Relational Physics: The Fundamental Structure of the Universe

Relational Physics: The Fundamental Structure of the Universe Benchekroun Mounssif Maarif, Casablanca, Morocco 2025 Abstract We develop a complete relational theory in which the universe is described as a discrete network indexed by the group (nZ,+). Physical objects have no intrinsic existence: only relations, organized as relational scenes (R, Ψ, ρ), constitute reality. The absence of a fundamental time variable forces a redefinition of physical quantities: velocity, energy, mass, gravitation and frequency emerge naturally from the variations (∆R, ∆E)between scenes. A constrained relational action S= Λ leads to a master unification equation. We also derive a relational blackbody spectrum free of ultraviolet divergences. This framework provides a full reconstruction of physics without time, waves or fields. Contents 1 Introduction 3 2 Relational Scenes: Foundation of the Discrete Foliation 3 2.1 Definition ....................................... 3 2.2 Discretefoliation ................................... 3 2.3 Relationaldistance .................................. 4 3 Relational Variations and the Rigorous Definition of an Event 4 4 Relational Action and Cosmological Constraint 4 5 Emergent Physical Quantities 4 5.1 Velocityandacceleration............................... 5 5.2 Relationalenergy................................... 5 5.3 Relationalgravitation................................. 5 5.4 Emergentmass .................................... 5 6 Photons and Maximal Coherence Speed 5 1 7 Absence of Time and Relational Causality 5 8 Relational Frequency 5 9 Application: Relational Young Interference 6 10 Relational Dispersion and Micro-Variations of c6 11 Relational Blackbody Spectrum 6 11.1 Energy of a relational mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.2 Optimal coherence distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 11.3Relationaldiscreteness ................................ 6 11.4Finalspectrum .................................... 7 12 Conclusion 7 2 1 Introduction Standard physics relies on three fundamental structures: (1) time as an evolution parameter, (2) continuous space as geometric support, and (3) fields and waves as carriers of interactions. In this work, we show that all three pillars can be entirely replaced by a single fundamental concept: relations. We propose a full reconstruction of physics within a strictly relational framework where the universe is not a spacetime continuum, but a discrete succession of relational scenes ordered by the group (nZ,+). This approach enables: •a rigorous and observer-independent definition of a physical event; •a natural unification of physical quantities (velocity, energy, mass, gravitation); •a reformulation of quantum phenomena without waves or particles; •a structural derivation of the blackbody spectrum; •automatic removal of ultraviolet divergences; •a purely geometric notion of causality, independent of absolute time. 2 Relational Scenes: Foundation of the Discrete Foliation 2.1 Definition We define a relational scene as the triplet: Sk= (R(k),Ψ(k), ρ(k)), where: •R(k)is a tensor encoding structural relations, •Ψ(k)is a relational amplitude, •ρ(k)=|Ψ(k)|2is a coherence density. A scene represents the global, timeless state of the universe. 2.2 Discrete foliation Scenes follow the discrete ordering: Sk→ Sk+n. This ordering is not temporal. The classical illusion of time arises from the ordering of scenes, not from a continuous variable t. 3 2.3 Relational distance We define an intrinsic distance between two scenes: d(S,S′)2=∥R−R′∥2 F+∥Ψ−Ψ′∥2 2. This relational geometry replaces spacetime metrics. 3 Relational Variations and the Rigorous Definition of an Event Variations between two scenes are defined as: ∆Rk=R(k+n)−R(k),∆Ek=E(k+n)−E(k). Definition. A physical event exists if and only if: (∆Rk,∆Ek)= (0,0). This relational definition replaces the quantum measurement problem entirely. 4 Relational Action and Cosmological Constraint We postulate a discrete action: S[Ψ, R] = X i,j Ψ K(Rij )Ψj+λ 2V(Rij )ρiρj, i associated with the fundamental constraint: S= Λ. Extremization leads to the master unification equation: X j K(Rij)Ψj+λΨiX j V(Rij)ρj=µΨi. This equation unifies linear coherence, nonlinear interactions and emergent quantities. 5 Emergent Physical Quantities Classical physical quantities are not fundamental; they emerge from (∆R, ∆E). 4 5.1 Velocity and acceleration vk∼∆Rk, ak∼R(k+ 2n)−2R(k+n) + R(k). 5.2 Relational energy Ek∝∆Ek. 5.3 Relational gravitation Gij ∼ − ∂E ∂Rij . 5.4 Emergent mass mk=⟨∆Ek,∆Rk⟩ ∥∆Rk∥2 F+η. 6 Photons and Maximal Coherence Speed A photon satisfies: ∆E≈0,∆R= 0. The maximal coherence speed is: c= max relational step ∥∆R∥. Thus, light is neither a particle nor a wave, but a pure coherence event. 7 Absence of Time and Relational Causality Causality arises from: Sk≺ Sk+n⇐⇒ ∆Rkconstrains ∆Rk+n. Causality is geometric, not temporal. 8 Relational Frequency We redefine frequency as a structural quantity: νrel =κF(∆R), where Fis a scalar projection. 5 Since waves do not exist fundamentally, frequency is purely relational. 9 Application: Relational Young Interference The total phase is: ∆Φ = ∆Φgeo + ∆Φrel, where: ∆Φrel =κ[F(∆RA)−F(∆RB)]. The detected intensity is: I=I0[1+Vcos(∆Φ)]. Interference arises from the relational network itself. 10 Relational Dispersion and Micro-Variations of c The coherence kernel imposes: ωrel =Kf(ν)ν. This leads to: •structure-dependent propagation, •micro-variations of c, •effective photon mass in extreme environments. 11 Relational Blackbody Spectrum 11.1 Energy of a relational mode ε(ν)=ℏ0Kf(ν)ν. 11.2 Optimal coherence distribution ρ(ν)=Z−1e−βε(ν). 11.3 Relational discreteness g(ν) = Aν2Ge(ν), Ge(ν)→0as ν→ ∞. 6 11.4 Final spectrum u(ν)=Aℏ0ν3Ge(ν)Kf(ν) [exp(βℏ0Kf(ν)ν)−1]−1. Ultraviolet divergences disappear structurally. 12 Conclusion We have formulated a complete relational theory in which: •time does not exist, •physical quantities emerge from relational variations, •quantum phenomena become geometric, •the speed of light is a structural limit, •ultraviolet divergences vanish naturally. This framework opens the way to a deep relational unification of physics, independent of spacetime, waves, or particles. 7