelia and to obtain a better solution to the Pioneer anomaly, in this study we proceed by describing gravity, both at micro- and macro-level, as well as galaxy formation, shapes and rotation, still according to a shear-thickening vacuum, also by means of high-resolution computational fluid dynamics simulations. We show that the resulting gravitational model allow us to describe gravitational waves and to obtain various galaxy shapes, along with the flat profile of galaxy-rotation velocity. As the previous investigations, also this study reinforces the evidence that space is a dilatant fluid, probably an unexpected hydrodynamic feature of the viscous Higgs field.

Keywords: dilatant vacuum, quantum gravity, spin, galaxy formation, flat profile, Higgs

Introduction by describing fundamental fermions as toroidal vortices (Sect. I). In the second section we derive an equation for The vacuum as a superfluid has been studied in several quantum gravity, without the classical Newtonian constant. approaches [1–3], whose recent evolution is for instance the The quantum aspects of this formula are then discussed in logarithmic BEC vacuum theory [4, 5]. Nevertheless, the Sect. III. In the following section we simulate a quadrupole hydrodynamic characteristics of the vacuum seem to be much via computational fluid dynamics (CFD) and we show that different in various cases. The Nobel laureate R.B. Laughlin, gravitational waves can be described as negative pressure states: waves in a dilatant vacuum (Sect. IV). Further CFD simulations demonstrate that a binary black hole located in Studies with large particle accelerators have now the center of a galaxy and in a dilatant vacuum, can generate led us to understand that space is more like various galaxy shapes (Sect. V) a well as the flat profile of a piece of window glass than ideal Newtonian the rotation velocity (Sect. VI). Eventually, we describe an emptiness. It is filled with stuff that is normally unexpected effect obtained in the simulations: a breath of transparent but can be made visible by hitting it galaxies (Sect. VII). sufficiently hard to knock out a part.

This behavior is typical of non-Newtonian shear-thickening I. Spin-generated Bernoulli force as the engine of gravity. fluids (dilatant fluids): they are fluid when low shear stress is applied and they progressively solidify as shear stress increases. The situation described by Laughlin, as regards Dilatant vacuum can be interpreted as a superfluid (dark large particle accelerators, has been in fact described as due energy [8]) doped with scattered dark-matter particles [9], to the existence of a dilatant vacuum [6], through which which together make up ∼ 95% of the mass-energy of charges accelerate. Not by case, by treating the vacuum as the universe: something like a cosmic oobleck [10]. The a shear-thickening fluid, the relativistic perihelion precession dilatant-vacuum model therefore goes in the direction of a of the planet Mercury has been solved by deriving Einstein unification of dark energy and dark matter into a single formula for the precession of perihelia in a simpler way dark fluid [11, 12]. At the same time, vacuum dilatancy and the Pioneer anomaly has been directly computed, with could be also interpreted as a hydrodynamic feature of the much greater accuracy, by using the Lorentz factor in Stokes’ viscous Higgs field. It has been proved that the apparent law, as a viscosity factor of the vacuum [7]. In this way viscosity of the vacuum obeys the Lorentz factor [7], thus, the curve expressed by the Lorentz factor turned out to be when shear stress is low, the vacuum is in a superfluid-like the rheogram of the vacuum. Vacuum dilatancy arises both regime. Just like classical mechanics is placed at the with particles and with macroscopic bodies (probes, planets) beginning of the curve expressed by the Lorentz factor. The traveling through a vacuum. temperature of the superfluid vacuum at 2.72K, i.e. that of In this study, the analysis of the vacuum as a dilatant fluid the cosmic microwave background (CMB), would match that is continued as regards gravity and galaxy formation and of laboratory superfluids. In superfluids, quantized vortices rotation. We start from the vacuum in superfluid regime, spontaneously arise and since a vortex tube must extend to a boundary or close in a loop (an interesting connection with loop quantum gravity [13] can be seen in this fact), obeying Helmholtz’s second principle, vortex rings (vortex ∗ E-mail: [email protected] tori) can form in a superfluid vacuum (Fig 1) [2, 3, 14]. † E-mail: [email protected] However, what is also important is the fact that only vortices 2

analogue of the Euler equation

∂ρ /0 + ∇ · (ρ v ) = 0 (2) ∂t /0 S

   2 2√  ∂ρ/0 h¯ ∇ ρ/0 m + vS · ∇ vS = ∇ µ0 − gρ + √ (3) ∂t 2m ρ/0

h¯ where vS = m ∇ϕ is the superfluid velocity and

2 2√ h¯ ∇ ρ/0 √ (4) 2m ρ/0 the quantum potential. The condensate must be a continuous function in space, so its phase is continuous modulo 2π. We define the quantized circulation (Γ) with the line integral

I 2πh¯ dx · vS = n ≡ Γ (n = 0,±1,±2,...) (5) C m where C is a close loop in space, encircling a vortex line, ψ = 0, and the superfluid’s density vanishes. The problem of ultraviolet divergence and of the radius of the fundamental particles is solved in this approach by considering toroidal vortices of vacuum’s quanta instead of point-like particles, whose radius is r = 2ξ, with ξ defined as the healing length, FIG. 1. A vortex line with healing length ξ, actually a vortex tube, closes r in a circle, becoming a vortex torus. Below: the Mobius-strip¨ trajectory of V ξ ≡ , (6) a hypothetical vacuum’s quantum in the vortex torus: the quantum returns 8πaN to any initial position after a rotation of 720° of the torus, so the ratio rotations/revolutions represents spin ½ (fermion). Below, on the right-hand where V is the volume in which the vortex arises, a the side, we see the paths of 12 quanta. scattering length and N a normalized number of vacuum’s quanta in the volume. A hydrodynamic reformulation of the Barut-Zanghi theory [17], which includes spin, was proposed in a doped superfluid show correct attraction-repulsion by Salesi and Recami, who suggested [18], similarly to us, interactions among them [15] (mathematically corresponding that quantum potential to gravity or electromagnetic interactions). Thus, to describe 1 ~2 1 a fundamental fermion as a toroidal vortex in a superfluid Q = − mv S − ∇ ·~vS (7) vacuum, it is necessary to specify that this superfluid has to 2 2 be doped. The dilatant-vacuum picture is therefore fit for (in natural units, h¯ = 1) totally arises from the internal motion the purpose, since a dilatant is made up of a (super)fluid and of a particle~vS ×~s, where scattered dopant particles. For example scattered dark matter in a sea of superfluid dark energy. 1 1 ∇R2 ~v = ρ−1∇ρ = , (8) Let us analyze the fundamental equations of what above S 2m 2m R2 described. Considering the physical vacuum as a doped Bose-Einstein condensate (BEC), we start from the being ~s the direction of spin. The link to the concept of Gross-Pitaevskii equation (GPE) to analyze its hydrodynamic vortex-particle is, in this way, quite immediate. If, to complete behavior [16] a turn in the poloidal direction, a quantum in the vortex needs the same time the vortex needs to complete two turns in the ∂ψ h2 toroidal direction, then the vortex has spin ½ (fermion), i.e. ih¯ = − ∇2ψ + gψ |ψ|2 − ψµ (1) ∂t 2m 0 the system returns to the same state after a rotation of 720° and after each quantum in the vortex has traveled along a where ψ is the condensate wave function, m the mass of a Mobius-strip¨ path (Fig. 1). By defining ω1, ω2 as the angular velocities (toroidal and poloidal), the spin angular momentum quantum of the fluid vacuum, µ0 the chemical potential and g = 4πah¯ 2/m a low-energy parameter, where a represents (S) is determined by the ratio ω1/ω2 = (nπ/dt)/(2π/dt) and the scattering length between vacuum’s quanta. In the phase after the cancellations √ representation ψ = ρ eiϕ , ρ = |ψ|2 is vacuum’s density. ω n /0 /0 1 = = S. (9) From (1) we can write the continuity equation and the ω2 2 3

Spin-0 vortices may arise through a further evolution of the interaction makes them weaker produce a pressure-related torus into a spheroidal vortex. The ratio of the angular Bernoulli effect, that is, in our opinion, the mechanism of velocities ω1 and ω2 of vacuum’s quanta in the vortex (Fig. quantum gravity. Eq. (13) mathematically equals Gauss’s law 1) satisfactorily represents all spin numbers. The parametric for gravity equation defining the position of a quantum in the torus vortex Z is Fg = g(r)n(r)dS. (14) S  x = (r + ξ cos(ω dt + φ ))cos(ω dt + θ )  1 0 2 0 For our approach, this equation is much more convenient y = (r + ξ cos(ω1dt + φ0))sin(ω2dt + θ0) (10) than Newton’s law (albeit it is equivalent to it), because  z = ξ sin(ω1dt + φ0) it describes the gravitational field as a flux entering into a spherical volume. We start by hypothesizing that this flux is where r ≥ 2ξ is the distance between the centers of the tube due to a real flow [20, 21] of fluid, dilatant vacuum, caused and of the torus, ω1 = dφ/dt, ω2 = dθ/dt and φ0 and θ0 are by the spin-mediated bernoulli effect. In short, the equation phases with arbitrary values between 0 and 2π. for gravity (14) is the expression of the Bernoulli force (13) The toroidal shape of fundamental particles also resembles the and this can lead, as discussed below, to the quantization loops of Loop Quantum Gravity [13, 19] but in our case on a of gravity. Of course, also macroscopic bodies, such as higher scale to describe a fundamental fermion. At the same a planet, which consist of vortex-particles (fundamental time, we can also notice a certain similarity between vortex fermions assembled in nuclei, atoms and molecules), produce lines and string theory. Vacuum’s hydrodynamics has in our a pressure gradient around themselves, due to the Bernoulli opinion the potential to reconcile different approaches into effect, which is called gravitational field (Fig. 2). It is a single model. Vortex chirality represents matter-antimatter interesting to consider that toroidal vortices, whose rotations parity (vortex-antivortex annihilation occurs and phonons are conflict, annihilate if they come into contact: parity symmetry generated) and an opportune change in the ratio ω1/ω2 and matter-antimatter annihilation would be then easily would transform a boson into a fermion, or vice versa: explained in hydrodynamic terms and since vortex-antivortex such a mechanism could be therefore important for better annihilation produces, as we know, phonon emission, the understanding the foundations of supersymmetry. photons emitted in matter-antimatter annihilation are in this It is known that quantum vortices exert Bernoulli force, picture phonons propagating through the dilatant vacuum’s generating attractive or repulsive forces [3, 15]. In particular, quasi-lattice: they propagate transversally and at a very high Pshenichnyuk proved that vortices in a doped superfluid frequency (two peculiar characteristics of light), thanks to the behave differently, compared to those in a standard superfluid shear-thickening feature of the vacuum. [15]. The correct attractive or repulsive force, obeying an inverse-square law, is observed only in doped superfluids. II. Equation of quantum gravity without classical The kinetic energy density, acting on the dopant’s particles gravitational constant. and caused by the superposition of the velocity fields of the superfluid vortices, can be written as The macroscopic Bernoulli effect, detectable as the
gravitational field of large bodies, is thence a pressure h¯ 2 ψ2 4R2 + d2 − 4Rd cosα ++ ∞ gradient which causes acceleration toward the center of the K = 2 2 2 2 2 , (11) 2mQ (R + ξ )(R + d + ξ − 2Rd cosα) massive object, according to the known hydrodynamic law h¯ 2 ψ2 d2 K+− = ∞ , (12) P 2m (R2 + ξ 2)(R2 + d2 + ξ 2 − 2Rd cosα) ~a = −~∇ (15) Q ρ where the + + / + − signs refer to same or different ~a P topological charges denoting repulsion or attraction, d is where is the acceleration, the pressure and ρ the density of the distance between the vortices, R the radius of a doping the fluid. By applying (15) to dilatant vacuum (denoted by the subscript /0)we can write the expression for the gravitational particle, ξ the healing length and α the azimuthal angle in cylindrical coordinates which is associated with the doping field particle. This mechanism is driven by the Bernoulli force, P ~g = −~∇ /0 (16) whose equation is ρ/0 Z It is fundamental to notice that, in this new expression of Fb = K(r)n(r)dS (13) S the gravitational field, the Newtonian constant of gravitation 2 disappears. Here we compare the formulas (with units) of the where K(r) = ρv /2 expresses the kinetic energy density, classical and quantum (Sect. III) gravitational potential which dominates on the vortex surface, while the density of  2   2  the superfluid drops to zero within the healing length, and M m P/0 m n(r) is a unit vector, normal to the surface S, over which the V = −G 2 ⇔ VQ = − 2 . (17) r s ρ/0 s integral is calculated. Bernoulli force arises in superfluids as a superposition of the velocity fields of the vortices. Generally in physics the presence of a constant refers to the Points where the fields reinforce each other and where their necessity of adjusting the scale and the measurement units. In 4

expression for the gravitational field ~g (16) in the formula of gravitation ~Fg = −m~g, we directly obtain an equation for quantum gravity

~ Q ~ P/0 Fg = −m∇ , (18) ρ/0

(the superscript Q means quantum) whose quantum nature is analyzed in the following section.

III. Quantum aspects

The classical gravitational potential energy U = −GMm/r becomes then

Q P/0 ~Fg UQ = −m = . (19) ρ/0 ~∇

Let us demonstrate that UQ is the quantum potential of (18). We first express (19) by resorting to momentum

2 P/0 kpk UQ = −m = −kpk · kuk = − , (20) ρ/0 m

where kuk = kpk/m. From (20), switching to the momentum operator, we can write a time-independent, one-dimensional, eigenvalue Schrodinger¨ equation (SE)

2 2 2   pˆ h¯ 2 h¯ 1 ∂ 2 ∂ ψ = − ∇ ψ = − 2 r ψ = m m m r ∂r ∂r (21) (Tˆψ=0) = Vˆ (r)ψ = Hˆ ψ = Eψ,

considering no initial velocity. The expectation value for the gravitational potential energy is FIG. 2. Above: gravitational field as pressure gradient in a fluid quantum 2 Z ∞  2   vacuum, which obeys the inverse-square law. Below: velocity field due to the p ∗ h¯ 1 ∂ 2 ∂ pressure gradient. Pressure and velocity scales are arbitrary. hEi = = ψ − 2 r ψ dr. (22) m 0 m r ∂r ∂r

The velocity field associated with the pressure field generated by the vortex-particles (grouped for instance in a macroscopic this case, Newton’s formula for universal gravitation resorts massive object), via Bernoulli force (13), in the fluid dilatant to quantities such as mass and distance that are not directly vacuum acts as a pilot wave in the vacuum, driving any involved in quantum gravity, a force that actually requires to particle through the gravitational field. It may be useful consider pressure and density of the vacuum (17). The fact to resort to Madelung’s quantum hydrodynamic approach. that such a constant is used by Einstein in general relativity Let us perform polar decomposition of the wavefunction, (GR) underlines a limitation of the theory, which is not indeed √ i S a quantum theory of gravitation. After a century from GR ψ = ρ/0e h¯ , being S the phase. By replacing into the SE and considering the correct alternative solution offered by the and introducing time dependency, we obtain the Madelung dilatant-vacuum approach to the first classical test of GR, equations, that is the continuity equation i.e. to the perihelion precession of the planet Mercury [7], 1 we believe the time has come to switch to pressure, density ∂t ρ/0 + ∇(ρ/0∇S) = 0, (23) and viscosity in a fluid dilatant vacuum, as a hydrodynamic m process occurring over time, instead of the mathematical that describes the flow of vacuum’s quanta along the current concept of curvature in a space-time, that, differently from in the fluid vacuum, produced by the Bernoulli effect, and the the dilatant-vacuum approach, cannot for instance justify the Hamilton-Jacobi equation flat profile of galaxy rotation velocity (Sect. VI): the various positive results achieved thus far as regards the existence of 1 ∂ S = − (∇S)2 −V(r) − Q, (24) a dilatant space cannot be ignored. By replacing the new t 2m 5 where

2 2√ h¯ ∇ ρ/0 Q = − √ (25) 2m ρ/0 is the quantum potential. Since we are considering no initial velocity and no classical potential (being the gravitational potential explained through the described quantum picture), the action S over time, i.e. the acceleration of a particle in the velocity field, reduces to

∂t S = −Q. (26)

Thus the quantum-hydrodynamic interpretation of the classical gravitational potential corresponds to double the quantum potential expressed in units of energy

2 2√ P/0 h¯ ∇ ρ/0 UQ = −m = 2Q = − √ , (27) ρ/0 m ρ/0 and (18) can be simplified to

~ Q Fg = 2∇Q, (28) as double the corresponding quantum force (∇Q), which can be expressed also in function of the quantum pressure tensor pQ, where pressure in a dilatant quantum vacuum replaces the curvature of Einstein’s geometrical space-time, specifically the curvature expressed by the Einstein tensor Gµυ = Rµυ − −1 2 gµυ R, of the field equations, where Rµυ is the Ricci tensor and R the scalar curvature. Resorting to the quantum pressure FIG. 3. CFD simulation of a binary black hole. Above: rotating CFD subdomain with sectors of 90° tensor, quantum gravity can be expressed as a hydrodynamic quantum force in a dilatant quantum vacuum as a wave with ×-polarization "  2 # ~ Q 2m h¯ Fg = − ∇ · − ρm∇ ⊗ ∇lnρm , (29) 1 G2 4m m   R ρm 2m h = − 1 2 (cosθ)sin 2ω t − . (30) × R c4 r c where ρm = mρ is mass density. By replacing G from (17), that is G = (rP/0/Mρ/0), where −1/2 −1/2 IV. Gravitational waves as negative pressure waves in a M = m1 + m2, and resorting to (ρ/0 j/0) = (ε0µ0) , dilatant vacuum where j/0 is the shear compliance of the shear-thickening vacuum and ε0 and µ0 are the permittivity and permeability Being in our model a non-Newtonian dilatant fluid, the (respectively) of the vacuum, we see that a gravitational vacuum can be treated as a quasi-classical quantum fluid. wave is a pressure oscillation propagating through the dilatant Quantum-like gravity waves in a classical fluid have been vacuum at the speed of light (phonon speed in a dilatant described by Nottale [22]. By resorting to CFD we simulated vacuum)) the action of a quadrupole in the fluid dilatant vacuum (3). r h = − (2P j )2 (m + m )−1 (cosθ)· The result is the propagation of negative pressure waves × R /0 /0 1 2 corresponding to gravitational waves, once the currently p ! (31) VQ  p  adopted geometrical explanation of GR is replaced by our · sin 2 t − R ρ/0 j/0 , hydrodynamic model. Indeed, if a distant test mass were hit r by these negative-pressure waves propagating in the vacuum, it would be expected to exactly reacts as if it were hit by whereas the polarization h+ reads gravitational waves. This because we justify the gravitational r h = − 2 (P j )2 (m + m )−1 (1 + cos2 θ)· force as the Bernoulli force in a fluid vacuum (Sections I). The + R /0 /0 1 2 simulation therefore shows gravitational waves as actually the p ! (32) propagation of negative pressure waves in the fluid vacuum VQ  p  · cos 2 t − R ρ/0 j/0 , (Fig. 4). Let us consider a supposed space-time deformation r 6

FIG. 4. CFD simulation of gravitational waves as pressure waves in a fluid dilatant vacuum (arbitrary pressure scale). The black dot in the center is structured to mime a binary system in the dilatant vacuum, according to Fig 3 FIG. 5. CFD simulations of different kinds of galaxies: (a),(b),(c) and (d) represent galaxies at different arbitrary pressure gradients in the fluid vacuum: -100 Pa, -50 Pa, -25 Pa, -11 Pa (respectively). In (e) and (f) pressure is -100 where R is the distance from the observer, t the elapsed time, Pa and sectors of 45°are used (differently from the 90°sectors showed in Fig. θ the angle between the perpendicular to the plane of the 3). In (f), with respect to (e), representing a grand-design galaxy, angular orbit and the line of sight of the observer, r the radius of the velocity has been halved, so an anemic galaxy appears. p quadrupole, VQ/r = ω its angular frequency, obtained by resorting to the identity in (17) in the Newtonian formula for p 3 constant angular velocity of a circular orbit G(m1 + m2)/r . vacuum show to possess a flat profile of the rotation velocity, A variation in pressure causes acceleration, acting for example according to observations (simulation of galaxy evolution at on LIGO’s test masses. [25]). This means that neither current dark matter theory [26, 27] nor MOND are necessary to explain the flat profile but it V. Galaxy formation and shapes is enough to replace geometrical Einsteinian space-time with a fluid, dilatant vacuum, in which hydrodynamic phenomena We have applied the above-described simulation also to driven by pressure and viscosity occur over time, which is galaxy formation, by considering a binary black hole in the relative also in our approach, due to clocks ticking in a more or galactic core. By varying pressure and angular velocity we less viscous vacuum (being apparent viscosity related to shear obtained different shapes for galaxies. This suggests that stress). A clock traveling faster through the dilatant vacuum also galaxy formation may be driven by pressure gradients undergoes greater apparent viscosity according to vacuum’s in a dilatant vacuum, due to a rotating, anisotropic mass rheogram, i.e. to the Lorentz factor. In short, GR is generally distribution, as in a binary system. We can therefore infer a correct quantitative tool (not for galaxy rotation for instance, that two or more black holes in the core can generate different whereas the dilatant vacuum works properly, see Sect. VI) but kinds of spyral galaxies [24], whereas a single supermassive its qualitative explanation (space-time deformations) actually black hole could be responsible for the shape of elliptical corresponds to the action of pressure gradients and apparent galaxies. viscosity in a dilatant vacuum, as demonstrated in [7]. By It is important to note that galaxies generated in the dilatant analyzing the different galaxy types showed in Fig. 5, we 7

FIG. 7. Velocity arrows showing the direction in which stars migrate, along with galactic gases, from an arm to another of a spyral galaxy, due to the warped pressure field in the dilatant space.

FIG. 6. Pressure in the dilatant vacuum along the X-direction of the simulations. The letters of the plots correspond to those of the galaxy shapes in Fig. 5. Animated plots at [25]

notice that a weaker pressure gradient, due to less baryon matter in the galaxy core, generates a smaller galaxy: Fig 5 c) represents a dwarf spyral (dS) and letter d) a pea galaxy. In e) FIG. 8. To confirm the flat profile of the rotation curve we monitored the we see a grand-design galaxy, whereas in f) an anemic galaxy, angular velocity of some probes at different arbitrary radii (R1 to R13). The obtained by halving the angular velocity of the galaxy. In a positive result highlighting the flat profile is shown in Fig. 9 spyral galaxy, stars tend to concentrate along the low-pressure corridors (the galaxy arms) in the dilatant vacuum, as the fluid flows toward the black holes situated in the galaxy core. This breath) of a spyral galaxy in a dilatant vacuum has been result from CFD is compatible with the findings presented by realized by us and it is available at [25]. Martin, O’Sullivan et al. [28], who described multifilament gas inflows fuelling galaxies: indeed, not only stars but of VI. Flat profile obtained in a dilatant vacuum course also gases can flow along such pressure corridors in the fluid vacuum. The specific black-holes configuration in The galaxies obtained in our simulations show a flat profile a core is another element which determines the shape of the of angular velocity, according to astronomical observations. galaxy. A 45° configuration (Fig. 5 letter e.) increases the This implies that the vacuum as a dilatant fluid (a unified dark grand-design feature of the galaxy and its size. But when the fluid that, due to its viscosity and ubiquity may correspond angular velocity is halved, the grand-design converts into a to the Higgs field itself, further extending the unification) is so-called anemic galaxy. Correct stars migration over time a sufficient condition to justify the flat profile. We simulated from an arm to another of a spyral galaxy has been also the motion of probes along concentric orbits in the galaxy, at observed in the CFD simulations (Fig. 7). A CFD simulation six different radius values (Fig. 8), to verify the dependence showing the birth, the evolution and the behavior (rotation, of the rotation velocity on the distance from the galactic core. 8

investigate the specific nature of physical vacuum as a dilatant fluid, are in our opinion of crucial importance.

VII. Breath of galaxies

In the video simulation in [25], one can watch the birth, evolution and behavior of a spyral galaxy in the fluid, dilatant vacuum. The animated diagram on the right-hand side of the video, shows the persistence of the flat profile of the rotation velocity over time, during the evolution and the life of the galaxy. It is also interesting to see that a breath of the galaxy emerges, as pressure waves traveling from the galaxy core cross the galactic disk that then continue to propagate through the intergalactic dilatant space. We enlarged the simulation domain enough to check whether the effect could vary and depend on the boundary conditions, but it persisted. FIG. 9. Pressure function of angle (for each radius). Since for the considered We believe this result could be linked to the phenomenon radii the wavelength is stable with good approximation, a flat profile of the described in [29]. rotation velocity is revealed. VIII. Conclusion

The theoretical and computational conclusions of this study reinforce the dilatant-vacuum model, suggesting that vacuum dilatancy could be a hydrodynamic property of the Higgs field itself, which is indeed endowed with a certain viscosity: if this is the case, we are thence showing something unexpected, i.e. that the Higgs field also interacts with macroscopic bodies (for instance with the planet Mercury and the Pioneer probes [7]) and that this ubiquitous fluid is enough to justify the formation and the specific rotation of galaxies, besides being a key to understanding quantum gravity, by describing the gravitational field as an inflow of dilatant vacuum (of Higgs fluid?) into massive particles, driven by the Bernoulli force produced by spin, as discussed in Sect. I. Alternatively, the dilatant vacuum can be seen as a unified, viscous dark fluid, i.e. superfluid dark energy and scattered dark matter particles FIG. 10. CFD simulation showing the velocity field of dilatant vacuum in that act as a dopant and cause space dilatancy as shear stress a spyral galaxy (arbitrary scale). Gases flow along these virtual corridors increases. What we actually believe is that a dilatant vacuum created by the distorsion of the central black holes’ gravitational fields could allow an extended unification of Higgs field, dark induced by galaxy rotation. See also [28] energy and dark matter. Our simulations also showed that a binary black hole can be the cause both of gravitational waves as negative pressure waves through the dilatant vacuum We confirm (Fig. 9) that the rotation frequency does not vary and of the formation of various types of spyral galaxies. A with distance, implying a flat profile of the rotation velocity. noteworthy effect similar to a breath of galaxies emerged Another important consideration concerning the existence from the simulations, whose cause is not yet clear but could of a dilatant vacuum, is that it can justify the expansion reconnect to previous studies concerning galactoseismology of the universe without bringing into play the concept of by Widrow, Gardner and collegues [29]. Studies based on the negative mass, because the energy density of the vacuum equations of dilatant vacuum should be therefore continued, corresponds to pressure (J/m3 = Pa), causing expansion. For also as far as other phenomena and unanswered questions are these reasons, this study and the previous ones [6, 7], that concerned.

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