A Scale-relative Approach to Unification: Analogical Equivalence of Fundamental Forces from Atomic to Cosmic Scales

International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 5, 2025 DOI: 10.5281/zenodo.17314701 Original Article ©2025 RS Publication, [email protected] 318 A Scale-relative Approach to Unification: Analogical Equivalence of Fundamental Forces from Atomic to Cosmic Scales Ankit Kumar Singh (Corresponding Author) New Delhi ARTICLE INFO ABSTRACT ©2025 RS Publication Paper ID: IJASTR68E4ECF39A3EC Received: 2025-09-10 Published: 2025-10-10 DOI: https://dx.doi.org /10.5281/zenodo.17 314701 Page No: 318-322 This paper proposes a theoretical framework suggesting that the four fundamental forces of nature, gravitational, electromagnetic, strong nuclear, and weak nuclear are interrelated through a principle of scale relativity. Drawing an analogy between the structure of atoms and the solar system, the study examines whether macroscopic gravitational and magnetic interactions are scaled analogues of microscopic nuclear and electromagnetic interactions. By exploring Kepler’s laws of planetary motion as a macro-scale reflection of atomic behavior, a correspondence model is proposed that maps force types across physical scales. The apparent separation of fundamental forces is hypothesized to be an artifact of observational scale rather than a true division in nature. The model is formalized mathematically using scaling functions and dimensional analysis, and illustrated through diagrams. Implications include the prospect for a scale-dependent grand unification, predictions of astrophysical resonances, and tests based on dimensionless invariants. Keywords: Scale Relativity, Unified Field Theory, Kepler’s Law, Atomic– Cosmic Analogy, Fundamental Forces, Gravity–Weak Equivalence, Mass– Charge Duality Cite This Paper: Ankit Kumar Singh (2025). "Scale-Relativity based unification of the four Fundamental Forces". INTERNATIONAL JOURNAL OF ADVANCED SCIENTIFIC AND TECHNICAL RESEARCH (IJASTR), vol. 15, no. 5, 2025, pp. 318-322. DOI: https://dx.doi.org/10.5281/zenodo.17314701 International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 5, 2025 DOI: 10.5281/zenodo.17314701 Original Article ©2025 RS Publication, [email protected] 319 Introduction The search for a unified description of nature’s fundamental forces is central in theoretical physics. Gravity, electromagnetism, the strong nuclear, and the weak nuclear force have traditionally been treated as distinct, but recurring structures on atomic to galactic scales suggest the possibility of underlying symmetries repeated across magnitudes of scale. This study introduces a scale-based equivalence hypothesis, positing that these forces are different manifestations of a single interaction, modulated by observational scale. The analogy between atomic and cosmic structures, especially in dynamics and configuration, forms the argument’s basis. Conceptual Framework Atom–Solar Analogy Just as a cricket stadium models the scale of an atom, with the nucleus as a tennis ball at the pitch and electrons near the outfield, so too does the solar system, with its massive, sun-like “nucleus” and planetary “electrons”, reflect this spatial and dynamical structure. The stability in both systems arises from a balance of centripetal and attractive forces, and elliptical orbits (planetary and atomic) are governed by analogous laws: Kepler's laws and the probabilistic shapes of atomic orbitals. Scale Relativity Principle The scale relativity principle asserts that the forms of physical laws are preserved, once appropriate scale factors are applied, across atomic, planetary, and galactic domains. The constants that dominate each regime (such as ℏ and G) shift with scale, but the underlying dynamics are structurally similar. This leads to the hypothesis that each force, at each scale, is the scale-relativistic image of another. Force Equivalence Macrocosmic Force , Microcosmic Analogue , Primary Characteristic , Proposed Equivalence Gravitational Force , Weak Nuclear Force , Long-range attraction via mass-energy curve , International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 5, 2025 DOI: 10.5281/zenodo.17314701 Original Article ©2025 RS Publication, [email protected] 320 Fg ∼ Fw Electromagnetic Force , Strong Nuclear Force , Binding via charge/magnetic effects , Fe ∼ Fs When mapped through appropriate scaling transformations, gravity mirrors the weak force, and electromagnetism orders as the strong force at reduced (nuclear) scales. These correspondences are proposed as the key symmetry underlying apparent force diversity. Mathematical Modeling Dimensional Setup Let scale ratios compare macro (capitalized) and micro (lowercase) quantities: SL = Lmacro / Lmicro, SM = Mmacro / Mmicro, ST = Tmacro / Tmicro Force dimension: [MLT−2]; therefore, the scaling factor for force is: SF = SM SL ST−2 Empirical Scale Estimates Bohr radius: a0 = 5.29 × 10−11 m Earth's orbital radius: RE = 1.49 × 10^11 m SL = RE / a0 ≈ 2.8 × 10^21 Proton radius: rp = 0.84 × 10−15 m Sun radius: R⊙ = 6.96 × 10^8 m SL(Sun/proton) = 8.3 × 10^23, SM(Sun/proton) = 1.19 × 10^57 Scaling Function Define the mapping: φ(S) = SM^p SL^(1−p), 0 ≤ p ≤ 1 Fmacro = φ(S) Fmicro This function adjusts fundamental constants for scale, e.g.: G′ = φ(S)G, ℏ′ = ℏ / φ(S) Mapping Between Forces Mass–Charge Analogy Let effective charge qeff = βm, giving a Coulomb-like force: FCeff = (1 / 4πε0) * β² * m1m2 / r² Relate to Newtonian gravity: Gm1m2 / r² = (β² / 4πε0) * m1m2 / r² ⇒ β = √(4πε0G) ≈ 8.62 × 10−11 C/kg Thus, gravity at one scale can be interpreted as a scaled image of electromagnetic interaction. International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 5, 2025 DOI: 10.5281/zenodo.17314701 Original Article ©2025 RS Publication, [email protected] 321 Weak–Gravity Scaling If the weak force satisfies Fw ∝ Gm² / r², then macro gravity and weak micro-interactions are scale-linked through φ(S). Kepler-Bohr Time Scaling Kepler’s 3rd law for orbital period: T² = 4π²a³ / GM Bohr’s orbital period: TBohr = (2πℏ³) / (Z e² m e²) Relating periods by scaling: ST ∼ SL^(3/2) SM^(−1/2) (Gmacro / Gmicro)^(1/2) This supplies the time scaling in φ(S). Implications and Tests Unification: Suggests a scale-dependent Grand Unified Theory (SD-GUT) pathway. Quantum Gravity: Points to gravitational quantization through inverse weak force scaling. Astrophysical Predictions: Magnetic or gravito-magnetic resonances in planetary systems may mirror atomic energy shells. Dimensionless Invariants: If specific force ratios are preserved across scale, this confirms self-similarity. Mass–Charge Proportionality: Compare q/m ratios of cosmic plasmas for proximity to β. Limitations and Future Work The exact form of the scaling function φ(S) must be rigorously derived from renormalization theory. Quantum and relativistic effects are beyond the present analysis. Large-scale simulations are necessary to validate scale-induced quantization and periodicity in astrophysical settings. Next Steps: - Construct renormalization-group (RG) models for coupling flows. International Journal of Advanced Scientific and Technical Research ISSN 2249-9954 Available online on http://www.rspublication.com/ijst/index.html volume 15, No. 5, 2025 DOI: 10.5281/zenodo.17314701 Original Article ©2025 RS Publication, [email protected] 322 - Simulate N-body gravitational systems with quantization imposed. - Analyze exoplanetary data for orbital period clustering. Conclusion A scale-relativity approach is introduced linking all fundamental forces through analogical and mathematical equivalence, suggesting that what appear as independent interactions are, in fact, scale-transformed facets of a single law. Parallels between atomic and cosmic structures provide a foundation for this hypothesis, opening avenues for scale-unifying formalism and simulation-based verification.