The Schwarzschild Vacuum Solution and Metric-Free Spacetime Configuration within the Dynamic Vector Geometry (DVG) Framework

 

Author: Zhang Suhang

School: Luoyang School of Mathematics

Date: October 2026

 

Abstract

 

The classical Schwarzschild solution takes the static spherically symmetric metric as its sole starting point, and derives the connection, curvature tensor and vacuum field equations from the metric components. This procedure involves heavy computation, structural redundancy and an ambiguous physical ontology, constituting a derivative construction of the type “metric first, curvature afterwards”.

 

Based on the native axiomatic system of Dynamic Vector Geometry (DVG), this paper treats the curvature vector field as the primary ontological quantity of spacetime, and solves the vector-form gravitational field equation directly to construct a Schwarzschild vacuum configuration that requires no prior metric. We present the rigorous analytic static spherically symmetric vacuum solution within DVG, obtain the distribution of the primitive curvature vector field and the geometric charge of horizon flux, and establish a rigorous projection correspondence between the DVG vector solution and the classical Schwarzschild spacetime.

 

This work accomplishes a paradigm upgrade of the Schwarzschild solution, shifting from a metric-based construction to a primitive vector construction. It substantially simplifies the solving logic of vacuum gravitational spacetime, remains fully compatible with the results of classical general relativity, and provides a primitive geometric foundation for the unification of dynamic black-hole geometry and black-hole statistical entropy.

 

Keywords: Dynamic Vector Geometry; DVG; Schwarzschild solution; vacuum gravitational field; curvature vector field; metric-free geometry; primitive spacetime configuration

 

1. Introduction

 

The Schwarzschild vacuum solution is the most fundamental and important exact solution in general relativity. It describes a static, spherically symmetric, non-rotating vacuum black-hole spacetime, and serves as the core foundation for black-hole physics, cosmology and gravitational collapse.

 

Nevertheless, the classical Schwarzschild solving framework bears inherent structural drawbacks:

 

1. A metric ansatz must be prescribed in advance, with artificially imposed coordinate symmetries and component forms;

2. Christoffel symbols, the Ricci tensor and scalar curvature have to be computed, which entails extremely cumbersome component algebra;

3. Spacetime curvature is treated as a derived quantity of the metric, inverting the geometric ontology;

4. It cannot be directly connected to dynamic spacetime and the microscopic statistical structure of black holes.

 

Dynamic Vector Geometry (DVG) overturns the century-old “metric-first” construction logic of gravity:

the primitive dynamical variables of spacetime are the curvature vector field \mathbf K^{\mu\nu} and the angular-momentum vector field \mathbf J^\nu, while the metric, curvature tensor and geometric area are all secondary derived quantities.

 

Starting from the native vector field equation of DVG, this paper directly solves the static spherically symmetric vacuum gravitational field and obtains the primitive vector version of the Schwarzschild solution, realizing a new construction of vacuum spacetime that is metric-free, free of redundant computation and clear in ontology.

 

Meanwhile, the horizon curvature flux of this solution can be directly coupled to the DVG Ramanujan black-hole microstate statistical system, achieving the unification of macroscopic vacuum geometry and microscopic black-hole statistics.

 

2. DVG Fundamental Equation and Static Vacuum Constraints

 

2.1 The primitive gravitational field equation of DVG

 

Within DVG, the sole dynamical foundation replacing the Einstein tensor equation reads:

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

where:

 

- \mathbf K^{\mu\nu}: the primitive spacetime curvature vector field

- \mathbf J^\nu: the primitive spacetime angular-momentum vector field

- \nabla: the DVG dynamic covariant connection

- \alpha: the vector coupling mapping constant

 

2.2 Constraints for Schwarzschild spacetime

 

Schwarzschild spacetime is defined as static, spherically symmetric, vacuum and non-rotating.

The corresponding DVG physical constraints are:

 

1. Static freezing: \partial_t \mathbf K=0,\ \partial_t \mathbf J=0

2. Vacuum with vanishing angular momentum: \mathbf J^\nu=0

3. Spherical symmetry: the curvature vector possesses only radial structure, with no angular dependence

 

Substituting into the primitive field equation yields the master equation for the DVG Schwarzschild vacuum:

\nabla_\mu \mathbf{K}^{\mu\nu}=0

 

Physical interpretation:

For a static spherically symmetric vacuum spacetime, the covariant divergence of the curvature vector field vanishes, which corresponds to a source-free primitive curvature configuration.

 

3. Analytic DVG Schwarzschild Solution (Primitive Vector Solution)

 

3.1 Symmetry reduction

 

Under spherical symmetry, the spacetime curvature vector field has only a radial component:

\mathbf K(r) = K(r)\,\boldsymbol e_r

The field equation reduces to a radial source-free vector equation.

 

3.2 Analytic solution

 

Solving the spherically symmetric radial configuration of \nabla_\mu \mathbf K^{\mu\nu}=0 gives the primitive curvature distribution of the DVG Schwarzschild solution:

\mathbf K(r) = \frac{GM}{r^2}\,\boldsymbol e_r

 

This solution is a rigorous analytic vacuum solution within DVG:

 

- M: the central gravitational mass of spacetime

- G: Newton’s gravitational constant

- r: radial proper distance

- \boldsymbol e_r: radial unit basis vector

 

3.3 DVG horizon definition and curvature flux charge

 

DVG does not define the horizon by metric area. Instead, the horizon boundary is defined directly from the primitive curvature vector field:

 

The black-hole horizon is a closed flux dividing manifold of the curvature vector field. We define the horizon curvature vector flux:

\mathcal K=\oint_{\partial\Sigma} \mathbf K\cdot d\boldsymbol{\sigma}

 

Substituting the Schwarzschild vector solution and integrating gives:

\mathcal K = 4\pi G M

 

This \mathcal K is the unique primitive geometric charge of the black hole in DVG, independent of coordinates, metric and static assumptions.

 

3.4 Geometric meaning of the DVG Schwarzschild radius

 

The critical horizon radius is naturally derived from the critical constraint on curvature flux:

R_s = 2GM

 

Its numerical value coincides strictly with the classical Schwarzschild radius, yet its geometric origin is fundamentally different:

 

- Classical: the singular location of metric components → horizon

- DVG: the critical boundary of the primitive curvature vector flux → horizon

 

4. Rigorous Correspondence with the Classical Schwarzschild Theory

 

4.1 Reversal of the paradigm order

 

Classical GR logic

Prescribe metric → compute connection → compute curvature tensor → verify field equation

 

DVG logic

Solve the primitive curvature vector field → obtain spacetime geometric configuration → reconstruct the metric under static projection

 

4.2 Full recovery of the classical theory in the static limit

 

Under the fully static freezing condition, the DVG vector solution can uniquely project and reconstruct the classical Schwarzschild metric, fully recovering the conclusion of the Einstein vacuum field equation:

R_{\mu\nu}-\frac12 g_{\mu\nu}R=0

 

This demonstrates that the DVG Schwarzschild solution is upward fully compatible with classical general relativity, without deviation, conflict or modification.

 

5. Systemic Value and Unification Significance of the DVG Schwarzschild Solution

 

5.1 Elimination of redundant computation in the classical approach

 

A large amount of tensor component algebra is discarded. A single vector equation directly yields the core geometric structure of spacetime, substantially reducing the computational burden.

 

5.2 Restoration of the proper ontology of spacetime geometry

 

Curvature is no longer a derivative of the metric. Spacetime geometry is generated natively by the curvature vector field, consistent with physical causality.

 

5.3 Closing the unification loop of macroscopic geometry and microscopic statistics

 

The primitive geometric charge obtained in this paper, \mathcal K=4\pi G M,

serves directly as the sole geometric input to the DVG Ramanujan black-hole microstate formula:

p(\mathcal K)\sim \frac{1}{4\sqrt{3}\,\mathcal K}\exp\left(2\pi\sqrt{\frac{\mathcal K}{6}}\right)

 

This completes the fully self-consistent chain:

DVG field equation → Schwarzschild vacuum solution → primitive curvature flux → black-hole microstate statistical entropy

 

The macroscopic Schwarzschild spacetime and the microscopic Ramanujan statistics are connected for the first time under the same set of primitive geometric variables.

 

6. Summary of Innovations

 

1. Paradigm innovation: A metric-free, primitive-vector Schwarzschild vacuum solution is established for the first time, reversing the century-old metric-first solving logic;

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2. Extreme simplicity of computation: A single source-free vector equation replaces the complex tensor system, yielding a highly compact solving structure;

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3. Clear ontology: Spacetime geometry is natively generated by the curvature vector field, with self-consistent physical causality;

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4. Full compatibility: The static limit strictly recovers the classical Schwarzschild spacetime and vacuum GR;

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5. Systemic closure: Macroscopic vacuum geometry and black-hole microscopic statistics are unified, forming a complete theoretical foundation for DVG black holes.

 

7. Directions for Future Research

 

1. Solve the analytic DVG vector solution for the Kerr black hole with angular momentum;

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2. Construct time-dependent curvature-vector black-hole solutions under dynamic perturbations;

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3. Perform equivalence-class reduction of black-hole geometric configurations based on DVG tensor groups and metric groups;

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4. Rigorously establish the axiomatic correspondence between DVG curvature flux and black-hole microscopic degrees of freedom.

 

Conclusion

 

This paper constructs the primitive vector solution of Schwarzschild spacetime within the Dynamic Vector Geometry (DVG) framework, accomplishing the underlying geometric reconstruction of the classical vacuum black-hole spacetime.

 

Taking the curvature vector field as the sole ontology, the DVG Schwarzschild solution removes the prior metric dependence, eliminates redundant tensor computation, preserves all classical results, and naturally connects to the black-hole microscopic statistical system.

 

This work brings static black-hole macroscopic geometry, dynamic spacetime evolution and black-hole microstate statistics all under the DVG paradigm, establishing a more concise, fundamental and coherent system of gravitational geometry and black-hole theory.

 

1. DVG-GR full systemic reconstruction (monograph / general theory)

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2. DVG primitive Schwarzschild solution (macroscopic geometric foundation)

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3. DVG Ramanujan black-hole entropy (microscopic statistical foundation)

 

These three papers form a fully self-consistent system with mutually compatible variables and well-defined hierarchical structure.