Gravitational Electromagnetic Gauge Field

Abstract

This paper proposes a gauge field theory of gravitation and electromagnetism based on principal and associated bundles, aiming to realize mutual transformations between electromagnetic and gravitational fields through Generalized Gauge Transformations (GGE) and explore its implications for unifying the four fundamental interactions. The theory is founded on the structural group G = S O ( 1 , 3 ) × U ( 1 ) × S U ( 2 ) × S U ( 3 ) , where S O ( 1 , 3 ) and U ( 1 ) govern gravitational and electromagnetic interactions, respectively. We construct the gauge potentials A μ a b and gauge tensors F μ ν a b for the gravitational-electromagnetic gauge field, verifying their compliance with the geralized gauge equations (GGEs) for connection and curvature. The study demonstrates that the “generalized” nature of GGE lies in its capacity to mediate cross-interaction transformations, enabling electromagnetic and gravitational fields to mutually convert in overlapping regions via principal bundle theory while preserving gauge invariance of connections and curvatures. A conversion model from optical solitons to gravitational solitons is established to validate GGE, accompanied by derivations of the relationship between the electromagnetic tensor and the Weyl tensor, revealing profound connections to solitonic transformations. Furthermore, we prove the gauge invariance of the gravitational-electromagnetic field and propose that the four fundamental interactions represent projected components of a unified cosmic gauge field across distinct domains, with their quantum or classical nature determined by gauge representations. This work establishes a theoretical framework for unifying fundamental interactions and highlights potential applications in high-energy physics, gravitational wave detection, and cosmology.