The Boukarroum Geometric Invariant: Emergence of Light, Gravity, and Spacetime from Conservation of Helical Energy Loops

Abstract

This work presents a unified geometric framework in which all physical phenomena—light, matter, gravity, spacetime curvature, and cosmic evolution—emerge from a single invariant: the conservation of helical energy loops N generated by a fundamental energy point. The fundamental equation defining the framework is: v_point^2 = c^2 + v_orb^2 where v_point is the intrinsic velocity of the fundamental energy point, c is the effective speed of light, and v_orb is the hidden orbital velocity of the photon's internal structure. This relation establishes the Boukarroum Geometric Invariant. Key elements: - Photon structure: photons are composed of helical loops; the total number of loops N is conserved, split between gravitationally active loops (N_g) and orbital loops (N_orb). - Emergence of gravity: gravitational effects are a geometric redistribution of loops; only N_g contributes to spacetime curvature. - Variable Planck length: ℓ_P = 1/N, reflecting the internal loop height. - Schwarzschild metric derivation: proper time and radial distance emerge from local loop density N_g(r). - Einstein equations: derived from ∇_μ N^μ = 0, with an effective energy-momentum tensor T_{μν}, producing G_{μν} = 8π G T_{μν}. - Gravitational waves: propagating perturbations of loop density, polarizations (+, ×); no graviton exists; waves are geometric distortions. - Cosmological redshift: photon wavelength scales with ℓ_P; 1+z = N_em / N_obs; the universe does not expand in time. - Cosmic time principle: the Cosmic Time is globally invariant; expansion occurs geometrically, not temporally. - Why spacetime does not expand: detailed comparison with ΛCDM; all observations are explained via redistribution of loop density, not metric expansion. Included in this record: - Full PDF of the book containing all chapters (1–15) and Appendices A–F. - Full LaTeX source (main.tex). - Integrated Priority Claim: "This record establishes scientific priority for the Boukarroum Geometric Invariant framework, timestamped and citable via DOI." Keywords / Subjects: Alternative Quantum Gravity, Emergent Gravity, Cosmic Redshift, Gravitational Waves, Planck-Scale Physics, Fixed Spacetime, Invariant Time. Instructions: 1. Upload PDF + LaTeX source. 2. Paste this Abstract/Description. 3. Set Publication type: Preprint, Access: Open. 4. Add ORCID. 5. Press Publish → DOI is generated immediately.