Geometric Reconstruction of the Nature of Spacetime under the MOC Framework: Curvature Configuration Theory

Author: Zhang Suhang, Luoyang

Abstract

Classical physics regards spacetime as a background stage independent of matter, while relativity unifies spacetime as a curved manifold. However, both presuppose the a priori existence of a “background spacetime.” Based on the Multi-Origin Curvature (MOC) paradigm, this paper proposes a curvature configuration theory of spacetime, completely abandoning the hypothesis of a background spacetime. Space is directly defined as the complete set of curvature and torsion configurations, and time as the ordered evolution sequence of those configurations. It is rigorously argued that spacetime is not an independent physical entity but the intrinsic manifestation of multi-origin curvature coupling relations. Without curvature, there is no space; without configuration evolution, there is no time. This theory dissolves the metaphysical presupposition of a “spacetime background” at its source, provides a brand new geometric perspective for understanding the nature of spacetime, and forms a complete, self-consistent closed loop with MOC three-body dynamics and field equations.

Keywords: Multi-Origin Curvature; MOC; Nature of spacetime; Curvature configuration; Geometric view of spacetime; Background independence

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1. Introduction

Since Newton proposed absolute spacetime, spacetime has always been regarded as the fundamental background of the physical world: in Newtonian mechanics, flat Euclidean spacetime is the rigid stage for material motion; in Einstein’s relativity, spacetime is unified as a pseudo‑Riemannian manifold that can be curved by matter. Although relativity achieves a correlation between spacetime and matter, it still retains the a priori assumption of a “background manifold”—that is, a pre‑existing spacetime structure within which matter and fields are distributed and evolve.

This “background‑dependent” view of spacetime has long constrained the exploration of the nature of spacetime in physics. In quantum gravity, the contradiction between background spacetime and quantum fluctuations remains unresolved. In classical mechanics, introducing an external coordinate system leads the three‑body problem into a chaotic projection predicament. Based on the MOC multi‑origin curvature paradigm, this paper proposes a thoroughly background‑independent theory of spacetime, reducing spacetime directly to a set and sequence of curvature configurations, without presupposing any background manifold or absolute reference frame.

The core objective of this paper is to give a fundamental definition of the nature of spacetime from the intrinsic geometry of the MOC framework, establish a direct mapping “curvature configuration ↔ space ↔ time”, and provide the underlying spatiotemporal foundation for subsequent unification of field theory, many‑body systems, and cosmology.

2. The Background‑Dependence Predicament of Classical Views of Spacetime

2.1 Limitations of Newton’s Absolute Spacetime

Newton held that spacetime is a uniform, flat, eternal, unchanging absolute background; material motion occurs within it without affecting spacetime itself. The core defects of this view are:

1. It presupposes the existence of “empty space” devoid of matter, contradicting the intrinsic nature that “no matter, no geometry.”

2. It introduces an absolute reference frame that transcends matter, contradicting the principle of relativity.

3. It treats time as an independent, uniformly flowing variable, decoupling spacetime from matter.

2.2 Residual Background Assumptions in Relativistic Spacetime

General relativity treats spacetime as a curved manifold determined by the distribution of matter, achieving a dynamic connection between spacetime and matter, but residual background dependence remains:

1. The manifold itself, as a topological background, is still a pre‑existing mathematical object.Certainly, GR successfully unifies gravity and geometry within the manifold, but the origin of the manifold itself still needs to be questioned.

2. The metric tensor is defined on the manifold; spacetime curvature is a property of the metric, not of matter itself.

3. It cannot explain the origin of the manifold itself—the metaphysical question “Where does the manifold come from?”

2.3 Core Contradiction of Background Dependence

Neither Newton’s nor Einstein’s view of spacetime can escape the logical presupposition “first a background, then matter.” This presupposition leads to:

· In quantum gravity, the continuity of background spacetime is incompatible with the discreteness of quantum fluctuations.

· In classical many‑body problems, introducing an external coordinate system causes projective distortion of geometric structure, leading to apparent chaos.

· The nature of spacetime remains a “metaphysical mystery” in physical theories, impossible to derive directly from intrinsic relations of physical quantities.

3. Definition of Spacetime under the MOC Framework: Curvature Configuration Theory

3.1 Basic Premises: Background‑Free, Intrinsic Geometry

The MOC framework strictly adheres to the following premises:

1. There is no background spacetime independent of matter; all geometric quantities are intrinsic properties of matter.

2. The essence of matter is multi‑origin curvature and torsion; curvature and torsion are the only fundamental quantities describing matter.

3. All physical relations are endogenously determined by couplings of curvature and torsion, requiring no external reference frame or pre‑existing manifold.

3.2 Fundamental Definition of Space: The Complete Set of Curvature Configurations

Definition 1 (Space): Under the MOC framework, space is the complete set of all multi‑origin curvature and torsion configurations in the system.

Mathematically:

\mathcal{S} = \left\{ \Omega \mid \Omega = (\kappa_I, \tau_I, \kappa_{IJ}), \ I,J=1,2,\dots,N \right\}

where \kappa_I is the intrinsic curvature of the I-th origin, \tau_I its spin torsion, and \kappa_{IJ} the coupling curvature between origins I and J.

Physical interpretation:

1. Space is not a pre‑existing container, but the set of all possible forms of curvature configurations.

2. The shape, dimension, and topological structure of space are entirely determined by the distribution and coupling relations of curvature configurations.

3. There is no “empty space”; absence of curvature configurations corresponds to the absence of space.

3.3 Fundamental Definition of Time: Ordered Evolution Sequence of Curvature Configurations

Definition 2 (Time): Under the MOC framework, time is the ordered evolution sequence of configurations that the system actually traverses within the curvature configuration set \mathcal{S}.

Mathematically:

\mathcal{T} = \left\{ \Omega_0, \Omega_1, \Omega_2, \dots, \Omega_n, \dots \right\}, \quad \Omega_k \in \mathcal{S}

where the order of the sequence is uniquely determined by the evolution relations among curvature configurations.

Physical interpretation:

1. Time is not an independently flowing variable, but a label for the evolution of curvature configurations.
2. If the curvature configuration remains unchanged (\Omega_k = \Omega_{k+1}), temporal evolution stagnates.
3. The flow rate of time is determined by the evolution rate of curvature configurations; the intrinsic time flow rate of different origins is determined by their own curvature change rates. There is no globally uniform absolute time.

3.4 Unification of Spacetime: Set and Sequence of Curvature Configurations

Under the MOC framework, space and time are not independent entities, but two aspects of the same curvature configurations:

· Space: The “synchronic” set of curvature configurations, describing all possible geometric states of the system.
· Time: The “diachronic” sequence of curvature configurations, describing the actual geometric state evolution experienced by the system.

The essence of spacetime is the intrinsic manifestation of multi‑origin curvature coupling relations. There is no spacetime independent of curvature configurations.

4. Core Corollaries of the Spacetime Curvature Configuration Theory

4.1 Corollary 1: Dissolution of Background Spacetime

The “background spacetime” in traditional physics is essentially an idealized abstraction of the curvature configuration set \mathcal{S}. When curvature configurations are uniformly distributed and coupling effects negligible, the configuration set approximates a flat, uniform “background”—this is the origin of Newton’s absolute spacetime view. When coupling effects of curvature configurations are significant, the configuration set manifests as a curved, dynamic structure, corresponding to the curved spacetime manifold in relativity.

Thus, background spacetime is not a physical reality but a low‑order approximation of curvature configurations. The background‑free MOC framework is a more fundamental description.

4.2 Corollary 2: Relativity and Intrinsicality of Time

According to Definition 2, the flow rate of time is determined by the evolution rate of curvature configurations, which is itself determined by the intrinsic curvature, torsion, and coupling relations of each origin. Different origins have different intrinsic time flow rates. This is essentially consistent with the time dilation effect in relativity: in a strong gravitational field (high intrinsic curvature), the curvature evolution rate is slower, corresponding to slower time flow; in a weak gravitational field (low intrinsic curvature), the curvature evolution rate is faster, corresponding to faster time flow.

Unlike relativity, the MOC framework derives the relativity of time directly from intrinsic relations of curvature configurations, without presupposing a spacetime metric, achieving a direct connection between time and matter.

4.3 Corollary 3: Geometric Nature of Chaos

The chaotic phenomenon in the classical three‑body problem is essentially the projective distortion of the curvature configuration sequence under a single‑origin external coordinate system. In MOC intrinsic geometry, the curvature configuration sequence is ordered, self‑consistent, and satisfies global conservation laws; there is no intrinsic chaos. When the evolution of multi‑origin curvature configurations is described using a single external coordinate, the high‑dimensional ordered configuration sequence is projected onto low‑dimensional irregular coordinate trajectories, appearing as apparent chaos.

This corollary completely demystifies chaos, proving that chaos is a product of the background‑dependent paradigm, not an intrinsic property of the physical system.

4.4 Corollary 4: Discreteness and Continuity of Spacetime

The curvature configuration set \mathcal{S} is discrete (each configuration is an independent geometric state). However, when the number of configurations is sufficiently large and the evolution sufficiently continuous, the configuration sequence \mathcal{T} manifests as a continuous flow of time, and the configuration set \mathcal{S} manifests as a continuous spatial structure. This property provides a new perspective for resolving the discrete‑continuous contradiction of spacetime in quantum gravity: the discreteness of spacetime originates from the discreteness of curvature configurations, while continuity is a statistical approximation of a large number of configurations.

5. Self‑Consistency Verification with the Existing MOC Framework

5.1 Consistency with Three‑Body Dynamics

In MOC three‑body dynamics, the evolution of the system is described by the coupling equations of curvature‑torsion configurations:

\frac{d}{dt}
\begin{pmatrix}
\boldsymbol{\kappa} \\
\boldsymbol{\tau} \\
\boldsymbol{\kappa}_{\text{orb}}
\end{pmatrix}
=
\mathbb{M}
\begin{pmatrix}
\boldsymbol{\kappa} \\
\boldsymbol{\tau} \\
\boldsymbol{\kappa}_{\text{orb}}
\end{pmatrix}
+
\mathcal{N}(\boldsymbol{\kappa},\boldsymbol{\tau},\boldsymbol{\kappa}_{\text{orb}})

Here, t is not an independent variable but an order parameter of the curvature configuration sequence. The essence of the equation is to describe how a curvature configuration evolves from \Omega_k to \Omega_{k+1}, which is fully consistent with the definition of time in this paper.

5.2 Consistency with Field Equations

The core constraint of the MOC field equation is:

\mathcal{D} \star \mathcal{R} = 0

where \mathcal{R} is the global curvature‑torsion tensor, corresponding to the overall structure of curvature configurations; \mathcal{D} is the evolution operator, describing the orderliness of the configuration sequence. This constraint guarantees the self‑consistency of the curvature configuration set and the lack of breakdown of the configuration sequence, forming a perfect closed loop with the definition of spacetime in this paper.

6. Conclusion and Outlook

Based on the MOC multi‑origin curvature paradigm, this paper has proposed a curvature configuration theory of spacetime, completely dissolving the a priori assumption of a background spacetime, defining space as the complete set of curvature and torsion configurations, and time as the ordered evolution sequence of configurations. The main conclusions are as follows:

1. Spacetime is not an independent physical entity but the intrinsic manifestation of multi‑origin curvature coupling relations.
2. The shape and structure of space are determined by the distribution of curvature configurations; there is no empty space without curvature.
3. Time is the order parameter of the evolution of curvature configurations; the intrinsic time flow rate of different origins is determined by their curvature evolution rate; there is no absolute time.
4. Background spacetime, chaotic phenomena, and the discrete‑continuous contradiction of spacetime can all be explained at their source by the curvature configuration theory.

The spacetime curvature configuration theory provides a unified spatiotemporal foundation for the MOC framework. Subsequent work will, based on this theory, further investigate the spacetime structure of many‑body systems, discrete spacetime models for quantum gravity, and alignment with the spacetime view of Yang–Mills gauge field theory, ultimately constructing a complete geometrized physical system.

Acknowledgments

This paper is dedicated to Gauss, Riemann, Einstein, and all pioneers who explored the geometric origin of spacetime; their work provided essential intellectual inspiration for the theoretical construction presented here.