Part I: A Scalar Field Approach to the Fundamental Principle of Gravity Structurable Spacetime, Quantum Consistency, and Observational Constraints

Part I: A Scalar Field Approach to the Fundamental Principle of Gravity Structurable Spacetime, Quantum Consistency, and Observational Constraints Mohamed Makraini1and Ali Mohamed Mustafa2 1Independent Researcher, M´alaga, Spain∗ 2Former Physics and Chemistry Student, University of Granada, Spain (Dated: July 19, 2025) We present a quantum formulation of spacetime as an emergent phenomenon from a scalar field ρ(x), whose expectation value ⟨ˆρ⟩governs its structural density. The equivalence principle arises as a statistical effect of the universal coupling between matter and ρ(x), eliminating the need for gravitons. Crucially, regions where ⟨ˆρ⟩ → 0 correspond to a pre-geometric Absolute Vacuum (AV), explaining JWST observations of early galaxies via non-standard light propagation. The theory predicts testable signatures in gravitational waves (high-frequency suppression) and colliders (heavy spin-0 resonances). Keywords: quantum spacetime, emergent gravity, absolute vacuum, JWST anomalies, nonmetric gravity I. INTRODUCTION The equivalence principle, cornerstone of general relativity, asserts the equality of inertial and gravitational mass. Despite high-precision experimental confirmations, its fundamental origin remains obscure [6]. We present a framework where this principle arises from dynamical properties of spacetime encoded in a scalar field ρ(x), termed the structural density. Unlike general relativity, where curvature is induced by mass-energy, our model postulates that geometry emerges from variations in ρ(x), analogous to modern aether theories but on a covariant basis. Anomalies in MICROSCOPE (2022) and discrepancies in ΛCDM suggest that a structural field like ρ(x) could resolve cosmological tensions, “similar to the emergent gravity paradigm [3,7], but with a covariant structural field”. A. Quantum Reformulation We extend the classical ρ(x) field to a quantum operator ˆρ(x) acting on a pre-geometric Hilbert space HVA: ˆρ(x)=ρ0+Zd3k (2π)3 1 √2ωkˆakeikx + ˆa† ke−ikx(I.1) where ˆa† kcreates spacetime structure quanta with dispersion relation ωk=qk2+m2 ρ. II. QUANTUM FOUNDATIONS A. Canonical Quantization The conjugate momentum operator: ˆπρ(x)=−iℏδ δρ(x)(II.1) with commutation relations: [ˆρ(x),ˆπρ(y)] = iℏδ(3)(x−y) (II.2) ∗[email protected]; This work was conceived and developed independently, with foundational reinforcement and critical discussion by Ali Makraini during the early stages. 2 B. Vacuum State Definition The Absolute Vacuum state |0⟩VA satisfies: ˆρ(x)|0⟩VA = 0 ∀x(II.3) while the structured spacetime emerges as coherent states: |ρ⟩= exp iZd3xρ(x)ˆπρ(x)|0⟩VA (II.4) III. MODIFIED FIELD EQUATIONS (QUANTUM) The Einstein equations become operator-valued: ˆ Gµν = 8πG ˆ T(ρ) µν +f(ˆρ)ˆ T(m) µν (III.1) with the stress-energy tensor including quantum fluctuations: ˆ T(ρ) µν =∂µˆρ∂νˆρ−1 2gµν(∂αˆρ)(∂αˆρ) + gµνV(ˆρ) (III.2) IV. RESOLUTION OF JWST ANOMALIES A. Light Propagation Model The observed redshift zobs relates to the cosmological redshift zcos via: 1+zobs = (1 + zcos) exp −Zpath dℓ λρ(IV.1) where λρ=⟨ˆρ⟩−1/2is the local structure scale. FIG. 1: Lightcone propagation through VA regions (ρ= 0) yields apparently older structures V. GRAVITON-FREE GRAVITY A. Two-Point Correlation Function The effective graviton propagator emerges as: Gµναβ(x, y) = ⟨ˆgµν(x)ˆgαβ(y)⟩−⟨ˆgµν(x)⟩⟨ˆgαβ(y)⟩(V.1) which decays exponentially for |x−y|> λρ, naturally regulating UV divergences. 3 VI. MODIFIED OBSERVABLES TABLE Phenomenon Prediction Test JWST early galaxies Apparent age reduction High-z luminosity function Gravitational waves f > ⟨ρ⟩1/2suppression LISA band cutoff CMB anomalies Power suppression at ℓ < 30 CMB-S4 TABLE I: Quantum VA theory predictions This approach eliminates the need for non-baryonic dark matter and vacuum energy by explaining all associated phenomena through the intrinsic structure of the field ρ(x), in the spirit of other emergent or modified gravity theories [14],[15],[16]. VII. ONTOLOGICAL FOUNDATIONS OF MASS AND SPACETIME The scalar field ρ(x) is not simply a dynamical field but a structural function that modulates the emergence of spacetime geometry itself. In this framework, we postulate that the fabric of spacetime arises from a modulation of vacuum structure, encoded in ρ(x), which defines the conformal factor relating the physical metric gµν to the background Minkowski metric ηµν : gµν(x)=ρ(x)ηµν.(VII.1) This scalar field embodies a structural vacuum – an ontological substrate responsible for generating gravitational and inertial effects. We interpret regions where ρ(x)→0 as approaching an ”Absolute Vacuum” (VA), in which spacetime loses its capacity to sustain physical processes. Conversely, higher values of ρ(x) correspond to structured regions where the geometry supports matter interactions. Gravitational mass is reinterpreted as a manifestation of local variations in ρ(x), while inertial mass reflects the resistance of this structure to deformation. The coupling between ρ(x) and the matter Lagrangian Lmunderpins the Principle of Equivalence: since all forms of matter couple to the same field ρ(x), they experience identical structural curvature, resulting in universal free fall. This ontological foundation allows us to rethink mass not as an intrinsic attribute of particles, but as a relational property between matter and the vacuum structure modulated by ρ(x). VA := {x∈M|ρ(x)=0,lim x→∂MTµν undefined}(VII.2) This contrasts with the classical vacuum in RG, which retains metric structure [1]. •Conceptual diagram: Include a figure showing: •Transition ρ(x)→0 as a ”dissolution” of spacetime. •Profile of ρ(r) around a point mass, compared to the Newtonian potential. 4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −5 −4 −3 −2 −1 0 1 r(radial distance) Gravitational Potential Φ(r) Comparison of Gravitational Potentials ΦNewton(r) = −1 r Φρ(r) = 1 2log r FIG. 2: Structural correction to Newtonian potential. Comparison of the classical Newtonian potential ΦNewton(r) and the structural potential Φρ(r) = 1 2log r, as predicted by the ρ-field model. The deviation becomes significant at short distances, suggesting an effective repulsion or scale regularization in strong-gravity regimes. The present framework stems from a radical ontological assumption: that the fabric of spacetime is not a pre-given geometric arena, but an emergent phenomenon arising from a fundamental scalar field ρ(x) which modulates its local structural density. This field plays a dual role—both geometrical and physical—serving as a measure of the existence and strength of spacetime structure. A. Postulate: The Absolute Vacuum (AV) We define the absolute vacuum (AV) as the primordial ontological state corresponding to ρ(x) = 0. In this regime: •No metric tensor gµν exists. •No causal structure is defined. •No propagation of fields or particles is possible. This state represents the pre-geometric condition, devoid of any spacetime ontology. The emergence of nonzero ρ(x) is the necessary and sufficient condition for a region of reality to admit geometric and dynamical structure. B. Emergence of Spacetime via Structural Activation When ρ(x)>0, the metric is defined as gµν(x)=ρ(x)ηµν,(VII.3) where ηµν is a fiducial Minkowski background in local inertial coordinates. This conformally flat structure implies that curvature, connections, and causal propagation arise entirely from the modulation of ρ(x). Thus, ρ(x) quantifies the degree of activation of spacetime itself. C. Structural Interpretation of Mass and Gravity In this picture, mass-energy does not “curve” spacetime in the traditional sense. Instead, mass locally thins the structural field ρ(x), reducing the effective geometric support. Gravitational attraction is thus reinterpreted as the 5 result of gradients in the structural density: Φρ(r) = 1 2log ρ(r), which plays the role of an effective gravitational potential. The structural field responds dynamically to matter via the coupling function f(ρ)Lmin the action. D. Theoretical Parallels This ontological perspective connects with several cutting-edge programs: •Emergent gravity approaches (e.g., Verlinde 2011 [3]), where geometry arises from entropic or informationtheoretic principles. •Relational dynamics (e.g., Barbour 2012 [5]), which posit that spacetime only exists where physical relations do. •Holographic and quantum gravity paradigms, where spacetime is emergent from quantum entanglement or microstate structure [4,8]. E. Foundational Implications This structural ontology: •Unifies spacetime and matter at the level of field activation. •Provides a clear ontological criterion for the existence of physical laws (i.e., ρ(x)>0). •Offers novel ways to interpret vacuum phases, cosmic singularities, and horizon transitions. The following sections formalize these principles and explore their observational consequences. CANONICAL ANALYSIS AND CONSERVATION EQUATIONS The canonical dynamics associated with the ρfield is derived through the energy-momentum tensor: T(ρ) µν =∂µρ ∂νρ−gµν 1 2∂αρ ∂αρ−V(ρ)(VII.4) and its interaction with matter and curvature gives rise to modified conservation equations: ∇µT(tot) µν =df dρ Lmatter +dh dρR∂νρ(VII.5) This implies that the geodesic trajectories of matter are modified by gradients of ρ, which represents a falsifiable prediction in astrophysical contexts. VIII. ANALYTICAL SOLUTIONS A. Static Spherically Symmetric Case (ρ(r)) We consider a vacuum, spherically symmetric configuration with the metric: ds2=ρ(r)−dt2+dr2+r2dΩ2,(VIII.1) 6 where dΩ2=dθ2+ sin2θdϕ2. In the absence of matter, the variation leads to the Laplace-like equation: 1 r2 d dr r2dρ dr = 0,(VIII.2) with general solution: ρ(r) = 1 −rs r,(VIII.3) where rsis an integration constant. The resulting line element becomes: ds2=1−rs r−dt2+dr2+r2dΩ2.(VIII.4) This solution mimics Schwarzschild-like behavior, but results from structural dilution rather than intrinsic curvature. r ρ(r) rs ρ(rs) = 0 B. Cosmological Dynamics (ρ(t), a(t)) We now consider a homogeneous, isotropic universe with: ds2=ρ(t)−dt2+a(t)2dr2 1−kr2+r2dΩ2,(VIII.5) where a(t) is the scale factor and kthe spatial curvature. The modified Friedmann-like equation [2], [8], in the absence of additional coupling terms, becomes: ˙a a2 +˙ρ ρ·˙a a+k a2=8πG 3 ρm ρ,(VIII.6) where ρmdenotes classical matter density. This formulation allows for expansion scenarios without a cosmological constant, and for structural oscillations affecting cosmic dynamics. IX. MATHEMATICAL FRAMEWORK AND LAGRANGIAN FORMULATION A. General Action and Metric Definition We consider a scalar field ρ(x) from which the effective metric emerges via gµν(x)=ρ(x)ηµν,(IX.1) 7 where ηµν is the Minkowski metric. The total action is given by: S=Zd4x√−g1 2κR+αρ(x)+β(∂µρ)(∂µρ)−V(ρ)+f(ρ)Lm,(IX.2) with κ= 8πG,Rthe Ricci scalar from gµν,V(ρ) a structural potential, and f(ρ) the matter coupling. B. Field Equations from Variations 1. Variation with respect to gµν The variation yields modified Einstein-type equations: Gµν =κhT(ρ) µν +f(ρ)T(m) µν i,(IX.3) with T(ρ) µν =∂µρ∂νρ−1 2gµν(∂λρ)(∂λρ)+gµν(αρ −V(ρ)).(IX.4) 2. Variation with respect to ρ(x) We obtain: β□ρ−3 2κ□ρ ρ−1 2 (∂ρ)2 ρ2+α−dV dρ +df dρLm= 0.(IX.5) C. Structural Conservation Law We postulate: ρ(x)+λLm=ρ0,(IX.6) and include it in the action via a Lagrange multiplier X(x): S=Zd4x√−g[Lgrav +Lm+X(x)(ρ+λLm−ρ0)] .(IX.7) - Matter-ρcoupling: f(ρ) = 1 + λρn(justified by conformal invariance with n= 2 for bosons) (IX.8) This form is motivated by scalar-tensor theories [2], where conformal invariance dictates the coupling. - Potential V(ρ): V(ρ)=µ2ρ2+λρ4+ Λ log ρ ρ0−1(IX.9) *Discuss the cases µ2>0 (stable VA) and µ2<0 (phase transition).* 8 Full Variational Framework To ensure the rigorous consistency of the model, we formalize the total action with all relevant couplings: S[ρ, gµν, ψ] = Zd4x√−g1 2∂µρ ∂µρ−V(ρ)+f(ρ)Lmatter(ψ, gµν)+h(ρ)R(IX.10) where ψcollectively represents the matter fields. D. Complete variational formalism We start from the generalized action: S=Zd4x√−g1 2∂µρ ∂µρ−V(ρ)+f(ρ)Lmatter +h(ρ)R(IX.11) The equation of motion for the scalar field ρis obtained by functional variation: □ρ+dV dρ =df dρLmatter +dh dρR(IX.12) Analysis canonical The canonically conjugate variable is: πρ=∂L ∂˙ρ= ˙ρ√−g(IX.13) And the density Hamiltonian is obtained as: H=1 2π2 ρ+1 2(∇ρ)2+V(ρ)−f(ρ)Lmatter −h(ρ)R(IX.14) Energy conservation is verified via ∇µTµν = 0 if f(ρ) and h(ρ) are sufficiently smooth. Classical limit In the regime where ρ≈ρ0+δρ, the equation linearizes as: □δρ +d2V dρ2ρ0 δρ =df dρρ0Lmatter +dh dρρ0 R(IX.15) This allows the study of stability and propagation of vacuum excitations. E. Emergent Equivalence Theorem Theorem IX.1 (Mass–Gravity Structural Equivalence).Let {Φi}be a system of fields coupled to the ρfield by f(ρ)Li and to spacetime by h(ρ)R. If ρ(x)is regular and its gradient induces an effective metric, then every inertial source coupled to f(ρ)will experience an effective curvature proportional to its mass, without needing a fundamental metric. Proof. The derivative action induces an effective metric geff µν ∼ηµν +∂µρ∂νρ, under which the mass and gravitational energy terms derive from the same functional coupling f(ρ). This functional symmetry guarantees that the inertial and gravitational mass emerge from the same scalar field ρ, fulfilling the equivalence principle in the emergent sense. 9 X. SOLUTIONS AND PREDICTIONS Table of observable corrections: Observable Predicted Correction Experimental Limit Perihelion precession ∆ϕ=6πGM a(1−e2)1 + ϵrs rϵ < 10−5(Cassini) [12] Light deflection δϕ =4GM c2b(1 + δ)δ≲10−4(Euclid) [11] CMB anisotropies (T/T0) Vacuum SSB via ρ(t) Planck 2018 data [8] TABLE II: Comparison of observational effects. Structural deviations from standard General Relativity predictions and corresponding experimental constraints. These corrections arise from the ρ-field coupling. For a direct comparison of the gravitational potentials, see Fig. 2, where the Newtonian potential ΦNewton(r) is contrasted with the structural potential Φρ(r) = 1 2log ρ(r). XI. OBSERVATIONAL SIGNATURES The structural scalar field ρ(x) introduces distinct gravitational effects that differ subtly from General Relativity (GR), enabling falsifiability through precision experiments and cosmological surveys [9] . These effects emerge due to the modulation of the conformal metric and the absence of full Riemannian curvature: •Gravitational Lensing and Light Bending: Since photons follow geodesics of the conformally scaled metric gµν =ρ(x)ηµν, the deflection of light near massive sources depends on the gradient of ρ(r). Unlike GR, where spacetime curvature dominates, in this model lensing is driven by structural inhomogeneities. This could result in either slightly enhanced or reduced deflection angles, testable via high-precision lensing surveys (e.g. Euclid, LSST) [11]. •Perihelion Precession and Orbital Dynamics: The effective radial profile ρ(r) alters the geodesic equation for test bodies. This generates small corrections to planetary perihelion shifts. Anomalies in Mercury’s orbit or tight binary systems could serve as probes for deviations from Newtonian potential derived from ρ(r)=1−rs/r. •Absence of Singularities: Regions where ρ(x)→0 act as dynamical boundaries for structured spacetime. Black hole interiors or big bang singularities may be replaced by structural vacua. This regularization should leave imprints in gravitational wave echoes or shadow profiles of compact objects. •Structure Formation and Power Spectrum: In the cosmological sector, the evolution of ρ(t) modulates the expansion rate and matter coupling. As a result [8], the growth function of density perturbations acquires time-dependent corrections. The linear matter power spectrum may present scale-dependent suppression or amplification not attributable to a cosmological constant. •CMB and ISW Effect: A nontrivial evolution of ρ(t) induces late-time variations in the gravitational potential, affecting the integrated Sachs-Wolfe effect. These changes can shift the low-ℓmultipoles of the CMB anisotropy power spectrum and may be observable in cross-correlations with galaxy surveys [8]. •Quantum fluctuations of ρ(x): ⟨δρ2⟩∼ℓ−3 Pl (Planck scale) (XI.1) •Relation to holography: “The area of horizons depends on ρ(A), suggesting an entropy S∼Rρ(x)√h d2x.” XII. DISCUSSION AND OUTLOOK We have proposed a gravitational theory in which the fundamental ontological structure of spacetime is encoded in a scalar field ρ(x). Unlike classical metric-based gravity, where curvature is a manifestation of mass-energy, here curvature is emergent and derived from the conformal modulation of vacuum structure. The equivalence principle is no longer assumed but results naturally from the universal coupling between all matter and the structural field ρ(x). Our results show that: