Structural Emergence Model of Space-Time from $\rho(x)$

Structural Emergence Model of Space-Time from ρ(x) Mohamed Makraini1,* and Ali Makraini2 1University of Granada, Granada, España *Corresponding author: [email protected] Abstract This work introduces the fundamental formulation of a new theory of spacetime, based on a structural scalar field ρ(x)whose dynamics determine the very existence of geometry, causality, and physical propagation. In contrast to the traditional paradigm, in which spacetime is treated as a preexisting geometric entity—whether classical or subject to quantization—this work proposes a more radical framework: that the spacetime structure is not fundamental, but emergent from a structural scalar field ρ(x), whose dynamics govern the very possibility of the existence of geometry, causality, and propagation. From this hypothesis, a Lagrangian formalism without an a priori metric is developed, in which gravity appears as an effect derived from the gradients of ρ(x), and matter does not curve spacetime, but rather destructures it. This framework allows us to reinterpret fundamental phenomena—such as quantum collapse, cosmic acceleration, or gravitational singularities—as structural transitions in ρ, and proposes a new conception of the absolute vacuum as a real physical limit: a state without space, without time, but structurally definable and quantizable. The model presented here constitutes a coherent, falsifiable, and radically innovative alternative to conventional geometric theories, and lays the groundwork for a complete reconstruction of fundamental physics from first structural principles. [8] Keywords: emergent gravity, Absolute Vacuum, scalar field ρ, quantum entanglement, geometric density, structural quantum computation, ontological decoherence, symmetry breaking, effective action. Fundamental axioms of the structural model 1. The basis space Mis a 4-dimensional differentiable topological manifold with no a priori metric. 2. There exists a structural scalar field ρ:M → R+ 0that locally determines the existence of physical structure. 3. The effective metric gµν =ρ ηµν is only defined in regions where ρ>0. 4. All physical dynamics (fields, causality, propagation) are restricted to physically active regions Uρ={x∈ M|ρ(x)>0}. 5. The limit ρ→0defines the Absolute Vacuum VA, a state without structure or propagation. 1. Diagnosis: The Foundational Error in the Conception of SpaceTime Modern physics has almost universally operated under the assumption that space-time has an ontological existence prior to and independent of the physical phenomena it contains. This assumption is manifested in: •General Relativity, where space-time is a dynamic geometric background curved by energy-matter. •Quantum gravity theories, where attempts are made to quantize this background as if it were just another physical field. 1 Although these formulations have been successful in certain phenomenological regimes, they are based on a possibly erroneous conception: that geometry is primary, and phenomena are secondary. In contrast, this work proposes an ontologically inverse thesis: that the geometry of spacetime is not a fundamental entity, but an emergent construct whose local validity depends on the value of a structural scalar field ρ(x). This approach does not attempt to patch up existing frameworks—such as semiclassical gravity or perturbative approximations of curved spacetimes—but rather replaces them with a more fundamental architecture. Just as thermodynamics emerges from microscopic degrees of freedom in statistical physics, here geometry emerges as an effective effect of the structural configuration of a scalar field ρ(x). This analogy is not merely formal: it implies that gravitational and quantum phenomena can share a common structural root, without the need for forced quantization of the background. [1] Matter does not curve space-time: it dissolves it. And gravity is not a deformation of the background, but the manifestation of structural gradients in ρ(x)that determine, point by point, how much "spacetime reality" can be sustained. This conceptual inversion demands a profound rewriting of theoretical formalism. It is not a matter of rejecting the observations that have validated current theories, but of relocating them within a more primitive and structurally coherent framework. Instead of unifying gravity with quantum mechanics by forcing geometry into the quantum domain, the proposal is to reconstruct both from a common origin: the dynamics of ρ(x). Thus, the objective of this work is not only to correct past theoretical errors, but to offer a new starting point from which the great unsolved enigmas of contemporary physics can be transformed into necessary consequences of a deeper structural principle. 1.1. Rigorous Mathematical Formalism of the Structural Theory of the Vacuum 1. Fundamental Definitions. Base space: Let Mbe a differentiable topological 4-dimensional manifold, with no a priori metric. Note: We do not start from a Lorentz space or a given metric structure. All geometry emerges from the field ρ(x). 2. Scalar structural field. Definition: ρ:M → R+ 0 ρrepresents the local density of structural existence, responsible for sustaining the possibility of: •Geometry •Causality [2] •Fields •Dynamics 3. Fundamental action (no prior metric) Let ϕibe a set of scalar matter fields defined on M[3][5]. We define: S[ρ, ϕi] = ZM d4x[ρ(x) (ηµν∂µϕi∂νϕi−V(ϕi)) + Λ(ρ)] (1.1) Integration is performed with respect to a fixed local coordinate basis until an effective metric is defined. 2 4. Field Equations Variation with respect to ϕi: ∂µ(ρ(x)∂µϕi)+ρ(x)dV dϕi = 0 (1.2) Variation with respect to ρ: ηµν∂µϕi∂νϕi−V(ϕi) + dΛ dρ = 0 (1.3) 5. Emergence of effective metrics Definition: gµν(x):=ρ(x)ηµν (1.4) From here we obtain: •det g=ρ4det η=−ρ4 •gµν(x) = 1 ρ(x)ηµν 6. Induced curvature The Levi-Civita connection tensor (twistless) for gµν =ρηµν gives: Γλ µν =1 2ρδλ µ∂νρ+δλ ν∂µρ−ηµνηλσ∂σρ(1.5) Scalar curvature: R=3 2□ρ ρ2−3 2 (∂µρ)(∂µρ) ρ3(1.6) 7. Rewriting the action in terms of emerging metrics [4] Once ρ>0, the action is rewritten: S=Zd4x√−ggµν∂µϕi∂νϕi−V(ϕi) + ˜ Λ(ρ)(1.7) Where: √−g=ρ2,˜ Λ(ρ) = Λ(ρ) ρ2(1.8) 8. Rigorous physical interpretation: Structural limit ρ→0(Absolute Vacuum) •gµν =ρ ηµν →0⇒There is no geometry: space-time collapses. •−g→0⇒There is no volumetric measurement; the action integral Rd4x√−gLvanishes. •R[ρ]→ ∞ ⇒ Formally divergent curvature, but physically harmless: there is no structure to support it. 3 Component Physical interpretation ρ(x)Scalar density quantifying the local structural existence of spacetime. gµν =ρ ηµν Effective metric induced by the structural field on Minkowski space. R[ρ]Emergent curvature associated with the gradients of ρ(x), with no a priori affine connection [6]. Λ(ρ)Absolute vacuum energy modulated by the local structure of spacetime. ϕiPhysical fields (matter, gauge, etc.) defined on the effective geometry induced by ρ(x). •Physical interpretation: The limit ρ→0describes the Absolute Vacuum (VA): There is no space, no time, no energy, no propagation. It is a state without structure, without fields, and without dynamics. 9. Final Formal Definitions Definition 1.1 (Absolute Vacuum).The Absolute Vacuum is defined as the set: VA:= x∈ Mρ(x)=0(1.9) In VA, every tensor, field, or dynamic cancels out. There is no metric, no curvature, no propagators: there is no physical structure of any kind. Definition 1.2 (Physically active region).The physically active region is defined as: Uρ:= x∈ Mρ(x)>0(1.10) In Uρ, the universe has local structure: there are metrics, fields, propagation and dynamics. Compatibility with observed phenomenological limits Despite its unconventional structure, the structural emergence model reproduces a Minkowski-type metric gµν = ρηµν in the limit, and under mild perturbations of ρ, a gravitational behavior analogous to General Relativity is recovered. [7] Likewise, in regions where ρ(x)>0and approximately constant, a perturbative expansion of the quantum fields ϕican be defined over an effective geometry, compatible with the standard predictions of quantum field theory in curved spacetime. Thus, this proposal does not contradict current empirical results, but rather extends them toward a more general and deeper structure. Physical Predictions and Falsifiability Criteria One of the pillars of any viable physical theory is its ability to generate predictions that are distinguishable from existing frameworks, as well as its openness to being refuted by experience. Although the model proposed here is based on a deep, non-perturbative ontological architecture, there are distinctive phenomenological consequences that allow for its eventual contrast with observation and experiment. Some of them are described below: 1. Structural Deactivation under Extreme Conditions In regions where the structural field ρ(x)tends to zero—such as inside a singularity, or in extreme gravitational collapse processes—the model predicts a total disappearance of the geometry, implying a real 4 transition to the Absolute Vacuum. Unlike GR, where the metric remains defined (albeit divergent), here it simply ceases to exist. This prediction suggests that black holes do not contain a "singularity," but rather a nucleus without physical structure, without space or time. Falsifiable implication: If any quantum or astrophysical model were able to demonstrate that space-time survives within a region with infinite energy, this model would be disproved. 2. Structural collapse during quantum measurement Under a complementary interpretation, the model allows us to understand the collapse of the wave function as a local transition of ρ(x), where the quantum superposition no longer supports a coherent spacetime structure. The act of measurement is not simply a statistical projection, but a physical restructuring of the density ρ(x)that defines what geometry can be sustained. Falsifiable implication: Correlations between decoherence measurements and structural discontinuities in weak gravitational fields could be sought through quantum interferometry-type experiments with high masses. 3. Structural Oscillations in the ρ(x)Field In quantum-cosmological settings, small fluctuations in ρ(x)induce fluctuating metrics, generating geometriclike oscillations without the need for gravitons or perturbations of gµν . These oscillations could leave observable signatures in the cosmic background radiation, in the form of non-Gaussian modes of pregeometric origin. Falsifiable Implication: If patterns are identified that cannot be explained by standard inflation, but that fit an oscillation in the structure density ρ, the model gains plausibility. 4. Boundary conditions between Uρand VAregions This model predicts anomalous physics at the boundaries between regions with ρ>0and regions with ρ→0, including discontinuities in field propagation or the appearance of "transparent" regions for certain modes. These transitions could mimic gravitational lensing effects, but without mass. Falsifiable implication: Detection of astrophysical events where the path of light is altered in the absence of detectable mass. 5. Delayed emergence of spacetime Under extreme initial conditions, such as those of the early Universe, it is possible that ρ(x)had negligible values throughout almost the entire region, allowing for a geometry-free model up to a certain critical threshold. Falsifiable implication: This hypothesis implies a pregeometric beginning and may give rise to unique signals in the polarization of the background radiation or observable topological structures on a large scale. 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