GWT - Gronau World Theory

GWT – Gronau World Theory (Version 2.0) A Geometric Theory of Gravitation and Quantum Mechanics in Relative Space — with 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 2.0 presents extensive quantitative calculations demonstrating that the gravitational evidence for dark matter can be explained geometrically—without invoking any new particles or fields. What's New in V2 This release includes complete numerical analyses for: 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: Level Embedding Regime Application DFT 3→4 r ≪ ℓ Spiral galaxies VFT 4→5 r ~ ℓ Galaxy clusters Tail ≥6D r ≳ ℓ, late epochs Large-scale structure ∞FT Continuous limit Full theory True nature DFT is presented to illustrate the structure; in principle, calculations should use the highest available level (VFT + Tail) for maximum accuracy. 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: QuantGWT-DFT: iℏ∂ₜψ = −(ℏ²/2m)Δψ + V_eff(χ,t)ψ QuantGWT-VFT: Scale-activated geometric corrections Quantitative Results Summary Spiral Galaxies (DFT Level) Galaxy Type v_flat [km/s] M_bar [M☉] Status DDO 154 Gas-dominated dwarf 47 4.6×10⁸ ✓ Reproduced IC 2574 Late-type spiral 78 9.4×10⁹ ✓ Reproduced NGC 2403 HSB spiral 134 2.8×10¹⁰ ✓ Reproduced NGC 2841 Massive early spiral 287 1.3×10¹¹ ✓ Reproduced NGC 5055 Massive spiral 179 5.0×10¹⁰ ✓ Reproduced NGC 6503 Compact disk 116 2.1×10¹⁰ ✓ Reproduced NGC 7331 Luminous Sb 239 8.9×10¹⁰ ✓ Reproduced NGC 7814 Bulge-dominated 231 8.3×10¹⁰ ✓ Reproduced Result: 8/8 rotation curves explained with baryons alone. Galaxy Clusters (VFT Level) Cluster σ_obs [km/s] DFT coverage VFT parameter K Result Virgo 700 31% 20 ✓ Match Coma 1100 60% 4 ✓ Match Abell 1689 (lensing) ~4% ~400 ✓ Match Result: VFT closes all dynamical deficits. Gravitational Lensing (DFT + VFT) System Scale R_E [kpc] DFT result VFT result Einstein Cross Galaxy 0.7 46% — Twin Quasar Galaxy+Cluster 31 39% — Horseshoe Galaxy 31 87% — Abell 370 Cluster 180 1.9% ✓ 100% SDSS J1004 Cluster 110 1.9% ✓ 100% 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 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: Flat rotation curves arise from χ-saturation at large radii—a purely geometric effect Cluster dynamics are explained by VFT scale-activation where DFT undercovers Lensing masses follow from the same geometric amplification mechanism No particle dark matter is required for any of these phenomena This does not prove that particle dark matter does not exist, but it shows that postulating new particles is not necessary to explain the gravitational anomalies that motivated the dark matter hypothesis. 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 — Extension to quantum mechanics 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: Correlated amplification: Dynamics and lensing should show coordinated VFT lifts at r ~ ℓ Scale dependence: DFT works for r ≪ ℓ; VFT activates at r ~ ℓ Universal parameters: K_lens ≈ 620 should hold across different cluster lenses No dark matter detection: Direct detection experiments should continue to find nothing Geometric signatures: Specific patterns in rotation curves and lensing that differ from NFW halos 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: [email protected] 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 License Creative Commons Attribution 4.0 International (CC BY 4.0) Citation Gronau, T. (2025). GWT – Gronau World Theory (Version 2.0). Zenodo. https://doi.org/10.5281/zenodo.17704764 Version History V1.0 (2025-11): Initial release with theoretical framework V2.0 (2025-11): Major update with quantitative calculations for galaxies, clusters, and lensing