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MOC Coordinate System: Geometric Definition of Multi-Origin Hierarchical Nesting and Classical Coordinate Degeneration

Author: Zhang Suhang

Founder of the Heluo Mathematical School

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Abstract

Based on the Multi-Origin Curvature (MOC) geometric framework, this paper presents a concrete implementation of the MOC coordinate system. By defining multi-level nested origins, independent curvature fields excited by each origin, and superposition rules for multi-stage spiral motions, a coordinate system for describing composite spacetime structures is constructed. When the system degenerates to a single origin with zero curvature, MOC coordinates naturally degenerate into inertial Cartesian/Minkowski coordinates; when a single origin carries non-zero curvature, they can be approximately reduced to spherically symmetric curved spacetime (such as the Schwarzschild metric). This paper lays a coordinate foundation for the formalization of MOC geometry and demonstrates its compatibility with traditional coordinate systems.

Keywords: MOC coordinates; multi-origin nesting; curvature superposition; spiral motion; coordinate degeneration

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I. Introduction

The core idea of MOC geometry is that physical spacetime is formed by the nested superposition of curvature fields excited by multiple local centers of mass (origins), with each origin also accompanied by its own rotational motion, creating a composite structure of multi-layered winding. To translate this idea into a computable coordinate system, this paper proposes an explicit definition of MOC coordinates, including the set of origins, curvature field functions, and superposition rules for motion trajectories. It is then demonstrated that under appropriate limits, MOC coordinates degenerate into classical coordinate systems.

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II. Definition of MOC Coordinates

2.1 Origins and Hierarchies

Consider a set of nested origins \{O_1, O_2, \ldots, O_N\}, arranged in order of spatial scale or gravitational binding hierarchy ( O_1 is the largest-scale origin, O_N the smallest-scale origin). Each origin O_i possesses:

· Position vector \mathbf{o}_i(t) (may vary with time or be fixed)

· Intrinsic curvature parameter R_i (scalar or tensor)

· Rotational angular velocity \boldsymbol{\Omega}_i

2.2 Axiom of Curvature Field Superposition

Each origin independently excites a curvature field \boldsymbol{\Phi}_i(\mathbf{x}, t), whose intensity decays with distance from the origin and is proportional to R_i at the origin. The total curvature field is the linear superposition of individual fields:

\boldsymbol{\Phi}_{\text{total}}(\mathbf{x}, t) = \sum_{i=1}^{N} \boldsymbol{\Phi}_i(\mathbf{x}, t)

Important property: Curvature fields do not cancel each other; even if they are oppositely directed, each contribution is preserved, manifesting as complex local geometric distortion.

2.3 Spiral Motion Nesting Formula

The trajectory of any point P in space is formed by the nested superposition of rotations from all origins. Let the relative position vectors with respect to each origin be given, using a recursive relation for involved motion:

\mathbf{r}_{\text{total}}(t) = \sum_{i=1}^{N} \mathbf{r}_i(t, \boldsymbol{\Omega}_i)

where \mathbf{r}_i is the motion component observed in the i-th rotating reference frame (typically taken as circular or spiral motion at angular velocity \boldsymbol{\Omega}_i). In particular, for an astronomical system (Galactic center → Sun → Earth), the above formula degenerates to:

\mathbf{r}_{\text{total}}(t) = \mathbf{r}_1(t,\boldsymbol{\Omega}_1) + \mathbf{r}_2(t,\boldsymbol{\Omega}_2) + \mathbf{r}_3(t,\boldsymbol{\Omega}_3)

This is the mathematical expression of the "spacetime twist."

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III. Mathematical Representation of MOC Coordinates

We denote MOC coordinates as a tuple:

\mathcal{M} = \left\{ \mathbf{x}, \{ \mathbf{o}_i \}, \{ R_i \}, \{ \boldsymbol{\Omega}_i \}, \boldsymbol{\Phi}_{\text{total}}, \mathbf{r}_{\text{total}} \right\}

Here, \mathbf{x} is the event coordinate in the background space (inertial Cartesian coordinates may be used as a reference basis). Unlike traditional coordinates, MOC coordinates explicitly include the parameters of all hierarchical origins and the structure of superposed fields.

3.1 Construction of the Metric (Conceptual Form)

Although this paper does not provide an explicit metric function, the general form of the metric can be defined as:

g_{\mu\nu}(\mathbf{x}) = \eta_{\mu\nu} + \sum_{i} \lambda_i(|\mathbf{x}-\mathbf{o}_i|) \cdot \mathcal{K}_{\mu\nu}(\boldsymbol{\Phi}_i) + \sum_{i<j} \mu_{ij}(\mathbf{x}) \cdot \mathcal{C}_{\mu\nu}^{(ij)}

where \eta_{\mu\nu} is the background Minkowski metric, \mathcal{K}_{\mu\nu} is a tensor constructed from the curvature field \boldsymbol{\Phi}_i, and \mathcal{C}_{\mu\nu}^{(ij)} are coupling terms between origins. The specific functional forms need to be selected according to the physical system but do not affect the degeneration analysis below.

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IV. Degeneration to Traditional Coordinates

4.1 Degeneration to Inertial Cartesian/Minkowski Coordinates

Conditions:

· Number of origins N = 1

· Curvature of that origin R_1 = 0

· Rotational angular velocity \boldsymbol{\Omega}_1 = 0 (or rotation ignored)

Then:

· Total curvature field \boldsymbol{\Phi}_{\text{total}} = \boldsymbol{\Phi}_1 = 0

· Trajectory \mathbf{r}_{\text{total}}(t) = \mathbf{r}_1(t), with no rotation, i.e., uniform linear motion (or rest)

· Metric degenerates to g_{\mu\nu} = \eta_{\mu\nu}

Thus, MOC coordinates are equivalent to Cartesian/Minkowski coordinates in an inertial background frame. The MOC coordinate system automatically includes classical spacetime as a special case.

4.2 Degeneration to Spherically Symmetric Curved Spacetime (Schwarzschild Metric)

Conditions:

· Number of origins N = 1

· Curvature R_1 = M (proportional to mass)

· Rotation ignored (\boldsymbol{\Omega}_1 = 0), and system static

· Choose background metric as Minkowski metric, and let the form of the curvature field \boldsymbol{\Phi}_1 be such that the metric becomes the Schwarzschild solution.

For example, taking \lambda_1(|\mathbf{x}-\mathbf{o}_1|) = \frac{2G R_1}{|\mathbf{x}-\mathbf{o}_1|} and adjusting the tensor structure yields:

g_{00} = -\left(1 - \frac{2GM}{r}\right),\quad g_{rr} = \left(1 - \frac{2GM}{r}\right)^{-1}

Here, MOC coordinates are equivalent to Schwarzschild coordinates. Therefore, MOC coordinates include curved spacetime descriptions from general relativity as a special case.

4.3 Degeneration to Polar Coordinates or Other Curvilinear Coordinates

When N=1, R_1=0, and the background space is Euclidean, by the coordinate transformation (x,y) \to (r,\theta), the MOC metric (i.e., \eta_{\mu\nu}) becomes dr^2 + r^2 d\theta^2. Hence, MOC coordinates allow arbitrary differentiable coordinate transformations and automatically include polar coordinates, spherical coordinates, etc.

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V. Discussion

1. Innovation of MOC Coordinates: Traditional coordinate systems are based on a single origin (coordinate origin), whereas MOC coordinates allow multiple independent origins to coexist, each contributing curvature and rotation, with superposition that does not cancel. This provides a natural language for describing multiple gravitational sources and multi-level rotating systems (e.g., galaxy-sun-earth).
2. Current Limitations: This paper does not provide an exact algebraic relation between the curvature field \boldsymbol{\Phi}_i and the metric, nor does it write out the specific functions for the coupling terms \mu_{ij}. These require modeling for specific physical systems, but do not affect the validity of MOC coordinates as a framework.
3. Future Work: Select appropriate \lambda_i and \mu_{ij} for different physical scenarios (e.g., binary black hole systems, many-body problems), and calculate observable effects (e.g., light deflection, time delay) to test the predictions of MOC coordinates.

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VI. Conclusion

1. Based on MOC geometry, this paper clearly defines the MOC coordinate system: composed of multi-level nested origins, superposition of independently excited curvature fields from each origin, and spiral motion nesting rules.
2. It is proven that for a single origin with zero curvature, MOC coordinates degenerate to inertial Cartesian/Minkowski coordinates; for a single origin with static spherically symmetric curvature, they can be reduced to the Schwarzschild metric; through coordinate transformations, they also include common curvilinear coordinates such as polar coordinates.
3. Therefore, the MOC coordinate system is a natural generalization of traditional coordinates, capable of unifying the description of flat spacetime, curved spacetime, and complex spacetime structures with multi-source rotational nesting.

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References

[1] Zhang Suhang. Multi-Origin Hierarchical Nesting Geometry: Curvature Superposition and Spiral Motion Formulas. 2026.

[2] Zhang Suhang. Fundamental Axioms and Physical Applications of MOC Geometry. 2026.

[3] Misner, C. W., Thorne, K. S., & Wheeler, J. A. Gravitation. Freeman, 1973.

[4] Wald, R. M. General Relativity. University of Chicago Press, 1984.

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Appendix: Concise Definition of MOC Coordinates

Total curvature: \displaystyle \boldsymbol{\Phi}_{\text{total}} = \sum_i \boldsymbol{\Phi}_i

Nested spiral trajectory: \displaystyle \mathbf{r}_{\text{total}}(t) = \sum_i \mathbf{r}_i(t,\boldsymbol{\Omega}_i)

Degeneration condition: N=1,\ \boldsymbol{\Phi}_1=0,\ \boldsymbol{\Omega}_1=0 → inertial coordinates.

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