GWT – Gronau World Theory (Version 3.0)

A Geometric Theory of Gravitation and Quantum Mechanics in Relative Space — with Complete Quantum Framework and Quantitative Evidence that Dark Matter is Unnecessary

Abstract

The Gronau World Theory (GWT) proposes a fundamental ontological shift in physics: the replacement of four-dimensional spacetime with three-dimensional relative space that is extrinsically curved into higher dimensions. In this framework, mass is identical to curvature (M ≡ K), and what appears as gravitational attraction is the projection of shorter paths through the higher-dimensional embedding.

Version 3.0 presents the complete quantum-mechanical framework of GWT, deriving quantum mechanics from geometric first principles and establishing the unified foundation for gravity and quantum phenomena.

What's New in V3

This release introduces the complete Quantum GWT framework:

Quantum GWT Foundation (8 new/expanded chapters)

• Q-I (QGWT-DFT): Schrödinger equation derived from curvature dynamics
• Q-II (QGWT-VFT): Complete VFT extension with three geometric operators
• Q-∞ (QGWT-∞FT): Full infinite embedding ladder formulation
• VFT Operators derived from 5D geometry: Λ_ℓ (dispersion), U_ℓ (potential), A_ℓ (drift)
• Self-adjointness and norm conservation proven
• Spectral analysis with explicit hydrogen atom calculations

Relativistic Extension

• Klein-Gordon equation in relative space with VFT corrections
• Dirac equation with geometric interpretation of spin
• Spin as intrinsic orientation in embedding plane
• Antimatter as mirrored embedding direction
• Pauli equation as non-relativistic limit

Unified Framework

• Gravity-quantum consistency established
• Parameter matching between sectors proven
• Equivalence principle preservation demonstrated
• Coupling hierarchy derived

Particle Physics Connection

• (3+3+1) sector structure → Standard Model gauge group
• Geometric confinement mechanism
• Electroweak symmetry breaking as sector reduction
• CKM and PMNS matrices from sector overlaps

Key Result: All VFT corrections to atomic observables are ~10⁻²⁵, explaining why standard quantum mechanics achieves precision of 10⁻¹⁵—the theory is self-consistent with all precision tests.

Previous Results (from V2.0)

Spiral Galaxies (8 systems)

DDO 154, IC 2574, NGC 2403, NGC 2841, NGC 5055, NGC 6503, NGC 7331, NGC 7814

• All rotation curves reproduced using only observed baryonic mass
• Single geometric scale parameter ℓ = 10 kpc
• No dark matter halos required

Galaxy Clusters (3 systems)

Virgo, Coma, Abell 1689

• Velocity dispersions matched through VFT scale-activation
• DFT alone provides 30–60%; VFT closes the gap completely
• No dark matter halos required

Gravitational Lensing (5 systems)

Galaxy-scale: Einstein Cross, Twin Quasar, Cosmic Horseshoe Cluster-scale: Abell 370, SDSS J1004+4112

• Universal lensing VFT parameter K_lens ≈ 620 discovered
• No dark matter halos required

Historical Context and Relation to General Relativity

GWT stands in deep respect to Albert Einstein and General Relativity. The fundamental insight that mass curves space remains the cornerstone of geometric gravity and will forever be associated with Einstein's name.

However, Einstein worked within the ontology available to his time—four-dimensional spacetime as the fundamental arena of physics. He did not have access to the galactic rotation curve data, cluster dynamics, or gravitational lensing observations that emerged decades later and led to the dark matter hypothesis.

GWT proposes that these phenomena do not require new matter, but rather a change in ontology: from intrinsically curved spacetime to extrinsically curved relative space. This is not a rejection of Einstein's work but its continuation in a new direction—just as Einstein continued and transformed Newton's gravity.

The credit for "mass curves space" belongs to Einstein. GWT adds: "and the curvature is extrinsic, into higher dimensions, with projection effects that explain what we call dark matter."

The Embedding Hierarchy

Nature is continuous. The mathematical description leads to a hierarchy of approximation levels:

Core Equations (Named for Citation)

Geometric quantities

χ(r) = r / (1 + r/ℓ)     — replacement coordinate
Π(r) = χ(r) / r          — projector

Gravitation

• GravG-I (DFT): g₍₁₎(r) = Π(r) · G·M(χ)/χ²
• GravG-II (VFT): g(r) = g₍₁₎(r) · [1 + η(ℓ)·β·(χ/ℓ)^p] + R_≥3

Lensing

• GLens-I (DFT): α₍₁₎(b) = C·G·M(χ)/(c²·b)
• GLens-II (VFT): α(b) = α₍₁₎(b) · [1 + η(ℓ)·β·(χ/ℓ)^p] + R^lens_≥3

Quantum (NEW in V3)

• Q-I (DFT): iℏ∂ₜψ = −(ℏ²/2m_geo)∇²ψ + V_eff·ψ
• Q-II (VFT): iℏ∂ₜψ = Ĥ_DFT·ψ + η(ℓ)·Δ̂_VFT·ψ + R̂_≥3·ψ
• Q-∞: iℏ∂ₜψ = Ĥ_DFT·ψ + Σₙ ηₙ(ℓ)·Δ̂₍ₙ₎·ψ

VFT Operators

• Λ_ℓ^ij = η·σ·Δ^ij — dispersion tensor (anisotropic kinetic energy)
• U_ℓ = η·σ·u — VFT potential (geometric potential)
• A_ℓ^i = η·σ·a^i — drift vector (geometric transport)

Relativistic (NEW in V3)

• Klein-Gordon-VFT: (□_g + η·D̂_VFT − m²c²/ℏ²)φ = 0
• Dirac-VFT: (iℏγᵃeₐᵘDᵤ − m_geo·c + η·Ŝ_VFT)ψ = 0

Ontological Foundation

In the GWT ontology:

• Space is not a passive stage but the fundamental substance of reality
• Mass ≡ Curvature: Matter is geometry, not "stuff in space"
• Time emerges from projection dynamics, not as a fundamental dimension
• c is a property of space itself, appearing constant in projection
• Particles are stable curvature patterns, not point-like objects
• Spin is intrinsic orientation in the embedding plane
• Antimatter is mirrored embedding direction
• Singularities do not exist in 3D; extreme curvature creates non-Cartesian expansion

Important caveat: All observational data compared in this work were obtained and interpreted within the spacetime ontology. In relative space, time flow and c may behave differently over cosmic timescales. The excellent agreement is encouraging, but a complete treatment requires re-deriving observables within the new framework.

Implications

For Dark Matter

The results demonstrate that the gravitational evidence for dark matter can be explained geometrically. No particle dark matter is required for rotation curves, cluster dynamics, or gravitational lensing.

For Quantum Mechanics

The Schrödinger equation is not postulated but derived from curvature dynamics. Standard QM emerges as the DFT limit, with VFT corrections suppressed by factors of ~10⁻²⁵ at atomic scales—explaining its extraordinary precision.

For Unification

Gravity and quantum mechanics share the same geometric origin. The unified VFT framework ensures parameter consistency across both sectors.

Structure of the Document

Part I: Foundations — Ontology, historical positioning, geometric intuition, mathematical framework

Part II: Gravitation — DFT, VFT, Tail levels of the Gronau equations

Part III: Lensing Theory — Core GWT lensing equations

Part IV: Quantum GWT (EXPANDED) — Complete quantum framework with 8 chapters:

• Ch. 11: QGWT-DFT (Gronau Quantum Equation I)
• Ch. 15: VFT Operators from 5D Geometry
• Ch. 12: QGWT-VFT (Gronau Quantum Equation II)
• Ch. 22: Spectral Consequences and Examples
• Ch. 25: Unified VFT Framework
• Ch. 27: Relativistic Extension (Klein-Gordon, Dirac)
• Ch. 28: Sector Structure and Particle Physics
• Ch. 30: QGWT-XFT and Summary

Part V: Galactic Dynamics — Methodology and calculations for 8 spiral galaxies

Part VI: Galaxy Clusters — Methodology and calculations for Virgo, Coma, Abell 1689

Part VII: Gravitational Lensing — Methodology and calculations for 5 lens systems

Part VIII: Special Topics — Black holes, fundamental constants, double slit, entanglement

Appendices — Notation, numerical framework, reference calculations, reproducibility

Testable Predictions

GWT makes specific, falsifiable predictions:

1. Correlated amplification: Dynamics and lensing should show coordinated VFT lifts at r ~ ℓ
2. Scale dependence: DFT works for r ≪ ℓ; VFT activates at r ~ ℓ
3. Universal parameters: K_lens ≈ 620 should hold across different cluster lenses
4. No dark matter detection: Direct detection experiments should continue to find nothing
5. Geometric signatures: Specific patterns in rotation curves and lensing that differ from NFW halos
6. Quantum precision: VFT corrections ~10⁻²⁵ at atomic scales, consistent with all precision tests
7. Universal scale exponent: Same p across all gravitational and quantum VFT phenomena

Reproducibility

The appendices provide complete specifications for independent verification:

• All constants and units
• Pseudocode for core algorithms
• Test cases with target values and tolerances
• Machine-readable input formats

Significance

Beyond its technical achievements, GWT offers a new perspective on humanity's place in nature: we are not in space but part of space—stable geometric forms curving together with the cosmos. This insight places everyday intuitions on a precise mathematical foundation and opens the path toward a unified description of gravity, quantum mechanics, and cosmology.

Author

Thomas Gronau Independent Researcher Broderstorf (Steinfeld), Germany

ORCID: 0009-0006-8614-4508 Email: gronauworldtheory@gmail.com

Keywords

relative space, extrinsic curvature, dark matter alternative, rotation curves, gravitational lensing, galaxy clusters, velocity dispersion, Einstein radius, embedding hierarchy, DFT, VFT, geometric gravity, Gronau World Theory, GWT, modified gravity, cosmology, quantum gravity, geometric quantum mechanics, Klein-Gordon, Dirac equation, spin geometry, antimatter, particle physics, Standard Model, sector structure

License

Creative Commons Attribution 4.0 International (CC BY 4.0)

Citation

Gronau, T. (2025). GWT – Gronau World Theory (Version 3.0). Zenodo. https://doi.org/10.5281/zenodo.17778008

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