Reconstruction of General Relativity and a Simplified Computational System under the Dynamic Vector Geometry (DVG) Framework

Author: Zhang Suhang

School: Luoyang School of Mathematics

October 2026

Abstract

Traditional general relativity takes the static metric tensor as its first-principles foundation and relies on the expansion of numerous connection, tensor, and curvature components for derivation, resulting in cumbersome structures, redundant computations, and high modeling costs. Based on the new paradigm of Dynamic Vector Geometry (DVG), this paper completes a fundamental reconstruction of general relativity: taking the curvature vector field and the angular momentum vector field as the native geometric quantities of spacetime, it establishes a minimal vector-type gravitational field equation to replace the traditional metric-tensor-priority construction.

This paper introduces a DVG-native tensor group and metric group computational system (distinct from the various transformation groups of classical differential geometry), structurally merging redundant geometric degrees of freedom and greatly simplifying the derivation and solution procedures of general relativity. This paper establishes a rigorous hierarchical closed loop: dynamic vector field dynamics → group-structural simplification operations → static-constraint reduction returning to classical GR. It not only preserves all valid conclusions of classical general relativity but also achieves a dual improvement in modeling simplicity and computational efficiency.

Keywords: Dynamic Vector Geometry; DVG; general relativity; curvature vector field; angular momentum vector field; metric group; tensor group

I. Introduction

The core construction logic of classical general relativity is “metric supremacy”: spacetime geometry, curvature structure, and gravitational dynamics are all derived from the second-order metric tensor. The entire theory requires repeated calculation of connections, curvature tensors, Ricci tensors, and scalar curvature, with enormous numbers of components, fragmented steps, and extremely redundant operations. Its fundamental limitation is that static geometry, as the underlying architecture, cannot naturally unify curvature and rotation, and artificially separates spacetime geometry from angular momentum dynamics.

Dynamic Vector Geometry (DVG) changes the underlying geometric paradigm: the ontology of geometry is no longer a static manifold and metric, but the dynamic evolutionary structure of the curvature vector field and the angular momentum vector field. Static Riemannian geometry and classical general relativity are merely special cross-sectional results after the evolution of dynamic spacetime is frozen.

Based on the DVG axiom system, general relativity can be modeled without starting from cumbersome metric tensor components, describing gravity through native vector field equations; at the same time, through the tensor group and metric group defined in this paper, a large number of equivalent redundant configurations are merged, achieving structural simplification of the GR computational system.

This paper focuses throughout on gravitational reconstruction and computational simplification, without superfluous extensional structures, maintaining the purity, rigor, and specificity of the theoretical system.

II. Fundamental Paradigmatic Differences between DVG and Classical GR

2.1 Defects of the Classical GR Paradigm

1. It takes the static metric as primary, and all dynamics are derived quantities;

2. Curvature, connections, and tensors all depend on metric expansion, making computations highly redundant;

3. Geometric curvature and spacetime rotational angular momentum have no native coupling, resulting in structural fragmentation.

2.2 The New DVG-GR Paradigm

1. It takes the curvature vector field and angular momentum vector field as the primary ontology of spacetime;

2. Metrics, tensors, and curvature tensors are all static secondary derived quantities;

3. Curvature and angular momentum are naturally coupled, and gravitational dynamics is directly described by vector field equations;

4. Redundancy can be reduced through group structure, greatly lowering derivation complexity.

III. Native GR Field Equations under the DVG Framework

3.1 Basic Gravitational Field Equation (DVG Core)

The native dynamic equation by which DVG replaces the Einstein equation is a vector coupling equation:

\nabla_\mu \mathbf{K}^{\mu\nu}=\alpha \mathbf{J}^\nu

where:

· \mathbf{K}^{\mu\nu}: spacetime curvature vector field (a DVG primitive geometric quantity)

· \mathbf{J}^\nu: spacetime angular momentum vector field

· \alpha: vector-space coupling mapping, which reduces to a scalar constant in the static case

· \nabla: DVG dynamic connection generated jointly by curvature and angular momentum

Physical interpretation:

The covariant divergence of the spacetime curvature vector field is equivalent to the coupling mapping output of the spacetime angular momentum vector field.

This equation is the sole basic equation of DVG gravitational theory. It requires no prior metric, no tensor stacking, and no presupposed spacetime symmetry, and its form is far simpler than the classical Einstein field equation.

The accompanying curvature superposition axiom is:

\mathbf K_{\text{total}}=\sum_i \mathbf K_i

Multiple components of curvature vectors superpose independently and do not undergo simple opposite cancellation, ensuring the completeness of the dynamic geometric structure.

3.2 Static Limit: Rigorous Return to Classical GR

When fully static frozen conditions are imposed:

\partial_t \mathbf K=0,\quad \partial_t \mathbf J=0

the dynamic connection reduces to the classical Riemannian connection, vector field evolution vanishes, and the DVG dynamic structure automatically reduces to static Riemannian geometry.

At this point, the DVG vector field equation can be projected, expanded, and reduced to the standard Einstein field equation:

R_{\mu\nu}-\frac12 g_{\mu\nu}R=8\pi G T_{\mu\nu}

This proves that classical general relativity is a special case of DVG-GR under static, non-evolving conditions. The new theory is fully compatible with the old theory and also covers regimes of dynamic spacetime that cannot be completely described by classical GR.

IV. DVG Tensor Group and Metric Group Simplified Computational System

4.1 Distinction of Conceptual Originality

The tensor group and metric group defined in this paper are original structures specific to DVG and are completely different from classical geometric groups:

1. Classical isometry groups and diffeomorphism groups: they act on coordinates and keep a fixed metric invariant;

2. DVG tensor groups and metric groups: they take tensor and metric geometric configurations as group elements, merging equivalent geometries and eliminating redundant expressions.

through group structure:

1. Equivalent geometric configurations are grouped into the same group equivalence class;
2. Non-physical redundant degrees of freedom are eliminated;
3. A large number of component-based algebraic operations are transformed into group-transformation structural operations.

This achieves structural complexity reduction rather than mere formula rearrangement and simplification.

4.3 Computational Hierarchical Closed Loop

1. Dynamic modeling: only \nabla_\mu \mathbf{K}^{\mu\nu}=\alpha \mathbf{J}^\nu is used, with no metric and no complex tensors;
2. Fine computation: tensor groups and metric groups are used for reduction and simplification, greatly reducing the amount of computation;
3. Classical comparison: under static constraints, the system returns to standard GR, ensuring that results are benchmarkable and verifiable.

V. Theoretical Innovation and Core Advantages

1. Paradigm innovation: it overturns the century-old metric-priority logic of gravitational construction, being vector-field-based and dynamics-first;
2. Minimal equations: a single vector coupling equation replaces complex tensor field equation systems;
3. Greatly simplified computation: the original group structure eliminates geometric redundancy at the root;
4. Fully self-consistent and compatible: the static limit rigorously returns to classical general relativity;
5. Suitable for dynamic spacetime: it naturally couples curvature and rotation, making it suitable for modeling rotating spacetimes and dynamically distorted spacetimes.

VI. Open Problems

1. Rigorously define the group axioms, operation rules, and equivalence classification of DVG tensor groups and metric groups;

2. Quantitatively compare the computational complexity of the DVG computational system with that of classical GR;

3. Derive the rigorous mathematical conditions for projecting the dynamic vector field equation onto the classical field equation;

4. Re-solve Schwarzschild and Kerr spacetimes using the group simplification method to verify the simplification effect.

VII. Conclusion

This paper completes a thorough reconstruction of general relativity under the Dynamic Vector Geometry (DVG) framework. It takes the curvature–angular momentum vector coupling equation as the primary gravitational equation, replacing the classical metric-tensor construction system; and it uses original tensor group and metric group structures to achieve structural simplification of gravitational computations.

The DVG-GR system resolves the problems of cumbersome derivation, component redundancy, and insufficient dynamic description in classical general relativity, while maintaining full compatibility with the classical theory, forming a simpler, more essential, and more general new system of gravitational geometry.