BeeTheory · Foundations · Technical Note XXII

Cavendish Revisited:
The Wave Mass of a Single Sphere

Returning to the simplest case — an isolated spherical mass — this note rebuilds the BeeTheory wave-mass calculation with a properly normalised kernel and a clear dimensional accounting. The Cavendish lead spheres and the Earth itself are used as concrete tests. The conclusion is a sharp scale separation: a galactic coherence length $\ell_0 \sim 1$ kpc makes the wave mass invisible at laboratory and planetary scales, while remaining the operative quantity at galactic scale.

2. The corrected kernel

In the previous galactic notes (XII–XXI), the wave kernel was written $\mathcal{K}(D) = K_0\,(1+\alpha D)e^{-\alpha D}/D^2$ with an unnormalised constant $K_0$. A clean dimensional accounting requires a normalised form. The kernel that produces a finite, dimensionally correct asymptotic wave mass is:

This form has dimension $[1/L^3]$ (since $\ell_0$ is a length and the kernel integrand involves $dV$). The convolution definition of the wave density becomes:

Dimensional check: $[\rho_\text{wave}] = [\text{kg/m}^3] = [\rho_\text{vis}] \cdot [\mathcal{K}] \cdot [dV] = [\text{kg/m}^3] \cdot [1/\text{m}^3] \cdot [\text{m}^3] = [\text{kg/m}^3]$ ✓

The previous $K_0 \approx 0.3759$ is now absorbed into the normalisation factor $1/(4\pi \ell_0^2)$. The free parameters reduce to two only:

Parameter	Dimension	Role
$\lambda$	Dimensionless	Fraction of wave mass to visible mass at $R \to \infty$
$\ell_0$	Length	Spatial extent over which the wave field deploys around a source

4. Visualisation: where the wave mass sits

6. Why this is consistent with Newton

The classical Newtonian law $F = G m_1 m_2 / r^2$, validated by Cavendish and by all planetary observations, requires the gravitational mass of each body to be a well-defined number. BeeTheory does not contradict this in any way:

(a) On the small scale $(R \ll \ell_0)$: the wave mass contribution to local gravity is at the $10^{-13}$ level for Earth, $10^{-21}$ for Cavendish. No experiment can detect such a deviation. The Newtonian relation $F = GM/r^2$ holds with $M$ being the visible mass alone.

(b) Spherical symmetry preserves the orbit. The wave mass generated by the Earth is spherically symmetric (because the Earth is). By the shell theorem, an external observer at any distance $r > R_oplus$ sees the Earth’s total mass (visible + the small amount of wave mass enclosed by $r$) acting as a point at the centre. The Moon’s orbit, the planets’, and any satellite trajectory are unaffected by the existence of the spreading wave field — only its enclosed-mass contribution matters, and it is negligible at planetary distances.

(c) The wave mass only matters where it has had room to deploy. The wave field requires distances comparable to $\ell_0 \sim 1$ kpc to fully form. Inside galaxies, where many massive objects ($10^{11}$ stars, gas, etc.) coexist within $\sim \ell_0$ of one another, the wave fields overlap and their cumulative enclosed mass becomes significant. This is where the rotation curves are affected — the topic of Notes XX and XXI.