Curvature of Field Corresponds to Field Strength

Author: Zhang Suhang, Luoyang School of Mathematics

Abstract

In 1975, Yang and Wu pointed out that gauge potentials correspond to connections on fibre bundles, and gauge field strengths correspond to curvatures of connections. This correspondence successfully describes the geometric structure of electromagnetic, strong and weak interactions. The gravitational field rests on a distinct geometric foundation: gravitation manifests as the Riemannian curvature of the spacetime manifold, a geometry on the base manifold, which occupies a different geometric hierarchy from the fibre bundle curvature in the internal space of gauge fields. This paper does not attempt to force gravity into the gauge-field framework; it only carries out a parallel geometric analogy. In gauge theories, the curvature of the connection is the field strength; in gravitation, the spacetime Riemannian curvature is the field strength of the gravitational field. This work is a conceptual paradigm summary. No quantitative field equations are constructed, no numerical derivations are performed, and no experimental predictions are made.

Keywords: curvature; field strength; gauge field; gravitational field; Yang–Wu correspondence; geometric paradigm in physics

1. Introduction

In 1975, Yang and Wu established the fibre-bundle correspondence for gauge fields, with core relations listed below:

- Gauge potential ↔ Fibre bundle connection

- Gauge field strength ↔ Curvature of connection

- Gauge transformation ↔ Fibre bundle parallel transport

- Gauge group ↔ Structure group

This correspondence unifies the geometric description of electromagnetic, strong and weak interactions. The three gauge fields share a common geometric picture: field strength is the curvature of the connection.

The geometric description of gravity is fundamentally different. General relativity attributes gravitational effects to the intrinsic curvature of the four-dimensional spacetime manifold, described by the Riemann curvature tensor defined on the base manifold. The two types of curvature differ in their geometric carriers:

- Gauge fields: curvature defined on abstract internal fibre spaces

- Gravitational field: curvature defined on physical spacetime itself

Some research programs seek to reformulate gravity as a type of gauge field. This paper rejects such forced unification. Instead, it draws a parallel analogy between the two geometric systems to extract shared geometric reasoning, without claiming they possess identical mathematical structures. The analogy states: the connection curvature of a gauge field constitutes its field strength; the spacetime Riemannian curvature constitutes the field strength of the gravitational field. While their geometric carriers differ, both follow the picture “curvature equals field strength”, without requiring mathematical isomorphism.

2. The Yang–Wu Correspondence (Gauge Field Sector)

The Yang–Wu correspondence table (1975)

Gauge Field Concept Fibre Bundle Concept 

Gauge potential Connection 

Gauge field strength Curvature 

Gauge transformation Fibre bundle parallel transport 

Gauge group Structure group 

Core correspondence: gauge field strength is equivalent to the curvature of the connection.

This framework unifies the three gauge interactions:

- Electromagnetic field: U(1) gauge field, whose field strength is the curvature of the U(1) connection;

- Weak field: SU(2) gauge field, whose field strength is the curvature of the SU(2) connection;

- Strong field: SU(3) gauge field, whose field strength is the curvature of the SU(3) connection.

The three gauge fields obey the same geometric logic: field sources excite gauge potentials, potentials define connections, and the curvature arising from connections is the field strength.

3. The Gravitational Field: Analogy, Not Identification

Geometric objects related to gravitation in general relativity:

Concept Mathematical Object Physical Meaning 

Gravitational potential Metric tensor Foundation of spacetime geometry 

Spacetime connection Christoffel symbols Connection derived from the metric 

Spacetime curvature Riemann tensor Spacetime bending, describing tidal effects 

Their mathematical structures are not identical. This paper makes only a formal analogy and does not identify gravity as a gauge field.

Analogy:

In gauge fields, the curvature of the connection is the gauge field strength.

In the gravitational field, the Riemannian curvature of the spacetime manifold is the gravitational field strength.

Important remark: This statement is a definition adopted within the analogy of this paper. In conventional terminology of general relativity, the Riemann tensor is not directly referred to as gravitational field strength. This naming is introduced solely to build the present parallel geometric analogy.

4. Core Proposition of the Analogy

The central thesis: gauge fields and gravitational fields do not share the same fibre bundle structure, but a parallel geometric analogy can be established.

1. Gauge fields (electromagnetic, strong, weak): field sources generate gauge potentials; potentials yield fibre bundle connections; the curvature associated with these connections is the gauge field strength.

2. Gravitational field: field sources (mass-energy) generate the metric field; the metric yields the spacetime connection; the Riemannian curvature associated with this connection is the gravitational field strength.

Unified picture from the analogy: Field sources excite a field; the field defines a connection; the curvature generated by the connection is the field strength.

Note: This remains a formal analogy. The two classes of curvature live in distinct geometric spaces and are not mathematically identical entities.

Field sources of the four fundamental interactions:

Interaction Field Source Field 

Gravitation Mass-energy Gravitational field 

Electromagnetism Electric charge Electromagnetic field 

Strong interaction Color charge Strong field 

Weak interaction Weak charge Weak field 

Causal chain of the analogy: Field sources excite the field → the field defines a connection → the connection produces curvature → curvature is the field strength.

5. Implications of the Analogous Paradigm

Conventional viewpoint:

- Gauge fields: curvature on internal fibre spaces, directly interpreted as field strength.

- Gravitation: curvature on the spacetime base manifold, conventionally not defined as field strength.

These two geometric objects are treated in separate frameworks.

Paradigm proposed here:

Gravity is not forced into the fibre bundle structure of gauge fields. Instead, a parallel analogy is extracted. Although the two fields have different geometric carriers, the picture “curvature is field strength” can be constructed for both. This analogy only supplies an alternative perspective and does not imply mathematical isomorphism.

6. Critical Remarks (To Mitigate Controversy)

1. This paper provides only conceptual analogy and summary, not a quantitative theory.

The text merely compares the underlying geometric images of the two theories. No unified differential field equations, unified tensor structures, numerical coefficients or observational predictions are provided.

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2. Relation to the Yang–Wu correspondence

The treatment of gauge fields follows the classic Yang–Wu correspondence. The new contribution of this paper is drawing the parallel analogy with gravity. It does not claim gravity belongs to the class of gauge fields. The two have fundamentally different geometric origins.

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3. Terminological note

In standard general relativity, the Riemann curvature tensor describes spacetime tidal effects and is not normally called gravitational field strength. Treating Riemannian curvature as gravitational field strength is a terminological choice within the present framework to construct the “curvature corresponds to field strength” analogy. It does not invalidate the standard terminology established in classical theories.

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4. No refutation of established classical theories

This analogous perspective offers a new way of understanding physics. It does not overturn the conclusions of general relativity and gauge field theory within their respective domains of validity.

7. Conclusion

1. The Yang–Wu correspondence supplies the geometric picture for three gauge fields: gauge field strength is equivalent to the curvature of fibre bundle connections.

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2. The gravitational field possesses its own independent geometric structure and is not a gauge field. A parallel analogy can be established: the spacetime Riemannian curvature of the gravitational field is the gravitational field strength.

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3. Within this analogous framework, the four fundamental interactions share a formal picture: field sources excite the field, the field induces a connection, the connection generates curvature, and curvature is field strength. It must be emphasized that this is merely an analogy of geometric images rather than a unified mathematical structure.

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4. This paper completes only the conceptual analogy. Construction of quantitative mathematical structures, rigorous field equations and observational verification are reserved for future research.

References

(Omitted)