A Superfluid Theory of Everything? Marco Fedi

A superfluid Theory of Everything? Marco Fedi

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Marco Fedi. A superfluid Theory of Everything?. 2016. hal-01312579v3

HAL Id: hal-01312579 https://hal.archives-ouvertes.fr/hal-01312579v3 Preprint submitted on 29 Jul 2016 (v3), last revised 3 May 2017 (v4)

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Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License A superuid Theory of Everything?

July 29, 2016

Marco Fedi1 Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR), Italy

Abstract If a real vacuum doesn't exist in quantum physics [1, 2, 3, 4, 5], by describing it as a superuid quantum space (SQS), quantum gravity can arise from attraction of space's quanta into massive particles, when they are considered superuid vortices, i.e. dynamic topological defects of SQS, given its non-zero viscosity. Gauss's law for gravity mathematically supports this hypothesis, since it denes the gravitational eld as an incoming ux, and the plausibility of this mechanism has been also conrmed through a set of CFD simulations. In this case, the gravitational eld would be an incoming ow of space's quanta and this leads to the formulation of a uid equivalence principle, able to show that gravity may be the origin of all relativistic eects, also in special relativity, producing a simplication in the theory of relativity. It is discussed how the hydrodynamics of a SQS would be able to replace Einstein's curved spacetime and to explain gravitational waves. Not less important, when considering a uid space, this hypothesis would not conict with the Michelson-Morley test and would open up a new scenario on the nature of light and on Hub- ble's law. A simple test is suggested to proof uid quantum gravity and its consequences. Among these, the mechanism of SQ absorption would automatically lead to the unication of the four fundamental interactions, making us reinterpret the universe as a uid phenomenon, from subatomic to cosmic scale.

[email protected]; [email protected]

1 Contents

1 Fluid quantum gravity and relativity. 3 1.1 Introduction ...... 3 1.2 Gravity as a hydrodynamic phenomenon in a superuid quantum space...... 3 1.3 The Michelson-Morley test: does the ether wind correspond to the gravitational eld? ...... 7 1.4 Simplifying the theory of relativity. Gravity as the only cause of all relativistic eects...... 7 1.4.1 Fluid principle of equivalence: is relativistic mass increase rather a matter of drag weight? ...... 7 1.5 Time dilation as the action of weight and drag weight on clocks. 10 1.6 Photon as a phonon through a superuid quantum space. . . . . 11 1.6.1 Reinterpretation of Lorentz factor, mass-energy equivalence and Hubble's law...... 13 1.7 Gravitational waves as negative pulses through a superuid quantum space...... 16 1.8 Twin paradox...... 17 1.9 Black holes...... 17 1.10 Verication ...... 18

2 Appearance of fundamental particles as vortices in a superuid quantum space and emergence of quantum potential. 20 2.1 Vacuum uctuations as vortices in a SQS? ...... 23

3 Unication of the fundamental interactions as a direct consequence of uid quantum gravity. 24 3.1 Gravity-electromagnetism unication...... 24 3.1.1 Interpretation of the ne-structure constant in SQS. . . . 29 3.2 Weak interaction...... 30 3.3 Strong interaction and quantum chromodynamics...... 31 3.4 Baryon asymmetry explained? ...... 33

4 Conclusion 34

2 1 Fluid quantum gravity and relativity.

1.1 Introduction A hypothesis has been formulated, according to which, vacuum is a quantum superuid [6, 7, 8, 9], here referred to as superuid quantum space (SQS), an ubiquitous fundamental scalar eld. Excitations in this eld would be vortices, similarly to nanovortices occurring in superuid helium-4 [12, 15], thus dynamic topological defects of SQS [6, 7, 10](2). These vortices would match the elementary particles of the Standard Model, whose spin would correspond to the circulation of space's quanta (SQ) occurring in the vortex (see for example g.14 for a visual representation of spin ½) and would be the origin of quantum potential [34](2). Because of the non-zero viscosity of quantum space, massive particles, as vortices, attract the surrounding SQ, generating a force eld which corresponds to the gravitational eld 2 (g.1). This hypothesis is supported by Gauss's law for gravity, which describes the gravitational eld as an incoming radial ux and allows us to replace general relativity's curved spacetime with a uid space (g. 2, 3, 4), whose hydrodynamics also determines time, as a chain of discrete uid phenomena on Planck scale. Such a uid quantum gravity, would imply a uid equivalence principle (1.4.1), according to which a moving body would be subject to an apparent gravitational eld acting in the opposite direction to motion. This would explain the role of velocity in special relativity as the action of gravity (drag weight, interpreted as mass increase) and would indicate that gravity is the cause of all relativistic eects. The speed of light would correspond to that of a pulse through SQS (1.6), without breaching the outcome of the Michelson-Morley test, since in this case the ether wind would exactly correspond to the gravitational eld and this would justify the bending of light and other known relativistic eects. To compensate the absorption of SQ in the vortices, emission of virtual photons would occur (g.14), capable of explaining the existence of charged particles (3.1). On the other hand, when no emission occurs we observe β- decay (3.2), due to energy imbalance. The exchange of discrete packets of SQ (interpreted as gluons) between two adjacent vortices with mutual spins would lastly justify the strong interaction (3.3), leading to the complete unication of the four fundamental forces.

1.2 Gravity as a hydrodynamic phenomenon in a superuid quantum space.

Once Gauss's law for gravity is applied to a SQS, we automatically obtain an eective, and the simplest, theory of quantum gravity and quantum relativity without resorting to strings, extra dimensions or gravitons. A set of CFD simulations (see further data in the annex) has been performed and has conrmed

2See also Consoli, Pappalardo [36] and Sarfatti, Levit [38] for similar points of view.

3 that the situation shown in g.1 agrees with Gauss's law for gravity and New- ton's law of universal gravitation. Navier-Stokes equations representing mass, momentum and energy have been used:

Figure 1: Two particles move the one toward the other since they're absorbing the uid (SQS) which they're immersed in (here in a 2D reduction). This phenomenon is in direct agreement with Gauss's law for gravity: . ∂V g · dA = −4πGM !

∂(u ) j = 0 (1) ∂xj

p ∂(u u ) ∂ ρ µ ∂ ∂u ∂u i j = − + i + j (2) ∂xj ∂xi ρ ∂xj ∂xj ∂xi ∂((ρE + p)u ) ∂ ∂T j = −k (3) ∂xj ∂xj ∂xj The condition of two stationary spheres immersed in an incompressible uid was set and the pressure integral of the forces acting on them was calculated. The analysis took into account the response to absorption velocity and to distance between the spheres. To simplify the simulations, the system was reduced as showed in g. 23 (annex). The attractive force produced by pressure forces and momentum, where A corresponds to the surface of the inner sphere and d~ is the unit vector for the distance between the spheres, is represented by the following equation:

~ ~ Fa = (p + ρ(~u · ~n)(~u · d))dA~ · d (4) ˆA The analysis of velocity and pressure, with respect to the distance (radius) from the absorbing sphere is illustrated in g. 17 and the diagrams in g. 25, 26, 27 and 28 (annex) show an inverse quadratic dependence on distance and a quadratic dependence on the ow velocity. Renement of computational grid and domain enlargement helped to reduce the curvature of the ow lines, up to a virtually radial ow (g. 24). The behavior of the attractive force shown by this analysis is concordant with Newton's law of universal gravitation, since the attractive force decreases with distance (radius) according to an inverse square law and quadratically grows

4 according to the velocity of the ux (g. 24, 25, 26). Two equal spheres have been considered, corresponding to equal masses in Newton's law. Moreover, a sphere absorbing the uid in which it is immersed generates a radial attraction eld equal to the Schwarzschild solution,

2Gm−1 2Gm ds2 = 1 − dr2 + r2(dθ2 + sin2 θdφ2) − c2 1 − dt2 (5) c2r c2r suggesting that the metric tensor of GR may be expressed through uid dynamic forces. Fluid quantum space whose hydrodynamics produces time and inuences it through gravity (likewise described as a uid phenomenon) instead of a deformable geometric spacetime (g. 2, 3, 4).

Figure 2: how the presence of a massive body curves spacetime (a) or absorbs uid quantum space (b), here in analogy with a bell-mouth spillway.

Figure 3: the Lense-Thirring eect according to Einstein's curved spacetime (a) and to uid dynamics (b). Here as an analogy with the Coriolis eect in a cyclone.

Fig.3 describes the strong uid dynamic analogy with a cyclone (Coriolis eect), where the gravitomagnetic eld related to the Lense-Thirring eect is expressed as

4 mωR2 B = − cos θ (6) 5 r3 and the Coriolis force can be written as:

5 FC = −2mω(ωR)uR. (7) where the dierence between a 3D (gravitomagnetic eld) and a 2D (Coriolis) model has to be however considered.

Figure 4: Gravitational lensing (a) and the motion of a satellite (b) according to SQS, still in analogy with a bell-mouth spillway representing the gravity of a star which curves space (in our case, which absorbs SQS).

Other eects which can be described by the uid dynamics of SQS are the gravitational lensing:

4G ~b α~ˆ(ξ~) = d2ξ0 dzρ(ξ~0, z) (8) c2 ˆ ˆ |~b|2 with b ≡ ξ~ − ξ~0, where ξ, z are coordinates and ~αˆ is the deection angle, which in g. 4.a is determined by vector interaction between light's and space quanta's momenta (gravitational ow), where their absorption is illustrated as water owing into a spillway (acting as an interposed star). While in g. 4.b SQS's uid dynamics describes orbital motion, since the angular velocity for any inverse square law, such as Gauss's law for gravity and (4), is given as µ u(θ) = − A cos(θ − θ ) (9) h2 0

where A and θ0 are arbitrary constants, h the angular momentum and µ the standard gravitational parameter. All this suggests that the found solutions to Einstein's eld equations could be fully replaced by uid dynamic solutions based on Navier Stokes equations. Consoli and Pappalardo [36] express a similar point of view. Einstein's spacetime as a single interwoven continuum is here described by the interdependence space↔time, since time would arise from the hydrodynamics of SQS (42)(2.1) and, also in this case, it wouldn't be absolute but inuenced by gravity (per se a uid phenomenon, we said), as discussed below (1.5).

6 1.3 The Michelson-Morley test: does the ether wind correspond to the gravitational eld?

When considering gravity as absorption of SQ (1.2), a gravitational eld corresponds then to a ow of SQ moving toward the center of the absorbing mass. This fact motivates us to reconsider the Michelson-Morley experiment [17] and to conclude that it has only demonstrated the nonexistence of relative motion Earth-ether but not the nonexistence of an ether, if we assume that the ether wind corresponds to the gravitational eld.

In this case the ether wind would not be dependent on Earth's orbital motion, while Earth's rotation would cause the Lense-Thirring eect by bending the incoming ether wind (g.3), i.e. the gravitational eld. Another eect caused by this kind of ether wind would be the gravitational lensing (g.4). Several hints which suggest to proceed with the hypothesis of a uid space and of quantum gravity as absorption of SQ. The consequences of a uid space as far as the propagation of light is concerned are likewise interesting and are discussed below (1.6). In the MM-experiment the ether wind would have therefore been investigated in an erroneous way. Moreover, the Michelson interferometer, though vertically positioned, would be inappropriate to prove this hypothesis for reasons which are discussed below (1.10).

1.4 Simplifying the theory of relativity. Gravity as the only cause of all relativistic eects.

1.4.1 Fluid principle of equivalence: is relativistic mass increase rather a matter of drag weight?

Because of the presence of SQS, also translational motion would put a moving body in the condition of being subject to a gravitational eld, as an apparent ow, which in this case acts in the opposite direction to motion. We can express that as a uid equivalence principle (FEP) (g. 5).

7 Figure 5: Fluid equivalence principle: it is impossible to distinguish between the two equivalent situations of a body moving at a given velocity through a stationary uid and a uid owing toward a stationary body at the same velocity. This would also occur as far as a body interacting with SQS is concerned, where the equivalence is between being stationary in a gravitational eld (being subject to an ether wind having a given velocity) or moving through SQS at the same velocity, i.e. being subject to a gravitational eld induced by motion which would increase according to Lorentz factor.

Which obtains as:

vΦ = vsq + v (10)

where vΦ is the velocity of the total resultant ow acting on the moving body, determined by the vector sum of the velocity at which SQ are absorbed

(vsq, drift velocity of SQ) in the point of the gravitational eld where the body is located at a given instant and of the body's translational velocity (v) through the uid space. According to the FEP, any translational velocity therefore provokes an apparent gravitational eld (gΦ) acting on the accelerated body and detected as a weight force opposite to motion (drag weight,WΦ) (g.6). Other cases of quantum vacuum friction have been discussed by several authors [29, 30, 31] and also Higgs eld is said to possess a certain viscosity: potential relationships between these elds, or a possible correspondence, should be then investigated.

Figure 6: Weight acting in the opposite direction to motion (drag weight, WΦ) due to the apparent gravitational eld caused by motion through SQS. At low, everyday speeds this eect wouldn't be noticed, since SQS's viscosity is quasi-zero, as for every superuid, but the eect of apparent viscosity would play a key role at relativistic velocities (extreme shear stress), increasingly opposing acceleration.

This is in agreement with the relativistic eect of mass increase, which would actually be a resistance to acceleration due to an increasing gravitational force

8 acting in the opposite direction to motion. This issue is clear if we suppose that, when dealing with accelerated particles in synchrotrons, we make a dimensional mistake, swapping kgf with kg, i.e. interpreting a weight force (WΦ) pointing in the opposite direction to the supplied acceleration as a mass increase. If drag weight grew according to Lorentz factor (1.6.1)(24), it could be the cause of the so-called relativistic mass increase:

F a = . (11) m+WΦ The new equation expressing the total weight of a body in uid quantum gravity would be:

Wtot = m(g + gΦ) (12)

where the accelerations g and gΦ may point in dierent directions, according to the presence of a gravitational eld and/or translational motion. Because of the viscosity of SQS (we know that no superuid has real zero-viscosity) a slight dierence in the weight of a body should moreover occur when we compare the gravitational force acting on a free falling body with that acting when it is in equilibrium in the gravitational eld: in the second case the weight should be slightly greater than W = mg but in the Earth's gravitational eld this dierence would be probably too small to be detected.

It is possible to proof the FEP in Einstein's relativity by equating the cause of time dilation in special relativity (velocity) to that in general relativity (gravity):

∆t ∆t ∆t0 = = (13) q 2 q v RS 1 − c2 1 − r hence

v2 R 2GM 1 2GM 1 2GM = S = =⇒ v2 = =⇒ v2 = (14) c2 r c2r c2 r c2 r and √ v = p2rg = 2Φ, (15) where (15) relates velocity and gravity (as gravitational potential Φ) in the formula of the second cosmic velocity, also meaning that a given velocity in SR corresponds to a certain gravity (drag weight), equivalent to that of a gravitational eld g in a point where the absorption of SQ occurs at the same velocity. Fig.7 indeed shows how a gravitational eld corresponds to an absorption velocity eld (velocity potential) analogous to that of g.17 from our CFD simulations (absorption of SQ ⇒ sink at the origin) and, on the right, the equivalence between translational velocity and absorption velocity of a gravitational eld, which explains the relativistic phenomenon of illusory mass increase as drag weight and unies the cause of time dilation of SR and GR (gravity in both cases).

9 All relativistic eects could be therefore explained through the sole action of gravity (g.7), described as a hydrodynamic force occurring in SQS. Besides what described in g.2, 3, 4, also mass increase (or better, what would be the current interpretation of drag weight), time dilation and consequently Lorentz-Fitzgerald contraction, which depends on time dilation and does not need to be discussed. What is relative in uid quantum

relativity is SQ's velocity with respect to a frame of reference (vΦ): it produces or varies the gravitational force acting in the frame, also modifying time, as discussed below.

Figure 7: equivalence of a gravitational eld (left) and a velocity eld (center), where the latter corresponds to the absorption velocity (vsq). In both cases we have sink at the origin and equipotential surfaces. On the right, a given translational velocity (SR) through SQS corresponds to being in a point of a gravitational eld (GR) where space's quanta are absorbed at the same velocity.

1.5 Time dilation as the action of weight and drag weight on clocks. If the eect of translational velocity in SR is explicable through that of gravity (drag weight, 1.4.1.), time dilation (13) would be therefore caused by gravity also in SR, since it interferes with clocks' motion. This would be a simplication of Einstein's relativity. Indeed, if the parts of a physical system (as a clock) become heavier, their movements are aected, unless more energy is provided to the system. We can say clocks in the broadest sense, since any physical system, also a biological organism, is not immune to the action of gravity, which inuences its dynamical aspects (macroscopic, molecular etc.) and consequently causes time dilation. If we had, for instance, a clock with period

Ff + Fg Tc = T0 = 1s (16) Fc

calibrated by balancing friction and gravity (Ff + Fg = Fc), where T0 = 1s is the period where g → 0, when we put out the accelerations acting on it,

10 m(af + g) Tc = T0 (17) mac it would be clear that any additional positive acceleration in the numerator would unbalance the mechanism, producing time dilation, both when the clock is in a stronger gravitational eld,

0 af + g + ∆g T = T0 > 1s (18) ac and when it is subject to drag weight,

0 af + g + gΦ T = T0 > 1s. (19) ac

We see that if gΦ increased according to Lorentz factor (24, 26)(g. 9), time dilation too would obey γ, until the clock is stopped by gravity. This explanation also conforms to Einstein's redshift., where clocks run dif- ferently depending on the intensity of the gravitational eld at the point where they are placed. To understand now why Lorentz factor is built on the ratio (β) between the translational velocity of a body and the speed of light, we need to discuss the nature of light in relation to the possible existence of a SQS. Let us hypothesize that a photon is a pulse (phonon) through a SQS. As shown below, equations and evidences seem to conrm that.

1.6 Photon as a phonon through a superuid quantum space. Once established that the MM-experiment didn't disprove the existence of a luminiferous ether (1.3) and since light would propagate through a uid medium if a SQS existed, we aim now to prove how a photon can be a pulse (a phonon) through a uid quantum medium. In this case light would be the sound of space. Let's consider the formula of the speed of sound through a uid. In the case of the propagation of a single photon as a phonon, we resort to Young's modulus σ instead of the bulk modulus. By setting , we obtain √1 . (E = ) σ = 1 a = ρ When 0 is seen as SQS's compressibility and ρ0 as its density, we have 1 a = √ , (20) 0ρ0 expressing the velocity of a pulse (then also of a phonon) through SQS, where 1 the numerator has unit Pa 2 . Mathematically analogous to 1 c = √ , (21) ε0µ0 resulting from Maxwell's equations. Hence, given the adimensional equivalence 0ρ0 = ε0µ0 (20, 21), photon's electromagnetic eld might be described as

11 a local density variation (⇒local energy variation) of SQS in time, due to the compression caused by the pulse, as illustrated in g. 8, whose spin (we know that also sound waves can spin [19, 22]) would play a role in the orthogonality of the magnetic eld with respect to the electric eld. The permeability and permittivity of free space would be therefore determined by SQS's properties such as its density and compressibility.

Figure 8: Density (ρ) and compressibility() of SQS at the origin of photon-phonon's trans- verse EM eld (b), whose oscillations are due to harmonic, orthogonal compressions of space's quanta, while the main pulse propagates along the z−axis. On the left (a), the probable quantum mechanism at the origin of transversality, due to compression and intrinsic angular momentum (spin) of SQ.

There are many analogies between photon and phonon. In the energy eigen- value of any eigenstate ψn of photon, expressed as 1 E = (n + ) ω (22) n 2 ~ 1 is vacuum (SQS) contribution, as well as for a phonon, where the har- 2 ~ω monic oscillator eigenvalues for the mode ωk (k, wave number) are

1 E = (n + ) ω n = 0, 1, 2, 3, ... (23) n 2 ~ k Both photons and phonons: are bosons [5]

possess wave-particle duality [20, 21]

obey the doppler eect z = femit−fobs fobs are symmetric under exchange |α, βi = |β, αi can be created by repeatedly applying the creation operator b† share the formula of momentum: h , where 2π pphonon ≡ ~k = λ k = λ

12 can produce photoelectric eect and Compton scattering thanks to their momentum h λ can have a spin [19, 22]. In this case photon would be a special spin-1 phonon.

can form squeezed coherent states [18] can interact via parametric down conversion [18] When a photon is described as a phonon in SQS, the energy it carries would be justied within the quantum phenomenon of second sound, occurring in this case in SQS. This would lead us to another simplication, since all waves in nature would be mechanical (quantum mechanical in the case of EM waves) and would need a medium to propagate.

1.6.1 Reinterpretation of Lorentz factor, mass-energy equivalence and Hubble's law. If SQS acted as a non-newtonian, dilatant uid under extreme shear stress (bodies accelerated through SQS at relativistic velocities), this fact would justify the asymptote at c to translational velocity. Being moreover a dilatant uid would conrm the granular [11] (and quantum) nature of space. When a photon is a phonon of space's quanta through SQS, the parameter β in Lorentz factor is intelligible as the ratio between the speed of a body and that of a pulse through SQS, where the accelerated body produces a greater shear stress and can't travel as fast as a pulse (photon): this because of the dilatancy it provokes in SQS when its speed approaches c. Indeed, the ratio 2 v is expressed as an adimensional parameter analogous to the ratio between c2 2 v the speed of a body and of sound (vs) through other superuids [23]: =⇒ vs v 2 2 c ≡ β . Within a SQS, Lorentz factor is then hydrodynamically interpretable as

0 vΦ 1 (24) γ ≡ arcsin = r vp 2 1 − vΦ vp indicating the asymptote at vp = c, where vp, is the speed of a pulse through the uid space and vΦ (10) is the relative velocity of SQ in a given frame of reference. By approaching vΦ = vp (i.e. v = c, if in the FEP formula vsq = 0), uid space's dilatancy would play more and more a strategic role in preventing further acceleration. Valid the analogy photon-phonon (20), c2 corresponds then to the square of the pulse velocity c2 = v2 = 1Pa . The similarity with v2 ≡ 1 in [23] is p ρ00 s ρaκ interesting, where vs is the speed of sound through a compressible superuid.

13 If this is true, we could state that light is the sound of space, inasmuch it would be a pulse (phonon) through a uid space.

Figure 9: motion-induced gravitational acceleration (gΦ) causing drag weight (WΦ = mgΦ) increase, currently interpreted as relativistic mass increase. This graph can also be read as a rheogram of SQS. As shear rate and velocity of the relative ow of SQ shift toward relativistic values, collisions among SQ intensify and viscosities, albeit initially near-zero, critically increase, until granular (quantum) uid space's dilatant behavior leads to an asymptote, corresponding to the speed of a pulse (vp = c) in SQS. Thus, exactly as sound has a given speed through a given uid, light would have one through SQS, corresponding to ∼ 299792458 m/s.

At this point E = mc2 would be intelligible as the ratio between the rest 3 3 mass-energy of a body and the energy of 1 m of SQS (ρ0) multiplied by the σ0 Young's modulus of SQS ( = Υ), where σ0 = 1Pa: 0

m σ m E = mc2 = 0 = Υ. (25) ρ0 0 ρ0

In other words, Einstein's mass-energy equation would indicate how much energy from SQS is entrapped in a body of mass m, given a specic density and compressibility of SQS. And Lorentz factor could be rewritten in the (still adimensional) form:

1 γΦ ≡ (26) p 2 −1 1 − vΦρ0Υ

where vΦ expresses the FEP in the considered frame of reference. It means that (26) could be used also in place of √ 1 if we knew the velocity of the Rs 1− r ow of SQ in a given point of a gravitational eld (g. 7). Furthermore:

3Obvious, after all, if fermions are superuid vortices of space's quanta.

14 Valid the analogy photon-phonon, Hubble's law would be compatible with an energy loss (⇒redshift) of photons due to the minimal viscosity of SQS, after they've traveled enormous distances, instead of representing a mys- terious acceleration of distant galaxies driven by dark energy. The further a galaxy, the greater the detected redshift. This theory therefore indicates that dark energy is not necessary and that it doesn't probably exist (the ∼ 68% of missing energy in the cosmo may be rather the SQS itself), in that the expansion of the universe would be not accelerating. When we therefore consider the role of an accelerated expansion in determining the age of our universe, astrophysicists should rst reconsider their reasonings

by taking into account the possible variation of SQS's density ( ρ0) in (20) in a younger and smaller universe, when that value was probably greater. We can now try to dene the formula for drag weight. While the formula for classical drag force is

1 F = −ˆv v 2ρC A (27) D 2 D

there is in this case no dependence on surface (A) or shape (CD) for a body traveling through SQS. According to the FEP, the same dynamics which is valid for standard weight has indeed to be valid also for drag weight, i.e. dependence only on mass and eld intensity. Hence it has to be true: . FDqs = WΦ = mgΦ Where the meaning of CDA in (27) is represented by mass, i.e. the quantity of matter exposed to the ow, regardless of shape and surface. Thus, we just need to dene the formula of gΦ. We know that it has to increase according to Lorentz factor, whose reformulation (26) precisely contains the dependence on velocity and medium's density as in (27). The formula for drag weight could be therefore written as: a WΦ = −ˆvm − 1 (28) p 2 −1 1 − vΦρ0Υ where a = 1 m/s2 and ˆv is a unit vector expressing the direction of motion. In conclusion, the gravitational eld is expressed as

1 v2 GM g = −ˆr e = −ˆr (29) 2 r r2 when gravity is caused by the presence of a mass (celestial body), and as a gΦ = −ˆv − 1 = −ˆvaγΦ − 1 (30) p 2 −1 1 − vΦρ0Υ when it is induced by motion through SQS.

15 1.7 Gravitational waves as negative pulses through a superuid quantum space.

Other consequences of a SQS and of uid quantum gravity would be: The known dissipation of energy occurring in the orbits of celestial bodies, which would be caused by SQS's minimal viscosity. Quantum vacuum friction has for instance been considered in pulsars [29, 30, 31]: when energy loss is indeed only related to magnetic dipole radiation, we have from the braking index ωω¨ , in disagreement with observations n = 3 n = ω˙ 2 which show that n < 3. Gravitational waves, which would be dened as a periodic variation in

SQS's local density (∆ρ0) caused by the variable position of a quadrupole in time (g. 10), corresponding to a variable rate in the absorption of SQ. In this case gravitational waves arise as negative pulses through SQS. Since also photons have been described as pulses (phonons) through SQS, it is obvious that gravitational waves travel at the speed of light (20). And, not surprisingly, also photons, as positive pulses that carry energy (like sound), can provide kinetic energy to a target (radiation pressure), exactly as for gravitational waves acting on LIGO's test masses, where the pressure is in that case negative. Also in this case it would be a hydrodynamic phenomenon and no deformation of a geometric spacetime would be necessary to explain gravitational waves.

Figure 10: Gravitational waves according to uid quantum gravity. A periodic variation in SQS's density due to a likewise periodic variation of the absorption velocity (due to the changing position of the quadrupole) may explain them as a hydrodynamic phenomenon, without the concept of spacetime deformation.

The changing rate in the absorption of space's quanta, occurring twice the orbital frequency (2ωq) , causes periodic decompressions (∆ρ0) in SQS (g.10.b),

16 currently interpreted as a deformation of spacetime (hij, in linear gravity: gij = ηij + hij).

1.8 Twin paradox. If we now analyze the twin paradox according to uid quantum relativity, we see that the paradox immediately vanishes. After the trip, the younger twin will be that who has been subject to a faster ow of SQ and he can be identied by comparing the ow of the gravitational eld on Earth, where his brother is, with the apparent ow induced by velocity on board the rocket. What is important here is therefore motion relative to SQS (⇒FEP), not that of a twin relative to the other. We see that the uid equivalence principle (10) would replace the strong equivalence principle as a quantum version of the former. Thus, if the apparent gravitational eld opposite to the direction of motion (gΦ) which acts in the rocket's frame of reference was stronger than Earth's gravitational eld (g) the younger twin at the end of the travel would be that who has traveled in the rocket. We can express how the period of a clock in the rocket's frame of reference varies with respect to a clock on Earth as

1 1 ∆Trocket = = (31) q p −1 ∆vΦ 1 − ∆vΦρ0Υ 1 − c2 where is the delta of the FEP applied to both ∆vΦ = vΦrocket − vΦEarth frames of reference. Of course, to use this quantum formula of the twin paradox we should know the drift velocity of SQ in the Earth's gravitational eld.

1.9 Black holes. Black holes can exist in uid quantum gravity as celestial bodies on which the incoming ow of SQ (1.2) has the same or a greater velocity than that of a pulse through the SQS (i.e. of light). Again, it wouldn't be a matter of curved spacetime. When a collapsing star increases its density, if it doesn't reach a point of equilibrium, becoming for instance a neutron star, the FEP predicts that at a certain point it would be in the same condition of a body traveling at the speed of light, i.e. when its absorption velocity became vsq = c. We could therefore think that SQ are never absorbed at that velocity but while in the case of particles accleration in synchrotrons it would be impossible to oppose gΦ(1.4.1) when particles approach the speed of light, nothing prohibits that the matter of the star falls in a gravitational eld where vsq > c, becoming more and more dense until the celestial body turns into a singularity. However, singularities are excluded in this theory by the fact that particles are vortices of SQ and a vortex needs spatial extent, it can't be adimensional as a singularity and this also agrees with the fact that quantum mechanics does not permit particles to inhabit a space smaller than their wavelengths. It is therefore likely that because of the increasing density, we would observe partial, gradual annihilation into SQ of the particles forming the celestial body. This annihilation might

17 help the black hole to maintain a state of equilibrium or lead it to a complete evaporation, depending on how much matter it absorbs. This decomposition of vortex-particles (2) into the SQ they're made of would solve the black hole information paradox. Black holes wouldn't be then singularities but celestial bodies (spheroids) with event horizon. Hawking radiation could be however possible while the system is in equilibrium at . vsq / c 1.10 Verication If it is true that:

1. vacuum is a quantum superuid (SQS). 2. An ether wind exists and points toward the center of the Earth (it is the gravitational eld), given our hypothesis of quantum gravity as absorption of space's quanta (1.2) into vortex-particles.

3. A photon is a phonon (pulse) through a SQS. we should measure, exactly as for sound in case of wind, a higher speed of light, and consequently a blueshift, for a laser beam propagating with the same orientation of the gravitational eld and a lower velocity when light is propagating upward, with opposite orientation with respect to the eld. The formula is:

c + (vφobs − vφsource ) fobs = (32) λsource where is the observed frequency and the dierence be- fobs vφobs − vφsource tween the velocity of the gravitational ow at the point of observation and that at the source (g. 7). Also Einstein's relativity predicts this eect (gravitational redshift), where the cause is however imputed to time dilation. According to Einstein, also this theory considers time dilation due to gravity and it moreover explains through gravity also the dilation of time due to velocity, as predicted in SR (⇒FEP (10), g.6, (11), derivation of (15), (28), 1.5). However, by synchronizing the clocks of the source and of the detector and thus by eliminating the time dilation contribution, we should be still able to detect a slight red-/blueshift or alternatively a slightly dierent speed of light, when a laser beam travels one way upward or downward with respect to the gravitational eld (g.11.a). While in a round trip (g.11.b) the speed of light remains unmodied when detected, because of the reciprocal compensation of the eects along the two trajectories. For this reason, a vertically placed Michelson interferometer would be useless to detect this eect, while the use of a Hanbury-Brown-Twiss interferometer (where light only travels one way paths) would be helpful. Thus, in an Earth-Moon round-trip test for measuring the speed of light, or in satellite communications, the slower velocity during the outward trip of EM waves, would be compensated by their faster velocity during the return trip and we wouldn't notice any change in the speed of light. This theory also predicts the

18 gravitational lensing (g.4) but as a hydrodynamic eect, similar to the bending of sound caused by wind, since it considers an ether wind corresponding to the gravitational eld (g. 11.c).

Figure 11: Possible test for verifying uid quantum gravity and photon as a phonon through a SQS. The clocks of source and detector are synchronized to exclude the contribution of gravitational time dilation and the speed of light (or its frequency) is measured while it travels up/downward with respect to the gravitational eld (a). A slightly lower speed should be detected while light travels upward and, vice versa, light should be faster, and this will correspond to a variation in frequency. The two opposite eects would be balanced in a round trip of light (b) giving vL = c. On the right (c) uid quantum gravity predicts the eect of gravitational lensing due to the gravitational eld as a ow of SQ and to photon as a phonon of SQ, without resorting to curved spacetime.

In the suggested test, a laser beam travels from a source till a detector, whose clock is synchronized with the one at the source to exclude the contribution of gravitational time dilation, and the speed of light, as the time needed to reach the detector (or as a frequency shift), is measured. While propagating upward light should have a slightly lower speed and a higher speed when traveling downward. We have to notice that the Pound-Rebka experiment [44] and later more precise tests, have detected the gravitational redshift without the use of lasers running parallel to the gravitational eld lines, only focusing on gravitational time dilation. The diculty in testing uid quantum gravity is related to the fact that most of its predictions coincide with those of Einstein: the main dierence is explaining them through the hydrodynamics of SQS, which also determines time, rather than through a curved spacetime.

19 2 Appearance of fundamental particles as vortices in a superuid quantum space and emergence of quantum potential.

The particles of the Standard Model could form as dynamic topological defects (superuid vortices) or pulses in a SQS. In this view, the superuid vacuum is a fundamental scalar eld with non-zero viscosity which gives mass to particles. There are several analogies with the Higgs eld, while Higgs boson would be a 0-spin vortex of SQ (a single vortex, then an elementary particle) whose re- markable mass of 125 GeV, given the low density and viscosity of SQS, would compel it to a short life, due to its instability, and to a quick decay into smaller vortices (lighter particles) and pulses (photons 1.6). We know that quantum vortices occur in other superuids such as those observed in helium-4 nanodroplets [12, 15]. It may be interesting to start from the analysis of vortices in Bose-Einstein condensates, for which the most simple model is the Gross-Pitaevskii equation:

∂ψ(r, t) 2 4π 2a i = − ~ ∇2ψ(r, t) + ~ s |ψ(r, t)|2 ψ(r, t) + V (r, t)ψ(r, t) (33) ~ ∂t 2m m ext From (33) Proment, Onorato and Barenghi [41], elaborate the continuity and linear momentum conservation equations for an inviscid, barotropic, compressible and irrotational uid:

∂ρ + ∇ · (ρv) = 0, (34) ∂t √ ∂v ∇ρ ∇2 ρ + (v · ∇)v = − + ∇ √ (35) ∂t 2 2 ρ where the last term in (35) is the quantum stress tensor, which represents an important dierence from classical Euler equation. Albeit the superuid is irrotational, quantized vortices can appear, with a quantized circulation which is analogous to that described in the Bohr model, as the wavefunction must return to its same value after an integral number of turns

2π v · dl ≡ ~n. (36) ˛ m C(t) where m is the mass of the superuid particle and 2πn the phase dierence around the vortex. Eq. (36) is also the additional condition to impose to the Madelung equations, as a hydrodynamic alternative to the Schrödinger equation, to describe a fundamental particle as a superuid vortex. These vortices behave as gaps in the medium where superuidity breaks down and the presence of a topological structure where pressure and density go to zero, would suggest the non-necessity of renormalization, since no ultraviolet diver- gence would occur. The use of hydrodynamic equations of vortices applied to

20 SQS to describe the fundamental particles and their force elds would be then advantageous under dierent aspects. We could, for instance, explain the appearance of particle-antiparticle pairs from quantum vacuum4 as a perturbative phenomenon analogous to that described in a Kármán vortex street (g. 12)

Figure 12: a computer simulation of a Kármán vortex street. Similar perturbations in a SQS might be responsible for the appearance of particle-antiparticle pairs as right- and left-handed superuid vortices of space's quanta.

where pairs formed by a right- and a left-handed vortex occur due to a perturbation of the ow. In our case the ow may be represented by the gravitational eld and the disturbance by other particles [25], force elds or stochastic perturbations of SQS. The trigger to the formation of vortex-antivortex pairs in uid quantum space, corresponding to matter-antimatter within our analogy, might besides be a phase transition similar to the Kosterlitz-Thouless transition, where bound vortex-antivortex pairs get unpaired at some critical temperature. In the case described by Kosterlitz and Thouless, the vortices can exist above a critical temperature of κ TC = (37) 2kB where κ is a parameter depending upon the system which the vortex is in and kB is Boltzmann's constant. Also the mathematics of Lamb-Chaplygin dipoles is interesting for describing the dynamics of symmetric vortices. The stream function [13] of the ow is:

( 2U − J1(kr) sin θ r < a kJ0(ka) (38) ψ = a2 U r − r sin θ r > a

where J0 and J1 are the zero- and rst-order Bessel functions. When the rst root of J1 is ka the solution corresponds to a dipole. Single isolated particles could be therefore considered as vortex singlets or monopole vortices, whose origin would be linked to another, symmetric, particle. The equations of superuid vortices may therefore be the new basis on which to describe the wave equations of fundamental fermions [24]. In this direction, Sbitnev's work [6] is also noteworthy: he considers quantum vacuum as a superuid and applies quantum

4Referring, for instance, to electron-positron pairs in the Casimir eect

21 considerations to Navier Stokes equations. The author describes vortex objects (vortex balls) which, unlike Hill's spherical vortices, show intersected stream- lines and satisfactorily reproduce fermions' spin by varying their orientation at each revolution. Also Volovik [7] accurately discusses the possible topology of quantum vacuum and the appearence of vortices.

Figure 13: vortex street phenomena (here from [27]) might help us to understand how particles and interactions may arise as perturbations in a SQS.

When fermions are dened as superuid vortices, Helmholtz's second theo- rem is respected, since vortex lines are extended to innity (indeed, SQ absorption and virtual photons emission are not conned in space), reecting gravity's innite radius of action in Gauss's and Newton's laws and of electrostatic force in Coulomb's law. Let us now see how the quantum potential of a vortex particle would orig- inate. Recami and Salesi [32] reect on the fact that fermions' spin can be the source of quantum potential. Salvatore Esposito [34], citing Recami and Salesi, denes two velocity elds related to a quantum particle, one external, 1 , with as the phase of the function in the Schrödinger equation ~vB = m ∇S S ψ ( ∂ψ ), and 1 1 1 ∇R2 as the internal velocity. Since we i ∂t = ψH ~vS = 2m ρ ∇ρ = 2m R2 can know the external initial conditions but not the initial conditions of internal motion, and since quantum mechanics is based on a probabilistic formulation which comes into play exactly when we deal with incognizable parameters, he asserts that the quantum potential of the particle is totally determined by its internal motion ~vS × ~s, where ~s is the direction of spin. From [34] we have 1 1 Q = − m~v − ∇ · ~v (39) 2 S 2 S We see then that the quantum potential of a particle is determined by the vortex itself, i.e. by its spin as a complex angular momentum. The mechanism of absorption (g.1) would be due to spin and permitted by non-zero viscosity, while SQS's pressure would play a fundamental role in keeping the vortex indef- initely active (see also [37]): its origin has to be sought in the Big Bang, when it had to be innitely high. The role of a pressure gradient in uid quantum gravity is discussed in 3.1.

22 Local variations in the density of SQS could explain dark matter as a sort of intermediate state between non-structured SQS and the emergence of topological defects, i.e. vortices, whose spin corresponds to internal kinetic energy and then to mass within the energy-mass equivalence. This would justify dark matter halos as dark matter nebulas which have led to the formation of matter. Agreement with Huang [8] about this issue is therefore expressed. After this discussion, we could state that the space's quantum (ς) would be the only very fundamental particle in the universe, since all other particles would arise as vortices or pulses (⇒photons) of SQ. It would be therefore necessary to add it into the standard model instead of graviton (whose existence is denied in this uid model as not necessary), but the issue concerning its properties still remains open.

2.1 Vacuum uctuations as vortices in a SQS? We now investigate the relationship among vacuum uctuations and vortices in SQS. We indeed know that vacuum uctuations can appear as particle- antiparticle pairs, which in this approach are described as vortices. Let's consider the relationship:

h 4E4t ≥ = (40) 2π ~ In natural units and we know that 2 From (25) ~ = 1EP tP EP = mP c . c2 = Υ , i.e. the ratio between compressibility and density of SQS. Hence ρ0 Υ EP = mP . Planck energy therefore corresponds to the maximum mass a ρ0 vortex which emits virtual photons (g.14) can have, multiplied by the ratio compressibility/density of SQS. We see that

Υ mP tP 4E4t ≥ ρ0 . (41) 2π Since 2π corresponds to a 360° turn, the reduced Planck constant is interpretable as the rotation a vortex with Planck mass can do within a Planck time, given a specic density and compressibility of SQS. On the other hand, time arises as a chain of elementary hydrodynamic phenomena in SQS. Once put (41) in (36) we have the circulation of SQ, Γς , (closely related to spin, g.14) in a vortex-particle expressed as

m Υ 4E4t Γ = v · dl ≡ P nt ≤ n, (42) ς ˛ m ρ P 2πm C(t) 0 which approximatively corresponds to n vacuum uctuations each complete turn (2π), divided by the mass of the vortex-particle whose life corresponds to ntP . If vacuum uctuations obey a model similar to Kármán vortex street (g.12), we have to consider two antisymmetrical vortices to match the appearance of electron-positron pairs from SQS. Also the annihilation of the pair would

23 occur on the basis of hydrodynamics because of the conicting circulation of a vortex with respect to another (g.20).

3 Unication of the fundamental interactions as a direct consequence of uid quantum gravity.

We analyze below the endofermionic consequences of SQ absorption (1.2). The rst one seems to be the unication of gravity with electromagnetism, which makes us reect on the fact that quantum gravity shouldn't be the last but the rst piece of the unication puzzle. The need to maintain energy equilibrium despite the absorption of SQ into the vortex would be satised through the emission of virtual photons (gravity-electromagnetism link). When no emission occurs (isolated neutral particle) the vortex is due to decay (⇒β-decay) and, nally, the exchange of SQ among adjacent vortices would correspond to the gluonic ow of quantum chromodynamics (strong interaction).

3.1 Gravity-electromagnetism unication.

As far as the most appropriate vortex geometry is concerned, it is interesting to consider the evolution: vortex tube → vortex torus → horn torus (g.14), since it could be able to account for the main mechanism suggested in this work, i.e. the absorption of SQ (gravity) and the consequent emission of virtual photons, which accounts for Coulomb's force and is necessary to maintain energy balance in spite of the absorption. Furthermore, referring to g.14, if a space's quantum (red dot in the gure) in the toroidal vortex needed the same time the vortex needs to complete two turns in the toroidal direction (σ1 spin component) to return in the same position, while the σ2 spin component completes a single turn in the poloidal direction, then the vortex would have spin½ (the system returns in the same state after a toroidal rotation of 720°, after each quantum forming the vortex has moved along a path similar to a Möbius strip). It is interesting to notice that a two-components spin can explain in mechanical terms any other type of spin, as the ratio between the two dierent rotations in the torus. We would have:

1σ 1 1σ 0σ 2 ⇒ spin ; 2 ⇒ spin 1; 2 ⇒ spin 0 (43) 2σ1 2 1σ1 1σ1 The geometry of the toroidal vortex would instead account for the charge of the particle: neutral if the vortex is not a horn torus, as there cannot be genesis of virtual photons. The theoretical mainstream describes the electrostatic eld of charged fermions as the emission and reabsorption of virtual photons. Also in the hypothesis of

24 gravity as absorption of SQ occurring in fermions (1.2) we have absorption and we need a subsequent emission to maintain energy balance. If we then hypothesize the emission of virtual photons as discrete packets of compressed SQ (g.14) we would start seeing how gravity can be related to electromagnetism. Unlike normal photons, virtual photons can have a mass, since they're packets of SQ not simple pulses. The strenght of their momentum is for instance evident when trying to bring together two magnets that repel each other and this would be due to a mechanism analogous to that depicted in g.18.

Figure 14: evolution of a torus vortex into a horn torus. This might justify the compression of space's quanta into virtual photons within the absorption-emission mechanism. On the right the possible mechanism corresponding to spin½, where the system returns in the same state after a rotation of 720° in the toroidal direction σ1, while each space's quantum in the vortex ows along a path similar to a Möbius strip. Such two-components model (σ1, σ2) may mechanically explain any other kind of spin as the ratio between the two rotations (43). In this case we have 1σ2 . 2σ1

25 Figure 15: current (a) and suggested (b) Feynman diagram which describes a charged particle's self-energy, where ς indicates a space's quantum and γς a virtual photon as a discrete packet of space's quanta.

The only dierence with the current mainstream would be having rst absorption and then emission. The Feynman diagram describing charged particles' self-energy might then change as suggested in g. 15.b. The absorption-emission mechanism could be possible if fermions were, as discussed, superuid vortices in SQS (also Sbitnev, [6] paper II, g.3, considers a periodic energy exchange between a vortex and the superuid quantum vacuum). The vortex dynamics illustrated in g.14 is interesting since it would allow partial emission and re-absorption according to the current Feynman diagram (g. 15.a), as shown in g.18: Coulomb force would be then determined by the recoil of the not re-absorbed component of SQ (emitted virtual photons). The absorption-emission mechanism would be periodic, harmonic, and would be in agreement with the undulatory nature of particles (g. 16).

Figure 16: sawtooth electrogravitational oscillator for a charged particle expressing its rest mass variation while producing gravitational pull and electrostatic eld. Vacuum contribution corresponds to the absorption of SQ in the vortex.

26 The time-dependent equation highlighting the periodic mass-energy oscillation at the basis of the link between gravity and a charged particle's electrostatic eld (g.16) would be

meff(t) = (t − btc)ka + m0, (44)

where meff(t) is the time-depending eective mass of the particle, which would rapidly oscillate between two values (m0, mmax), and ka is a costant of mass-energy absorption expressed in , whose value is , kg/s ka = mγς /temission i.e. the ratio between the mass of a virtual photon and the necessary time to emit it from the vortex. The proper mass of a fermion (this would be also valid for neutral particles as discussed in 3.2) would therefore minimally oscillate and this fact would agree with the indeterminacy of quantum mechanics. The oscillatory behavior of superuid vortices shown in g.16, might also account for the phenomenon of zitterbewegung (trembling motion). In this respect see also [40] for a possible link with nutation occurring in toroidal vortices. From (36) and (44) we have

2πn u · dl= ~ , (45) ˛ m C(t) eﬀ (t) the time-depending circulation in the vortex also expressing the mass-energy oscillation due to absorption of SQ (gravity) and emission of virtual photons (electrostatic force). In 2 we have pointed out that the quantum potential may arise from the spin of the superuid vortex. Now, if the spin thanks to a minimal viscosity of SQS is responsible for the absorption of SQ and therefore for quantum gravity, g.17 (from the performed CFD simulations) shows that a velocity potential (left) is causally linked to a pressure potential (right), which could be expressed by resorting to the quantum pressure tensor of Madelung equations5:

2 ~ PQ = − ρ∇ ⊗ ∇ ln ρ (46) 2mς 5see also [37] for a pressure mediated link between quantum vacuum and matter

27 Figure 17: absorption velocity and pressure gradient from the performed CFD simulations.

whose relation with quantum potential is

∇ · PQ = ρ∇Q (47) and from (44) and (39) obtains

1 1 ∇ · P = ρ∇ − ((t − btc)k + m )~v − ∇ · ~v (48) Q 2 a 0 S 2 S This would mean that the quantum potential of a superuid vortex (driven by the spin [34]), which causes SQ absorption, is directly linked both to a negative pressure potential (g.17 right) and to the absorption velocity potential (g.17 left) which corresponds to a mathematically analogous quantum force,

Fq = −∇Q, interpreted as gravitational eld (in this case a mass-induced gravitational eld, since we have discussed also the case of apparent gravitational eld induced by motion in 1.4.1). We can then reect on the following relations:

vsq = −∇φ ⇔ Fq = −∇Q ⇔ g = −∇Φ (49) Now, to describe the dynamics of the electrostatic force, an anisotropic interaction between superuid vortices, instead of the exchange of virtual photons in isotropic conditions, could explain both attraction and repulsion between two charges as a mechanism driven by recoil (g. 18). On the contrary, the current isotropic model based on virtual photons exchange is weak in clarifying the mechanism of attraction. The conguration of two same charges in g.18 (right) shows Pauli exclusion principle (antisymmetric spins), which also accounts for the validity of the anisotropic hypothesis for the electrostatic interaction. The positioning of a charge with respect to another would generate the illusion of a radial electrostatic force. Such a positioning mechanism would be coherent with the interactions occurring also in systems containing more particles. The compression of

28 a given amount of absorbed SQ into a single virtual photon, which is unidirec- tionally emitted, could explain why, for particles, the electrostatic interaction is 39 orders of magnitude stronger than the gravitational interaction.

Figure 18: Electrostatic attraction and repulsion due to the recoil caused by the emission of virtual photons (γς ) from superuid vortex-particles.

A positioning of the particles according to their charge would occur (anisotropic interaction) on hydrodynamic basis and the emitted virtual photons are not ex- changed but disperse obeying an inverse-square law, according to Coulomb. The conguration of same charges (right) corresponds to Pauli exclusion principle (antisymmetric spins). The analogy with ferromagnetism is very strong and interesting and may be in favor of the hypothesis of anisotropy in the electrostatic interaction, since also within that phenomenon a repositioning of the particles occurs when the spin of the electrons (thus also the axis of the vortices as in g.18) changes its orientation under the external inuence of a magnetic eld.

3.1.1 Interpretation of the ne-structure constant in SQS.

2 If we consider the ne-structure constant, e.g. in the form α = kee , since ~c q Υ Υ c = (25), = EP tP and EP = mP , we obtain a further denition, ρ0 ~ ρ0 which shows the relationships with SQS's properties:

2 ρ0 α = kee (50) q Υ mP Υ tP ρ0

where ρ0 is SQS's density, Υ its elastic modulus (we consider unidirectional compression occurring in a pulse, see 1.6 and eq.25), mP the maximum mass

29 for a vortex-particle, e the elementary charge, which we have just interpreted (3.1) as virtual photons emitted from a vortex-particle and ke the electrostatic force constant. The reciprocal of Planck time can be interpreted as a frequency, related to the rotation of the vortex-particle. Moreover, we see that (50) contains the reciprocal of the circulation in a quantum vortex (42), , multiplied by n , Γς m so we can rewrite (50) as: e

k e2n α = e . (51) q Υ Γς me ρ0 The constant of the ne structure of an atom emerges then as strictly related to a quantum vortex occurring in SQS and (51) shows it in particular as the ratio between the emission of virtual photons and the fundamental properties q of SQS ( Υ ) multiplied by the mass of the vortex (here an electron) and its ρ0 quantum circulation. We can also argue that, as for the speed of light and other constants, a variation in SQS's density would change the value of α. Now, since the universe is expanding and this implies a variation of ρ0 in time, it is plausible that these constants have not had the same value billions of years ago and that they will change in a distant future. For the same reason, it is not excluded that SQS's density and the constants which depend on it have dierent values in very distant regions of the universe.

3.2 Weak interaction.

From 3.1, it may be inferred that if no emission of virtual photons occurs (charge neutrality) a particle would be compelled to decay, because of the increase of its internal energy due to absorption of SQ. We indeed observe in unbound neutral particles such as isolated neutrons (g. 19), whose mean lifetime is 881s.

30 Figure 19: β-decay in unbound neutrons, (a) caused by vacuum energy absorption (SQ), − being τn the neutron's mean lifetime. Here the W boson corresponds to the sum of the exceeding absorbed SQ. (b) A comparison with Feynman diagram for neutron's decay.

A prediction of this theory would be a greater mass of isolated neutrons before they decay, if compared with the mass of bound neutrons in a nucleus and, for instance, also that of a faster decay of neutral pions (indeed 8.4 · 10−17s) if compared with charged pions (2.6 · 10−8s), as it actually occurs. Decay would also occur in unstable charged particles, where the equilibrium absorption-emission is imperfect (g 20.d).

3.3 Strong interaction and quantum chromodynamics.

When describing the elementary fermions as superuid vortices of SQ, it is immediately clear that their spins can be conicting or mutual when two particles approach. Two adjacent vortices with conicting spins would annihilate each other. This fact could explain several known issues of atomic physics (g. 20). Only electrons with antiparallel spin (g. 20.b) can be in the same orbital: as vortices, they have in this circumstance mutual spins, explaining Pauli exclusion principle, and this is precisely the conguration described in g.18 for same charges (antisymmetric spin). If they didn't repel each other and came in contact, two electrons with parallel spin (g. 20.a) would annihilate like a pair electron-positron (g. 20.c) because of their conicting spins. While matter- antimatter annihilation would be then simply due to the conicting spins of left- and right-handed vortices in antiparallel

31 Figure 20: Conicting (C) or mutual (M) spin would explain: Pauli exclusion principle (a, forbidden; b, allowed → anti-parallel spin, according to g. 18); particle-antiparticle annihilation (c); mesons, such as π0 (d) or exotic atoms as ortho-positronium. Light grey vortices correspond to matter (left-handed) and dark grey ones to antimatter (right-handed). Without electrostatic repulsion, annihilation between two same charges with parallel spins would occur (a) even without involving antiparticles, due to conicting rotations.

Figure 21: Quantum chromodynamics as a system of superuid vortices of SQ in mutual-spin conguration (here the hypothetical structure of a proton). The gluonic octet is represented as the exchange of discrete amounts of SQ among the vortices (i.e. as a continuous ow of SQ). The charge of the whole system is determined by the emission of virtual photons from the central vortex (down quark), while the SQ expelled from the other quarks are reuptaked by the central quark (b¯r, rb)¯ . This uid model would explain why most of the mass of the quarks in a proton is expressed as binding energy (gluonic ow): SQ, as gluons, continuously ow among the quarks (they exchange mass the one with the other) and this determines their continuous change of color. The geometry in b) is forbidden since it causes conicting spins.

spin conguration, when they have parallel spin they can form unstable particles such as mesons (for instance a neutral pion π0, g. 20.d) or ortho- positronium.

32 Conicting spins may also explain the decay occurring in nucleon resonances. Quantum chromodynamics and the strong interaction as exchange of gluons (seen as discrete amounts of space's quanta passing from a vortex to another) can be therefore described in terms of adjacent superuid vortices in mutual-spin conguration, as shown in g. 21. The continuous exchange of discrete amounts of SQ (interpreted as gluons) among the three vortices would account for the fact that most of the mass- energy of bound quarks is in the form of force-eld energy and that color mi- grates from a quark to another. A uid dynamic, quantum explanation seems to be clear and necessary. The so-called residual strong interaction can be likewise described as exchange of SQ between vortices which are close together, when a tube arises, whose break perturbs the uid space and simmetrically generates a quark-antiquark pair (π0) as a vortex-antivortex pair (2), within a self-sustainable process.

3.4 Baryon asymmetry explained?

The decay of B mesons is predicted to favor the production of matter over antimatter, but not enough to explain the huge preponderance of matter in the present universe. Baryon asymmetry could be however explained by the fact that particles and antiparticles have more than one way to interact. By referring to g.20, the dierent, possible congurations that the superuid vortices can assume, might explain the origin of baryon asymmetry, starting from equal amounts of matter and antimatter in a quark-gluon-lepton plasma (g. 22). The statistically prevalent production of unstable and annihilation groupings would account for the large number of photons in the cosmic background radiation for each existing baryon. The dynamics illustrated in the gure could be extended up to even 1 billion photons each baryon.

An asymmetry may therefore arise as a mere probability from all the dif- ferent combinations and spin congurations that vortex-particles can assume. In g. 22, by starting from a perfect symmetry (S), particles (light grey) and antiparticles (dark grey) in a quark-gluon-lepton plasma (primordial universe) get stochastically mixed up and merge. Phenomena of annihilation and decay are statistically prevalent. The gure shows only one of the possible, innite combinations, which produces in this case a single proton, two electrons, many photons and totally erases antimatter. This means that the Big Bang (or rather a cascade perturbation occurred in a calm, pre-existing SQS?) may have produced equal amounts of matter and antimatter but the ways they have stochastically combined may have led to baryon asymmetry. As we can see, this model also predicts, in every possible combination of particles, the existence in the universe of more leptons than baryons.

33 Figure 22: Baryon asymmetry as a mere probability which arises from all dierent combinations and spin congurations that fundamental vortex-particles might assume, starting from a perfect symmetry (S).

4 Conclusion

We can rstly summarize the theory of uid quantum space, gravity and relativity in 12 points: 1. Particles of the Standard Model are vortices in a superuid quantum space (superuid quantum vacuum for some authors) or pulses (as for photons, 1.6). 2. Due to the non-zero viscosity of SQS, these vortices may attract the surrounding space's quanta. This would be the fundamental mechanism of quantum gravity (g.1), on a hydrodynamic basis, where the quantum potential comes from spin and is linked to the pressure gradient due to absorption. 3. Absorption in the vortex is balanced by an outcoming ux of compressed space's quanta (virtual photons) which produces the electrostatic eld of charged particles (g.14), where attraction and repulsion are due to recoil (g.18). 4. When no emission occurs (e.g. isolated neutrons), the vortex is due to decay because of its unsustainable energy increase due to the absorption

34 of SQ. This would explain the β-decay (g.19). 5. Vortices can interact through mutual or conicting spins, causing strong interaction in the rst case (through exchange of discrete packets of SQ, as gluons, from a vortex to another, see g.21), or annihilation in the second case, as in the case of an electron-positron pair.

6. Depending on their pairing conguration, two particles (or two antiparticles) could also annihilate, if electrostatic repulsion could be overwhelmed and a pair particle-antiparticle could also not annihilate (as in the case of π0). Thus, in the primordial universe, at extremely high energy and density, more scattering congurations might have occurred than those currently observed, causing baryon asymmetry on a mere statistical basis, as one of many possible outcomes (g.22) which, however, would always lead to a preponderance of photons and leptons over baryons. 7. General relativity is determined by the absorption of SQ (point #2), i.e. by quantum gravity, and the relativistic eects it describes are hydro- dynamical. The hydrodynamics of uid space also determines time. No curved spacetime exists. Gravity slows down clocks (meaning also a cell, an atom's decay etc.) since it acts by increasing the weight of their components. 8. The MM-test doesn't confute the existence of the ether (SQS in our case) if the ether wind corresponds to the gravitational eld (i.e. to a ow of SQ directed toward the centre of the Earth, due to the absorption mechanism of uid quantum gravity, 1.2). In this case, the assumptions of that experiment become wrong. 9. Special relativity is determined by the viscosity of SQS, as apparent viscosity due to shear stress. Indeed, SQS would behave as a dilatant uid for extreme shear stress, i.e. for translational velocity approaching the speed of sound (of a pulse) in SQS (⇒speed of light). A dilatant behavior would also conrm the granular (quantum) nature of space. Dierent translational velocities in dierent frames of reference correspond to dif- ferent interactions with SQS and determine the relativistic eects of SR. We have then: speed ⇒ shear stress and apparent viscosity ⇒ apparent gravitational force opposite to motion (drag weight, 1.4.1), justifying the illusory ef- fect of mass increase⇒ time dilation due to drag weight acting on a clock's components, as for point #7 (see the uid equivalence principle: 1.4.1, eq.10 and g.5, 7), and unication of the causes of time dilation in GR (gravity) and SR (velocity) into gravity only⇒ Lorentz-Fitzgerald contraction as a direct consequence of time dilation, as in the current approach. 10. Equations and evidences suggest that a photon can be a special spin-1 phonon propagating through a SQS (1.6). Its velocity c would simply correspond to the speed of a pulse in a specic medium (SQS in this

35 case), as every other medium allows a specic propagation velocity for a pulse through it. 11. Redshift in Hubble's law doesn't increase with the galaxies distance because of an accelerated expansion of the universe but due to photons' energy loss, according to the distance they travel through SQS, more precisely, according to the distance covered by the pulses, assumed they are phonons. There would be therefore no reason for the existence of dark energy and for an accelerated expansion of the universe. 12. Gravitational waves can be described as periodic variations (negative pulses) in the local density of SQS, due to a variable absorption of SQ (from point #2), depending on the orientation of a quadrupole.

If the single hypothesis of fermions as vortices of space's quanta in a superuid quantum vacuum, which obeys Gauss's law for gravity (incoming ux ⇒ absorption of space's quanta, determining the relationship vsq = −∇φ ⇔ Fq = −∇Q ⇔ g = −∇Φ), generates a series of consequences which range from describing and simplifying special and general relativity up to unifying the four fundamental interactions, justifying baryon asymmetry, Hubble's law without resorting to dark energy and up to producing a general simplication of various other issues, it should be then considered for mathematical and experimental in-depth analysis. Furthermore, the hypothesis of SQ absorption solves the incompatibility of a uid space with the outcome of the Michelson-Morley experiment, since an ether wind corresponding to the gravitational eld is possible and conforms to the known experimental evidences, such as light deection and gravitational redshift. Finally, the description of photons as phonons propagating through a superuid space would reduce waves which exist in nature to only one type (medium-dependent). We could therefore state that entia non sunt multiplicanda praeter necessitatem, as it seems that a superuid approach may be able to explain nature without extra dimensions, strings or gravitons. The suggested test concerning the speed of light (1.8) is simple and might demonstrate the existence of a superuid quantum space, of gravity as absorption of SQ and of light as a pulse (phonon) through a uid space. Within this superuid reconsideration of the universe, it is impossible to keep fundamental forces separeted and to keep them separated from particles, since we see that everything is expression of the hydrodynamics of SQS. From vortices of SQ we have indeed mass, charge, fundamental forces etc. and nothing appears to be really fundamental but space's quanta.

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39 Annex.

Other images and charts from the performed CFD simulations.

Figure 23: Simulation settings

Figure 24: radial ow obtained in the simulations

40 Figure 25: Test for force dependence on absorption velocity: sphere diameter 1mm, distance 2mm. Tested velocities: 50, 100, 200, 500, 700, 1000 m/s. Other tested conditions (50, 100, 200, 500 m/s) are shown in g. 6 and 7.

Figure 26:

41 Figure 27:

Figure 28: Test for force dependence according to the distance between the spheres.