Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury’s perihelion precession
Abstract
By using the Lorentz factor as a viscosity term in Stokes’ law for objects traveling in a vacuum, Mercury’s perihelion precession and the Pioneer anomaly are directly and exactly solved, demonstrating that physical vacuum is a shear-thickening (dilatant) fluid. The modified Stokes’ equation also correctly indicates that planetary orbits are stable (over trillions of years). This unexpected feature of physical vacuum may help in achieving quantum relativity and implies interesting consequences for various fields of modern physics.
Résumé
En utilisant le facteur de Lorentz comme terme de viscosité dans la loi de Stokes pour des objets voyageant dans le vide, la précession du périhélie de Mercure et l’anomalie Pioneer sont directement et exactement solutionnées, démontrant que le vide est un fluide antithixotropique. L’équation modifiée de Stokes indique précisément que les orbites planétaires sont stables (sur des trillions d’années). Cette caractéristique inattendue du vide physique peut aider à réaliser la relativité quantique et introduit des conséquences pour différents champs de la physique moderne. [Traduit par la Rédaction]
1. Introduction
Some authors considered the possibility that the physical vacuum may be a superfluid, a special Bose–Einstein condensate [1–9]. Here, by exactly solving two known anomalies, along with other correct results, it is demonstrated that the physical vacuum rather behaves as a dilatant fluid, as shear stress increases. Section 2 introduces a modified Stokes’ formula for motion in a shear-thickening vacuum. In Sect. 3, by applying this formula, the Pioneer anomaly, the orbital stability of the planets, and Mercury’s perihelion precession are correctly calculated. The Pioneer anomaly is currently considered solved after thermal simulations, whose results, approximate and based on various assumptions and scenarios [10–16], are now challenged by the simpler, direct, and precise result presented in Sect. 3.1, which exactly correspond to the acceleration of −8.74 × 10−10 m·s−2 measured by NASA. The modified Stokes’ formula, put into Newton’s second law of motion, says that due to the large masses of planets (unlike the case of the Pioneer probes), planetary orbits are stable over billions of years, averting what would otherwise be a major objection to the existence of a shear-thickening vacuum (i.e., its effect on orbital stability). Section 3.3 shows that the vacuum’s apparent (shear-dependent) viscosity emerges as the real cause of the anomalous precession of perihelia, suggesting that the quantum foundations of relativity are based on a fluid, shear-thickening, quantum vacuum. In fact, it is shown that Einstein’s formula for the precession of perihelia can be directly derived from the modified Stokes’ formula. The conclusion also suggests possible reasons for vacuum dilatancy, building a bridge to the dark sector and to the Higgs field, while further research in this direction is expected.
2. Methods: modified Stokes’ law for a dilatant vacuum
Stokes’ law, derived in 1851 to calculate the viscous force acting on a body traveling through a viscous Newtonian fluid [17], reads
(1)
where v is the translational velocity, r the radius of the object (the law refers to spherical shape), and η is a coefficient of dynamic viscosity expressed in Pa·s. However, for a shear-thickening vacuum, the viscosity coefficient η, in (1), is not appropriate, because it is valid only for Newtonian fluids. To express a dilatant vacuum, we need a nonlinear law. In the present investigation it is demonstrated (Sect. 3) that the correct mathematical behavior of shear stress for the physical vacuum is expressed by the Lorentz factor, therefore reinterpreted as vacuum’s rheogram, in which the asymptote at the speed of light consequently refers to a transient, solid-like condition of the vacuum, which occurs when approaching a certain level of shear stress. Let us then replace η with the Lorentz factor, using the expression γ − 1, which also expresses the speed v of (1). We arrive at a modified Stokes’ equation
(2)
where κ is a unitary constant expressed in kg·s−2. Let us also define
(3)
as a dimensionless term of vacuum dilatancy. The formula is simply written Fvac = −6πrDκ. If we use this formula for bodies traveling in a vacuum, such as probes or planets, the viscous force it refers to is of course that of the physical vacuum, so (2) is the formula for the viscous force exerted by a dilatant vacuum: and its applications and validity are presented in the following sections, by precisely solving — as announced — two known anomalies and by obtaining other correct results, such as the stability of planetary orbits.
3. Results: shear-thickening vacuum
3.1. Exact value for the Pioneer acceleration
The anomalous negative acceleration of Pioneer spacecraft 10 and 11 being well-known (concrete investigations of that anomaly started in 1994 [12]), it is not necessary to summarize this issue here. In light of the exact result presented below, it appears evident that this problem has not been correctly solved yet, despite copious investigations based on thermal simulations [10–16], which produced approximate results from several assumptions and different scenarios. In 2012 [10] a value of 7.4 (± 2.5) × 10−10 m·s−2 was proposed. On the contrary, the exact solution (without models and assumptions) is directly offered by (2), that is, via the interaction of the Pioneer probes with a shear-thickening vacuum. Let us put (2) in Newton’s second law, using the known data of the Pioneer 10 and we directly obtain the exact negative acceleration of the Pioneer (aP) detected by NASA
(4)
where mP = 222 kg is the mass of the spacecraft (258 kg) minus that of the burned fuel (36 kg hydrazine) after the Jupiter flyby; rP = 1.371 m is the radius of the antenna (diameter is 9 ft); and vmax = 36 737 m·s−1 is the maximum speed of the probe, as indicated by NASA’s Scientific and Technical Information Office [18] after the swing-by caused by Jupiter. This exact and direct result cannot be ignored and the Pioneer issue has to be reopened. Moreover, NASA should now consider Doppler tracking other probes. Unfortunately this was not the case for the New Horizons spacecraft, for which data are missing. Testing other probes (maybe a dedicated probe) with (2) would be useful to check this result.
3.2. Second test: stability of planetary orbits
The existence of a shear-thickening vacuum does immediately raise an objection as regards orbital stability. However, considering the second law of motion in the form a = F/m and putting the large mass of a planet in the denominator and the modified Stokes’ equation in the numerator, we see that, despite the existence of a dilatant vacuum, planetary orbits are stable over trillions of years. For instance, the deceleration of the Earth (a⊕) corresponds to the following negligible value:
(5)
using (2) in the numerator, with mean radius and mean orbital velocity of the Earth and the mass of the Earth in the denominator. Subscript ⊕ refers to the Earth. Such a negative acceleration corresponds to a decrease in orbital speed < −3.13 × 10−9 m·s−1 per billion years, making orbits stable for thousands of billions of years. For Jupiter, orbital deceleration is −6.59 × 10−28 m·s−2. One therefore concludes that planetary orbits are stable despite the presence of a shear-thickening vacuum.
3.3. Deriving Einstein’s formula for the precession of perihelia
Net of classical gravitational contributions, perihelia precessions show an anomalous positive contribution, which is particularly evident for the planet Mercury. The correct calculation of this anomaly is one of the classical tests for general relativity (GR). Here Einstein’s formula for the precession of perihelia is differently derived via the modified Stokes’ equation presented above. In this way, it is suggested that the long-awaited quantum foundations of GR are situated in a shear-thickening quantum vacuum. In GR [19], the anomalous perihelia precession is represented by a formula that can be observed in three equivalent forms
(6)
where 
expresses the relativistic contribution to perihelia precessions in radians per revolution corresponding, using the data of Mercury, to the known value of 42.98″ per century (or 5.0186 × 10−7 rad/rev), a = r is the semi-major axis, T the orbital period, and e = 0.205 the orbital eccentricity. The expression in the center of (6) is obtained via the equivalence T2 = 4π2a2/v2, resorting to mean orbital velocity, and in the expression on the right-hand side the stable second cosmic velocity, 
, is used. As for the case of the Pioneer, the modified Stokes’ equation for a dilatant vacuum is used below. In this case, the treatment of the planet as a point mass is respected, as in GR, so the direct proportionality to planetary radius is not taken into account, and by using the dimensionless norm of (2) we obtain
expresses the relativistic contribution to perihelia precessions in radians per revolution corresponding, using the data of Mercury, to the known value of 42.98″ per century (or 5.0186 × 10−7 rad/rev), a = r is the semi-major axis, T the orbital period, and e = 0.205 the orbital eccentricity. The expression in the center of (6) is obtained via the equivalence T2 = 4π2a2/v2, resorting to mean orbital velocity, and in the expression on the right-hand side the stable second cosmic velocity, , is used. As for the case of the Pioneer, the modified Stokes’ equation for a dilatant vacuum is used below. In this case, the treatment of the planet as a point mass is respected, as in GR, so the direct proportionality to planetary radius is not taken into account, and by using the dimensionless norm of (2) we obtain (7)
which can be expressed in radians. Invoking Taylor, let us proceed via the approximation
(8)
and (7) now reads
(9)
where, on the right-hand side, we see again the stable second cosmic velocity (here squared), as in the rightmost expression in (6). Because we are considering an elliptic orbit, we have to use the elliptic parameter, correcting a into a(1 − e2) and we obtain a formula that gives exactly 1/2 the result of GR found in (6),
(10)
This 1/2 result can be considered as the precession occurring in a semi-orbit and is due to the use of mean orbital velocity. Indeed, in the elliptic orbit, the orbital speed actually varies as in Fig. 1 on the left-hand side. Because the mean orbital velocity is given as (vmax/2) + (vmin/2), let us adopt the reduced model on the right-hand side of Fig. 1 (i.e., one semi-orbit at maximum orbital speed and one at minimum speed). The full precession (6) is therefore given by
(11)
where vmax and vmin, each referring to a semi-orbit, recombine in the mean orbital velocity v and the rightmost equivalence comes from (6). Now, by merging the steps above into a single formula, we can look at the relationship between the modified Stokes’ equation, which expresses the viscous force in a dilatant vacuum, and the contribution to the precession of perihelia. The formula reads
(12)
where D is the term of vacuum dilatancy (3). By testing (12) with the parameters of the planet Mercury, we see that it exactly gives the well-known value of GR
(13)
where, in D, 
is used, as in (6). The positive contribution to the precession of perihelia treated in GR is in this way revealed as a phenomenon driven by a shear-thickening vacuum. We see that nothing, in the present study, contradicts GR: this investigation, by rederiving Einstein’s formula for perihelia precession, only highlights that the quantum foundations of relativity are in a dilatant quantum vacuum, as (12) shows. After all, this is compatible with the stress–energy tensor of the field equation, in which T00 is vacuum’s energy–mass density (ρvac), also present in the cosmological constant Λ = κρvac, and the remaining components of the tensor can be as well hydrodynamically interpreted, being pressure, shear stress, momentum flux, and momentum density. Interestingly, also an investigation by Marcucci and Conti [20], following the present findings [21], shows that the interaction at Planck-scale with a quantum fluid may be the cause of precession.
is used, as in (6). The positive contribution to the precession of perihelia treated in GR is in this way revealed as a phenomenon driven by a shear-thickening vacuum. We see that nothing, in the present study, contradicts GR: this investigation, by rederiving Einstein’s formula for perihelia precession, only highlights that the quantum foundations of relativity are in a dilatant quantum vacuum, as (12) shows. After all, this is compatible with the stress–energy tensor of the field equation, in which T00 is vacuum’s energy–mass density (ρvac), also present in the cosmological constant Λ = κρvac, and the remaining components of the tensor can be as well hydrodynamically interpreted, being pressure, shear stress, momentum flux, and momentum density. Interestingly, also an investigation by Marcucci and Conti [20], following the present findings [21], shows that the interaction at Planck-scale with a quantum fluid may be the cause of precession.Fig. 1.
4. Conclusion
The modification of Stokes’ law, obtained by replacing η with the Lorentz factor (in the form γ − 1), reinterpreted as the rheogram of a fluid, shear-thickening vacuum, has produced a new formula expressing the viscous force exerted by physical vacuum. This formula has been confirmed by solving two well-known anomalies, the relativistic contribution to perihelia precession, rederiving Einstein’s equation, and the Pioneer anomaly, for which a direct and exact result has been obtained, to be therefore preferred to the more approximate and complicated solutions so far presented. Deriving Einstein’s formula for the precession of perihelia from the viscous force of physical vacuum suggests that the quantum foundations of relativity are rooted in the existence of a shear-thickening quantum vacuum. We can conclude by considering that such a vacuum may correspond to the dark sector (i.e., to 95% of the universe’s mass–energy), in which diffused dark matter particles (a granular component) could play the role of a dopant in a sea of superfluid dark energy, determining vacuum dilatancy. Nevertheless, also the Higgs field, as a ubiquitous, scalar, viscous field, could be the reason for the existence of a dilatant vacuum. For sure, the shear-thickening aspect of the physical vacuum helps us to better understand its nature, along perhaps with that of dark energy and dark matter, soliciting further specific research.
Acknowledgements
The author thanks Lorenzo Dominici for discussions concerning the validity of the modified Stokes’ equation, also for computing orbital decay, Ali Taani for references and comments, and Lorenzo Brandi for comments.
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Information & Authors
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Published In
Canadian Journal of Physics
Volume 97 • Number 4 • April 2019
Pages: 417 - 420
History
Received: 7 October 2018
Accepted: 11 January 2019
Accepted manuscript online: 13 February 2019
Version of record online: 13 February 2019
Copyright
© 2019.
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Marco Fedi. 2019. Physical vacuum as a dilatant fluid yields exact solutions to Pioneer anomaly and Mercury’s perihelion precession. Canadian Journal of Physics.
97(4): 417-420. https://doi.org/10.1139/cjp-2018-0744
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