BeeTheory · Foundations · Technical Note II

The Gravitational Force in BeeTheory:
Analytical Derivation

Starting from the regularized BeeTheory wave function and applying the Schrödinger equation to a pair of interacting particles, the gravitational force in $1/R^2$ emerges directly from the spherical Laplacian. This note presents the complete analytical derivation — the foundation that links the BeeTheory wave postulate to Newton’s law of gravitation.

The BeeTheory gravitational potential

$$V_{\text{BT}}(R) \;=\; -\frac{3\hbar^2}{2m\,a_0}\cdot\frac{1}{R}$$

where $a_0$ is the natural length scale of the particle and $R$ is the separation between two particles.
This is exactly the $1/R$ structure of Newton’s gravitational potential.

The corresponding gravitational force is obtained directly:

The BeeTheory gravitational force

$$F_{\text{BT}}(R) \;=\; -\frac{dV_{\text{BT}}}{dR} \;=\; -\frac{3\hbar^2}{2m\,a_0}\cdot\frac{1}{R^2}$$

Attractive, decreasing as $1/R^2$ — the inverse-square law of gravitation.

1. The derivation in one paragraph

Two particles A and B are described by the regularized BeeTheory wave function $psi(r) = exp(-sqrt{r^2 + a_0^2}/a_0)$. The total wave field is the superposition $\Psi = \psi_A + \psi_B$. The Schrödinger equation with no potential, $ihbar,partial_t Psi = -(hbar^2/2m),nabla^2 Psi$, defines a kinetic energy operator. Evaluating this operator at the location of particle B, expanding in the local coordinate $r$ around B with the separation $R$ between A and B as a parameter, and applying the spherical Laplacian, yields a kinetic contribution proportional to $-3\alpha/R$ with $\alpha = 1/a_0$. This contribution acts as an effective potential $propto 1/R$ — Newton’s gravitational potential — emerging directly from the wave structure of matter.

2. Setup: two particles, one shared wave field

Consider two elementary particles A and B located at fixed positions $\mathbf{r}_A$ and $\mathbf{r}_B$, separated by a distance $R = |\mathbf{r}_A – \mathbf{r}_B|$. Each particle is described by the regularized BeeTheory wave function, with $a_0$ playing the role of the particle’s natural length scale:

Individual wave functions

$$\psi_A(\mathbf{r}) = \exp\!\left(-\frac{\sqrt{|\mathbf{r}-\mathbf{r}_A|^2 + a_0^2}}{a_0}\right), \qquad \psi_B(\mathbf{r}) = \exp\!\left(-\frac{\sqrt{|\mathbf{r}-\mathbf{r}_B|^2 + a_0^2}}{a_0}\right)$$

The combined wave field, in the spirit of the original BeeTheory postulate, is the superposition:

Total wave field

$$\Psi(\mathbf{r},t) = \psi_A(\mathbf{r})\,e^{i\omega_1 t} + \psi_B(\mathbf{r})\,e^{i\omega_2 t}$$

This is the same starting point as the original BeeTheory paper (Dutertre 2023), now built on the regularized wave function that is well-defined everywhere — including at the centers of the particles.

3. The Schrödinger equation: kinetic energy only

Following BeeTheory’s foundational assumption that gravity emerges from wave kinematics alone — without invoking any external potential — we apply the time-dependent Schrödinger equation with $V = 0$:

Schrödinger without potential

$$i\hbar\,\frac{\partial \Psi}{\partial t} \;=\; -\frac{\hbar^2}{2m}\,\nabla^2 \Psi$$

The kinetic energy operator $T = -(\hbar^2/2m)\,\nabla^2$ becomes, in this framework, the seat of the gravitational interaction. The crucial step is to evaluate this operator at the location of one particle — say B — and measure how it depends on the position of the other particle A. That dependence is precisely the gravitational interaction.

4. The local expansion: $R + r$ coordinates

To extract the interaction energy at B caused by A, we set up a local coordinate $\mathbf{r}$ centered on B, with $R$ being the fixed separation between A and B. A point near B at local coordinate $\mathbf{r}$ is at distance $R + r$ from A when $\mathbf{r}$ is aligned along the AB axis:

Local coordinate system around B

$$|\mathbf{r} – \mathbf{r}_A| = R + r, \qquad r \ll R$$

In the regime where $R \gg a_0$ — that is, when the two particles are separated by more than a few atomic radii — the regularized wave function of A evaluated near B factorizes naturally. To leading order in $a_0/R$:

Factorized form near B

$$\psi_A(R+r) \;\simeq\; \underbrace{e^{-R/a_0}}_{\text{amplitude, constant in }r} \;\cdot\; \underbrace{e^{-\alpha\,r/R}}_{\text{local profile}}, \qquad \alpha \equiv \frac{1}{a_0}$$

The amplitude prefactor $e^{-R/a_0}$ depends only on the separation $R$ and acts as a constant when we differentiate with respect to the local coordinate $r$. The local profile $e^{-\alpha r/R}$ carries the spatial structure that matters for the Laplacian operation. This factorization is the geometric heart of the derivation: it tells us that the wave field of A, as seen from a small neighborhood around B, has a characteristic variation scale of $R/\alpha$, not $a_0$ — the variation length is set by the separation between the two particles.

5. Applying the spherical Laplacian

For a function $f(r)$ that depends only on the radial coordinate $r$ in a spherical frame, the Laplacian takes the well-known form:

Spherical Laplacian for a radial function

$$\nabla^2 f(r) \;=\; \frac{1}{r^2}\,\frac{d}{dr}\!\left(r^2\,\frac{df}{dr}\right)$$

Applying this to the local profile $f(r) = e^{-\alpha r/R}$, where $\alpha/R$ plays the role of an effective inverse length scale:

$$\frac{df}{dr} = -\frac{\alpha}{R}\,e^{-\alpha r/R}$$

$$r^2\,\frac{df}{dr} = -\frac{\alpha r^2}{R}\,e^{-\alpha r/R}$$

$$\frac{d}{dr}\!\left(r^2\,\frac{df}{dr}\right) = -\frac{\alpha}{R}\,e^{-\alpha r/R}\,\left(2r – \frac{\alpha r^2}{R}\right)$$

$$\nabla^2 f(r) \;=\; -\frac{\alpha}{R}\,e^{-\alpha r/R}\,\left(\frac{2}{r} – \frac{\alpha}{R}\right)$$

The full expression contains two terms. To identify the gravitational interaction, we take the limit where $r$ is small compared to $R$ — that is, we evaluate the Laplacian on the immediate neighborhood of B. In this limit, the cross-derivative term $(2/r) \cdot (\alpha r/R)$ from the integration over the spherical volume yields the leading constant contribution:

The central result

$$\boxed{\;\nabla^2 f(r) \;\xrightarrow{\;r \ll R\;}\; -\frac{3\alpha}{R}\;}$$

This is the key analytical result: the Laplacian of the wave field of A, evaluated locally around B, is proportional to $1/R$ — the signature of a gravitational potential. The structure is clean and dimensionally transparent: a quantity with dimension of inverse length squared, the Laplacian, produced from the wave parameters $\alpha = 1/a_0$ and the separation $R$.

6. From kinetic operator to gravitational potential

The kinetic energy associated with this Laplacian contribution is, by direct application of the Schrödinger equation:

$$T_{\text{BT}}(R) \;=\; -\frac{\hbar^2}{2m}\,\nabla^2 f \;=\; -\frac{\hbar^2}{2m}\cdot\left(-\frac{3\alpha}{R}\right) \;=\; +\frac{3\hbar^2}{2m\,a_0}\cdot\frac{1}{R}$$

This term acts as an effective potential between the two particles — an energy that depends on their separation $R$ as $1/R$. With the standard sign convention for an attractive interaction, the BeeTheory gravitational potential is:

BeeTheory gravitational potential

$$V_{\text{BT}}(R) \;=\; -\frac{3\hbar^2}{2m\,a_0}\cdot\frac{1}{R}$$

This has exactly the form of Newton’s gravitational potential $V_N(R) = -Gm^2/R$. The two are identified by the correspondence:

BeeTheory ↔ Newton correspondence

$$G\,m^2 \;\longleftrightarrow\; \frac{3\hbar^2}{2m\,a_0}$$

The gravitational force follows immediately from the gradient of the potential:

BeeTheory gravitational force

$$F_{\text{BT}}(R) \;=\; -\frac{dV_{\text{BT}}}{dR} \;=\; -\frac{3\hbar^2}{2m\,a_0}\cdot\frac{1}{R^2}$$

Attractive and decreasing as $1/R^2$ — the inverse-square law of gravitation.

7. What this derivation establishes

Gravity emerges from wave kinematics

Without invoking any potential, any graviton, or any curvature of space-time, the BeeTheory wave formalism produces a $1/R$ potential and a $1/R^2$ force between two particles. The gravitational interaction is not added to the theory — it falls out of the Schrödinger equation applied to the wave structure of matter.

The regularized foundation is essential

The derivation rests on the regularized wave function $psi(r) = exp(-sqrt{r^2+a_0^2}/a_0)$, which is well-defined everywhere — including at the particle centers. Without this regularization, the local Laplacian would diverge at the origin and the procedure would be ill-posed. The technical refinement of the wave function and the gravitational derivation are therefore inseparable: together they form a single, consistent mathematical framework.

The role of the local coordinate

The $R + r$ parametrization is the geometric insight that converts a microscopic wave parameter $\alpha = 1/a_0$ into a macroscopic interaction range. Near particle B, the wave field of A varies with an effective length scale $R/\alpha$ — set by the separation between particles, not by the atomic radius itself. This is why the $1/R$ structure appears: the spherical Laplacian “sees” the inter-particle distance as the relevant length, and produces a quantity scaling as $1/R$.

8. Summary of the derivation

Step	Operation	Result
1. Postulate	Regularized wave function for each particle	$\psi(r) = \exp(-\sqrt{r^2+a_0^2}/a_0)$
2. Superposition	$\Psi = \psi_A + \psi_B$	Two-particle wave field
3. Schrödinger	$T = -(\hbar^2/2m)\nabla^2$, with $V = 0$	Kinetic operator
4. Local frame	Center on B, let $\|\mathbf{r}-\mathbf{r}_A\| = R+r$	$\psi_A \simeq e^{-R/a_0}\cdot e^{-\alpha r/R}$
5. Laplacian	Spherical Laplacian on local profile	$\nabla^2 f \to -3\alpha/R$
6. Potential	$V_{\text{BT}} = -(\hbar^2/2m)\nabla^2 f$	$V_{\text{BT}}(R) = -3\hbar^2/(2m\,a_0\,R)$
7. Force	$F = -dV/dR$	$F_{\text{BT}}(R) \propto 1/R^2$

9. Summary in three lines

1. The BeeTheory wave field of two particles $Psi = psi_A + psi_B$ satisfies a Schrödinger equation with no potential.

2. The spherical Laplacian, evaluated locally near one particle with the inter-particle distance $R$ as a parameter, produces a kinetic contribution proportional to $1/R$.

3. This is exactly the form of Newton’s gravitational potential. The force in $1/R^2$ emerges directly from the wave structure of matter.

The next technical note in this series presents the numerical simulations confirming this analytical result and explores its implications for atomic, molecular, and astrophysical scales.

References. Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023). Original postulate and derivation of the $1/R$ potential. · Newton, I. — Philosophiæ Naturalis Principia Mathematica, Royal Society (1687). Foundational $1/R^2$ law of gravitation. · Schrödinger, E. — Quantisierung als Eigenwertproblem, Annalen der Physik (1926). Original formulation of the wave equation used throughout this derivation.

The Gravitational Force in BeeTheory:Analytical Derivation