Structural Gravitational Principle: Scalar gradient between space-time and matter through the non-spacetime Absolute Vacuum

Structural Gravitational Principle: Scalar gradient between space-time and matter through the non-spacetime Absolute Vacuum Mohamed Makraini1,* and Ali Makraini2 1Universidad De Granada, Granada, España *Corresponding author: [email protected] Abstract We propose that gravity is not a force, but rather an emergent effect of an imbalance between space-time and a deeper substratum (the Absolute Vacuum). This unifies quantum entanglement, mass, and geometry under a single principle. And while General Relativity masterfully describes the geometric mechanism of gravity as the curvature of space-time, it does not explain what that curvature is or its ontological origin. In this proposal, it is argued that matter does not act as a direct source of curvature, but rather as a disruptor of symmetry between space-time and a deeper structure: the Absolute Vacuum (AV). This AV represents the limit of the geometric scaffolding of the quantum vacuum, where the density of space-time tends to zero and geometry ceases to exist continuously. Thus, gravity does not originate as a local deformation caused by matter, but as an emergent structural gradient between the material content and the collapse of space-time toward the AV, understood as the final frontier of geometric existence. Keywords: emergent gravity, Absolute Vacuum, scalar field ρ, quantum entanglement, geometric density, structural quantum computation, ontological decoherence, symmetry breaking, effective action. 1. Introducción General Relativity has successfully established a geometric framework for understanding gravity as a manifestation of the curvature of space-time induced by the presence of energy and matter. However, this approach fails to answer a more fundamental question: what is the ontological nature of such curvature? In particular, it remains unclear what structure underlies space-time itself, nor how geometry arises when the energy density tends to zero. This work proposes a deeper structural gravitational principle, in which the curvature of space-time does not arise directly from matter as a source, but rather from a symmetry breaking between geometry and a fundamental entity: the Absolute Vacuum (AV). This AV is defined as a non-spacetime substrate that represents the lower limit of structural density in the universe. Matter, when interacting with geometry, induces an imbalance that manifests as a scalar gradient toward the AV, thus generating the curvature we interpret as gravity. At the quantum level, the mass of quarks is dominated not by their rest mass but by the confinement energy (QCD), which creates local imbalances in energy density. These imbalances generate an exclusion zone: a kind of ontological incompatibility between confined matter and pure spacetime. This exclusion zone acts as a connecting channel to the Absolute Vacuum (AV). The exclusion principle applies not only to fermions (as in the Pauli case), but also between confined matter and the metric structure. Matter does not "curve" space directly, but by becoming incompatible with space, it opens a structural gap that deforms the geometric environment in the direction of the AV. This scalar gradient (matter ↔space-time) is what generates the “motion” or gravitational attraction. What we call macroscopic gravity is not a “force”ora“curvature” in itself, but the emergent effect of this partial dissociation between mass and space-time, compensated by the VA–quantum vacuum gradient. In other words, mass locally reduces the density of space-time to maintain an internal measure (minimal geodesic) between that object and the rest of the universe. 1 At Planck scales [15], what breaks down is not gravity, but space-time itself. That is why gravity is not observed at these scales, but characteristic quantum phenomena do manifest themselves, such as: •Entanglement without a defined path •Oscillation without a determined classical energy •Superposition without localizable coordinates •Absence of classical causality This perspective redefines the relationship between matter, spacetime, and gravity, proposing that quantum entanglement, the emergence of mass, geometric dissolution at the Planck scale [15], and the very causal structure of the universe can be explained by the behavior of a fundamental scalar field ρ(x), which measures the “density of geometric existence” at every point in the universe. The field ρ(x)acts as a thermometer of geometric density: high values indicate classical spacetime, while ρ→0 marks the transition to the Absolute Vacuum [20]. 1.1. Formal Framework of the Scalar Field The field ρ(x)quantifies the structural density of spacetime relative to the AV (Figure 1). Its action (1.1) generalizes scalar-tensor theories, but with a potential V(ρ)that imposes an asymmetry between geometry and AV (Appendix A). Note that, unlike Brans-Dicke, ρis not an auxiliary gravitational field, but an ontological mediator. Let ρ(x)be a real scalar field, defined on a Lorentzian manifold (M, gµν ), whose magnitude represents the structural density of spacetime relative to the Absolute Vacuum. The following effective action [7][8] is proposed: S=Zd4x√−g1 2∂µρ ∂µρ−V(ρ)+f(ρ)Lmatter +Lgravity(gµν , ρ)(1.1) donde: *V(ρ)It is a scalar potential that imposes asymmetry between regions of high and low geometric density, with a typical shape: V(ρ)=−α ρ ln ρ ρ0 +ε(1.2) with α > 0,ρ0a normalization scale, and ε≪1a regulation to avoid logarithmic divergences. *f(ρ)is a coupling function between the VA field and the matter fields (fermions, gauge bosons, Higgs), allowing a seamless transition between emergent geometry and physical content. *Lgravity can initially be the Einstein-Hilbert term: Lgravity =1 2κR(1.3) or extended to include nonlocal or ρ-dependent terms, as in scalar-tensor gravity theories, and even extended to include holographic effective geometries as in AdS/CFT-type theories [5]. The equation of motion for ρ, obtained by functional variation, is: □ρ+dV dρ =f′(ρ)Lmatter +δLgravity δρ (1.4) The ρfield transforms as a scalar under general coordinates and guarantees the Lorentz invariance of the model. In the ρ→0limit, geometry ceases to have physical meaning, recovering the concept of the Absolute Vacuum as the ontological limit of space-time. 2 This formulation establishes the foundations for understanding gravity as an emergent phenomenon, not from energy, but from the structural imbalance between space-time and its lower limit: the Absolute Vacuum. 1.2. Dynamic consistency and couplings of the ρ(x)field The scalar field ρ(x), proposed as a structural mediator between matter and the Absolute Vacuum (AV) [20], has internal dynamics and couplings that must be consistent with the principles of General Relativity and Quantum Field Theory. Its fundamental dynamical properties are developed below. 1. Energy-momentum tensor of the field ρFrom the scalar Lagrangian: Lρ=1 2∂µρ∂µρ−V(ρ),(1.5) the canonical energy-momentum tensor is obtained: T(ρ) µν =∂µρ ∂νρ−gµν 1 2∂αρ ∂αρ−V(ρ).(1.6) This tensor couples to the geometry in the modified Einstein-Hilbert term: Gµν =κT(matter) µν +T(ρ) µν .(1.7) 2. Functional form of coupling f(ρ)To preserve stability and invariance under scaling transformations in the limit ρ→0, an analytic function of the type is proposed: f(ρ) = 1 −e−λρ,(1.8) where λis a universal coupling parameter with inverse dimension of ρ. This coupling applies to the standard matter Lagrangian: Lint =f(ρ)Lmatter,(1.9) suggesting that ρdynamically regulates the effectiveness of material content in generating geometry. 3. Broken Symmetry The ρfield breaks symmetry under translational transformations in the local geometric density. In particular, it breaks homogeneous scalar symmetry: ρ(x)→ρ(x)+δρ, con δρ =constante.(1.10) This implies that the presence of ρintroduces an ontological gradient that distinguishes regions of spacetime. 4. Dynamical Conservation If the curvature coupling is minimal and not explicitly dependent on derivatives of ρ, then the total energy-momentum tensor is conserved: ∇µT(matter)µν +T(ρ)µν= 0.(1.11) In extended versions with curvature feedback or non-local sources, an effective source may appear that modifies this conservation, as modeled in the equation: □ρ+dV dρ =βTµ µ+γR, (1.12) where Tµ µis the trace of the energy-momentum tensor and Rthe scalar curvature. 3 2. Physical and observational predictions From the proposed framework, several predictions are derived that can, in principle, be tested experimentally: 1. Modifications to the Higgs boson mass spectrum at colliders The coupling f(ρ)·Lmatter implies that the scalar field ρacts as an emergent mass regulator. This may induce a nonlinear dependence of the Higgs effective mass on the local ρdensity: mef H∼m0 H·1+ξ·∂ρ ρ0(2.1) where ξrepresents a VA-Higgs coupling coefficient. This implies a possible variation in the decay channels and Higgs width observed at the LHC, especially in regions with high energy concentration. For ξ∼10−3and ∂ρ/ρ0∼0.1in LHC collisions, ∆mH/mH∼10−4, detectable in the Higgs widths. 2. Neutrino Oscillations Without Intrinsic Mass The presence of ρ(x)as a structural gradient can explain neutrino flavor oscillations without assuming Dirac or Majorana masses, through couplings of the type: Lν=ρ(x)·¯νLγµ∂µνL(2.2) This generates an effective geometric phase that can drive oscillations through the VA channel without resorting to explicit mass terms, a prediction that could be verified by precise measurements of anomalies in the PMNS matrix. 3. Cosmological Deviations in the Deep Microwave Background (CMB) In regions of the early Universe where ρexhibits steep gradients, non-Gaussian spectral fluctuations in the CMB, associated with regions of partial geometric dissociation, could have been generated. A local excess of anisotropies at low angular scales (low multipoles ℓ), as well as a residual "fall-off toward the VA" signal [16][19], are expected. 4. Nonclassical behavior in black holes and compact objects The region ρ→0suggests a new type of internal structure for black holes, in which the singularity is replaced by a controlled geometric dissolution [9][10]. This could be expressed as: •Hawking evaporation delay •Discrete emission from VA-validatable states •Small but detectable violations of the no-hair theorem As demonstrated in recent studies of analog gravity [23], ρ(x)gradients could be measured in Bose-Einstein condensates. Furthermore, the link to holographic theories [24] suggests that VA could correspond to a nongeometric bulk limit in AdS/CFT dualities. These predictions constitute an exploratory, but falsifiable, framework for validating the structural gravitational principle and the existence of the Absolute Vacuum as a deep physical component of the universe. 3. Quantum Entanglement as Structural Validation in the Absolute Vacuum In the conventional framework, quantum entanglement [17] is interpreted as a statistical correlation between subsystems that cannot be explained by hidden local variables. However, this view does not specify the ontological basis of such a correlation, nor how its structural coherence is preserved beyond space-time. From the perspective of the Absolute Vacuum [20], we propose that entanglement is not an emergent property of the composite system, but rather a simultaneous structural validation of two or more physical entities 4 within the same region of ρ= 0, mediated by the VA substrate. This validation functions as a shared quantumtemporal fingerprint, analogous to a cryptographic signature or a blockchain-like integrity structure. Formally, let |Ψ⟩AB be an entangled state of two particles A and B: |Ψ⟩AB =X i,j cij |i⟩A⊗|j⟩B(3.1) We define the VA structural footprint as a non-spatiotemporal validation function: ρAB(x)=ρ0δ(tA−tB) Ω(|Ψ⟩)(3.2) where: •ρ0is the minimum density for structural validation, •δ(tA−tB)guarantees ontological simultaneity of events, •Ωis a coherence operator on the entangled subspace. Measuring one of the particles breaks the structural symmetry: MA: Ω(|Ψ⟩)→ΩA⊗ρred B⇒ρAB →0(3.3) That is, breaking VA validation replaces the concept of state collapse. The change does not propagate through space, but rather the common validation in the VA substrate is deactivated. This mechanism: •Preserves nonlocality without violating relativity, •Prohibits cloning due to the impossibility of duplicating Ω, •Justifies that entangled states cannot be observed locally if the entire validated structure is not measured. Entanglement, in this context, is not a statistical property of a system, but an irreducible ontological unit validated outside of space-time, sustained by the ρ(x)field as its fundamental structure. 4. Formal Justification of Compatibilities This phase develops the compatibility of the Absolute Vacuum (AV) model with existing theoretical frameworks, particularly General Relativity (GR) and Quantum Field Theory (QFT). We explore its correspondence with scalar-tensor theories such as Brans-Dicke, [21], its invariance under Lorentz transformations, and the construction of a Lagrangian that unifies geometry, quantum fields, and the ontological effect of the AV. As explained in Section 1.1, the scalar field ρ(x)represents the structural density of spacetime relative to the Absolute Vacuum. To ensure that this principle has physical and not just interpretative validity, it is necessary to establish its formal compatibility with these frameworks, ensuring both mathematical consistency and the possibility of experimental testing. SBD =Zd4x√−gϕR −ω ϕ∂µϕ∂µϕ+Lm(4.1) The VA model proposes a fundamental scalar field ϕV A, equivalent to ρ(x), with similar behavior, but with a deeper ontological interpretation. A Lagrangian of the type is proposed: LV A =√−gf(ϕV A)R−1 2∂µϕV A∂µϕV A −V(ϕV A)+Lmatter(ψ, Aµ, ϕV A)(4.2) where f(ϕV A)can take the form 1+αϕ2 V A and V(ϕV A)represents a self-interaction potential. 5 5. Comparison with Brans-Dicke Theory Brans-Dicke theory [21] introduces a scalar field ϕthat modifies the coupling between the metric and matter. Its action is expressed as: SBD =Zd4x√−gϕR −ω ϕ∂µϕ∂µϕ+Lm(5.1) The VA model proposes a fundamental scalar field ϕV A, equivalent to ρ(x), with similar behavior, but with a deeper ontological interpretation. A Lagrangian of the type is proposed: LV A =√−gf(ϕV A)R−1 2∂µϕV A∂µϕV A −V(ϕV A)+Lmatter(ψ, Aµ, ϕV A)(5.2) where f(ϕV A)can take the form 1+αϕ2 V A and V(ϕV A)represents a self-interaction potential. Unlike entropy-based emergent gravity [19], our model attributes gravity to a measurable ontological gradient ρ(x), not to statistical information. 6. Lorentz Invariance of the Field ϕV A To be compatible with Special Relativity, the field ϕV A must transform as a scalar under the Lorentz group: ϕ′ V A(x′)=ϕV A(x),con x′µ= Λµ νxν(6.1) VA excitations do not violate this symmetry; on the contrary, they can represent vacuum perturbations that propagate while respecting relativistic causality. 7. Unified Lagrangian with GR and QFT We propose a full Lagrangian that incorporates geometry, VA field, and quantum fields: Ltotal =√−g1 2κf(ϕV A)R−1 2∂µϕV A∂µϕV A −V(ϕV A)+Lmatter(ϕV A)(7.1) The term Lmatter may include: •Fermion mass modulation: Lf=¯ ψ(iγµ∂µ−m(ϕV A))ψ •Correction to the gauge coupling constant: LY M =−1 4g2(ϕV A)FµνFµν •Influence on the Higgs potential: V(H, ϕV A)=λ|H|4−µ2(ϕV A)|H|2 8. Variation of Lagrangian and Field Equations From the full Lagrangian, the field equations are obtained varying with respect to gµν and ϕV A. These equations generalize General Relativity by coupling ϕV A to curvature and matter. The derivation (Appendix A) shows how the term ∇µ∇νf(ϕV A)emerges from the nonminimal variation of the coupling. 6 8.1. Modified Einstein Equation Variation with respect to the metric gµν gives: 1 2κ[f(ϕV A)Gµν + (gµν□−∇µ∇ν)f(ϕV A)]=Tmatter µν +T(ϕV A) µν (8.1) where the energy-momentum tensor of the field ϕV A is: T(ϕV A) µν =∂µϕV A∂νϕV A −gµν 1 2(∂ϕV A)2+V(ϕV A)(8.2) 8.2. Modified Klein-Gordon equation for ϕV A Variation with respect to ϕV A leads to: □ϕV A −dV dϕV A +1 2κ df dϕV A R=∂Lmatter ∂ϕV A (8.3) This equation shows how ϕV A couples to both the curvature and the material content of the universe. 9. Reduction to the Limit of Classical GR When the ϕV A field stabilizes at a constant value ϕ0, the model must be reduced to conventional GR: ϕV A →ϕ0⇒f(ϕV A)→1, V (ϕV A)→Λ(9.1) So: LV A →√−g1 2κR−Λ+Lm(9.2) 10. Physical Implications of the VA •Corrections to the Newtonian potential: modified MOND or Yukawa effects [19]. •Variable effective mass of particles in regions of different VA states [20]. •Influence on inflation and cosmic expansion [15]. •Possible quantum refraction of the vacuum (interferometry, neutrons) [16]. These consequences open up the possibility of designing critical experiments where the presence of the VA can be indirectly detected through geometric-quantum observables, consolidating its role as a new fundamental component of the universe. 11. Simplified Examples: Solutions with a Rigorously Controlled Formalism 11.1. Homogeneous Field in Flat Space We consider a flat universe with a Minkowski-type metric gµν =ηµν , and a homogeneous scalar field VA dependent only on time: ϕV A(t)[22]. The scalar action reduces to: 7 S=Zd4x−1 2(∂tϕV A)2−V(ϕV A)+Lmatter(ϕV A)(11.1) The dynamic equation becomes: ¨ ϕV A +dV dϕV A =∂Lmatter ∂ϕV A (11.2) This represents a Klein-Gordon-type evolution system with an effective source, where the right-hand side can represent feedback from the material content (e.g., fermion or boson mass). 11.2. Effective mass generation in a coupled fermion We consider the Lagrangian for a Dirac spinor ψwith mass modulated by ϕV A: Lf=¯ ψ(iγµ∂µ−m(ϕV A))ψ(11.3) Let’s assume a coupling function of the type: m(ϕV A)=m0+ξϕn V A,con ξ∈R, n ∈N(11.4) So the effective mass depends on the structural environment: mef =m(ϕV A(x)) (11.5) In regions where ϕV A →0, we recover the reference mass m0. In VA-modified regions, the mass is perturbed depending on the local structural configuration. This type of coupling can have consequences for neutrino oscillations, the Higgs width, or the initial conditions of the universe. 11.3. Simple cosmological solution: FLRW space We consider a flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric: ds2=−dt2+a(t)2dx2+dy2+dz2(11.6) The field ϕV A(t)generates a density and pressure given by: ρϕ=1 2˙ ϕ2 V A +V(ϕV A)(11.7) Pϕ=1 2˙ ϕ2 V A −V(ϕV A)(11.8) The modified Friedmann equations are: H2=κ 3(ρmatter +ρϕ)(11.9) ˙ H=−κ 2(ρmatter +Pmatter +ρϕ+Pϕ)(11.10) These solutions show how the VA can contribute to the dynamics of the universe as a structural dark field. 12. Field Equations The fundamental equations of the model are (see Appendix ?? for details): Gµν =κT(total) µν (12.1) 8 A. Derivations of the Lagrangian A.1. Variation with respect to the metric The variation of the modified Einstein-Hilbert term is: δ√−gf(ϕV A)R=f(ϕV A)δ(√−gR)+R√−gdf dϕV A δϕV A.(A.1) The details of the derivative terms ∇µ∇νf(ϕV A)are obtained by integrating by parts... x(espacio) ρ(x) Densidad estructural ρ(x) Materia VA (ρ→0) Gradiente ∇ρGradiente ∇ρ El gradiente de ρ(x)genera gravedad como efecto emergente A.2. Variation with respect to ϕV A The modified Klein-Gordon equation (8.3) follows from: δS δϕV A =□ϕV A −dV dϕV A +1 2κ df dϕV A R−∂Lmatter ∂ϕV A = 0.(A.2) A.3. Variation with respect to ϕV A The modified Klein-Gordon equation (8.3) follows from: δS δϕV A =□ϕV A −dV dϕV A +1 2κ df dϕV A R−∂Lmatter ∂ϕV A = 0.(A.3) Full mathematical details are in the Online Appendix. B. Implications for Quantum Computing: From Formal Qubit to VA Structural Validation The traditional approach to quantum computing is based on the formal interpretation of the qubit as a vector in a Hilbert space, subject to unitary transformations and measurement collapses. This mathematical representation, although operationally efficient, lacks a clear physical ontology of what a quantum state actually is and how its coherence is preserved.[11][12] From the perspective of the Absolute Vacuum,[20], it is proposed to reinterpret the qubit as a structurally validated unit in the ρ(x)field. Quantum superposition is not described as an abstract coexistence of states, 9