BeeTheory · Foundations · Technical Note XXX

From Points to Densities:
Extending BeeTheory to Galaxies

For the Sun–Earth system, two point-masses suffice: the Sun carries its regularized wave function, the Earth feels the local Laplacian at its position, and Newton emerges. For a galaxy, the visible mass is no longer localized — it is a continuous density $\rho_\text{vis}(\mathbf{r}’)$ spread across the disk. Each volume element carries its own wave field, and the visible mass at a distant point responds to the gradient of the collective wave field. The mathematical extension is direct; the physical consequences are profound.

1. The result first

The transition in one equation

Two points (solar system):

$$U(r) \;=\; -K \cdot \frac{2}{a\,r} \;=\; -\frac{G\,M_\odot\,M_\oplus}{r}$$

Extended density (galaxy):

$$\Phi(\mathbf{r}) \;=\; -G\int\frac{\rho_\text{vis}(\mathbf{r}’)}{|\mathbf{r}-\mathbf{r}’|}\,d^3r’$$

The galactic potential is the integral of the $T_2$ term from every volume element of visible matter, each carrying its own regularized wave function. The Newtonian $1/|\mathbf{r}-\mathbf{r}’|$ kernel emerges naturally from this sum.

2. The point-mass case revisited

In Note XXIX we established that, for the Sun and the Earth treated as point-masses, the gravitational attraction emerges from the Laplacian of the Sun’s regularized wave function $psi^odot(r)$ evaluated at the Earth’s position. The dominant term of this Laplacian — call it $T_2$ — has the form $-2/(ar)$, which is precisely the spatial structure of the Newtonian $1/r$ potential.

With the coupling $K = G M_\odot M_\oplus \cdot a/2$ (where $a$ is the Bohr radius, fixed by atomic physics), the interaction energy reproduces Newton’s law exactly:

$$U_\text{Sun-Earth}(r) \;=\; -K \cdot \frac{2}{a\,r} \;=\; -\frac{G\,M_\odot\,M_\oplus}{r}\,, \qquad F(r) = \frac{G\,M_\odot\,M_\oplus}{r^2}$$

The key features of this point-mass formulation are:

  • The visible mass of the Sun ($M_\odot$) is treated as a single point.
  • The Sun’s visible mass generates the regularized wave function $\psi^\odot$ throughout space.
  • The visible mass of the Earth ($M_\oplus$) is also a single point.
  • The Earth’s visible mass responds to the Laplacian of $psi^odot$ at its location — specifically to the $T_2$ term, which has the Newtonian structure.

This works perfectly when the visible masses are well-localized and far from each other compared to their own physical extent — as is the case in the solar system.

3. The transition: from points to densities

From point-point to density-density: the same mechanism, two scales Solar system — two points M_⊙ (visible)ψ_⊙(r) regularizedM_⊕point massF = -∇U(r) via T₂r ≈ 1 AUU(r) = -K · 2/(a·r) = -GM_⊙M_⊕/r Galaxy — extended density star at rdm′ = ρ_vis(r′)dV′|r-r′|visible density ρ_vis(r′) — wave field ψ extends beyondΦ(r) = -G ∫ ρ_vis(r′)/|r-r′| dV′ extendto ρ_vis
Left: The solar system. The Sun is a single point carrying its wave field $\psi^\odot$. The Earth, also a point, feels the local gradient of this field at its position. Right: The galaxy. Visible mass forms a continuous density $\rho_\text{vis}(\mathbf{r}’)$. Every volume element $dm’ = \rho_\text{vis}(\mathbf{r}’)\,dV’$ carries its own wave function. The collective wave field $\psi_\text{galaxy}$ extends beyond the visible matter (red halo), and its gradient acts on any other visible mass located at $\mathbf{r}$.

For a galaxy, the visible matter cannot be reduced to a point. It is distributed as a continuous density: $\rho_\text{vis}(\mathbf{r}’)$, where $\mathbf{r}’$ ranges over the disk, the bulge, the gas layer, and so on. The transition from points to densities follows two natural principles:

  • Each volume element $dm’ = \rho_\text{vis}(\mathbf{r}’)\,dV’$ behaves like an elementary point-mass. It carries its own regularized wave function, centered on $\mathbf{r}’$.
  • The total wave field at any point $\mathbf{r}$ is the superposition of contributions from every volume element of the source. This collective wave mass has its own characteristic spatial decay — slower than that of the visible density itself, because waves from many sources overlap.

4. The Newtonian limit emerges naturally

For each pair of volume elements separated by a distance $|\mathbf{r}-\mathbf{r}’|$, the Sun–Earth derivation of Note XXIX applies: the $T_2$ term of the Laplacian of the wave function centered on $\mathbf{r}’$, evaluated at $\mathbf{r}$, has the form:

$$T_2(\mathbf{r},\mathbf{r}’) \;=\; -\frac{2}{a\,|\mathbf{r}-\mathbf{r}’|}$$

Summing over all source elements with coupling coefficient $K(\mathbf{r}’) = G\,\rho_\text{vis}(\mathbf{r}’)\,dV’ \cdot a/2$ per element, the gravitational potential at $\mathbf{r}$ becomes:

$$\boxed{\Phi(\mathbf{r}) \;=\; \int\rho_\text{vis}(\mathbf{r}’)\cdot T_2(\mathbf{r},\mathbf{r}’)\cdot\frac{a}{2}\cdot G\,d^3r’ \;=\; -G\int\frac{\rho_\text{vis}(\mathbf{r}’)}{|\mathbf{r}-\mathbf{r}’|}\,d^3r’}$$

This is exactly Newton’s potential for an extended mass distribution. The factor of $a$ from each wave function cancels against the factor $1/a$ in $T_2$, leaving the standard Newtonian convolution. The Poisson equation follows:

$$\nabla^2\Phi(\mathbf{r}) \;=\; 4\pi G\,\rho_\text{vis}(\mathbf{r})$$

Standard Newtonian gravity for extended distributions is therefore recovered as the limit of point-by-point BeeTheory applied to every infinitesimal volume element of visible matter. The mathematical structure of the regularized Laplacian guarantees this.

5. The wave field extends beyond the visible

The subtle physical content of BeeTheory at galactic scale lies not in recovering Newton — it does so automatically. It lies in the recognition that the collective wave field generated by visible matter extends spatially beyond the visible density itself.

The collective wave mass decreases more slowly than visible density The wave field’s outer tail produces the gradient that pulls visible mass at large r bulk of visible matterwave field tail acts here 05101520253010⁻⁵10⁻⁴10⁻³10⁻²10⁻¹10⁰ distance from galactic center r (kpc) normalized density ρ_vis(r) — visible mass densityψ_galaxy(r) — collective wave mass (decreasing tail)
Comparison of the visible mass density $rho_text{vis}(r)$ (gold, decreases as $e^{-r/R_d}$ with $R_d = 2.6$ kpc for the Milky Way) and the collective wave mass field $psi_text{galaxy}(r)$ (red, decreases more slowly because it integrates contributions from all source elements). Beyond $\sim 3 R_d$, visible matter is nearly gone, but the wave field still has a significant tail. It is the gradient of this tail that creates an attractive force on any visible mass located there.

This is the physically distinctive prediction of BeeTheory at galactic scale: at radii where visible matter is sparse, the gravitational attraction is dominated by the gradient of the wave field’s outer tail, not by the residual visible density itself.

Standard Newtonian gravity assumes that the source of the field is the visible density — and concludes that orbital velocities should decline beyond the bulk of the visible matter. Observations show otherwise: rotation curves stay flat far past the optical disk. BeeTheory’s natural explanation is that the wave field, which extends further than the visible density, continues to produce a gradient (and therefore an attractive force) at large radii.

6. Side-by-side comparison

Solar system (point-point)Galaxy (density-density)
Visible mass sourceSingle point at $\mathbf{r}_\odot$ with mass $M_\odot$Continuous density $rho_text{vis}(mathbf{r}’)$ over the disk and bulge
Wave functionOne $\psi^\odot(r)$ centered on the SunSum of $\psi(\mathbf{r}-\mathbf{r}’)$ over every volume element $dm’$
Coupling coefficient$K = G M_\odot M_\oplus a/2$$K(\mathbf{r}’) = G\,\rho_\text{vis}(\mathbf{r}’)\,dV’\cdot a/2$ per element
Active term$T_2 = -2/(a\,r)$ at Earth$T_2(\mathbf{r},\mathbf{r}’) = -2/(a\,|\mathbf{r}-\mathbf{r}’|)$, integrated
Resulting potential$U = -GM_\odot M_\oplus/r$$\Phi(\mathbf{r}) = -G\int\rho_\text{vis}(\mathbf{r}’)/|\mathbf{r}-\mathbf{r}’|\,d^3r’$
Field equationPoint charge: $\nabla^2 U = 4\pi GM_\odot M_\oplus\,\delta(\mathbf{r})$Poisson: $\nabla^2\Phi = 4\pi G\,\rho_\text{vis}(\mathbf{r})$
Spatial extent of the wave fieldSame as the visible mass (point-like)Larger than the visible density — extends beyond the optical disk
Where the gradient actsAt the Earth’s position onlyEverywhere — including at radii where visible density is negligible
The mathematical structure is identical: in both cases, the visible mass at the field point responds to the Laplacian of the wave field generated by the visible mass at the source point. Only the spatial structure of the source changes — from a single point to a continuous distribution.

7. Why this matters for rotation curves

The standard Newtonian calculation of rotation curves uses only the visible density: the circular velocity at radius $R$ is determined by the visible mass enclosed within that radius. For an exponential disk, this gives a velocity that declines beyond $\sim 3 R_d$ — because almost no visible mass remains at larger radii.

Observed rotation curves remain flat well beyond $3R_d$. The standard interpretation invokes a dark matter halo to supply the missing gravitational pull. BeeTheory provides a different account, derived from first principles:

  • Each volume element of visible matter generates its own wave function with characteristic decay scale $a$.
  • The collective wave field at radius $R$ integrates contributions from all source elements within the galaxy. Even at $R = 10 R_d$, source elements at every $\mathbf{r}’$ inside the disk contribute their $T_2$ component.
  • The result is a wave field whose effective decay length is much longer than $R_d$ — it is determined by the geometry of the entire visible distribution, not by the local density at $R$.
  • The gradient of this extended wave field, acting on a star or gas parcel orbiting at radius $R$, produces an additional gravitational pull beyond what standard Newtonian calculation gives.

The physical statement

In BeeTheory, the “missing mass” inferred from flat rotation curves is not a separate species of matter. It is the natural consequence of the wave field extending beyond the bulk of the visible density. The gradient of this outer wave field produces an attractive force on visible matter at large radii, exactly mimicking what dark matter would do — but without invoking any new particle.

8. Summary

1. In the solar system, the visible masses (Sun, planets) are well-localized points. Each point generates its own regularized wave function; each point feels the Laplacian of the others. The $T_2$ term reproduces Newton’s force exactly.

2. In a galaxy, visible matter is a continuous density $\rho_\text{vis}(\mathbf{r}’)$. Every volume element $dm’$ carries its own wave function. The collective wave field at any point $\mathbf{r}$ is the sum of contributions from all source elements.

3. Integrating the $T_2$ kernel over the visible density automatically recovers the standard Newtonian potential $\Phi(\mathbf{r}) = -G\int\rho_\text{vis}(\mathbf{r}’)/|\mathbf{r}-\mathbf{r}’|\,d^3r’$ and the Poisson equation $\nabla^2\Phi = 4\pi G\rho_\text{vis}$.

4. The physical distinction of BeeTheory is that the collective wave field extends beyond the visible density, with a slower decay determined by the geometry of the entire visible distribution.

5. Visible matter located at large radii feels the gradient of this outer wave field — a gravitational pull that standard Newtonian calculation (which uses only the local visible density) does not predict.

6. This is the BeeTheory mechanism for flat rotation curves and for the so-called “dark matter” inferred from galactic kinematics: a wave field that extends naturally beyond the visible source it springs from.

References. Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023). · Note I — A Regularized Wave Function for BeeTheory, BeeTheory.com (2026). ·. & Tremaine, S. — Galactic Dynamics, 2nd ed., Princeton University Press (2008), §2.6 (potential of an exponential disk). · Freeman, K. C. — On the Disks of Spiral and S0 Galaxies, ApJ 160, 811 (1970).

BeeTheory.com — Wave-based quantum gravity · From point-mass to extended density · © Technoplane S.A.S. 2026