BeeTheory

BeeTheory — also written Bee Theory or Bee Theory™ — is a proposed wave-based model of gravity developed by Xavier Dutertre. The model describes elementary particles as localized wave functions with exponentially decreasing radial amplitudes and interprets gravitational attraction as an emergent effect of wave superposition rather than as the exchange of gravitons.<ref name="Dutertre2023">Xavier Dutertre, Bee Theory: Wave-Based Modeling of Gravity — Application of the Schrödinger Equation to the Sum of Two Exponential -r Waves, 2023. English PDF hosted on CollaborativeBee Wiki: 20231226_BeeTheory_v2_EN.pdf.</ref><ref name="BeeWave">BeeTheory.com, A Wave-Based Model of Gravity: beetheory.com/a-wave-based-model-of-gravity/.</ref>

The central idea of BeeTheory is that gravity can be modeled as a wave interaction phenomenon. Instead of treating gravity primarily as spacetime curvature, as in general relativity, or as a force mediated by hypothetical gravitons, BeeTheory applies the Schrödinger equation to the sum of two exponentially decreasing wave functions.<ref name="Dutertre2023" />

Basic information

Field	Information
Name	BeeTheory / Bee Theory™
Main topic	Wave-based modeling of gravity
Main author	Xavier Dutertre
Reference article	Bee Theory: Wave-Based Modeling of Gravity
Core mathematical tools	Schrödinger equation, wave functions, radial exponential functions, spherical Laplacian
Central claim	A gravitational-like attraction can emerge from the superposition of particle wave functions.
Alternative to	Graviton-mediated gravity and purely geometric spacetime-curvature descriptions
Research status	Theoretical model under development; requires further independent mathematical validation and experimental comparison.

Overview

BeeTheory proposes that elementary particles may be represented by wave functions whose amplitudes decrease exponentially with distance from their centers. When two such waves are superposed, the combined wave structure is claimed to shift the effective localization peaks of the particles toward one another. BeeTheory interprets this reciprocal displacement as the origin of an attractive gravitational interaction.<ref name="Dutertre2023" />

In this view, gravity is not introduced as an external force at the beginning of the model. Instead, it is expected to arise from the mathematics of wave overlap, wave gradients, and the Laplacian operator applied to radially decreasing wave functions.

Scientific context

Classical Newtonian gravity describes attraction between masses through an inverse-square law. General relativity replaces the Newtonian force picture with a geometric model in which mass-energy curves spacetime. Gravitational waves, directly detected by LIGO in 2015, are one of the major predictions of general relativity and are understood as propagating ripples in spacetime.<ref name="LIGO">LIGO Laboratory, What are Gravitational Waves?: ligo.caltech.edu/page/what-are-gw.</ref><ref name="Nobel2017">Nobel Prize, The Nobel Prize in Physics 2017 — Press release: nobelprize.org/prizes/physics/2017/press-release/.</ref>

Quantum gravity research attempts to reconcile general relativity with quantum mechanics. In many quantum-field approaches, the graviton is introduced as a hypothetical quantum of the gravitational field. BeeTheory takes a different route by modeling gravitational attraction through wave-function interaction rather than through graviton exchange.<ref name="Dutertre2023" />

Core assumptions

BeeTheory is based on the following working assumptions:

Elementary particles can be modeled as localized wave structures.
The radial amplitude of a particle wave can be approximated by an exponentially decreasing function.
Two particle waves can be superposed into a combined wave field.
The effective attraction between particles can be derived from the behavior of this combined field under the Schrödinger equation.
A potential proportional to <math>-1/R</math> leads to a force proportional to <math>-1/R^2</math>, structurally matching the Newtonian inverse-square dependence.

Mathematical formulation

Particle wave functions

A simplified BeeTheory representation of two localized particles, centered at positions <math>\mathbf{x}_A</math> and <math>\mathbf{x}_B</math>, may be written as:

<math> \psi_A(\mathbf{x},t) = A\,e^{-\alpha \lVert \mathbf{x}-\mathbf{x}_A\rVert} e^{i\omega_A t} </math>

<math> \psi_B(\mathbf{x},t) = B\,e^{-\beta \lVert \mathbf{x}-\mathbf{x}_B\rVert} e^{i\omega_B t} </math>

where:

<math>A</math> and <math>B</math> are complex amplitudes;
<math>\alpha</math> and <math>\beta</math> are radial decay parameters;
<math>\omega_A</math> and <math>\omega_B</math> are angular frequencies;
<math>\mathbf{x}_A</math> and <math>\mathbf{x}_B</math> are particle-center positions.

The combined wave field is then:

<math> \Psi(\mathbf{x},t)=\psi_A(\mathbf{x},t)+\psi_B(\mathbf{x},t) </math>

This superposition is the starting point of the BeeTheory gravity model.<ref name="Dutertre2023" />

Schrödinger equation

BeeTheory applies the Schrödinger equation to the combined wave field:

<math> i\hbar\frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi </math>

In the simplified derivation presented in the reference article, the model considers the kinetic operator and examines how the Laplacian of the exponential radial structure contributes to an effective gravitational potential.<ref name="Dutertre2023" />

Spherical Laplacian

For a radially symmetric function <math>f(r)</math>, the Laplacian is:

<math> \nabla^2 f(r) = \frac{1}{r^2} \frac{d}{dr} \left( r^2\frac{df}{dr} \right) </math>

BeeTheory focuses on the local behavior of one wave near the second particle, with the inter-particle distance denoted by <math>R</math>. In the article’s simplified radial approximation, the calculation identifies a term proportional to:

<math> \nabla^2 f(r) \sim -\frac{1}{R} </math>

This is interpreted as an effective potential of the form:

<math> V_{\mathrm{eff}}(R) = -\frac{k_{\mathrm{B}}}{R} </math>

where <math>k_{\mathrm{B}}</math> is a BeeTheory coupling parameter that must carry the correct physical dimensions.

The corresponding force is:

<math> F(R) = -\frac{dV_{\mathrm{eff}}}{dR} = -\frac{k_{\mathrm{B}}}{R^2} </math>

Thus, the model aims to recover an inverse-square attraction from wave mathematics rather than by assuming Newton’s law at the outset.<ref name="Dutertre2023" />

Relation to Newtonian gravity

Newtonian gravity is commonly written as:

BeeTheory seeks to reproduce the same distance dependence:

<math> F_{\mathrm{Bee}} \propto -\frac{1}{R^2} </math>

The scientific challenge is to determine whether the BeeTheory coupling term can be rigorously connected to mass, energy, frequency, phase, and known constants such as <math>G</math>, <math>\hbar</math>, and <math>c</math>.

Relation to gravitons

In standard quantum-field reasoning, the graviton is usually introduced as a hypothetical spin-2 quantum associated with the gravitational field. No direct detection of individual gravitons has yet been established. BeeTheory argues that a graviton is not necessary if gravitational attraction can be derived from direct wave-field interaction.<ref name="NatureGraviton">G. Tobar et al., Detecting single gravitons with quantum sensing, Nature Communications, 2024: nature.com/articles/s41467-024-51420-8.</ref>

This does not by itself disprove the graviton hypothesis. It defines BeeTheory as an alternative modeling route that must be tested against both existing gravitational observations and quantum-mechanical consistency requirements.

Comparison with established models

Model	Description of gravity	Mathematical emphasis	Status
Newtonian gravity	Force between masses	Inverse-square law	Highly accurate in weak-field, low-speed regimes
General relativity	Curvature of spacetime caused by mass-energy	Differential geometry and Einstein field equations	Experimentally successful at astronomical and relativistic scales
Graviton-based quantum gravity	Gravity mediated by hypothetical quanta	Quantum field theory	Theoretical; no direct graviton detection
BeeTheory	Emergent attraction from wave-function superposition	Schrödinger equation, exponential radial waves, spherical Laplacian	Theoretical model under development

Validation requirements

For BeeTheory to become a complete physical model, it must address several validation points:

Derive the inverse-square law with full dimensional consistency.
Show how mass and inertia emerge from the wave parameters.
Recover Newtonian gravity in the classical weak-field limit.
Reproduce key tests of general relativity, including gravitational lensing, time dilation, orbital precession, and gravitational-wave propagation.
Provide falsifiable predictions that differ from general relativity or standard quantum gravity.
Compare numerical simulations with observational data from gravitational-wave observatories such as LIGO, Virgo, and KAGRA.
Clarify the mathematical behavior of the radial Laplacian near <math>r=0</math>, including regularization, normalization, and boundary conditions.

Potential research directions

BeeTheory can be developed through several research tracks:

symbolic verification of the radial derivation;
numerical simulation of two-wave and many-wave systems;
comparison with Newtonian and relativistic gravitational potentials;
study of phase effects in gravitational attraction;
extension to many-body systems and astrophysical structures;
analysis of possible deviations at microscopic or quantum-coherent scales.

Limitations and open questions

BeeTheory remains a theoretical proposal. Its main open questions include:

whether the exponential wave ansatz is physically justified for all particle types;
whether the proposed effective potential survives a complete treatment of the Schrödinger equation;
whether the model can reproduce the full tensor structure of general relativity;
whether it can explain gravitational waves with the same precision as general relativity;
whether it produces new measurable predictions;
whether its coupling constants can be derived from known physical constants rather than fitted.

These questions are essential for presenting BeeTheory in a scientific and falsifiable way.

Reference article

The main reference document is:

Xavier Dutertre, Bee Theory: Wave-Based Modeling of Gravity — Application of the Schrödinger Equation to the Sum of Two Exponential -r Waves, 2023.

 English PDF on CollaborativeBee Wiki

A related internal wiki file may also be referenced as:

File:20231226 BeeTheory v2.pdf

External links

References

Beetheory

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