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A Theory of Space as the Fundamental Oscillator, Collapse as Geometric Phase Locking, and Gravity as Space Flow

Lakshmi & ChatGPT
Date: September 30, 2025

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Abstract

We develop a framework in which the quantum wavefunction does not represent a particle, but space itself. In this view, the complex amplitude Ψ(x,t) is a field intrinsic to spatial geometry. Matter and radiation are excitations bound to this space-wave; gravity arises as the flow of space toward concentrations of mass-energy; and measurement collapse is a nonlinear geometric phase locking of Ψ into a localized configuration.

We present a Schrödinger-type dynamical equation for Ψ with Laplace–Beltrami kinetic term, curvature coupling, and nonlinear self-interaction. We demonstrate how the Young double-slit experiment emerges naturally as interference of space modes, and estimate gravitational self-energy effects on collapse using the Diósi–Penrose heuristic. We connect this “wave-geometry” program with objective collapse models, canonical quantum gravity, the river model of black holes, and nonlinear Schrödinger analogies. Experimental implications and open problems are discussed.

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1. Introduction

The foundational puzzles of physics — quantum measurement, spacetime structure, and gravity — remain unresolved despite a century of progress.

• Quantum mechanics provides a predictive algorithm but leaves ambiguous the ontological status of the wavefunction. Is Ψ a real physical field, a probability amplitude, or something else?
• Gravity, in general relativity, is geometry itself: matter tells space how to curve, and space tells matter how to move. Yet this geometric description resists unification with quantum theory.
• Measurement collapse introduces non-unitary “events” into otherwise deterministic dynamics. The question of how and why remains central.

Existing proposals include objective collapse theories (GRW, CSL, Diósi–Penrose) [Bassi et al. 2013], canonical quantum gravity (Wheeler–DeWitt), and heuristic models such as Painlevé–Gullstrand’s “river model” of black holes.

In this paper we propose a unified reinterpretation: the wavefunction is not a function of particles in space, but a function of space itself. Collapse corresponds to geometric phase locking of this field, and gravity corresponds to the inward flow of space toward dense concentrations.

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2. Formalism

2.1 Ontological postulate

Let M be the three-dimensional spatial manifold. Define a complex scalar field:

Ψ : M × R → C.

Interpretation: ρ(x,t) ≡ |Ψ(x,t)|² is the amplitude density of space itself. Excitations of this field are what we perceive as particles and photons.

2.2 Dynamical law

We postulate a nonlinear Schrödinger-type equation:

iħ ∂t Ψ(x,t) = [ – (ħ² / 2μ) Δg + Vcurv(Ψ) + Λ f(|Ψ|²) ] Ψ(x,t). (1)

• Δg: Laplace–Beltrami operator associated with metric g.
• μ: effective inertia of space.
• Vcurv: curvature-coupled potential.
• Λ f(|Ψ|²): nonlinear collapse-driving term.

Equation (1) is nonlinear and self-consistent.

2.3 Metric coupling

To close the system, |Ψ|² is mapped to curvature. Options:

1. Scalar curvature ansatz: R(x,t) = κ ρ(x,t).
2. Metric perturbation ansatz: gij = ηij + α Gij[ρ] + …, where Gij is a Green’s function convolution of ρ.

Thus Ψ determines curvature, which in turn governs Ψ.

2.4 Collapse term

Collapse is modeled as a focusing nonlinearity or stochastic CSL-type term. A simple deterministic choice:

f(ρ) = ρ, with Λ > 0.

This allows attractor-like soliton solutions, representing localized “spots” on measurement.

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3. Young’s Double-Slit as Geometry Interference

Consider two slits A and B at separation d. At a screen a distance L away, approximate:

Ψ(x,t) = cA ΦA(x) + cB ΦB(x),

with Φ modes carrying phase terms ~ e^{ikr}/√r.

Projecting Eq. (1) onto this two-mode subspace yields:

iħ d/dt (cA, cB)ᵀ = H₂×₂ (cA, cB)ᵀ,

with Hamiltonian H containing energies EA, EB and coupling K.

The screen intensity is:

I(x) ∝ |ΦA(x) + e^{iΔφ} ΦB(x)|²
= |ΦA|² + |ΦB|² + 2 Re{ΦA ΦB* e^{iΔφ}}.

Thus, interference fringes emerge as geometry-mode interference. Collapse is the transition of Ψ into a localized soliton — the observed “hit.”

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4. Worked Example: Gravitational Perturbation from Collapse

Following the Diósi–Penrose heuristic, collapse timescale τ relates to gravitational self-energy EG:

τ ≈ ħ / EG,
with EG ≈ (3 G m_eff²) / (5 R).

Take m_eff = 10⁻¹⁵ kg, R = 10⁻⁶ m:

EG ≈ 4 × 10⁻⁵⁰ J.
τ ≈ 2.6 × 10¹⁵ s ≈ 80 million years.

Thus for microscopic systems collapse is negligible, consistent with coherence. At mesoscopic scales, τ decreases, predicting new physics.

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5. Relation to Existing Programs

• Objective collapse (Diósi, Penrose): Our collapse term aligns with gravitational self-energy heuristics.
• Wheeler–DeWitt: Ψ(x,t) resembles a reduced wavefunction of 3-geometry. A covariant extension could connect directly.
• River model of black holes: Gravity as “space flow” is a natural extension here.
• Nonlinear Schrödinger/Gross–Pitaevskii: Collapse dynamics mirror soliton formation in condensates.

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6. Predictions and Tests

1. Mesoscopic interference: Molecule interferometry should show deviations from standard quantum predictions at masses approaching 10⁻¹⁶ – 10⁻¹⁴ kg.
2. Local metric fluctuations: Tiny metric perturbations may accompany detection events — potentially observable by ultra-sensitive interferometers.
3. Quantum-gravity crossovers: Experiments combining superposition with gravitational effects could discriminate this model from standard quantum mechanics.

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7. Open Problems

• Covariant relativistic generalization.
• Derivation from a deeper pre-geometric substrate.
• Ensuring Lorentz invariance and causality.
• Parameter bounds from interferometry and collapse-model constraints.

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8. Conclusion

In the wave-geometry picture, quantum amplitudes belong not to particles but to space. Collapse is a geometric phase transition, and gravity is the flow of space itself.

This program unifies strands from objective collapse, canonical quantum gravity, river models, and nonlinear Schrödinger analogies. It yields a concrete PDE framework, rich with mathematical and experimental challenges.

If correct, this reframing suggests that the true universal constant is not the speed of light, but the speed of space.

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References (selected)

• Bassi, A., Lochan, K., Satin, S., Singh, T.P., Ulbricht, H. Models of Wave-function Collapse: A Review. Rev. Mod. Phys. 85, 471 (2013).
• Penrose, R. On Gravity’s Role in Quantum State Reduction. Gen. Rel. Grav. 28, 581 (1996).
• Diósi, L. Gravity-related wave function collapse: mass density resolution. Phys. Lett. A 120, 377 (1987).
• Kiefer, C. Quantum Gravity. Oxford University Press (2012).
• Hamilton, A.J.S., Lisle, J.P. The river model of black holes. Am. J. Phys. 76, 519 (2008).
• Gross, E.P., Pitaevskii, L.P. Nonlinear Schrödinger equation for Bose condensates. JETP (1961).

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