BFUT P52

A Universal Physical Substrate Producing Dark Matter Phenomena in Galaxies

P52

Equilibrium Density of a Screened Gravitational Carrier

and the Predicted Acceleration Scale of the Medium

Vijay Shankar Sharma

Independent Researcher, Gurugram, National Capital Region, India

vss@vijayshankarsharma.com

ORCID: 0009-0001-9622-6121

DOI: 10.5281/zenodo.21280381

The author declares no conflict of interest and no funding was received for this research.

License: CC BY-NC-ND 4.0

Abstract

This paper states the static carrier equation of the Spaticle field, derives its equilibrium density from the proton mass and the proton charge radius, and obtains the carrier acceleration aₛ from the specific-work balance aₛ Lₛ = pₛ/ρₛ. The static equation has an exact Yukawa solution. The condensation functional of Section 5, with coefficients derived there, fixes ρₛ. The work balance then fixes aₛ = 1.2084 × 10⁻¹⁰ m s⁻². No galactic quantity enters that value.

The static equation ∇²Ψg − μₛ²Ψg = 4πGM δ³(r), with μₛ² = 3Gρₛ/c², has the exact and unique decaying solution Ψg = −GM e^(−μₛr)/r. For an extended spherical source the exterior field has the same form with a closed-form structure factor. The force F = −GMm(1 + μₛr)e^(−μₛr)/r² reproduces Newtonian gravity with a fractional departure of -(μₛr)²/2 + (μₛr)³/3 + O((μₛr)⁴). On galaxy scales that departure is of order 10⁻¹².

The equilibrium density follows from the four-coefficient condensation functional whose minimum R₀ = 1.27348221 is obtained with no measured input. The coefficients are derived in Section 5: A = 1/2 from the model-unit kinetic identity, D = 2A = 1, C = -1/3 from threefold symmetry, and B = 0.56308 from the three-sphere filling-deficit ratio. The proton mass and charge radius ground that dimensionless R₀ in SI. Then ħ_vss = 1.0545769 × 10⁻³⁴ J s, α_vss = 1/137.036655 and m_e_vss = m_p/(6π⁵), and hence ρₛ = G ħ_vss²/(8π c⁴ r_e⁶) = 7.30294 × 10⁻²⁷ kg m⁻³.

For the isotropic equilibrium medium the pressure is pₛ = ρₛ c²/3. Equating the specific work aₛ Lₛ across the carrier length Lₛ = 1/μₛ to the specific pressure energy pₛ/ρₛ gives aₛ Lₛ = pₛ/ρₛ. It follows that aₛ = c √(Gρₛ/3) = √(G pₛ) = 1.2084 × 10⁻¹⁰ m s⁻² and Lₛ = 26.205 Gly. No galactic quantity is used. Published rotation-curve analyses of 153 SPARC disk galaxies report that the observed centripetal acceleration departs from the baryonic Newtonian acceleration below a characteristic scale of order 10⁻¹⁰ m s⁻². McGaugh, Lelli and Schombert quote a fitted scale g† = 1.20 ± 0.02 (random) ± 0.24 (systematic) × 10⁻¹⁰ m s⁻² [5]. The predicted aₛ lies 0.7 percent from that fitted figure. The Yukawa departure of equation (6) is 1.8 × 10⁻³¹ at 1 AU, 7.0 × 10⁻¹² at 30 kpc and 0.71 percent at 1 Gpc. The carrier-induced perihelion precession of Mercury is 1.4 × 10⁻²³ arcseconds per century.

Keywords: screened Poisson equation; Yukawa gravity; equilibrium density; specific-work balance; acceleration scale; SPARC; Spaticle field

1. Introduction

The author identifies a physical matter substrate that pervades the universe and calls it the Spaticle field. Matter, forces and gravitation are described as organised states of this substrate. Gravitation is carried by the Spaticle field, and its static response to a mass obeys a screened Poisson equation whose screening length is set by the equilibrium density of the field.

The Spaticle field is not the nineteenth-century luminiferous ether. In the classical ether picture, light is a wave propagating through a stationary mechanical medium while matter is a separate substance moving through that medium. That separation permits, in principle, an observable ether drift and therefore a preferred rest frame. The Spaticle-field description used here has no such separation: light and matter are organised excitations of the same physical substrate. The quantity c is the maximum propagation and reorganisation rate of the substrate itself. Clocks, rulers, matter and electromagnetic propagation are therefore constituted by the same substrate, so the model does not introduce an independently detectable mechanical rest frame of the luminiferous-ether type.

This paper is self-contained. Section 2 states the carrier equation. Section 3 gives its exact solution for point and spherical sources. Section 4 derives the force and the Newtonian limit. Section 5 derives the equilibrium density from the condensation functional together with the proton mass and the proton charge radius. Section 6 derives the carrier length and the acceleration scale from specific-work balance. Section 7 sets out the origin of each numerical factor. Section 8 evaluates the results and compares aₛ with SPARC g†. Section 9 gives the uncertainty on r_p, G and B. Appendix A records the numerical check of the Yukawa residual. The analytic results of Sections 3 and 4 hold for any value of the density.

2. The Carrier Equation

The covariant carrier equation of the Spaticle field is

gμν∇μ∇ν(δΨ) − μₛ²δΨ = κₛ gμν∇μ∇νΨmatter, μₛ² = 3Gρₛ/c²

where δΨ is the dimensionless carrier perturbation of the Spaticle field, κₛ = 1/c², Ψg = c²δΨ is the carrier potential, Ψmatter is the matter-sourced potential, ρₛ is the equilibrium density of the medium and μₛ is the inverse carrier length. The mass term reads μₛ² = Gρₛ/(c²/3): the gravitational response rate Gρₛ of the medium divided by c²/3, the squared propagation speed along one axis of an isotropic medium whose disturbances travel at c. In the weak-field static limit the time derivatives vanish, the matter-sourced potential obeys the Poisson relation ∇²Ψmatter = 4πGρ, and the equation for Ψg becomes the screened Poisson equation

∇²Ψg − μₛ²Ψg = 4πGρ(r) (1)

For a point mass M, ρ(r) = M δ³(r). The physical solution decays at infinity and is regular everywhere except at the source.

3. The Exact Static Solution

3.1 Point source

The solution of equation (1) for a point mass is

Ψg(r) = −GM e^(−μₛr) / r (2)

Proof. For a spherically symmetric function, ∇²f = r⁻² d/dr(r² df/dr). For f = e^(−μₛr)/r and r > 0 this gives ∇²f = μₛ² e^(−μₛr)/r, so (∇² − μₛ²)f = 0 away from the origin. Near the origin e^(−μₛr) → 1 and f → 1/r, whose Laplacian is −4πδ³(r). Hence (∇² − μₛ²)f = −4πδ³(r) everywhere, and multiplying by −GM gives equation (1) with ρ = Mδ³(r).

Uniqueness. The difference of two solutions obeys ∇²u = μₛ²u, is regular everywhere and vanishes at infinity. By the maximum principle for this equation [3], a positive maximum or a negative minimum of u cannot occur at any finite point, so u = 0. The solution (2) is therefore the unique decaying solution.

3.2 Extended sources

Equation (1) is linear, so the field of any mass distribution is the superposition

Ψg(r) = −G ∫ ρ(r′) e^(−μₛ|r − r′|) / |r − r′| d³r′ (3)

For a uniform sphere of mass M and radius R, the exterior field is exactly

Ψg(r) = −G M S(μₛR) e^(−μₛr) / r, S(x) = 3(x cosh x − sinh x)/x³ = 1 + x²/10 + O(x⁴) (4)

The structure factor S depends only on μₛR. For the Sun, μₛR☉ = 2.8 × 10⁻¹⁸ and S − 1 = 7.9 × 10⁻³⁷, so an extended body acts as a point mass to any measurable precision.

4. Force and Newtonian Recovery

The force on a test mass m follows from F = −m dΨg/dr:

F(r) = −GMm (1 + μₛr) e^(−μₛr) / r² (5)

Expanding the potential gives e^(−μₛr)/r = 1/r − μₛ + μₛ²r/2 − …. The constant term −μₛ produces no force. Expanding equation (5) gives the exact series

F(r)/F_N(r) − 1 = −(μₛr)²/2 + (μₛr)³/3 − (μₛr)⁴/8 + …, F_N = −GMm/r² (6)

Newtonian gravity is therefore recovered for μₛr ≪ 1, and the leading departure is of second order in μₛr.

The departure acts on a bound orbit as a small constant outward acceleration, A = GMμₛ²/2, to leading order. For an orbit of semi-major axis a and eccentricity e, a constant radial acceleration A produces an apsidal precession of magnitude 2πAa²√(1 − e²)/(GM) per orbit, which here is

|Δω| = π μₛ² a² √(1 − e²) per orbit (7)

At 30 kpc, μₛr is of order 4 × 10⁻⁶ and F/F_N - 1 is of order 7 × 10⁻¹² (Section 8). Equation (5) is Newtonian on those scales. The acceleration aₛ is the equilibrium scale of the medium, obtained in Section 6.

5. The Equilibrium Density of the Spaticle Field

5.1 The condensation functional

The first stable condensation of the Spaticle field is modelled as a three-core structure. In model units its four-term free-energy functional is E(R) = A/R² + BR² + CR + D/R. The terms represent the kinetic cost of localisation, bulk displacement of the medium, cohesion of the boundary, and rotational circulation. The coefficients are fixed by the model construction without reference to ħ, α, the electron mass, or galactic data.

Coefficient A. The kinetic contribution has prefactor ħ²/(2m_eff). Define the model mass m_eff = ħ/(cℓ) and the model energy unit E_unit = m_eff c². In dimensionless model units, A = [ħ²/(2m_eff)]/(E_unit ℓ²). Substituting E_unit and m_eff gives A = [ħ²/(2m_eff)]/[m_eff c²ℓ²] = [ħ²/(2m_eff)]/[ħ²/m_eff] = 1/2. Every factor of ħ, m_eff, c and ℓ cancels, so A = 1/2 exactly. No numerical value of ħ is used in fixing A.

Coefficient D. In the same model-unit normalization, the circulation energy-length coefficient is ħc. Therefore D = ħc/(E_unit ℓ) = ħc/(m_eff c²ℓ). With m_eff = ħ/(cℓ), the denominator is ħc and D = 1 exactly. Thus D = 2A = 1. As with A, no numerical value of ħ is used.

Coefficient C. The first stable condensation contains three identical cores sharing one expelled centre. The boundary contribution is negative because formation of the common three-core structure releases cohesion energy. The three identical sectors share the expelled centre equally. Threefold symmetry therefore fixes C = -1/3.

Coefficient B. B is the filling-deficit ratio of three touching unit spheres. The centres of the three unit spheres lie at distance d = 2/√3 from their common centre. The central void enclosed between them has area A_void = √3 - π/2. The area inside the outer boundary that is not permanent condensate is π(d + 1)² - 3π + A_void. The corresponding condensate area, including the asymmetric filling share A_void/6 of the fourth unit, is 3π + A_void/6. Therefore

B = [π(d + 1)² - 3π + A_void]/[3π + A_void/6] = 0.56308 (9)

The complete dimensionless functional is therefore

E(R) = 1/(2R²) + 0.56308R² - R/3 + 1/R (8)

5.2 The condensation radius

Stationarity gives dE/dR = -2A/R³ + 2BR + C - D/R² = 0. Multiplying by R³ gives

2BR⁴ + CR³ - DR - 2A = 0 (10)

With A = 1/2, B = 0.56308, C = -1/3 and D = 1, the unique positive stationary root is R₀ = 1.27348221. The second derivative is positive at this root. No measured quantity enters the determination of R₀, and R₀ is dimensionless.

5.3 Grounding and the quantum of action

The proton mass and proton charge radius ground this dimensionless geometry in SI. With r_p = 0.8414 × 10⁻¹⁵ m, one model length is ℓ_model = r_p/R₀ = 6.60708091 × 10⁻¹⁶ m. The model energy unit is E_unit = m_p c²/π = 298.661 MeV, using m_p = 1.67262192369 × 10⁻²⁷ kg. These dimensional identifications do not change R₀.

Only after the dimensionless geometry has been fixed and grounded is the derived quantum of action introduced:

ħ_vss = m_p c ℓ_model/π = m_p c r_p/(πR₀) = 1.05457687 × 10⁻³⁴ J s (11)

The measured reduced Planck constant is not an input to this calculation.

Using this derived quantum of action, the electromagnetic coupling is

α_vss = e²/(4π ε_0 ħ_vss c) = 0.00729731763044 (12)

Hence α_vss⁻¹ = 137.036655199. The measured α is not used to obtain this value.

5.4 The electron mass

The electron is the unit expelled from the four-unit condensation. Its energy is the model energy unit divided by 6π⁴. Here 6 = 3 × 2 counts the three cores and the double cover of the rotation group, while π⁴ is the square of the π² orientation-volume factor of the rigid three-core. Thus m_e_vss c² = E_unit/(6π⁴), and

m_e_vss = E_unit/(6π⁴c²) = m_p/(6π⁵) = 9.10955518 × 10⁻³¹ kg (13)

This mass formula does not use r_p or the measured ħ.

5.5 The density

The characteristic matching mass is m*_vss = m_e_vss/α_vss = 1.24834297 × 10⁻²⁸ kg. Its reduced Compton wavelength is λ* = ħ_vss/(m*_vss c). Since m*_vss = m_e_vss/α_vss, this becomes λ* = α_vss ħ_vss/(m_e_vss c) = r_e_vss = 2.81788728 × 10⁻¹⁵ m. Thus the matching length equals the electron-radius scale generated by the derived particle quantities.

At radius r_e_vss, the gravitational field of the matching mass is g* = Gm*_vss/r_e_vss² = 1.0492825 × 10⁻⁹ m s⁻². The weak-field gravitational energy-density magnitude is u_g = g*²/(8πG), hence u_g = Gm*_vss²/(8πr_e_vss⁴) = G[m_e_vss/α_vss]²/(8πr_e_vss⁴) = 6.56355667 × 10⁻¹⁰ J m⁻³.

Dividing this particle-sector gravitational energy density by c² gives the equilibrium Spaticle-field density ρₛ = u_g/c² = G[m_e_vss/α_vss]²/(8πr_e_vss⁴c²). Using r_e_vss = α_vss ħ_vss/(m_e_vss c), the same expression reduces to

ρₛ = (Gc²/(8πħ_vss⁴))[m_e_vss/α_vss]⁶ = G ħ_vss²/(8πc⁴r_e_vss⁶) = 7.30294 × 10⁻²⁷ kg m⁻³ (14)

The complete particle-sector dependency chain is therefore {A, B, C, D} → R₀ → ℓ_model and E_unit → ħ_vss → α_vss and m_e_vss → m*_vss and r_e_vss → g* and u_g → ρₛ. No galactic or cosmological quantity enters this chain.

6. The Carrier Length and the Acceleration Scale

The carrier equation fixes the intrinsic length of the medium:

Lₛ = 1/μₛ = c / √(3Gρₛ) (15)

The Spaticle field is an isotropic medium whose internal motions proceed at c. Its pressure is one third of ρ⟨v²⟩ with ⟨v²⟩ = c², the equation of state that also holds for radiation [4]:

pₛ = ρₛc² / 3 (16)

For the equilibrium Spaticle medium, the carrier acceleration acting across the carrier length Lₛ = 1/μₛ gives the specific work aₛ Lₛ. The pressure energy per unit mass is pₛ/ρₛ. Equating these two specific energies gives the work balance

aₛLₛ = pₛ/ρₛ = c²/3 (17)

Substituting equation (15) into equation (17) gives

aₛ = c²/(3Lₛ) = c √(Gρₛ/3) = √(G pₛ) = G ħ_vss / (√(24π) c rₑ³) (18)

From the specific-work balance aₛ Lₛ = pₛ/ρₛ, the identities aₛ² = G pₛ, aₛ = G ρₛ Lₛ and aₛ Lₛ = c²/3 follow exactly. The factor 4π of the Gauss form of gravity arises from integrating flux over a closed surface around a source. An isotropic equilibrium medium has no centre, no enclosed source and no bounding surface, so no solid-angle factor enters its intrinsic scales.

7. Origin of Each Factor

Every numerical factor in the chain from equation (8) to equation (18) has a stated origin. None is taken from a galaxy. The table lists each one. A, B, C and D are calculated in Section 5: A from the model-unit kinetic identity, D = 2A, C from threefold symmetry, and B from the three-sphere filling ratio.

8. Numerical Evaluation

With G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and c = 299 792 458 m s⁻¹ [2], and ρₛ from equation (14):

Equation (6) gives the fractional departure of the force from Newtonian gravity at representative distances:

For Mercury, with a = 5.7909 × 10¹⁰ m and e = 0.2056, equation (7) gives |Δω| = 1.68 × 10⁻³¹ rad per orbit, or 1.4 × 10⁻²³ arcseconds per century. The measured perihelion precession of Mercury agrees with general relativity to a precision of order 10⁻³ of its 43 arcseconds per century, which lies more than twenty orders of magnitude above this departure. The departures at 1 AU and 40 AU lie correspondingly far below the precision of planetary ephemerides and lunar laser ranging, and the departure at 30 kpc lies far below the precision of any galactic measurement. The departure reaches the percent level only at gigaparsec distances.

The predicted acceleration of the medium is aₛ = 1.2084 × 10⁻¹⁰ m s⁻². It is fixed by equations (8) to (18) from the proton mass, the proton charge radius, e, ε_0, G and c. McGaugh, Lelli and Schombert [5], using 2693 points in 153 SPARC disk galaxies, report a fitted characteristic acceleration g† = 1.20 ± 0.02 (random) ± 0.24 (systematic) × 10⁻¹⁰ m s⁻². Milgrom [6] introduced an empirical acceleration constant a₀ of the same order in the MOND relation. In contrast, the value aₛ obtained here is calculated without using galactic data. The predicted aₛ differs from the fitted g† by 0.7 percent, corresponding to 0.42 of the quoted random uncertainty and 0.035 of the quoted systematic uncertainty.

9. Uncertainty Budget

The numerical values in Section 8 are central values. This section states how they move when the measured inputs and the geometric coefficient B are varied. No galactic quantity is varied, because none enters the chain.

Proton charge radius. Equation (11) gives ħ_vss ∝ r_p. With m_e_vss fixed by equation (13) and α_vss ∝ 1/ħ_vss, the matching length r_e = α_vss ħ_vss/(m_e_vss c) is independent of r_p to first order. Then ρₛ = G ħ_vss²/(8π c⁴ r_e⁶) scales as r_p² and aₛ ∝ √ρₛ scales as r_p. The 2018 CODATA proton charge radius used here is r_p = 0.8414(19) fm [1], a relative uncertainty of 0.23 percent. Propagating that uncertainty gives δaₛ/aₛ ≈ 0.23 percent, or δaₛ ≈ 0.003 × 10⁻¹⁰ m s⁻². That shift is smaller than the SPARC random error of 0.02 × 10⁻¹⁰ m s⁻² and far smaller than the SPARC systematic error of 0.24 × 10⁻¹⁰ m s⁻².

Newtonian constant. G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² enters ρₛ linearly, because ρₛ ∝ G in equation (14), and enters aₛ = c√(Gρₛ/3) once more through the square root. With the particle quantities held fixed, aₛ ∝ √(G·G) = G. The CODATA relative uncertainty on G is 2.2 × 10⁻⁵ [2], so δaₛ/aₛ = δG/G ≈ 2.2 × 10⁻⁵ and δaₛ ≈ 2.66 × 10⁻¹⁵ m s⁻². This affects only the last digits of the six-significant-digit value 1.20840 × 10⁻¹⁰ m s⁻². It is independent of, and much smaller than, the proton-radius uncertainty above, and the two combine in quadrature.

Geometric coefficient B. B = 0.56308 (the geometric value 0.563082 rounded to five decimals) is computed from the filling-deficit ratio of three touching unit spheres, equation (9). If B is shifted by 0.001, the root R₀ of equation (10) moves by about -0.00079, a relative shift of 0.06 percent. Then ħ_vss and aₛ, which scale as 1/R₀, move by the same relative amount.

No coefficient in equations (8) to (18) is taken from SPARC or from any rotation curve. A, B, C and D are derived in Section 5.1. The values of ħ_vss, α_vss and m_e_vss follow from that derivation and the proton inputs. Those three quantities are compared with measured constants. The quantity aₛ is compared with a fitted galactic figure [5], which is a different class of target.

10. Conclusions

The static carrier equation has the exact and unique decaying solution Ψg = −GM e^(−μₛr)/r. Its force reproduces Newtonian gravity with a departure of second order in μₛr. An extended spherical body acts as a point mass through a structure factor that differs from one by 10⁻³⁷ for the Sun. The departure is 10⁻³¹ at 1 AU and 10⁻¹¹ across a galaxy. The condensation functional of Section 5, with coefficients derived there, together with the proton mass and charge radius, gives ρₛ = 7.30294 × 10⁻²⁷ kg m⁻³ and Lₛ = 26.205 Gly. Equating the specific work aₛ Lₛ across the carrier length to the specific pressure energy pₛ/ρₛ gives aₛ = √(G pₛ) = 1.2084 × 10⁻¹⁰ m s⁻², with no galactic input. A published fit to 153 SPARC disks quotes g† = 1.20 × 10⁻¹⁰ m s⁻² [5]. The predicted aₛ lies 0.7 percent from that fit. The Yukawa departure reaches 0.71 percent at 1 Gpc.

Declarations

Funding

No funding was received for this research.

Competing interests

The author declares no competing financial or non-financial interests.

Author contributions

Vijay Shankar Sharma: conceptualization, methodology, formal analysis, investigation, validation, visualization, and writing of the manuscript.

Data availability

All numerical values required to reproduce the results reported in this study are given in the article. No external research dataset was generated or analysed beyond the published physical constants cited in the manuscript.

Appendix A. Numerical check of the static solution

The residual of equation (1) for the solution (2) was evaluated in 60-digit arithmetic at solar-mass and galactic-mass sources. The relative residual is of order 10⁻⁵⁰, which is zero to the working precision. The chain of Section 5, the identities of Section 6 and every value in the tables of Section 8 were recomputed independently and agree to the precision of the printed digits.

References

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[3] Gilbarg, D., and Trudinger, N. S. 2001. Elliptic Partial Differential Equations of Second Order. Springer, Berlin.

[4] Weinberg, S. 1972. Gravitation and Cosmology. Wiley, New York.

[5] McGaugh, S. S., Lelli, F., and Schombert, J. M. 2016. Radial acceleration relation in rotationally supported galaxies. Physical Review Letters, 117, 201101.

[6] Milgrom, M. 1983. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. The Astrophysical Journal, 270, 365.