Qiao Bi
Physics Department, Science School, Wuhan University of Technology, Wuhan, China.
DOI: 10.4236/jmp.2025.1611079   PDF    HTML   XML   60 Downloads   484 Views   Citations

Abstract

This paper develops a Generalized Gauge Equation (GGE) framework within the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ to achieve geometric unification of fundamental interactions. The core innovation lies in establishing precise transformation mechanisms between gauge fields, particularly demonstrating how electromagnetic interactions can be geometrically mapped onto gravitational configurations through well-defined gauge transformations. The mathematical foundation integrates connections, curvature, and gauge transformations into a unified description where different interactions emerge as projections of underlying spacetime geometry. Crucially, we derive the Weyl-electromagnetic relation $C_{\mu \nu \rho \sigma }=\kappa \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)$ from first principles, with the conversion coefficient $\kappa =8\pi \alpha$ determined through variational analysis of the coupled Lagrangian $ℒ\supset \alpha R_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma \nu }$ , rather than dimensional arguments alone. We demonstrate that two optical solitons transform into gravitational solitons via rotational gauge transformation, corresponding to photon-graviton conversion in weak-field limits. These gravitational solitons form localized curvature structures capable of superluminal propulsion, with derived effective velocities reaching $3c$ . The geometric framework naturally supports Closed Timelike Curves, enabling feasible interstellar travel distances. This work extends established gauge theory without speculative assumptions, providing experimentally testable predictions for unified field theory and advanced propulsion concepts.

Keywords

Generalized Gauge Transformation, Gravitational Solitons, Weyl Tensor, Field Unification, Warp Drive

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1. Introduction

The quest for quantum gravity, aiming to unify general relativity with quantum mechanics, represents a central challenge in theoretical physics. String theory describes the graviton through higher-dimensional spacetime, yet it faces computational complexity and a lack of experimental verification, as evidenced by the non-observation of supersymmetric particles at the LHC [1]. Loop Quantum Gravity (LQG) proposes spacetime discretization, applied by Rovelli and others to cosmology [2] [3]. Research at MIT explores the quantum information basis of spacetime geometry through holographic duality, addressing the black hole information paradox [4] [5]. The Perimeter Institute advances LQG and Causal Dynamical Triangulation (CDT), simulating the early universe [6]. Experimentally, Bose and colleagues proposed testing gravitational superpositions using quantum entanglement [7].

However, quantizing gravity encounters the fundamental problem of non-renormalizability. This arises from the infinite-dimensional diffeomorphism gauge group of gravity contrasting with the finite-dimensional gauge groups of the Standard Model (e.g., SU (3) × SU (2) × U (1)), compounded by experimental limitations at the Planck scale (10⁻³⁵ m, ~10¹⁹ GeV) hindering the development of a unified field theory [8] [9]. Approaches like locally covariant quantum field theory [10] and holographic duality [11] offer geometric pathways attempting to resolve non-renormalizability. Recent work within a gauge-theoretic framework explores geometric unification of gravity with the electromagnetic, weak, and strong interactions, achieving renormalizability via BRST symmetry [12] [13]. Nevertheless, the question of whether the universe unifies all four fundamental interactions on a geometric foundation remains unresolved [14].

This paper proposes a framework based on Generalized Gauge Equation (GGE) within the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ [15] [16]. This approach circumvents the direct quantization of gravity, instead unifying the four interactions geometrically. GGE transformations enable cross-group conversion of gauge potentials, mapping the electromagnetic, weak, or strong force onto the gravitational gauge field. Because the unification of electromagnetic force and gravity is the main difficulty in the grand unification of physics, we focus on the key point of the transformation of electromagnetic force into gravity through generalized gauge transformation. We derive the transformation of the electromagnetic field strength to the Weyl curvature, facilitating the generation of gravitational solitons [17]. Leveraging optical solitons to manipulate spacetime curvature, we design a curvature-bubble spacecraft capable of superluminal propulsion (achieving effective velocities up to $3c$ ) and the formation of Closed Timelike Curves (CTCs). This provides a potential means for interstellar travel (e.g., to Tau Ceti, 13.1 light-years distant).

Crucially, this research constitutes a natural extension of established gauge field theory, requiring no unconventional assumptions. The results strongly indicate that the fundamental structure of the universe is unified within the geometry of the principal bundle.

Paper Structure: Section 2-3 establish the principal bundle theory and cosmic structure. Sections 4-8 develop the GGE connection, curvature equations, and Lagrangian invariance. Sections 9-10 validate the transformation of the weak and strong forces. Section 11 derives the mapping from the electromagnetic field to the Weyl curvature. Section 12 details the design of the superluminal spacecraft and CTC mechanism. This work provides novel perspectives for unified field theory and interstellar travel.

2. Unification of Fundamental Forces and Principal Bundle Theory

This section frames the unified theory of the four fundamental interactions (gravitation, electromagnetism, weak interaction, strong interaction) within the framework of principal bundle theory. We analyze its mathematical and physical descriptions, relating it to the Standard Model of particle physics and the Generalized Gauge Equivalence (GGE) model. It essentially serves as an outline for subsequent sections.

2.1. Gauge Symmetry Groups in Principal Bundle Theory

In principal bundle theory, gauge field theories can be described by a principal bundle $P\left(M,G\right)$ , used to formulate the gauge field theories for the four fundamental interactions. Here:

• $M$ is the base manifold, a four-dimensional pseudo-Riemannian spacetime (Minkowski spacetime or curved spacetime in general relativity), representing our physical universe.

• $G$ ($G=GL\left(n,ℂ\right)$ ) is the structure group, the group of ($n\times n$ ) invertible complex matrices. It contains the gauge groups of the Standard Model ($SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ ) and the local Lorentz group $SO\left(1,3\right)$ of gravity as subgroups:

• $SU\left(3\right)\subset GL\left(3,ℂ\right)$ , describing the strong interaction (Quantum Chromodynamics, QCD), mediated by 8 gluons.

• $SU\left(2\right)\times U\left(1\right)\subset GL\left(2,ℂ\right)\times GL\left(1,ℂ\right)$ , describing the electroweak interaction, mediated by the photon $U{\left(1\right)}_{em}$ , $W^{pm}$ , and Z bosons.

• $SO\left(1,3\right)\subset GL\left(4,ℝ\right)\subset GL\left(n,ℂ\right)$ , describing gravitation, represented by the spin connection ${\omega }_{\mu }^{ab}$ or the orthonormal frame (tetrad/vierbein) $e_{\mu }^a$ .

• The fiber of the principal bundle $P$ is $GL\left(n,ℂ\right)$ , locally trivialized as $U\times GL\left(n,ℂ\right)$ , where $U\subset M$ is an open set on the base manifold.

Unified Theory Principal Bundle:

• The Standard Model unifies electromagnetism and the weak force via the Higgs mechanism, breaking $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)$ down to $U{\left(1\right)}_{em}$ .

• Grand Unified Theories (GUTs) [7] [18] (e.g., $SU\left(5\right)$ or $SO\left(10\right)$ ) unify the strong interaction with the electroweak force at high energies (~10¹⁶ GeV), embedding into $GL\left(n,ℂ\right)$ .

• This study employs $GL\left(n,ℂ\right)$ as the structure group, encompassing the Standard Model subgroups, the gravitational representation, and coupled subgroups (e.g., $SO\left(1,3\right)\times SU\left(3\right)$ ), supporting the generalized transformations of the Gravito-Electromagnetic Gauge (GGE) field theory.

Local Sections and Gauge Fields:

• A local section ${\sigma }_U:U\to P$ of the principal bundle $P$ defines a local gauge field. The connection form is:

${\omega }_U={\sigma }_U^{*}\tilde{\omega }$ (1)

where $\tilde{\omega }\in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ is the global gauge potential (connection form) on $P$ , whose components, upon quantization, correspond to the gauge bosons.

• In different regions $U$ of the base manifold $M$ , choosing different sections describes the gauge fields of specific interactions:

• Electromagnetism: $U\left(1\right)\subset GL\left(n,ℂ\right)$ , connection $A_{\mu }$ , curvature $F_{\mu \nu }$ .

• Weak Force: $SU\left(2\right)\times U\left(1\right)\subset GL\left(n,ℂ\right)$ , connections $W_{\mu }^a$ , $B_{\mu }$ .

• Strong Force: $SU\left(3\right)\subset GL\left(n,ℂ\right)$ , connection $G_{\mu }^a$ .

• Gravitation: $SO\left(1,3\right)\subset GL\left(n,ℂ\right)$ , connection ${\omega }_{\mu }^{ab}$ , orthonormal frame $e_{\mu }^a$ .

• The generality of $GL\left(n,ℂ\right)$ allows the transition functions $g_{UV}\in GL\left(n,ℂ\right)$ to encompass subgroup couplings (e.g., gravito-electromagnetic interaction), describing gauge transformations across interactions.

2.2. Base Manifold and the Universe

The base manifold $M$ is the four-dimensional pseudo-Riemannian spacetime, representing the physical universe. Physical fields (e.g., electromagnetic, weak, strong, gravitational fields) are either the connection form $\tilde{\omega }\in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ on $P$ or sections of associated bundles, defined on $M$ . The components of the connection $\tilde{\omega }$ incorporate the gauge fields of the Standard Model and the gravitational connection, unified within the Lie algebra $gl\left(n,ℂ\right)$ .

2.3. GGE and Transformations on Overlapping Regions

On overlapping regions of the base manifold $U\cap V\ne \varnothing$ , local sections ${\sigma }_U$ , ${\sigma }_V$ are related by a transition function $g_{UV}:U\cap V\to GL\left(n,ℂ\right)$ :

${\sigma }_V\left(x\right)={\sigma }_U\left(x\right)g_{UV}\left(x\right),x\in U\cap V$ (2)

The corresponding connection forms ${\omega }_U={\sigma }_U^{*}\tilde{\omega }$ , ${\omega }_V={\sigma }_V^{*}\tilde{\omega }$ satisfy the Generalized Gauge Equivalence (GGE), see Ref. [19] [20]:

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$ (3)

The curvature form $F=d\omega +\omega \wedge \omega$ transforms as, see Refs. [21] [22]:

$F_V=g_{UV}^{-1}{F}_U{g}_{UV}$ (4)

GGE describes the gauge transformation of the connection and curvature between different regions. $g_{UV}\in GL\left(n\right)$ can include subgroup couplings (e.g., SO (1, 3) × U (1)), supporting a unified description across interactions.

2.4. GGE and Unified Theory

The GGE framework uses $GL\left(n,ℂ\right)$ as the structure group to uniformly describe gravitation, electromagnetism, weak, and strong interactions:

• Gravitation: Via $SO\left(1,3\right)\subset GL\left(n,ℂ\right)$ , connection ${\omega }_{\mu }^{ab}$ , curvature $R_{\mu \nu }^{ab}$ , orthonormal frame $e_{\mu }^a$ , satisfying the metric $g_{\mu \nu }=e_{\mu }^a{e}_{\nu }^b{\eta }_{ab}$ .

• Electromagnetism: Via $U\left(1\right)\subset GL\left(n\right)$ , connection $A_{\mu }$ , curvature $F_{\mu \nu }$ .

• Weak Force: Via $SU\left(2\right)\times U\left(1\right)\subset GL\left(n\right)$ , connections $W_{\mu }^a$ , $B_{\mu }$ .

• Strong Force: Via $SU\left(3\right)\subset GL\left(n\right)$ , connection $G_{\mu }^a$ .

The choice of section in different regions of the base manifold $M$ emphasizes the contribution of specific subgroups. GGE ensures consistent transformations across regions, supporting the gauge invariance of the Lagrangian:

$ℒ=\frac{1}{16\pi G}\sqrt{-g}R+\frac{1}{4}\sqrt{-g}{F}_{\mu \nu }{F}^{\mu \nu }+\text{coupling terms}$ (5)

We will discuss the harmony between the gravitational-electromagnetic Lagrangian gauge invariance and the GGE in Section 8 later.

2.5. Ricci Tensor and Invariance of the Gravitational Term

To verify the physical consistency of the GGE framework with gravitational gauge fields [23], we can check the gauge invariance of the Ricci tensor $R_{\mu \nu }$ and the gravitational term $\frac{1}{16\pi G}\sqrt{-g}R$ under transformations within $SO\left(1,3\right)\subset GL\left(n\right)$ . The Ricci tensor is defined as:

$R_{\mu \nu }=R_{\mu \nu }^{ab}{e}_a^{\rho }{e}_b^{\sigma }{\eta }_{\rho \sigma }$ (6)

Under a gauge transformation $g\in SO\left(1,3\right)$ :

${R^{\prime }}_{\mu \nu }={R^{\prime }}_{\mu \nu }^{ab}{e^{\prime }}_a^{\rho }{e^{\prime }}_b^{\sigma }{\eta }_{\rho \sigma }$ (7)

where the transformed curvature is ${R^{\prime }}_{\mu \nu }^{ab}={\left(g^{-1}\right)}_c^a{R}_{\mu \nu }^{cd}{g}_d^b$ , and the transformed orthonormal frame is ${e^{\prime }}_a^{\rho }={\left(g^{-1}\right)}_a^e{e}_e^{\rho }$ . Substituting these into (7):

${R^{\prime }}_{\mu \nu }={\left(g^{-1}\right)}_c^a{R}_{\mu \nu }^{cd}{g}_d^b{\left(g^{-1}\right)}_a^e{e}_e^{\rho }{\left(g^{-1}\right)}_b^f{e}_f^{\sigma }{\eta }_{\rho \sigma }$ (8)

Utilizing the orthogonality conditions of the frame and the $SO\left(1,3\right)$ group:

$e_e^{\rho }{\eta }_{\rho \sigma }{e}_f^{\sigma }={\eta }_{ef}$ (9)

${\left(g^{-1}\right)}_a^e{\left(g^{-1}\right)}_b^f{\eta }_{ef}={\eta }_{ab}$ (10)

Combined orthogonality of $g$ and frame:

${\left(g^{-1}\right)}_c^a{g}_d^b{\eta }_{ab}={\eta }_{cd}$ (11)

Combined orthogonality of $g$ , Equation (8) then simplifies to:

${R^{\prime }}_{\mu \nu }=R_{\mu \nu }^{cd}{\eta }_{cd}=R_{\mu \nu }$ (12)

Therefore, the gravitational term $\frac{1}{16\pi G}\sqrt{-g}R$ remains invariant under the transformation, consistent with the GGE framework.

3. The Structure of the Cosmological Principal Bundle and Its Physical Significance

This section explores the principal bundle $P\left(M,G\right)$ as the mathematical framework for a unified description of the four fundamental interactions (gravitation, electromagnetism, weak interaction, strong interaction). We analyze its structure and physical significance, elucidating the connection between gauge field theory and Generalized Gauge Equivalence (GGE) through associated bundles and the frame bundle.

3.1. Mathematical Structure of the Principal Bundle and Associated Bundles

The principal bundle $P\left(M,G\right)$ is defined as follows:

• Base Manifold $M$ : A four-dimensional pseudo-Riemannian manifold, representing the physical universe (Minkowski spacetime or curved spacetime in general relativity).

• Structure Group $G$ : Taken as $G=GL\left(n,ℂ\right)$ , the group of $n\times n$ invertible complex matrices. It contains the subgroups:

• $SO\left(1,3\right)$ (The indefinite special orthogonal group of signature (1,3)): Describing gravitation.

• $U\left(1\right)$ (The unitary group of degree 1): Describing electromagnetism.

• $SU\left(2\right)$ (The special unitary group of degree 2): Describing the weak interaction.

• $SU\left(3\right)$ (The special unitary group of degree 3): Describing the strong interaction.

• Principal Bundle Structure: The fiber is $G=GL\left(n,ℂ\right)$ , locally trivialized as $U\times GL\left(n,ℂ\right)$ , where $U\subset M$ is an open set.

The principal bundle satisfies the following conditions:

1) Free Right Action: $GL\left(n,ℂ\right)$ acts freely on the right on $P$ , denoted $R:P\times GL\left(n,ℂ\right)\to P$ , $R\left(p,g\right)=pg$ , with no fixed points.

2) Projection Map: There exists a smooth projection $\pi :P\to M$ , with fiber ${\pi }^{-1}\left(x\right)\cong GL\left(n,ℂ\right)$ .

3) Local Trivialization: For an open cover $\left\{U_i\right\}$ of $M$ , there exist diffeomorphisms $T_{U_i}:{\pi }^{-1}\left(U_i\right)\to U_i\times GL\left(n,ℂ\right)$ of the form, $T_{U_i}\left(p\right)=\left(\pi \left(p\right),S_{U_i}\left(p\right)\right)$ , satisfying

$S_{U_i}\left(pg\right)=S_{U_i}\left(p\right)g,\forall p\in {\pi }^{-1}\left(U_i\right),g\in GL\left(n,ℂ\right)$ (13)

Associated Bundles: Let the fiber be a manifold $F={ℂ}^n$ . $GL\left(n,ℂ\right)$ acts linearly on $F$ via a representation $\rho :GL\left(n,ℂ\right)\to GL\left({ℂ}^n\right)$ , i.e., $\rho \left(g\right)\psi =g\psi$ . Define a right action on $P\times {ℂ}^n$ :

${\xi }_g\left(p,\psi \right)=\left(pg,g^{-1}\psi \right),p\in P,\psi \in {ℂ}^n,g\in GL\left(n,ℂ\right)$ (14)

The associated bundle $Q=\left(P\times {ℂ}^n\right)/~$ is the quotient manifold under this action, with orbits denoted $q=p\cdot \psi$ . The projection map is:

$\widehat{\pi }:Q\to M,\widehat{\pi }\left(p\cdot \psi \right)=\pi \left(p\right)$ (15)

Local trivializations ${\widehat{T}}_{U_i}:{\pi }^{-1}\left(U_i\right)\to U_i\times {ℂ}^n$ correspond to those of the principal bundle. The representation space ${ℂ}^n$ is equipped with a Hermitian inner product $\langle \phi ,\psi \rangle ={\phi }^{†}\psi =\overline{\phi }\psi$ . Subgroup representations are:

• $SO\left(1,3\right)$ : Acts on ${ℝ}^4\subset {ℂ}^n$ , with inner product given by the Minkowski metric ${\eta }_{ab}$ .

• $U\left(1\right)$ : Acts on $ℂ\subset {ℂ}^n$ via phase transformations.

• $SU\left(2\right)$ : Acts on ${ℂ}^2\subset {ℂ}^n$ (weak isospin space).

• $SU\left(3\right)$ : Acts on ${ℂ}^3\subset {ℂ}^n$ (color triplet space).

3.2. Physical Significance of the Principal Bundle Structure

The principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ provides a unified description of fundamental interactions:

• Base Manifold $M$ : The four-dimensional pseudo-Riemannian spacetime, equipped with a metric $g_{\mu \nu }$ , represents the observable universe.

• Structure Group $GL\left(n,ℂ\right)$ : Encompasses $SO\left(1,3\right)$ , $U\left(1\right)$ , $SU\left(2\right)$ , $SU\left(3\right)$ , providing the underlying symmetry and supporting subgroup couplings.

• Connection and Gauge Potentials: The connection form $\tilde{\omega }\in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ decomposes into subgroup components (e.g., ${\omega }_{\mu }^{ab}$ , $A_{\mu }$ , $W_{\mu }^a$ , $G_{\mu }^a$ ), corresponding to the gauge potentials.

• Curvature and Field Strengths: The curvature form $F=d\tilde{\omega }+\tilde{\omega }\wedge \tilde{\omega }$ corresponds to the physical field strengths (e.g., $R_{\mu \nu }^{ab}$ , $F_{\mu \nu }$ , $F_{\mu \nu }^a$ ).

• Associated Bundle Sections: Represent physical fields defined by the gauge structure (e.g., orthonormal frame $e_{\mu }^a$ , electromagnetic potential $A_{\mu }$ ).

• Gauge Transformations: On overlapping regions $U\cap V\ne \varnothing$ , the transition function $g_{UV}\in GL\left(n,ℂ\right)$ reconciles the connection via GGE, Equation (3); the curvature transforms as Equation (4).

3.3. Base Manifold and the Physical World

The base manifold $M$ , as a four-dimensional pseudo-Riemannian manifold, describes both quantum physics and gravitational phenomena:

• In the Standard Model, gauge transformations of $SU\left(3\right)\times SU\left(2\right)\times U\left(1\right)\subset GL\left(n,ℂ\right)$ leave the Lagrangian invariant, describing electromagnetic, weak, and strong interactions.

• In General Relativity, $M$ is equipped with a metric $g_{\mu \nu }$ . Gravitation is described via the connection $SO\left(1,3\right)$ (associated with $SO\left(1,3\right)\subset GL\left(4,ℝ\right)$ ) and the Ricci tensor $R_{\mu \nu }$ .

• The generality of $GL\left(n,ℂ\right)$ allows for coupling terms (e.g., $R_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ ), supporting the exploration of unified theories.

3.4. Advantages of the Frame Bundle

The frame bundle $FM\left(M,GL\left(4,ℝ\right)\right)\subset P\left(M,GL\left(n,ℂ\right)\right)$ is defined as:

$FM=\left\{\left(x,e\right)|x\in M,e:{ℝ}^4\to T_{x}M\text{is}\text{a}\text{linear}\text{isomorphism}\right\}$ (16)

Its structure group $GL\left(4,ℝ\right)$ contains $SO\left(1,3\right)$ . The metric constraint is enforced via:

$g_{\mu \nu }=e_{\mu }^a{e}_{\nu }^b{\eta }_{ab}$ (17)

Its advantages include:

• Connection and Derivative Operator: A connection $\tilde{\omega }$ on the frame bundle $FM$ induces a derivative operator $\nabla$ on the base manifold $M$ . There is a one-to-one correspondence:

$\left(M,\nabla \right)\leftrightarrow \left(FM,\tilde{\omega }\right)$ (18)

Proof Sketch:

• From $\left(M,\nabla \right)\to \left(FM,\tilde{\omega }\right)$ : Given $\nabla$ , define horizontal subspaces $H_p\subset T_{p}FM$ satisfying the properties of a connection.

• From $\left(FM,\tilde{\omega }\right)\to \left(M,\nabla \right):\tilde{\omega }$ induces $\nabla$ .

• Physical Significance: The $SO\left(1,3\right)$ component of $\tilde{\omega }$ corresponds to the gravitational spin connection ${\omega }_{\mu }^{ab}$ , and its curvature $R_{\mu \nu }^{ab}$ describes the gravitational field strength.

3.5. Construction of Associated Bundles and Representations

The associated bundle $Q=P{\times }_{GL\left(n,ℂ\right)}{ℂ}^n$ provides a unified description of gauge fields. Subgroup representations yield specific associated bundles:

• Gravitation (SO (1,3)): Fiber ${ℝ}^4\subset {ℂ}^n$ , representation $\rho {\left(g\right)}_v=g_v$ , inner product ${\eta }_{ab}$ . The associated bundle $Q_{SO\left(1,3\right)}\cong TM$ (tangent bundle). Sections are orthonormal frames $e_{\mu }^a$ . Connection ${\omega }_{\mu }^{ab}$ , curvature $R_{\mu \nu }^{ab}$ .

• Electromagnetism (U (1)): Fiber $ℂ\subset {ℂ}^n$ , representation $\rho \left({\text{e}}^{i\theta }\psi \right)={\text{e}}^{iq\theta }\psi$ (where $q$ is charge). The associated bundle is $Q_{U\left(1\right)}$ . Sections correspond to the electromagnetic potential $A_{\mu }$ , curvature is $F_{\mu \nu }$ .

• Weak Interaction (SU (2)): Fiber ${ℂ}^2\subset {ℂ}^n$ , representation $\rho \left(g\right)\psi =g\psi$ . The associated bundle is $Q_{SU\left(2\right)}$ . Sections correspond to the weak gauge fields $W_{\mu }^a$ .

• Strong Interaction (SU (3)): Fiber ${ℂ}^3\subset {ℂ}^n$ , representation $\rho \left(g\right)\psi =g\psi$ . The associated bundle is $Q_{SU\left(3\right)}$ . Sections correspond to the strong gauge fields $G_{\mu }^a$ .

Sections of the overall associated bundle $Q$ (fiber ${ℂ}^n$ ) provide a unified description of gravitational, electromagnetic, weak, and strong gauge fields. The GGE transformation (Equation (3)) ensures consistent transformations across overlapping regions.

3.6. Discussion: Choice of Structure Group and Frame Bundle

Structure Group $GL\left(n,ℂ\right)$ :

• Advantages: Encompasses $SO\left(1,3\right)$ , $U\left(1\right)$ , $SU\left(2\right)$ , $SU\left(3\right)$ , supporting subgroup couplings (e.g., gravito-electromagnetic interaction). The GGE transformation (Equation (2)) is universally applicable; $g_{UV}\in GL\left(n,ℂ\right)$ allows complex transformations adaptable to unified theories.

• Disadvantages: Non-compactness may introduce unphysical degrees of freedom, requiring constraints via the metric (e.g., ${\eta }_{ab}$ ) and Hermitian inner product. The physical interpretation of $n$ and the representation space must be clarified, and coupling terms are constrained by experiment.

Frame Bundle $FM\left(M,GL\left(4,ℝ\right)\right)$ :

• Advantages: Directly corresponds to gravitation; the connection ${\omega }_{\mu }^{ab}$ and curvature $R_{\mu }^{ab}$ are consistent with General Relativity. Associated bundles can be extended to describe particle physics gauge fields.

• Limitations: Redundancy in $GL\left(4,ℝ\right)$ necessitates constraint to $SO\left(1,3\right)$ . Describing couplings requires explicitly defining cross-terms (e.g., $SO\left(1,3\right)\times SU\left(3\right)$ ) within the larger $GL\left(n,ℂ\right)$ framework.

4. Transition Functions across Fundamental Interactions

This section explores gauge transformations across fundamental interactions defined by transition functions $g_{UV}$ within the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ framework. We analyze their mathematical structure and physical significance, focusing on the unification of electromagnetic and gravitational fields through Generalized Gauge Equivalence (GGE), which reconciles connections and curvatures across different interactions.

4.1. Definition of Gauge Transformations

In the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ :

• Base manifold $M$ : 4D pseudo-Riemannian spacetime;

• Structure group $P\left(M,GL\left(n,ℂ\right)\right)$ : Contains subgroups $SO\left(1,3\right)$ , $U\left(1\right)$ , $SU\left(2\right)$ , $SU\left(3\right)$ for gravitation, electromagnetism, weak, and strong interactions;

• Local trivialization over open cover $\left\{U_i\right\}$ :

$T_{U_i}:{\pi }^{-1}\left(U_i\right)\to U_i\times GL\left(n,ℂ\right),T_{U_i}\left(p\right)=\left(\pi \left(p\right),S_{U_i}\left(p\right)\right),p\in {\pi }^{-1}\left(U_i\right)$ (19)

satisfying

$S_{U_i}\left(pg\right)=S_{U_i}\left(p\right)g,g\in GL\left(n,ℂ\right)$ (20)

On overlaps $U\cap V\ne \varnothing$ , for $p\in {\pi }^{-1}\left(U\cap V\right)$ :

$T_U\left(p\right)=\left(x,g_U\right),T_V\left(p\right)=\left(x,g_V\right),x=\pi \left(p\right)\in U\cap V$ (21)

where $g_U=S_U\left(p\right)$ , $g_V=S_V\left(p\right)\in GL\left(n,ℂ\right)$ . The transition function is defined as:

$g_{UV}\left(x\right)=S_U\left(p\right)S_V^{-1}\left(p\right),x\in U\cap V$ (22)

implying:

$g_U=g_{UV}\left(x\right)g_V$ (23)

Cross-Interaction Gauge Transformation:

When $g_U$ and $g_V$ project to different subgroups (e.g., $g_U\in U\left(1\right)$ , $g_V\in SO\left(1,3\right)$ , $g_{UV}$ defines a gauge transformation between distinct interactions (e.g., electromagnetism → gravitation). When $g_U$ , $g_V$ belong to the same subgroup (e.g., $SU\left(3\right)$ ), it reduces to conventional gauge transformations.

4.2. Equivalent Bundle Representation

The principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ is reconstructed from disjoint union $\Sigma ={\displaystyle {\cup }_i\left(U_i\times GL\left(n,ℂ\right)\right)}$ via equivalence relation:

$\left(x,g\right)~(x^{\prime },g^{\prime }),\text{iff}x=x^{\prime }\text{and}g=g_{UV}\left(x\right)g^{\prime }$ (24)

where $\left(x,g\right)\in U\times GL\left(n,ℂ\right)$ , $\left(x^{\prime },g^{\prime }\right)\in V\times GL\left(n,ℂ\right)$ . The quotient $P=\Sigma /~$ ensures consistency.

Example (EM-to-Gravity):

• $g_U\in U\left(1\right)\subset GL\left(n,ℂ\right)$ : EM potential $A_{\mu }$ ;

• $g_V\in SO\left(1,3\right)\subset GL\left(n,ℂ\right)$ : Spin connection ${\omega }_{\mu }^{ab}$ ;

• Transition function: $g_{UV}\left(x\right)=g_U{g}_V^{-1}$ defines EM → gravity gauge transformation.

4.3. Cross-Interaction Gauge Transformation and GGE Equations

Cross-interaction transformations relate physical fields (e.g., $F_{\mu \nu }$ , $R_{\mu \nu }^{ab}$ ) via $g_{UV}\in GL\left(n,ℂ\right)$ through GGE (3) and (4):

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

$F_V=g_{UV}^{-1}{F}_U{g}_{UV}$

EM-Gravity Case:

• ${\omega }_U=A_{\mu }d{x}^{\mu }\in u\left(1\right)\subset gl\left(n,ℂ\right)$ ;

• ${\omega }_V={\omega }_{\mu }^{ab}d{x}^{\mu }\in so\left(1,3\right)\subset gl\left(n,ℂ\right)$ ;

• $g_{UV}=W=W_2{W}_1^{-1}\left(W_1\in U\left(1\right),W_2\in SO\left(1,3\right)\right)$ .

GGE reconciles EM-gravity transformations and enables subgroup couplings (e.g., $SO\left(1,3\right)\times U\left(1\right)$ ).

4.4. Unification of Electromagnetic and Weyl Tensors

To unify EM and gravity fields, consider:

• EM tensor $F_{\mu \nu }$ ($U\left(1\right)$ curvature);

• Weyl tensor *$C_{\mu \nu }^{ab}$ ($SO\left(1,3\right)$ curvature component).

Eigenvalue alignment via subgroup elements:

$W_2^{-1}{C}_{\mu \nu }^{ab}{W}_2=W_1^{-1}{F}_{\mu \nu }{W}_1$ (25)

where $W_1\in U\left(1\right)$ , $W_2\in SO\left(1,3\right)\subset GL\left(n,ℂ\right)$ .

Dediagonalizing to reconstruct the tensor:

$C_{\mu \nu }^{ab}=W{F}_{\mu \nu }{W}^{-1},W=W_2{W}_1^{-1}$ (26)

Here $W=g_{UV}$ represents the gauge transformation mediating EM-gravity unification, consistent with GGE and enabling subgroup couplings.

4.5. Physical Significance and Discussion

Cross-interaction gauge transformations demonstrate that EM, gravitational, weak, and strong fields unify under $P\left(M,GL\left(n,ℂ\right)\right)$ via $g_{UV}$ . Key advances:

1) $GL\left(n,ℂ\right)$ enables subgroup couplings (e.g., gravito-electromagnetic) beyond single-interaction limits;

2) GGE provides mathematical foundation for cross-interaction transformations;

3) Potential applications in curvature-based propulsion and CTC engineering.

The unification mechanism (Equation 26) will be leveraged in Section 12 for superluminal spacecraft design.

5. Gauge Field Theory on Principal and Associated Bundles

This section examines the role of the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ and the associated bundle $Q=P{\times }_{GL\left(n,C\right)}{ℂ}^n$ in gauge field theory. We analyze the mathematical structure and physical significance of gauge choices, local transformations, and gauge-invariant fields, focusing on how gauge transformations across fundamental interactions unify gravitation, electromagnetism, weak, and strong interactions via Generalized Gauge Equation (GGE).

5.1. Bundle Structure and Gauge Choice

The principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ has:

• Base Manifold $M$ : 4D pseudo-Riemannian spacetime.

• Structure Group $GL\left(n,ℂ\right)$ : Contains subgroups $SO\left(1,3\right)$ (gravitation), $U\left(1\right)$ (electromagnetism), $SU\left(2\right)$ (weak), $SU\left(3\right)$ (strong).

• Representation Space: ${ℂ}^n$ with Hermitian inner product $\langle \phi ,\psi \rangle ={\phi }^{†}\psi =\overline{\varphi }\psi$ . Subgroup representations and inner product constraints:

• Gravitation (SO (1, 3)): Subspace ${ℝ}^4\subset {ℂ}^n$ , inner product ${\eta }_{ab}=diag\left(-1,1,1,1\right)$ . Orthonormal frame $e_{\mu }^a$ satisfies:

$g_{\mu \nu }=e_{\mu }^a{e}_{\nu }^b{\eta }_{ab}$ (27)

The Minkowski inner product $\left(\varphi ,\psi \right)={\varphi }_a{\psi }^b{\eta }^{ab}$ , $\varphi \in {\left({ℝ}^4\right)}^{*}$ , $\psi \in {ℝ}^4$ reflects local Lorentz symmetry for frames like $e_{\mu }^a$ . This corresponds to the pseudo-Riemannian structure of tangent spaces in GR, ensuring metric consistency.

• Electromagnetism (U (1)): Subspace $ℂ\subset {ℂ}^n$ , Hermitian inner product $\langle \varphi ,\psi \rangle =\overline{\varphi }\psi$ , $\forall \varphi ,\psi \in ℂ$ , describing phase transformations of charged particles.

• Weak Interaction (SU (2)): Subspace ${ℂ}^2\subset {ℂ}^n$ , Hermitian inner product $\langle \varphi ,\psi \rangle =\overline{\varphi }\psi ={\displaystyle {\sum }_{i=1}^2{\overline{\varphi }}_i{\psi }_i}$ , $\forall \varphi ,\psi \in {ℂ}^2$ , describing weak isospin doublets.

• Strong Interaction (SU (3)): Subspace ${ℂ}^3\subset {ℂ}^n$ , Hermitian inner product $\langle \varphi ,\psi \rangle =\overline{\varphi }\psi ={\displaystyle {\sum }_{i=1}^3{\overline{\varphi }}_i{\psi }_i}$ , $\forall \varphi ,\psi \in {ℂ}^3$ , describing quark color triplets.

These inner products ensure orthogonality and physical field symmetries: ${\eta }_{ab}$ aligns with GR, while the Hermitian inner products align with the Standard Model.

Principal Bundle Action:

$P$ has a free right action:

$R:P\times GL\left(n,ℂ\right)\to P,R_{g_1}\left(x,g_2\right)=\left(x,g_2{g}_1\right)$ (28)

Local Sections and Gauge Choice:

Local sections ${\sigma }_U:U\to P$ , ${\sigma }_V:V\to P$ are related on overlaps $U\cap V\ne \varnothing$ by transition functions $g_{UV}:U\cap V\to GL\left(n,ℂ\right)$ ) via Equation (2):

${\sigma }_V\left(x\right)={\sigma }_U\left(x\right)g_{UV}\left(x\right),\forall x\in U\cap V$

For cross-interaction transformations, $g_{UV}\left(x\right)$ can connect different subgroups (e.g., EM $U\left(1\right)\to SO\left(1,3\right)$ or EM → strong $SU\left(3\right)$ ). The section ${\sigma }_U:U\to P$ represents a gauge choice: an $SO\left(1,3\right)$ section corresponds to the orthonormal frame $e_{\mu }^a$ , while a $U\left(1\right)$ section corresponds to the EM potential $A_{\mu }$ .

5.2. Local Transformation of Gauge Fields

Gauge fields (particle fields) $\psi \left(x\right)\in {ℂ}^n$ transform under the representation group $\widehat{G}$ via $U\left(x\right)=\rho \left(g\left(x\right)\right)\in \widehat{G}$ :

${\psi }^{\prime }\left(x\right)=U\left(x\right)\psi \left(x\right)=\rho \left(g\left(x\right)\right)\psi \left(x\right)$ (29)

where $g\left(x\right)\in GL\left(n,ℂ\right)$ . For $g\left(x\right)=\exp \left({\theta }^r{e}_r\right)$ ($e_r$ basis of $Lie\left(GL\left(n,ℂ\right)\right)$ , ${\theta }^r$ parameters), the pushforward ${\rho }_{\ast }:Lie\left(GL\left(n,ℂ\right)\right)\to Lie\left(\widehat{G}\right)$ is defined by:

$\rho \left(\exp \left(A\right)\right)=U\left(\theta \right)=\exp \left({\rho }_{\ast }\left(A\right)\right)=\exp \left(-i{L}_r{\theta }^r\right)$ (30)

where $A\in T_{e}GL\left(n,ℂ\right)$ generates the 1-parameter subgroup $\exp \left(tA\right)$ . Examples:

• $U\left(1\right)$ : $A=i\theta$ , $\exp \left(i\theta t\right)={\text{e}}^{i\theta t}$ , generates phase transformations.

• $SO\left(1,3\right)$ : $A={\theta }^{ab}{J}_{ab}$ , $\exp \left(t{\theta }^{ab}{J}_{ab}\right)$ generates Lorentz rotations.

Here, $L_r=-i{\rho }_{\ast }\left(e_r\right)$ are generators of $Lie\left(\widehat{G}\right)$ , represented as $n\times n$ matrices,$U\left(\theta \right)={\text{e}}^{-iL\cdot \theta }$ . Thus, principal bundle section transformations induce local gauge transformations (Equation 29) of $\psi \left(x\right)$ , and vice versa.

5.3. Associated Bundles and Gauge-Invariant Entities

Sections $\widehat{\sigma }:U\to Q$ of the associated bundle $Q$ represent gauge fields (particle fields) on $M$ . The entity $\tilde{\psi }\left(x\right)\in Q$ is gauge-invariant. Let fiber $F={ℂ}^n$ , with left action:

$\chi :GL\left(n,ℂ\right)\times {ℂ}^n\to {ℂ}^n,{\chi }_{g_1}\left(f_1\right)=\rho \left(g_1\right)f_1,\forall g_1\in GL\left(n,ℂ\right),f_1\in {ℂ}^n$ (31)

Given a section $\sigma \left(x\right)\in {\pi }^{-1}\left(x\right)\subset P$ and an $F$ -valued function $f:U\to {ℂ}^n$ , construct the associated bundle section:

$\tilde{\psi }\left(x\right)=\sigma \left(x\right)\cdot f\left(x\right)\in {\widehat{\pi }}^{-1}\left(x\right)\subset Q$ (32)

Under gauge transformation (new section ${\sigma }^{\prime }\left(x\right)=\sigma \left(x\right)g^{-1}\left(x\right)$ , new function $f^{\prime }\left(x\right)={\chi }_{g\left(x\right)}f\left(x\right)=\rho \left(g\left(x\right)\right)f\left(x\right)$ :

${\tilde{\psi }}^{\prime }\left(x\right)={\sigma }^{\prime }\left(x\right)\cdot f^{\prime }\left(x\right)=\left(\sigma \left(x\right)g^{-1}\left(x\right),\rho \left(g\left(x\right)\right)f\left(x\right)\right)=\left(\sigma \left(x\right),f\left(x\right)\right)=\tilde{\psi }\left(x\right)$ (33)

This invariance holds because ${\sigma }^{\prime }\left(x\right)\cdot f^{\prime }\left(x\right)$ and $\sigma \left(x\right)\cdot f\left(x\right)$ lie on the same orbit in $P\times F$ under the equivalence defining $Q$ [19].

Define $\psi \left(x\right)=f\left(x\right)\in {ℂ}^n$ as the component of the gauge field relative to the section $\sigma \left(x\right)$ . Under gauge transformation (Equation 29):

$\psi \text{'}\left(x\right)=\rho \left(g\left(x\right)\right)\psi \left(x\right)$

However, the associated bundle section $\tilde{\psi }\left(x\right)=\sigma \left(x\right)\cdot \psi \left(x\right)=\left(\sigma \left(x\right),\psi \left(x\right)\right)\in Q$ is invariant. Therefore, the section $\widehat{\sigma }:U\to Q$ , given by $\widehat{\sigma }\left(x\right)=\tilde{\psi }\left(x\right)$ , represents a gauge-invariant entity on $M$ . Only its component $\psi \left(x\right)\in {ℂ}^n$ changes under gauge transformations. Mathematically, the fiber $Q_x={\widehat{\pi }}^{-1}\left(x\right)$ is ($P_x\times V/G$ ), and orbit equivalence ensures $\tilde{\psi }\left(x\right)$ ’s invariance. Physical Significance: $\tilde{\psi }\left(x\right)$ is the intrinsic gauge field entity (e.g., EM field, gravitational field). Its components $\psi \left(x\right)$ change with the section $\sigma \left(x\right)$ , but the entity itself remains fixed in $Q$ , reflecting the intrinsic symmetry of gauge field theory.

5.4. GGE Equations and Cross-Interaction Transformation

Cross-interaction gauge transformations reconcile gauge fields of different subgroups via transition functions $g_{UV}$ . For example, transforming the EM field $A_{\mu }$ (U (1)) to the gravitational field ${\omega }_{\mu }^{ab}$ (SO (1,3)) is described by GGE Equations (3) and (4):

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

$F_V=g_{UV}^{-1}{F}_U{g}_{UV}$

Here, $g_{UV}$ projects onto the $U\left(1\right)$ and $SO\left(1,3\right)$ subgroups to reconcile EM and gravity. The invariance of associated bundle sections $\tilde{\psi }\left(x\right)$ ensures field entity consistency. Examples:

• EM field component $\psi \left(x\right)\in ℂ\subset {ℂ}^n$ transforms as ${\psi }^{\prime }\left(x\right)={\text{e}}^{iq\theta \left(x\right)}\psi \left(x\right)$ .

• Gravitational orthonormal frame $e_{\mu }^a\in {ℝ}^4\subset {ℂ}^n$ transforms as ${e^{\prime }}_{\mu }^a={\left(g^{-1}\right)}_b^a{e}_{\mu }^b$ , $g\in SO\left(1,3\right)$ ).

Sections $\tilde{\psi }\left(x\right)$ of $Q$ provide a unified description, ensuring Lagrangian invariance (e.g., $ℒ=\frac{1}{16\pi G}\sqrt{-g}R+\frac{1}{4}\sqrt{-g}{F}_{\mu \nu }{F}^{\mu \nu }+\alpha \sqrt{-g}{R}_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ ) under gauge transformations.

5.5. Physical Significance

• Principal Bundle Section $\sigma :U\to P$ : Represents a choice of gauge (e.g., Lorentz frame, EM gauge).

• Associated Bundle Section $\widehat{\sigma }:U\to Q$ : Represents the intrinsic, gauge-invariant field entity $\tilde{\psi }\left(x\right)$ .

• Gauge Transformations: Change only the field components $\psi \left(x\right)$ relative to the section $\sigma \left(x\right)$ , leaving the intrinsic field entity $\tilde{\psi }\left(x\right)$ invariant. This reflects the core invariance principle of gauge theories.

• Generalization via $GL\left(n,ℂ\right)$ : Allows cross-interaction transformations (e.g., gravito-electromagnetic coupling) via $g_{UV}$ .

• GGE Unification: Provides the framework to unify gravitational, electromagnetic, weak, and strong fields.

• Application Foundation: Underpins potential applications like curvature-based propulsion engines.

6. Generalized Gauge Equations (GGE) for Connections

This section explores the connection $\tilde{\omega }$ on the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ and its Generalized Gauge Equations (GGE). We prove that when transition functions $g_{UV}$ involve transformations across fundamental interactions (e.g., electromagnetism to gravitation), the GGE and related structures (connection, curvature, Cartan’s second structure equation) remain valid. This provides the mathematical-physical foundation for unifying gravitation, electromagnetism, weak, and strong interactions.

6.1. Definition of the Connection

In the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ :

• Base Manifold $M$ : 4D pseudo-Riemannian spacetime.

• Structure Group $GL\left(n,ℂ\right)$ : Contains subgroups $SO\left(1,3\right)$ (gravitation), $U\left(1\right)$ (electromagnetism), $SU\left(2\right)$ (weak), $SU\left(3\right)$ (strong).

• Connection $\tilde{\omega }$ : A smooth $gl\left(n,ℂ\right)$ -valued 1-form on $P$ satisfying:

1. Vertical Restoration: For any $p\in P$ , $A\in gl\left(n,ℂ\right)$ , and its induced vertical vector field $A_p^{*}\in T_{p}P$ :

${\tilde{\omega }}_p\left(A_p^{*}\right)=A$ (34)

2. Adjoint Invariance: For any $g\in GL\left(n,ℂ\right)$ , $p\in P$ , $X\in T_{p}P$ , and right action $R_g:P\to P$ :

${\tilde{\omega }}_{pg}\left(R_{g*}X\right)=A{d}_{g^{-1}}{\tilde{\omega }}_p\left(X\right)$ (35)

where $A{d}_g:gl\left(n,ℂ\right)\to gl\left(n,ℂ\right)$ is the adjoint $A{d}_g\left(A\right)=I_{g*}\left(A\right)=gA{g}^{-1}$ .

Local Sections and GGE:

Given local sections ${\sigma }_U:U\to P$ , ${\sigma }_V:V\to P$ on overlap $U\cap V\ne \varnothing$ , related by Equation (2):

${\sigma }_V\left(x\right)={\sigma }_U\left(x\right)g_{UV}\left(x\right),\forall x\in U\cap V$

where $g_{UV}:U\cap V\to GL\left(n,ℂ\right)$ may involve cross-subgroup transformations (e.g., projecting to $U\left(1\right)$ and $SO\left(1,3\right)$ ). The pullback connections are:

${\omega }_U={\sigma }_U^{*}\tilde{\omega },{\omega }_V={\sigma }_V^{*}\tilde{\omega }$ (36)

both $gl\left(n,ℂ\right)$ -valued 1-forms on $U\cap V$ . The GGE describes their transformation:

${\omega }_V\left(Y\right)=A{d}_{g_{UV}^{-1}\left(x\right)}{\omega }_U\left(Y\right)+L_{g_{UV}\left(x\right)*}^{-1}{g}_{UV*}\left(Y\right)$ (37)

$\forall x\in U\cap V,Y\in T_{x}M$ . Equation (37) is the general GGE form, valid for any $gl\left(n,ℂ\right)$ -valued connection, including cross-subgroup cases (e.g., ${\omega }_U\in u\left(1\right)$ , ${\omega }_V\in so\left(1,3\right)$ ). Its matrix form is Equation (3):

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

6.2. Proof of the Connection GGE

Proof Strategy: Use the pullback via sections ${\sigma }_U$ , ${\sigma }_V$ to project $\tilde{\omega }$ to ${\omega }_U$ , ${\omega }_V$ , verifying Equation (37) and its matrix form (3). The key is analyzing the pushforward ${\sigma }_{V*}Y$ , combined with section transformation (2) and adjoint action [19].

Steps:

1. Pullback Connection: For $Y\in T_{x}M$ , by definition:

${\omega }_V\left(Y\right)=\tilde{\omega }\left({\sigma }_{V*}Y\right)$ (38)

where ${\sigma }_{V*}:T_{x}M\to T_{{\sigma }_V\left(x\right)}P$ is the pushforward. Compute ${\sigma }_{V*}Y$ using ${\sigma }_V\left(x\right)={\sigma }_U\left(x\right)g_{UV}\left(x\right)$ .

2. Pushforward Decomposition: Let $\eta :I\to U\cap V$ be a curve ($I\subset ℝ\ni 0$ ), $\eta \left(0\right)=x$ , ${\frac{\text{d}}{\text{d}t}|}_{t=0}\eta \left(t\right)=Y$ . Then:

${\sigma }_{V*}Y={\sigma }_{V*}{\frac{\text{d}}{\text{d}t}|}_{t=0}\eta \left(t\right)={\frac{\text{d}}{\text{d}t}|}_{t=0}{\sigma }_V\left(\eta \left(t\right)\right)={\frac{\text{d}}{\text{d}t}|}_{t=0}\left[{\sigma }_U\left(\eta \left(t\right)\right)g_{UV}\left(\eta \left(t\right)\right)\right]$

Apply Leibniz rule:

$\begin{array}{c}{\sigma }_{V*}Y=\left({\frac{\text{d}}{\text{d}t}|}_{t=0}{\sigma }_U\left(\eta \left(t\right)\right)\right)g_{UV}\left(x\right)+\left({\frac{\text{d}}{\text{d}t}|}_{t=0}{\sigma }_V\left(x\right)L_{g_{UV}\left(x\right)}^{-1}{g}_{UV}\left(\eta \left(t\right)\right)\right)\\ =R_{g_{UV}\left(x\right)*}{\sigma }_{U*}Y+R_{{\sigma }_V\left(x\right)*}{\frac{\text{d}}{\text{d}t}|}_{t=0}{L}_{g_{UV}\left(x\right)}^{-1}{g}_{UV}\left(\eta \left(t\right)\right)\\ =R_{g_{UV}\left(x\right)*}{\sigma }_{U*}Y+{\frac{\text{d}}{\text{d}t}|}_{t=0}{g}_{UV}{\left(\eta \left(t\right)\right)}_{{\sigma }_V\left(x\right)}^{*}\end{array}$ (39)

where the first term arises from the pushforward of the right multiplication by the group element $g_{UV}\left(x\right)$ , and the second term is induced by the mapping $R_p:G\to P$ with $p={\sigma }_V\left(x\right)$ , via the relation $A_p^{*}=R_{p*}A$ .

3. Term 1 Calculation:

$\tilde{\omega }\left(R_{g_{UV}\left(x\right)*}{\sigma }_{U*}Y\right)=A{d}_{g_{UV}^{-1}\left(x\right)}\tilde{\omega }\left({\sigma }_{U*}Y\right)=A{d}_{g_{UV}^{-1}\left(x\right)}{\omega }_U\left(Y\right)$ (40)

4. Term 2 Calculation: The vertical vector $\left(g_{UV*}\left(Y\right)\right){ }_{{\sigma }_V\left(x\right)}^{*}\in T_{{\sigma }_V\left(x\right)}P$ is generated by $g_{UV*} \left(Y\right)\in T_{g_{UV}\left(x\right)}GL\left(n,ℂ\right)$ at ${\sigma }_V\left(x\right)$ . The connection acts on the vertical vector as:

$\tilde{\omega }\left({\left(g_{UV*}\left(Y\right)\right)}_{{\sigma }_V\left(x\right)}^{*}\right)=\left(L_{g_{UV}\left(x\right)*}^{-1}{g}_{UV*}\right)\left(Y\right)$ (41)

Combining (40) and (41) we get Equation (37):

${\omega }_V\left(Y\right)=\tilde{\omega }\left({\sigma }_{V*}Y\right)=A{d}_{g_{UV}^{-1}\left(x\right)}{\omega }_U\left(Y\right)+\left(L_{g_{UV}\left(x\right)*}^{-1}{g}_{UV*}\right)\left(Y\right)$

Matrix Form Justification: For $GL\left(n,ℂ\right)$ , ${\omega }_U,{\omega }_V\in {\Lambda }^1\left(U\cap V,gl\left(n,ℂ\right)\right)$ , $g_{UV}\in {\Lambda }^0\left(U\cap V,GL\left(n,ℂ\right)\right)$ . Expanding $g_{UV}=f^r{e}_r$ ($e_r$ as basis of $gl\left(n,ℂ\right)$ ):

$d{g}_{UV}\left(Y\right)=e_r\left(d{f}^r\right)\left(Y\right)$ ,

$g_{UV}^{-1}d{g}_{UV}\left(Y\right)=g_{UV}^{-1}\left(x\right)e_r\left(d{f}^r\right)\left(Y\right)=\left(L_{g_{UV}\left(x\right)*}^{-1}{g}_{UV*}\right)\left(Y\right)$

Thus, the matrix form (3) holds. For cross-subgroup transformations (e.g., ${\omega }_U\in u\left(1\right)$ , ${\omega }_V\in so\left(1,3\right)$ ), with $g_{UV}$ projecting to $U\left(1\right)\times SO\left(1,3\right)$ , Equation (37) ensures consistent transformation.

6.3. Implications of the $GL\left(n,ℂ\right)$ Framework

The single structure group $GL\left(n,ℂ\right)$ impacts GGE and cross-interaction transformations:

1. Unified Connection: $\tilde{\omega }\in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ incorporates subgroup components (e.g., ${\omega }_{\mu }^{ab}$ , $A_{\mu }$ , $W_{\mu }^a$ , $G_{\mu }^a$ ), providing a unified description. Pullback ${\omega }_U={\sigma }_U^{*}\tilde{\omega }$ contains all subgroup Lie algebra components.

2. Cross-Subgroup Transformation: $g_{UV}\in GL\left(n,ℂ\right)$ enables cross-interaction transformations (e.g., EM $U\left(1\right)$ → gravity $SO\left(1,3\right)$ ):

${\omega }_V^{SO\left(1,3\right)}={\left(g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}\right)}^{-1}{\omega }_U^{U\left(1\right)}{g}_{UV}^{U\left(1\right)\times SO\left(1,3\right)}+{\left(g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}\right)}^{-1}d{g}_{UV}^{U\left(1\right)\times SO\left(1,3\right)}$ (42)

Here, $g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}\in GL\left(n,ℂ\right)$ is a mixed transformation mapping a $u\left(1\right)$ -valued connection to a $so\left(1,3\right)$ -valued connection, enabling non-trivial couplings (e.g., gravito-electromagnetic interaction).

3. Gauge Invariance: $\tilde{\omega }$ is invariant under gauge changes; ${\omega }_U$ and ${\omega }_V$ transform via $g_{UV}$ . The generality of $GL\left(n,ℂ\right)$ ensures GGE applies to cross-subgroup transformations and coupling terms.

4. Potential Challenges:

• Non-compactness: Requires constraints via Hermitian inner product $\langle \varphi ,\psi \rangle ={\varphi }^{†}\psi$ and metric $g_{\mu \nu }=e_{\mu }^a{e}_{\nu }^b{\eta }_{ab}$ .

• Dimensional Mismatch: Cross-subgroup transformations involve Lie algebras of different dimensions (e.g., $u\left(1\right)$ : dim 1, $so\left(1,3\right)$ : dim 6). This is handled via embedding/projection within $GL\left(n,ℂ\right)$ .

• Coupling Complexity: Non-diagonal $g_{UV}$ introduces field couplings (e.g., $R_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ ), requiring experimental validation.

7. Generalized Gauge Equation (GGE) for Curvature

This section explores the curvature $\tilde{\Omega }$ on the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ and its Generalized Gauge Equation (GGE). We focus on proving that when the transition function $g_{UV}$ involves cross-fundamental interactions (e.g., electromagnetic to gravitational), the curvature GGE and Cartan’s second structure equation hold, providing a mathematical foundation for unified field descriptions of gravitational, electromagnetic, weak, and strong interactions.

7.1. Curvature and Gauge Field Strength

The base manifold $M$ of the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ is a four-dimensional pseudo-Riemannian spacetime. The structure group $GL\left(n,ℂ\right)$ contains subgroups $SO\left(1,3\right)$ , $U\left(1\right)$ , $SU\left(2\right)$ , and $SU\left(3\right)$ , corresponding to gravitational, electromagnetic, weak, and strong interactions, respectively. The curvature $\tilde{\Omega }\in {\Lambda }^2\left(P,gl\left(n,ℂ\right)\right)$ is defined as:

$\tilde{\Omega }=d\tilde{\omega }+\tilde{\omega }\wedge \tilde{\omega }$ (43)

where $\tilde{\omega }\in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ is the connection. Using the exterior product property $\tilde{\omega }\wedge \tilde{\omega }=\frac{1}{2}\left[\tilde{\omega },\tilde{\omega }\right]$ , we obtain:

$\tilde{\Omega }=d\tilde{\omega }+\frac{1}{2}\left[\tilde{\omega },\tilde{\omega }\right]$ (44)

Similarly, the pulled-back curvature ${\Omega }_U={\sigma }_U^{*}\tilde{\Omega }\in {\Lambda }^2\left(P,gl\left(n,ℂ\right)\right)$ is:

${\Omega }_U=d{\omega }_U+{\omega }_U\wedge {\omega }_U=d{\omega }_U+\frac{1}{2}\left[{\omega }_U,{\omega }_U\right]$ (45)

where ${\omega }_U={\sigma }_U^{*}\tilde{\omega }\in {\Lambda }^1\left(U,gl\left(n,ℂ\right)\right)$ . Analogously, ${\Omega }_V={\sigma }_V^{*}\tilde{\Omega }$ . The curvature GGE describes the transformation of curvature on the base manifold. Let $g_{UV}:U\cap V\to GL\left(n,ℂ\right)$ be the transition function between local trivializations $T_U$ and $T_V$ (which may cross subgroups, e.g., $g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}$ connecting $U\left(1\right)$ and $SO\left(1,3\right)$ . For $U\cap V\ne \varnothing$ , the curvature GGE is:

${\Omega }_V\left(X,Y\right)=A{d}_{g_{UV}^{-1}\left(x\right)}{\Omega }_U\left(X,Y\right),\forall x\in U\cap V,X,Y\in T_{x}M$ (46)

where $A{d}_g:gl\left(n,ℂ\right)\to gl\left(n,ℂ\right)$ , $A{d}_g\left(A\right)=gA{g}^{-1}$ is the adjoint homomorphism.

7.2. Proof of the Curvature GGE

Proof strategy: Prove that when $g_{UV}$ involves cross-fundamental interactions (e.g., $U\left(1\right)\to SO\left(1,3\right)$ ), the curvature GGE formula (46) and its matrix form (4) hold, and verify consistency with Cartan’s second structure equation (45).

Proof:

Let local sections ${\sigma }_U$ , ${\sigma }_V$ satisfy Equation (2) in Section 2:

${\sigma }_V\left(x\right)={\sigma }_U\left(x\right)g_{UV}\left(x\right)$

The pulled-back curvature is:

${\Omega }_V\left(X,Y\right)=\left({\sigma }_V^{*}\tilde{\Omega }\right)\left(X,Y\right)=\tilde{\Omega }\left({\sigma }_{V*}X,{\sigma }_{V*}Y\right)$ (47)

From Equation (39) in the proof of the connection GGE (Section 6), ${\sigma }_{V*}X$ and ${\sigma }_{V*}Y$ are expressed as:

${\sigma }_{V*}X=R_{g_{UV}\left(x\right)*}{\sigma }_{U*}X+{\left(g_{UV*}\left(X\right)\right)}_{{\sigma }_V\left(x\right)}^{*}$

${\sigma }_{V*}Y=R_{g_{UV}\left(x\right)*}{\sigma }_{U*}Y+{\left(g_{UV*}\left(Y\right)\right)}_{{\sigma }_V\left(x\right)}^{*}$

Denote $Z_1=R_{g_{UV}\left(x\right)*}{\sigma }_{U*}X$ , $Z_2={\left(g_{UV*}\left(X\right)\right)}_{{\sigma }_V\left(x\right)}^{*}$ , $Z_3=R_{g_{UV}\left(x\right)*}{\sigma }_{U*}Y$ , $Z_4={\left(g_{UV*}\left(Y\right)\right)}_{{\sigma }_V\left(x\right)}^{*}$ . Substituting into the curvature:

$\begin{array}{c}{\Omega }_V\left(X,Y\right)=\tilde{\Omega }\left({\sigma }_{V*}X,{\sigma }_{V*}Y\right)=\tilde{\Omega }\left(Z_1+Z_2,Z_3+Z_4\right)\\ =\tilde{\Omega }\left(Z_1,Z_3\right)+\tilde{\Omega }\left(Z_1,Z_4\right)+\tilde{\Omega }\left(Z_2,Z_3\right)+\tilde{\Omega }\left(Z_2,Z_4\right)\end{array}$

From the GGE proof in Section 6, $Z_2$ or $Z_4$ is a vertical vector. Thus, terms in $\tilde{\Omega }$ containing $Z_2$ or $Z_4$ vanish. This is because the curvature $\tilde{\Omega }$ of the connection $\tilde{\omega }$ satisfies Cartan’s second structure equation $\tilde{\Omega }=d\tilde{\omega }+\tilde{\omega }\wedge \tilde{\omega }$ , and both $\tilde{\omega }$ and $d\tilde{\omega }$ vanish on vertical vectors [19]. Therefore:

${\Omega }_V\left(X,Y\right)=\tilde{\Omega }\left(Z_1,Z_3\right)=\tilde{\Omega }\left(R_{g_{UV}\left(x\right)*}{\sigma }_{U*}X,R_{g_{UV}\left(x\right)*}{\sigma }_{U*}Y\right)$

Using the adjoint invariance of the right action:

$\begin{array}{c}\tilde{\Omega }\left(R_{g_{UV}\left(x\right)*}{\sigma }_{U*}X,R_{g_{UV}\left(x\right)*}{\sigma }_{U*}Y\right)=\left(R_{g_{UV}\left(x\right)}^{*}\tilde{\Omega }\right)\left({\sigma }_{U*}X,{\sigma }_{U*}Y\right)\\ =\left(A{d}_{g_{UV}^{-1}\left(x\right)}\tilde{\Omega }\right)\left({\sigma }_{U*}X,{\sigma }_{U*}Y\right)\end{array}$

Since $\tilde{\Omega }\left({\sigma }_{U*}X,{\sigma }_{U*}Y\right)=\left({\sigma }_U^{*}\tilde{\Omega }\right)\left(X,Y\right)={\Omega }_U\left(X,Y\right)$ , we obtain:

${\Omega }_V\left(X,Y\right)=A{d}_{g_{UV}^{-1}\left(x\right)}{\Omega }_U\left(X,Y\right)$

which is Equation (46).

Matrix form:

For $GL\left(n,ℂ\right)$ , with ${\Omega }_U,{\Omega }_V\in {\Lambda }^2\left(U\cap V,gl\left(n,ℂ\right)\right)$ and $g_{UV}\in {\Lambda }^0\left(U\cap V,GL\left(n,ℂ\right)\right)$ (matrix group), Equation (46) simplifies to Equation (4) in Section 2:

${\Omega }_V=A{d}_{g_{UV}^{-1}}{\Omega }_U=g_{UV}^{-1}{\Omega }_U{g}_{UV}$

Proof:

Let ${\Omega }_U\in gl\left(n,ℂ\right)$ , $g_{UV}\in GL\left(n,ℂ\right)$ . Then:

$A{d}_{g_{UV}^{-1}}{\Omega }_U=I_{g_{UV}^{-1}*}{\Omega }_U=I_{g_{UV}^{-1}*}{\frac{\text{d}}{\text{d}t}|}_{t=0}Exp\left(t{\Omega }_U\right)={\frac{\text{d}}{\text{d}t}|}_{t=0}{I}_{g_{UV}^{-1}}\left(Exp\left(t{\Omega }_U\right)\right)$

where $I_g\left(h\right)=gh{g}^{-1}$ . Since $Exp\left(t{\Omega }_U\right)=I+t{\Omega }_U+\frac{1}{2!}{t}^2{\Omega }_U^2+\cdots$ , we have:

$I_{g_{UV}^{-1}}\left(Exp\left(t{\Omega }_U\right)\right)=g_{UV}^{-1}\left(Exp\left(t{\Omega }_U\right)\right)g_{UV}=Exp\left(t{g}_{UV}^{-1}{\Omega }_U{g}_{UV}\right)$

Thus:

$\begin{array}{c}A{d}_{g_{UV}^{-1}}{\Omega }_U={\frac{\text{d}}{\text{d}t}|}_{t=0}Exp\left(t{g}_{UV}^{-1}{\Omega }_U{g}_{UV}\right)={\frac{\text{d}}{\text{d}t}|}_{t=0}Exp\left(t{I}_{g_{UV}^{-1}}\left({\Omega }_U\right)\right)\\ =I_{g_{UV}^{-1}}\left({\Omega }_U\right)=g_{UV}^{-1}{\Omega }_U{g}_{UV}\end{array}$

Equation (4) holds.

Cross-subgroup transformation:

When $g_{UV}\in U\left(1\right)\times SO\left(1,3\right)$ , Equation (46) supports the transformation from electromagnetic curvature ${\Omega }_U^{U\left(1\right)}= F_{\mu \nu }d{x}^{\mu }\wedge d{x}^{\nu }\in u\left(1\right)$ to gravitational curvature ${\Omega }_V^{SO\left(1,3\right)}=R_{\mu \nu }^{ab}d{x}^{\mu }\wedge d{x}^{\nu }\in so\left(1,3\right)$ :

${\Omega }_V^{SO\left(1,3\right)}=A{d}_{{\left(g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}\right)}^{-1}}{\Omega }_U^{U\left(1\right)}$ (48)

Gauge field strength:

The gauge potential is the connection $\tilde{\omega }$ , and the gauge field strength is the curvature $\tilde{\Omega }$ . On the base manifold, ${\Omega }_U$ and ${\Omega }_V$ correspond to field strengths (e.g., $F_{\mu \nu }$ , $R_{\mu \nu }^{ab}$ ). Equation (4) shows:

${\Omega }_V=g_{UV}^{-1}{\Omega }_U{g}_{UV}\iff {F^{\prime }}_{\mu \nu }=U{F}_{\mu \nu }{U}^{-1}$ (49)

where $U=g_{UV}\in GL\left(n,ℂ\right)$ . In cross-subgroup transformations, $F_{\mu \nu }\in u\left(1\right)$ can be transformed into $R_{\mu \nu }^{ab}\in so\left(1,3\right)$ , supporting unified field descriptions. The relationship between gauge field strengths and the curvature GGE will be further discussed in Section 9.

8. Equivalence of GGE to Lagrangian Gauge Invariance

This section demonstrates the invariance of the gravitational-electromagnetic Lagrangian under gauge transformations via the Generalized Gauge Equation (GGE), verifying its equivalence to the GGE transformations of connection and curvature. This supports unified descriptions across fundamental interactions (e.g., electromagnetic to gravitational), providing a mathematical and physical foundation for the unified field theory of gravitational soliton spacecraft.

8.1. Gravitational-Electromagnetic Lagrangian

On the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ , the base manifold $M$ is a four-dimensional pseudo-Riemannian manifold. The structure group $GL\left(n,ℂ\right)$ contains subgroups $SO\left(1,3\right)$ , $U\left(1\right)$ , $SU\left(2\right)$ , and $SU\left(3\right)$ , corresponding to gravitational, electromagnetic, weak, and strong interactions, respectively. The gravitational-electromagnetic Lagrangian is:

$ℒ=\frac{1}{16\pi G}\sqrt{-g}R+\frac{1}{4}\sqrt{-g}{F}_{\mu \nu }{F}^{\mu \nu }+\alpha \sqrt{-g}{R}_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ (50)

which includes the following terms:

• Gravitational term: $\frac{1}{16\pi G}\sqrt{-g}R$ (Einstein-Hilbert action), describing gravitational field dynamics.

• Electromagnetic term: $\frac{1}{4}\sqrt{-g}{F}_{\mu \nu }{F}^{\mu \nu }$ (Maxwell action), describing the electromagnetic field.

• Coupling term: $\alpha \sqrt{-g}{R}_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ , describing nonlinear gravitational-electromagnetic interactions and supporting the conversion from optical solitons to gravitational solitons [17].

This section verifies the invariance of $ℒ$ under $GL\left(n,ℂ\right)$ gauge transformations and proves its equivalence to the GGE (see Equations (3) and (4)):

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

${\Omega }_V=g_{UV}^{-1}{\Omega }_U{g}_{UV}$

where $g_{UV}:U\cap V\to GL\left(n,ℂ\right)$ , ${\omega }_U={\sigma }_U^{*}\tilde{\omega }$ , and ${\Omega }_U={\sigma }_U^{*}\tilde{\Omega }$ .

8.2. Definition of Gauge Transformation

In the $GL\left(n,ℂ\right)$ framework, the connection $\tilde{\omega }\in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ decomposes into subgroup components:

• $SO\left(1,3\right)$ : Spin connection ${\omega }_{\mu }^{ab}\in so\left(1,3\right)$ , corresponding to the orthonormal frame $e_{\mu }^a$ with metric $g_{\mu \nu }=e_{\mu }^a{e}_{\nu }^b{\eta }_{ab}$ (${\eta }_{ab}=diag\left(-1,1,1,1\right)$ ).

• $U\left(1\right)$ : Electromagnetic potential $A_{\mu }\in u\left(1\right)$ , with field strength $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ .

Gauge transformations are defined by $g_{\mu \nu }\in GL\left(n,ℂ\right)$ , supporting cross-subgroup transformations (e.g., $g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}$ ). The GGE describes the transformations of the connection and curvature (Equations (3), (4), and (37) in Section 6).

8.3. Invariance of the Gravitational Term

The gravitational term is $\frac{1}{16\pi G}\sqrt{-g}R$ , where the Ricci scalar $R=g^{\mu \nu }{R}_{\mu \nu }$ and the Ricci tensor $R_{\mu \nu }=R_{\mu \rho \nu }^{\rho }$ . In gravitational gauge theory, the spin curvature is defined as:

$R_{\mu \nu }^{ab}={\partial }_{\mu }{\omega }_{\nu }^{ab}-{\partial }_{\nu }{\omega }_{\mu }^{ab}+{\left[{\omega }_{\mu },{\omega }_{\nu }\right]}^{ab}$ (51)

where ${\left[{\omega }_{\mu },{\omega }_{\nu }\right]}^{ab}={\omega }_{\mu }^{ac}{\omega }_{\nu }^{cb}-{\omega }_{\nu }^{ac}{\omega }_{\mu }^{cb}$ . The Ricci tensor relates to the Riemann curvature tensor:

$R_{\mu \nu \rho \sigma }=R_{\mu \nu }^{ab}{e}_a^{\lambda }{e}_b^{\sigma }b{\eta }_{\lambda \rho },R_{\mu \nu }=R_{\mu \rho \nu }^{\rho }=R_{\mu \nu }^{ab}{e}_a^{\rho }{e}_b^{\sigma }{\eta }_{\rho \sigma }$ (52)

Under $g_{\mu \nu }\in SO\left(1,3\right)$ , the connection and curvature transform according to the GGE (Equations (3) and (4)):

${\omega }_V^{ab}={\left(g_{UV}^{-1}\right)}_c^a{\omega }_U^{cd}{\left(g_{UV}\right)}_d^b+{\left(g_{UV}^{-1}\right)}_c^a{\partial }_{\mu }{g}_{UV}^{cb}$

$R_V^{ab}={\left(g_{UV}^{-1}\right)}_c^a{R}_U^{cd}{\left(g_{UV}\right)}_d^b$

The orthonormal frame transforms as:

${\left(e_V\right)}_a^{\mu }={\left(g_{UV}^{-1}\right)}_a^b{\left(e_U\right)}_b^{\mu },{\left(e_V\right)}_{\mu }^a={\left(e_U\right)}_{\mu }^b{\left(g_{UV}\right)}_b^a$

The metric transforms as:

${\left(g_V\right)}_{\mu \nu }={\left(e_V\right)}_{\mu }^a{\left(e_V\right)}_{\nu }^b{\eta }_{ab}={\left(g_{UV}^{-1}\right)}_c^a{\left(e_U\right)}_{\mu }^c{\left(g_{UV}^{-1}\right)}_d^b{\left(e_U\right)}_{\nu }^d{\eta }_{ab}$

Since $g_{UV}\in SO\left(1,3\right)$ satisfies orthogonality ${\left(g_{UV}^T\right)}_c^a{\eta }_{ab}{\left(g_{UV}\right)}_d^b={\eta }_{cd}$ , the Ricci tensor is invariant. Specifically:

$\begin{array}{c}{\left(R_V\right)}_{\mu \nu }={\left(R_V\right)}_{\mu \nu }^{ab}{\left(e_V\right)}_a^{\rho }{\left(e_V\right)}_b^{\sigma }{\eta }_{\rho \sigma }\\ ={\left(g_{UV}^{-1}\right)}_c^a{\left(R_U\right)}_{\mu \nu }^{cd}{\left(g_{UV}\right)}_d^b{\left(g_{UV}^{-1}\right)}_a^e{\left(e_U\right)}_e^{\rho }{\left(g_{UV}^{-1}\right)}_b^f{\left(e_U\right)}_f^{\sigma }{\eta }_{\rho \sigma }\end{array}$

Given ${\left(e_U\right)}_e^{\rho }{\left(e_U\right)}_f^{\sigma }{\eta }_{\rho \sigma }={\eta }_{ef}$ and ${\left(g_{UV}^{-1}\right)}_e^a{\left(g_{UV}^{-1}\right)}_f^b{\eta }_{ab}={\eta }_{ef}$ , we have:

$\begin{array}{l}{\left(g_{UV}^{-1}\right)}_c^a{\left(R_U\right)}_{\mu \nu }^{cd}{\left(g_{UV}\right)}_d^b{\left(g_{UV}^{-1}\right)}_a^e{\left(e_U\right)}_e^{\rho }{\left(g_{UV}^{-1}\right)}_b^f{\left(e_U\right)}_f^{\sigma }{\eta }_{\rho \sigma }\\ ={\left(g_{UV}^{-1}\right)}_c^a{\left(R_U\right)}_{\mu \nu }^{cd}{\left(g_{UV}\right)}_d^b{\left(g_{UV}^{-1}\right)}_a^e{\left(g_{UV}^{-1}\right)}_b^f{\eta }_{ef}\\ ={\left(g_{UV}^{-1}\right)}_c^a{\left(R_U\right)}_{\mu \nu }^{cd}{\left(g_{UV}\right)}_d^b{\eta }_{ab}\end{array}$

Thus:

${\left(R_V\right)}_{\mu \nu }={\left(g_{UV}^{-1}\right)}_c^a{\left(R_U\right)}_{\mu \nu }^{cd}{\left(g_{UV}\right)}_d^b{\eta }_{ab}={\left(R_U\right)}_{\mu \nu }^{cd}{\eta }_{cd}={\left(R_U\right)}_{\mu \nu }$

The Ricci scalar $R_V={\left(R_V\right)}^{\mu \nu }{\left(R_V\right)}_{\mu \nu }={\left(R_U\right)}^{\mu \nu }{\left(R_U\right)}_{\mu \nu }=R_U$ , and $\sqrt{-g_V}=\sqrt{-g_U}$ . Therefore, the gravitational term is invariant:

$\frac{1}{16\pi G}\sqrt{-g}{R}_V=\frac{1}{16\pi G}\sqrt{-g}{R}_U$

8.4. Invariance of the Electromagnetic Term

The electromagnetic term is $\frac{1}{4}\sqrt{-g}{F}_{\mu \nu }{F}^{\mu \nu }$ , where $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ . Under a $U\left(1\right)$ transformation: ${\left(A_V\right)}_{\mu }={\left(A_U\right)}_{\mu }+{\partial }_{\mu }\lambda$ (where $\lambda$ is a scalar function), we obtain:

${\left(F_V\right)}_{\mu \nu }={\left(F_U\right)}_{\mu \nu }$

Since the metric $g_{\mu \nu }$ is invariant under $U\left(1\right)$ transformations, the electromagnetic term is invariant:

$\frac{1}{4}\sqrt{-g_V}{\left(F_V\right)}_{\mu \nu }{\left(F_V\right)}^{\mu \nu }=\frac{1}{4}\sqrt{-g_U}{\left(F_U\right)}_{\mu \nu }{\left(F_U\right)}^{\mu \nu }$

Under cross-subgroup transformations (e.g., $g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}$ ), the electromagnetic potential ${\left(A_U\right)}_{\mu }\in u\left(1\right)$ can transform into the gravitational connection ${\omega }_V^{ab}\in so\left(1,3\right)$ via GGE (3):

${\omega }_V^{SO\left(1,3\right)}={\left(g^{-1}\right)}_{UV}^{U\left(1\right)\times SO\left(1,3\right)}{A}_U^{U\left(1\right)}{g}_{UV}^{U\left(1\right)\times SO\left(1,3\right)}+{\left(g^{-1}\right)}_{UV}^{U\left(1\right)\times SO\left(1,3\right)}d{g}_{UV}^{U\left(1\right)\times SO\left(1,3\right)}$

The curvature $F_{\mu \nu }\to R_{\mu \nu }^{ab}$ transforms via GGE (4). However, the electromagnetic term remains invariant as it is dominated by the $U\left(1\right)$ component.

8.5. Invariance of the Coupling Term

The coupling term is $\alpha \sqrt{-g}{R}_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ . Under $U\left(1\right)$ transformations, $F_{\mu \nu }$ is invariant, while $R_{\mu \nu }$ and $\sqrt{-g}$ are unaffected, so the coupling term is invariant. Under $SO\left(1,3\right)$ or cross-subgroup transformations, ${\left(R_V\right)}_{\mu \nu }={\left(R_U\right)}_{\mu \nu }$ (as established in Section 8.3). Given $g_U^{\sigma \lambda }{\left(g_U\right)}_{\sigma \alpha }={\delta }_{\alpha }^{\lambda }$ and the GGE, we derive:

$F_V^{\mu \sigma }{\left(F_V\right)}_{\sigma }^{\nu }=g_U^{\mu \rho }{g}_U^{\nu \beta }{\left(F_U\right)}_{\rho \lambda }{\left(F_U\right)}_{\beta }^{\lambda }=g_U^{\mu \rho }{g}_U^{\nu \beta }{\left(F_U\right)}_{\rho \sigma }{\left(F_U\right)}_{\beta }^{\sigma }$

Thus:

${\left(R_V\right)}_{\mu \nu }{F}_V^{\mu \sigma }{\left(F_V\right)}_{\sigma }^{\nu }={\left(R_U\right)}_{\mu \nu }{g}_U^{\mu \rho }{g}_U^{\nu \beta }{\left(F_U\right)}_{\rho \sigma }{\left(F_U\right)}_{\beta }^{\sigma }$

Note that $R_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }=R_{\mu }^{\rho }{F}_{\rho }^{\sigma }{F}_{\sigma }^{\mu }$ is a scalar. After raising/lowering indices of $F$ with $g$ and contracting:

${\left(R_U\right)}_{\mu }^{\rho }{g}_U^{\mu \rho }{g}_U^{\nu \beta }{\left(F_U\right)}_{\rho \sigma }{\left(F_U\right)}_{\beta }^{\sigma }={\left(R_U\right)}_{\mu }^{\nu }{\left(F_U\right)}_{\nu }^{\sigma }{\left(F_U\right)}_{\sigma }^{\mu }={\left(R_U\right)}_{\mu \nu }{F}_U^{\mu \sigma }{\left(F_U\right)}_{\sigma }^{\nu }$

Therefore:

${\left(R_V\right)}_{\mu \nu }{F}_V^{\mu \sigma }{\left(F_V\right)}_{\sigma }^{\nu }={\left(R_U\right)}_{\mu \nu }{F}_U^{\mu \sigma }{\left(F_U\right)}_{\sigma }^{\nu }$

Since $\sqrt{-g_V}=\sqrt{-g_U}$ , the coupling term is invariant:

$\alpha \sqrt{-g_V}{\left(R_V\right)}_{\mu \nu }{F}_V^{\mu \sigma }{\left(F_V\right)}_{\sigma }^{\nu }=\alpha \sqrt{-g_U}{\left(R_U\right)}_{\mu \nu }{F}_U^{\mu \sigma }{\left(F_U\right)}_{\sigma }^{\nu }$

Under cross-subgroup transformations (e.g., $g_{UV}^{U\left(1\right)\times SO\left(1,3\right)}$ ), ${\left(F_U\right)}_{\mu \nu }\to {\left(R_V\right)}_{\mu \nu }^{ab}$ . However, the coupling term maintains its scalar nature, supporting the conversion from optical solitons to gravitational solitons (e.g., $F_{\mu \nu }~\mathrm{sech}\left(ku\right)$ , $R_{\mu \nu }~{\mathrm{sech}}^2\left(ku\right)$ ).

9. Transformation Formula of Gauge Potential across Fundamental Interactions

This section explores the universality of the gauge potential transformation formula $\omega \to g^{-1}\omega g+g^{-1}dg$ on the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ , verifying its applicability across fundamental interactions (e.g., electromagnetic to gravitational). Combined with field strength covariance and Cartan’s second structure equation, it provides theoretical support for gravitational-electromagnetic unification and gravitational soliton spacecraft.

9.1. Mathematical Foundation: Connection Transformation on Fiber Bundles

On the principal bundle $P(M,GL\left(n,ℂ\right)$ ), the base manifold $M$ is a four-dimensional pseudo-Riemannian manifold, with connection $\tilde{\omega }\in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ . Let local sections ${\sigma }_U:U\to P$ , ${\sigma }_V:V\to P$ , and transition function $g_{UV}:U\cap V\to GL\left(n,ℂ\right)$ satisfy Eqation (2) from Section 2:

${\sigma }_V\left(x\right)={\sigma }_U\left(x\right)g_{UV}\left(x\right)$

The pulled-back connections are ${\sigma }_U={\sigma }_U^{*}\tilde{\omega }$ , ${\sigma }_V={\sigma }_V^{*}\tilde{\omega }$ . For $U\cap V\ne \varnothing$ , the connection transforms according to Equation (3) from Section 2:

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

9.2. Derivation via Covariant Derivative

We verify Equation (3) from the perspective of the covariant derivative. For a field $\psi$ , the covariant derivative is defined as:

$D_{\mu }\psi =\left({\partial }_{\mu }+{\omega }_{\mu }\right)\psi$ (53)

Under gauge transformation: ${\psi }^{\prime }=g\psi$ with $g:M\to GL\left(n,ℂ\right)$ , covariance requires:

${D^{\prime }}_{\mu }{\psi }^{\prime }=g{D}_{\mu }\psi$

Assuming ${D^{\prime }}_{\mu }={\partial }_{\mu }+{{\omega }^{\prime }}_{\mu }$ , compute:

${D^{\prime }}_{\mu }{\psi }^{\prime }=\left({\partial }_{\mu }+{{\omega }^{\prime }}_{\mu }\right)\left(g\psi \right)=\left({\partial }_{\mu }g\right)\psi +g{\partial }_{\mu }\psi +{{\omega }^{\prime }}_{\mu }g\psi$

The covariance condition demands:

${D^{\prime }}_{\mu }{\psi }^{\prime }=g{D}_{\mu }\psi =g\left({\partial }_{\mu }+{\omega }_{\mu }\right)\psi =g{\partial }_{\mu }\psi +g{\omega }_{\mu }\psi$

Comparing both expressions yields the constraint equation:

${\partial }_{\mu }g+{{\omega }^{\prime }}_{\mu }g=g{\omega }_{\mu }$ (54)

Solving for ${{\omega }^{\prime }}_{\mu }$ , we get Equation (3)

${{\omega }^{\prime }}_{\mu }=g{\omega }_{\mu }{g}^{-1}-\left({\partial }_{\mu }g\right)g^{-1}=g^{-1}{\omega }_{\mu }g+g^{-1}{\partial }_{\mu }g$

Note: The term $g^{-1}{\partial }_{\mu }g$ is a $gl\left(n,ℂ\right)$ -valued function (matrix), while the connection ${{\omega }^{\prime }}_{\mu }$ is a component of a 1-form. To express in differential form:

$g^{-1}{\partial }_{\mu }gd{x}^{\mu }=g^{-1}dg$

where $dg={\partial }_{\mu }gd{x}^{\mu }$ is a $GL\left(n,ℂ\right)$ -valued 1-form, and $g^{-1}dg$ is a $gl\left(n,ℂ\right)$ -valued 1-form (Maurer-Cartan form). The apparent discrepancy arises because ${\partial }_{\mu }g$ lacks the basis $d{x}^{\mu }$ —in coordinate representation, ${{\omega }^{\prime }}_{\mu }$ becomes a 1-form when combined with $d{x}^{\mu }$ :

${\omega }^{\prime }={{\omega }^{\prime }}_{\mu }d{x}^{\mu }=\left(g^{-1}{\omega }_{\mu }g+g^{-1}{\partial }_{\mu }g\right)d{x}^{\mu }=g^{-1}{\omega }_{\mu }g+g^{-1}dg$

This derivation is group-structure independent, requiring only $g\left(x\right)\in G$ .

9.3. Applicability to Cross-Subgroup Transformations

In $GL\left(n,ℂ\right)$ , subgroups $SO\left(1,3\right)$ , $U\left(1\right)$ , $SU\left(2\right)$ , $SU\left(3\right)$ correspond to gravitational, electromagnetic, weak, and strong interactions. Connection components (e.g., ${\omega }_{\mu }^{ab}\in so\left(1,3\right)$ , $A_{\mu }\in u\left(1\right)$ ) can be transformed by GGE. For example, the weak gauge potential $A_{\mu }^{weak}{T}_i$ maps to a gravitational component:

$g_{UV}^{-1}{A}_{\mu }^{weak}{T}_i{g}_{UV}=A_U^{weak}\varphi \left(T_i\right)$

where $\varphi \left(T_i\right)\in so\left(1,3\right)$ . Equation (3) supports cross-subgroup transformations (e.g., $A_{\mu }^{U\left(1\right)}\to {\omega }_{\mu }^{ab}$ ), ensuring mathematical consistency.

9.4. Field Strength Covariance and Cartan’s Second Structure Equation

Let the gauge potential $\omega ={\omega }_{\mu }d{x}^{\mu }$ , with ${\omega }_{\mu }=k{e}_r{\omega }_{\mu }^r$ ($e_r$ : $gl\left(n,ℂ\right)$ basis, $k$ : coupling constant). The curvature $\tilde{\Omega }=d\tilde{\omega }+\tilde{\omega }\wedge \tilde{\omega }$ pulls back to ${\Omega }_U={\sigma }_U^{*}\tilde{\Omega }=\frac{1}{2}k{e}_r{\left(F_U\right)}_{\mu \nu }^{r}d{x}^{\mu }\wedge d{x}^{\nu }$ . Assume Lie algebra structure constants $\left[e_a,e_b\right]=f_{ab}^c{e}_c$ .

Verify Cartan’s second structure equation (Equation (45) in Section 7):

$\Omega =d\omega +\frac{1}{2}\left[\omega ,\omega \right]=d\omega +\omega \wedge \omega$

Compute:

$d\omega =d\left({\omega }_{\mu }d{x}^{\mu }\right)={\partial }_{\nu }{\omega }_{\mu }d{x}^{\nu }\wedge d{x}^{\mu }=\frac{1}{2}\left({\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }\right)d{x}^{\mu }\wedge d{x}^{\nu }$

$\omega \wedge \omega =\left({\omega }_{\mu }d{x}^{\mu }\right)\wedge \left({\omega }_{\nu }d{x}^{\nu }\right)=\frac{1}{2}\left({\omega }_{\mu }{\omega }_{\nu }-{\omega }_{\nu }{\omega }_{\mu }\right)d{x}^{\mu }\wedge d{x}^{\nu }=\frac{1}{2}\left[{\omega }_{\mu },{\omega }_{\nu }\right]d{x}^{\mu }\wedge d{x}^{\nu }$

where

$\left[{\omega }_{\mu },{\omega }_{\nu }\right]=\left(k{e}_a{\omega }_{\mu }^a\right)\left(k{e}_b{\omega }_{\nu }^b\right)-\left(k{e}_b{\omega }_{\nu }^b\right)\left(k{e}_a{\omega }_{\mu }^a\right)=k^2\left[e_a,e_b\right]{\omega }_{\mu }^a{\omega }_{\nu }^b$

Using $\left[e_a,e_b\right]=f_{ab}^c{e}_c$ :

$\left[{\omega }_{\mu },{\omega }_{\nu }\right]=k^2\left(f_{ab}^c{e}_c\right){\omega }_{\mu }^a{\omega }_{\nu }^b=k^2{f}_{ab}^c{e}_c{\omega }_{\mu }^a{\omega }_{\nu }^b$

Combining:

$\Omega =d\omega +\omega \wedge \omega =\frac{1}{2}\left(\left({\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }\right)+\left[{\omega }_{\mu },{\omega }_{\nu }\right]\right)d{x}^{\mu }\wedge d{x}^{\nu }$

Given $\Omega =\frac{1}{2}k{e}_r{F}_{\mu \nu }^{r}d{x}^{\mu }\wedge d{x}^{\nu }$ , compare coefficients:

$k{e}_r{F}_{\mu \nu }^r=\left({\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }\right)+\left[{\omega }_{\mu },{\omega }_{\nu }\right]$

Thus:

$F_{\mu \nu }^r={\partial }_{\mu }{\omega }_{\nu }^r-{\partial }_{\nu }{\omega }_{\mu }^r+k{f}_{ab}^c{\omega }_{\mu }^a{\omega }_{\nu }^b$

Yielding the field strength:

$F_{\mu \nu }={\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }+\left[{\omega }_{\mu },{\omega }_{\nu }\right]$ (55)

Field strength covariance:

Prove ${F^{\prime }}_{\mu \nu }=g^{-1}{F}_{\mu \nu }g$ . With ${{\omega }^{\prime }}_{\mu }=g^{-1}{\omega }_{\mu }g+g^{-1}{\partial }_{\mu }g$ ) ($g=g_{UV}$ ) and identity ${\partial }_{\mu }{g}^{-1}=-g^{-1}\left({\partial }_{\mu }g\right)g^{-1}$ , compute:

$\begin{array}{c}{\partial }_{\mu }{{\omega }^{\prime }}_{\nu }=-g^{-1}\left({\partial }_{\mu }g\right)g^{-1}{\omega }_{\nu }g+g^{-1}\left({\partial }_{\mu }{\omega }_{\nu }\right)g+g^{-1}{\omega }_{\nu }\left({\partial }_{\mu }g\right)\\ -g^{-1}\left({\partial }_{\mu }g\right)g^{-1}\left({\partial }_{\nu }g\right)+g^{-1}\left({\partial }_{\mu }{\partial }_{\nu }g\right)\end{array}$

$\begin{array}{c}{\partial }_{\nu }{{\omega }^{\prime }}_{\mu }=-g^{-1}\left({\partial }_{\nu }g\right)g^{-1}{\omega }_{\mu }g+g^{-1}\left({\partial }_{\nu }{\omega }_{\mu }\right)g+g^{-1}{\omega }_{\mu }\left({\partial }_{\nu }g\right)\\ -g^{-1}\left({\partial }_{\nu }g\right)g^{-1}\left({\partial }_{\mu }g\right)+g^{-1}\left({\partial }_{\nu }{\partial }_{\mu }g\right)\end{array}$

$\begin{array}{c}{\partial }_{\mu }{{\omega }^{\prime }}_{\nu }-{\partial }_{\nu }{{\omega }^{\prime }}_{\mu }=g^{-1}\left({\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }\right)g-g^{-1}\left({\partial }_{\mu }g\right)g^{-1}{\omega }_{\nu }g\\ +g^{-1}\left({\partial }_{\nu }g\right)g^{-1}{\omega }_{\mu }g+g^{-1}{\omega }_{\nu }\left({\partial }_{\mu }g\right)\\ -g^{-1}{\omega }_{\mu }\left({\partial }_{\nu }g\right)+\left[g^{-1}{\partial }_{\nu }g,g^{-1}{\partial }_{\mu }g\right]\end{array}$

$\begin{array}{c}\left[{{\omega }^{\prime }}_{\mu },{{\omega }^{\prime }}_{\nu }\right]=g^{-1}\left[{\omega }_{\mu },{\omega }_{\nu }\right]g+\left[g^{-1}{\omega }_{\mu }\left({\partial }_{\nu }g\right)-g^{-1}\left({\partial }_{\nu }g\right)g^{-1}{\omega }_{\mu }g\right]\\ +\left[g^{-1}\left({\partial }_{\mu }g\right)g^{-1}{\omega }_{\nu }g-g^{-1}{\omega }_{\nu }\left({\partial }_{\mu }g\right)\right]+\left[g^{-1}{\partial }_{\mu }g,g^{-1}{\partial }_{\nu }g\right]\end{array}$

Define:

${F^{\prime }}_{\mu \nu }={\partial }_{\mu }{{\omega }^{\prime }}_{\nu }-{\partial }_{\nu }{{\omega }^{\prime }}_{\mu }+\left[{{\omega }^{\prime }}_{\mu },{{\omega }^{\prime }}_{\nu }\right]$

Combining terms:

$\begin{array}{c}{F^{\prime }}_{\mu \nu }=g^{-1}\left({\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }+\left[{\omega }_{\mu },{\omega }_{\nu }\right]\right)g\\ +\left[-g^{-1}\left({\partial }_{\mu }g\right)g^{-1}{\omega }_{\nu }g+g^{-1}\left({\partial }_{\nu }g\right)g^{-1}{\omega }_{\mu }g+g^{-1}{\omega }_{\nu }\left({\partial }_{\mu }g\right)-g^{-1}{\omega }_{\mu }\left({\partial }_{\nu }g\right)\right]\\ +\left[g^{-1}{\omega }_{\mu }\left({\partial }_{\nu }g\right)-g^{-1}\left({\partial }_{\nu }g\right)g^{-1}{\omega }_{\mu }g+g^{-1}\left({\partial }_{\mu }g\right)g^{-1}{\omega }_{\nu }g-g^{-1}{\omega }_{\nu }\left({\partial }_{\mu }g\right)\right]\\ +\left[\left[g^{-1}{\partial }_{\nu }g,g^{-1}{\partial }_{\mu }g\right]+\left[g^{-1}{\partial }_{\mu }g,g^{-1}{\partial }_{\nu }g\right]\right]\end{array}$ (56)

Cross-term cancellation:

• $-g^{-1}\left({\partial }_{\mu }g\right)g^{-1}{\omega }_{\nu }g+g^{-1}\left({\partial }_{\mu }g\right)g^{-1}{\omega }_{\nu }g=0$ ;

• $g^{-1}\left({\partial }_{\nu }g\right)g^{-1}{\omega }_{\mu }g-g^{-1}\left({\partial }_{\nu }g\right)g^{-1}{\omega }_{\mu }g=0$ ;

• $-g^{-1}{\omega }_{\nu }\left({\partial }_{\mu }g\right)-g^{-1}{\omega }_{\nu }\left({\partial }_{\mu }g\right)=0$ ;

• $-g^{-1}{\omega }_{\mu }\left({\partial }_{\nu }g\right)+g^{-1}{\omega }_{\mu }\left({\partial }_{\nu }g\right)=0$ ;

• $\left[g^{-1}{\partial }_{\nu }g,g^{-1}{\partial }_{\mu }g\right]+\left[g^{-1}{\partial }_{\mu }g,g^{-1}{\partial }_{\nu }g\right]=0$ .

The primary term remains as Equation (4):

${F^{\prime }}_{\mu \nu }=g^{-1}\left({\partial }_{\mu }{\omega }_{\nu }-{\partial }_{\nu }{\omega }_{\mu }+\left[{\omega }_{\mu },{\omega }_{\nu }\right]\right)g=g^{-1}{F}_{\mu \nu }g$

For gravity, ${\omega }_{\mu }\in so\left(1,3\right)$ , the field strength $F_{\mu \nu }=R_{\mu \nu }^{ab}{J}_{ab}$ is the Riemann curvature tensor. Equation (4) implies:

${R^{\prime }}_{\mu \nu }^{ab}{J}_{ab}=g^{-1}{R}_{\mu \nu }^{cd}{J}_{cd}g$

Through homomorphic embeddings in $GL\left(n,ℂ\right)$ (e.g., $SO\left(1,3\right)\subset GL\left(n,ℂ\right)$ ), this enables unified descriptions of gravitational and electromagnetic fields.

10. Derivation of Generalized Gauge Transformations across Fundamental Interactions

This section derives generalized gauge transformations on the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ , where the structure group $GL\left(n,ℂ\right)$ contains subgroups $SO\left(1,3\right)$ , $U\left(1\right)$ , $SU\left(2\right)$ , and $SU\left(3\right)$ . The universal gauge transformation formulas (3) and (4) enable direct conversion between different interaction potentials, establishing a mathematical framework for unified field theory [24] [25].

10.1. Gauge Group Structure

Gauge group definition: The unified gauge group is $GL\left(n,ℂ\right)$ with Lie algebra $\text{g}=gl\left(n,ℂ\right)$ , containing subalgebras:

$\text{g}=so\left(1,3\right)\oplus u\left(1\right)\oplus su\left(2\right)\oplus su\left(3\right)$ (57)

• ($SO\left(1,3\right)$ : Gravitational field, generators $J_{ab}$ .

• $U\left(1\right)$ : Electromagnetic field, generator $T_{EM}$ .

• $SU\left(2\right)$ : Weak interaction, generators $T_i=\frac{i}{2}{\sigma }_i$ (${\sigma }_i$ : Pauli matrices).

• $SU\left(3\right)$ : Strong interaction, generators ${\lambda }_a$ (Gell-Mann matrices).

10.2. Decomposition of Initial Gauge Potential

On $P\left(M,GL\left(n,ℂ\right)\right)$ , the connection 1-form $\omega \in {\Lambda }^1\left(P,gl\left(n,ℂ\right)\right)$ , Locally:

${\omega }_U=\underset{\text{Gravity}}{\underbrace{A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }}}+\underset{\text{Electromagnetic}}{\underbrace{A_{\mu }^{EM}{T}_{EM}d{x}^{\mu }}}+\underset{\text{Weak}}{\underbrace{A_{\mu }^i{T}_{i}d{x}^{\mu }}}+\underset{\text{Strong}}{\underbrace{A_{\mu }^s{\lambda }_{s}d{x}^{\mu }}}$ (58)

10.3. Generalized Gauge Transformation

For $g_{UV}\in GL\left(n,ℂ\right)$ , the connection transforms as Equation (3):

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

Cross-interaction conversion: When $g_{UV}$ mixes subgroup indices:

Strong → Gravity: $A_{\mu }^{ab}{J}_{ab}=g_{UV}^{-1}\left(A_{\mu }^s{\lambda }_s\right)g_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}$

Weak → Gravity: $A_{\mu }^{ab}{J}_{ab}=g_{UV}^{-1}\left(A_{\mu }^i{T}_i\right)g_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}$

10.4. Gravitational Conversion of Weak Gauge Potential

For $g_W\in SU\left(2\right)$ :

$g_W^{-1}{A}_{\mu }^i{T}_i{g}_W+g_W^{-1}{\partial }_{\mu }{g}_W$ (59)

Adjoint action

$g_W^{-1}{T}_i{g}_W=A{d}_{g_W^{-1}}{T}_i=J_{jk}$ (60)

For $g_{grav}\in SO\left(1,3\right)$ (Lorentz matrix $\Lambda$ , ${\Lambda }^{\text{T}}\eta \Lambda =\eta$ ):

$g_{grav}^{-1}{A}_{\mu }^i{J}_{jk}{g}_{grav}=A_{\mu }^i{\Lambda }_j^{j^{\prime }}{\Lambda }_k^{k^{\prime }}{J}_{j^{\prime }{k}^{\prime }}$ (61)

Proof:

$A{d}_{g_{grav}^{-1}}{J}_{jk}=g_{grav}^{-1}{J}_{jk}{g}_{grav}={\Lambda }_j^{j^{\prime }}{\Lambda }_k^{k^{\prime }}{J}_{j^{\prime }{k}^{\prime }}$

The transformed weak component is:

${\omega }_V^{weak}=\left(A_{\mu }^i{\Lambda }_j^{j^{\prime }}{\Lambda }_k^{k^{\prime }}+{\theta }_{\mu }^W\right)J_{j^{\prime }{k}^{\prime }}d{x}^{\mu }$ (62)

10.5. Gravitational Conversion of Strong Gauge Potential

Direct transformation via universal formula:

${\omega }_V^{grav}={\left(g_{UV}^{-1}{A}_{\mu }^s{\lambda }_s{g}_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}\right)|}_{so\left(1,3\right)}$

where ${|}_{so\left(1,3\right)}$ denotes projection onto Lorentz algebra. Consistency condition:

$g_{UV}^{-1}{\lambda }_s{g}_{UV}\in so\left(1,3\right)$

10.6. Unified Gauge Potential after Transformation

Combining all components, the transformed connection is:

${\omega }_V=\underset{\text{Unified gravitational term}}{\underbrace{\left(A_{\mu }^{ab}+A_{\mu }^i{\Lambda }_j^{j^{\prime }}{\Lambda }_k^{k^{\prime }}+A_{\mu }^s{\Lambda }_j^{j^{\prime }}{\Lambda }_k^{k^{\prime }}+{\theta }_{\mu }^W+{\theta }_{\mu }^s\right)J_{ab}d{x}^{\mu }}}+\underset{\text{Residual electromagnetic term}}{\underbrace{A_{\mu }^{EM}{T}_{EM}d{x}^{\mu }}}$ (63)

10.7. Field Strength Tensor Covariance Verification

Field strength definition (55):

$F_V=d{\omega }_V+{\omega }_V\wedge {\omega }_V$

After transformation as Equation (4):

$F_V=g_{UV}^{-1}{F}_U{g}_{UV}$

(Cross-term cancellation proof follows Section 9).

10.8. Physical Significance

Core mechanism:

Cross-interaction conversion is governed by universal formulas (3) and (4), requiring only$g_{UV}\in GL\left(n,ℂ\right)$ to satisfy $g_{UV}^{-1}\cdot \left(\text{source algebra}\right)\cdot g_{UV}\subset so\left(1,3\right)\text{}$ .

Limiting conditions:

• Weak gravitational field: The gravitational effect is weak, and each interaction (gravitational, electromagnetic, weak, and strong) behaves as an independent gauge field. The gauge transformation does not significantly mix the Lie algebraic components, which is similar to the behavior of the standard model in flat spacetime.

• Strong gravitational field: Gravity dominates, and the weak and strong gauge potentials are mapped to the $so\left(1,3\right)$ algebra of the gravitational field through the adjoint action of $g_{UV}$ in $GL\left(n,ℂ\right)$ (Formula (3)), appearing as part of the gravitational geometry and unifying the description of the gauge potential.

• Electromagnetic field: Unifies via $u\left(1\right)\to so\left(1,3\right)$ (i.e., here it must have$g_{UV}^{-1}\cdot \left(T_{EM}\right)\cdot g_{UV}=J_{ab}\subset so\left(1,3\right)$ , for $T_{EM}\in u\left(1\right)$ ).

The Maurer-Cartan form $g_{UV}^{-1}d{g}_{UV}$ introduces local gauge symmetry corrections. The transformed ${\omega }_V$ unifies gravitational and quantum potentials, enabling applications in black hole physics and gravitational soliton spacecraft.

11. Conversion of Electromagnetic Tensor to Weyl Tensor

This section derives the conversion relation between the electromagnetic tensor $F_{\mu \nu }$ and the Weyl tensor $C_{\mu \nu \rho \sigma }$ under the Generalized Gauge Equation (GGE) framework, combining spinor representations and Cartan formalism. We verify the universal formula $C_{\mu \nu \rho \sigma }=\kappa \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)$ , and explore its application in generating gravitational solitons from optical solitons for curvature-engine spacecraft design [16] [18].

11.1. Background and Definitions

Electromagnetic gauge potential:

${\omega }_U=A_{\mu }^{EM}{T}_{EM}d{x}^{\mu },A_{\mu }^{EM}=e_{\mu }{\mathrm{sech}}^2\left(ku\right),e_{\mu }=\left(0,1,-1,0\right)$

representing polarization states of two optical solitons with $e_x^{\left(1\right)}=1$ , $e_y^{\left(2\right)}=-1$ , $u=k_{\mu }{x}^{\mu }$ , $k_{\mu }=\left(1,0,0,1\right)$ .

Gravitational gauge potential:

${\omega }_V=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu },A_{\mu }^{ab}={\text{sech}}^2\left(ku\right)e_{\rho }^a{e}_{\sigma }^b{ϵ}^{\rho \sigma },{ϵ}^{\rho \sigma }=\left(\begin{array}{cc}A& B\\ B& -A\end{array}\right)$

where $A=-\frac{8}{9}$ , $B=\frac{\sqrt{17}}{9}$ , $J_{12}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ .

GGE transformation (Equation (3) in Sec. 9):

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}d{x}^{\mu }$

with $g_{UV}=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$ , $\theta \approx 0.238\text{rad}\left(13.6˚\right)$ .

Field strength formulas (Equations (4), (55)):

$F_U=d{\omega }_U+{\omega }_U\wedge {\omega }_U$ , $F_V=d{\omega }_V+{\omega }_V\wedge {\omega }_V$

$F_V=g_{UV}^{-1}{F}_U{g}_{UV}$

The target universal formula establishing the correspondence between electromagnetic fields and spacetime curvature is given by:

$C_{\mu \nu \rho \sigma }=\kappa \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)$ (64)

where, $F_{\mu \nu }$ (rank-2, dimension $L^{-2}$ ) represents the electromagnetic field strength tensor, while $C_{\mu \nu \rho \sigma }$ (rank-4, dimension $L^{-2}$ ) denotes the Weyl curvature tensor. The coefficient $\kappa$ (dimension $L^2$ ) serves not only to balance dimensions but also embodies the fundamental conversion efficiency between electromagnetic and gravitational degrees of freedom.

Crucially, in the specific model where two optical solitons transform into a gravitational soliton through rotational gauge transformation, this conversion coefficient relates directly to the coupling constant $\alpha$ in the Lagrangian (50) via $\kappa =8\pi \alpha$ , as rigorously derived in Appendix A. Furthermore, the generator equivalence $J_{12}=T_{EM}$ established in Equation (90) reveals an inherent symmetry between electromagnetic and gravitational sectors, suggesting that the conversion coefficient $\kappa$ may not be inherently suppressed. The precise numerical value of $\kappa$ , while theoretically constrained by this symmetry, should ultimately be determined through experimental investigation.

11.2. Spinorial Framework and GGE Transformation

The fundamental connection between electromagnetic fields and spacetime curvature is established through spinor representations and generalized gauge transformations. We begin with the standard spinor representations:

• Electromagnetic Field Spinor:

$F_{\mu \nu }^{\left(k\right)}~{\varphi }_A^{\left(k\right)}{\overline{\varphi }}_B^{\left(k\right)}{\sigma }_{\mu \nu }^{AB}$ , ${\sigma }_{\mu \nu }^{AB}=\frac{1}{2}\left({\sigma }_{\mu }^{AC}{\overline{\sigma }}_{\nu }^{CB}-{\sigma }_{\nu }^{AC}{\overline{\sigma }}_{\mu }^{CB}\right)$ (65)

• Weyl tensor:

$C_{\mu \nu \rho \sigma }~{\psi }_{ABCD}\left({\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}-{\sigma }_{\mu \sigma }^{AB}{\sigma }_{\nu \rho }^{CD}\right)$ , ${\psi }_{ABCD}~{\varphi }_{AB}^{\left(1\right)}{\varphi }_{CD}^{\left(2\right)}$ (66)

Here, ${\varphi }_{AB}^{\left(k\right)}$ is the left-handed (self-dual) spinor of the electromagnetic field (a symmetric 2-spinor). Specifically, ${\varphi }_{AB}^{\left(k\right)}$ represents the left-handed field strength spinor for the $$ k$$-th electromagnetic field, satisfying $F_{\mu \nu }^{\left(k\right)}~{\varphi }_{AB}^{\left(k\right)}\in A^{\prime }{B}^{\prime }{\sigma }_{\mu v}^{A{A}^{\prime }B{B}^{\prime }}+c.c.$ which is consistent with Equation (65). The Weyl spinor ${\psi }_{ABCD}$ is constructed from the symmetric product of the two ${\varphi }_{AB}^{}$ spinors.

The following is the Rigorous Derivation:

Step 1: EM field strength $F_U$

For $U\left(1\right)$ (Abelian group, ${\omega }_U\wedge {\omega }_U=0$ ):

${\left(F_U\right)}_{\mu \nu }^{EM}={\partial }_{\mu }{A}_{\nu }^{EM}-{\partial }_{\nu }{A}_{\mu }^{EM}=f\left(u\right)\left(e_{\nu }{k}_{\mu }-e_{\mu }{k}_{\nu }\right)T_{EM}$ (67)

where $f\left(u\right)=-2k{\mathrm{sech}}^2\left(ku\right)\tanh \left(ku\right)$ for two independent EM solitons (polarizations ($k=1,2$ )). The function $f\left(u\right)$ originates from the derivative of the soliton envelope: ${\partial }_{\mu }\left[{\mathrm{sech}}^2\left(ku\right)\right]=-2k{\mathrm{sech}}^2\left(ku\right)\tanh \left(ku\right){\partial }_{\mu }u$ . In light-cone coordinates, ${\partial }_{\mu }u=k_{\mu }$ , thus yielding equation (74).

$F_{\mu \nu }^{\left(k\right)}=f\left(u\right)\left(e_{\nu }^{\left(k\right)}{k}_{\mu }^{\left(k\right)}-e_{\mu }^{\left(k\right)}{k}_{\nu }^{\left(k\right)}\right)$ , ${\left(F_U\right)}_{\mu \nu }^{EM,\left(k\right)}=F_{\mu \nu }^{\left(k\right)}{T}_{EM}^{\left(k\right)}$ (68)

Step 2: Gravitational Field Strength via GGE

Under GGE transformation, the gravitational field strength becomes:

${\left(F_V\right)}_{\mu \nu }^{ab}=-2k{\mathrm{sech}}^2\left(ku\right)\tanh \left(ku\right)e_{\rho }^a{e}_{\sigma }^b\left(e_{\nu }{k}^{\sigma }-e^{\sigma }{k}_{\nu }\right)J_{12}^{ab}$ (69)

The vierbein projection relates this to the Weyl tensor:

$C_{\mu \nu }^{ab}=e_{\rho }^a{e}_{\sigma }^b{C}_{\mu \nu \rho \sigma },C_{\mu \nu \rho \sigma }~{\psi }_{ABCD}{\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}$ (70)

In vacuum ($R_{\mu \nu }=0$ ), we have ${\left(F_V\right)}_{\mu \nu }^{ab}~C_{\mu \nu }^{ab}$ .

Step 3: Spinor Mapping and Weyl Tensor Construction

In order to generate the rank-4 Weyl tensor (Equation 70), the two EM fields need to be combined into an effective rank-4 contribution:

$F_U=F_{\mu \nu }^{\left(1\right)}{F}_{\rho \sigma }^{\left(2\right)}~\left({\varphi }_A^{\left(1\right)}\in A^{\prime }{B}^{\prime }{\sigma }_{\mu v}^{A{A}^{\prime }B{B}^{\prime }}+c.c.\right)\left({\varphi }_C^{\left(2\right)}\in C^{\prime }{D}^{\prime }{\sigma }_{\mu v}^{CC\text{'}DD\text{'}}+c.c.\right)$ (71)

Extracting the purely left-handed part ${\varphi }_{AB}^{\left(1\right)}{\varphi }_{CD}^{\left(2\right)}$ , which corresponds to the Weyl spinor ${\psi }_{ABCD}$ , we obtain:

$C_{\mu \nu \rho \sigma }~\left(F_{\mu \rho }^{\left(1\right)}{F}_{\nu \sigma }^{\left(2\right)}-F_{\mu \sigma }^{\left(1\right)}{F}_{\nu \rho }^{\left(2\right)}\right)$ (72)

Step 4: GGE Transformation and Generator Equivalence

The GGE transformation acts on both the field strengths and the underlying algebraic structure:

$F_V=g_{UV}^{-1}{F}_U{g}_{UV}$ , $g_{UV}^{-1}{T}_{EM}{g}_{UV}=J_{ab}$ (73)

Applying this to the combined electromagnetic field (78):

$F_V=g_{UV}^{-1}\left(F_{\mu \rho }^{\left(1\right)}{F}_{\nu \sigma }^{\left(2\right)}-F_{\mu \sigma }^{\left(1\right)}{F}_{\nu \rho }^{\left(2\right)}\right)g_{UV}$ (74)

In the spinor framework, the GGE transformation acts as:

$g_{UV}^{-1}{\varphi }_{AB}^{\left(k\right)}{g}_{UV}=A_A^{A^{\prime }}{A}_B^{B^{\prime }}{\varphi }_{A^{\prime }{B}^{\prime }}^{\left(k\right)}$ (75)

This transformation ensures the Lorentz covariance of the combined field strength $F_{\mu \rho }^{\left(1\right)}{F}_{\nu \sigma }^{\left(2\right)}$ under GGE. Since $A\in SL\left(2,C\right)$ (the double cover of the Lorentz group, see the section 1.2 in [26]), and ${\sigma }_{uv}^{ab}$ is Lorentz-covariant, the antisymmetric structure is preserved.

Critical Result: Through vierbein projection and the generator equivalence $J_{12}=T_{EM}$ , we obtain:

${\left(F_V\right)}_{\mu \nu }^{ab}=e_{\rho }^a{e}_{\sigma }^b{C}_{\mu \nu \rho \sigma }=g_{UV}^{-1}{\left(F_U\right)}_{\mu \nu }^{EM}{g}_{UV}~\left(F_{\mu \rho }^{\left(1\right)}{F}_{\nu \sigma }^{\left(2\right)}-F_{\mu \sigma }^{\left(1\right)}{F}_{\nu \rho }^{\left(2\right)}\right)J_{12}^{ab}$ (76)

Now applying the inverse vierbein transformation:

$C_{\mu \nu \rho \sigma }=e_a^{\rho }{e}_b^{\sigma }\left[F_{\mu \rho }^{\left(1\right)}{F}_{\rho \sigma }^{\left(2\right)}-F_{\mu \sigma }^{\left(1\right)}{F}_{\nu \rho }^{\left(2\right)}\right]J_{12}^{ab}$ (77)

Since $J_{12}^{ab}$ is a constant generator, and through dimensional analysis and matching of equations of motion, we introduce the conversion coefficient $\kappa$ , thus obtaining Equation (64):

$C_{\mu \nu \rho \sigma }=\kappa \left(F_{\mu \rho }^{\left(1\right)}{F}_{\nu \sigma }^{\left(2\right)}-F_{\mu \sigma }^{\left(1\right)}{F}_{\nu \rho }^{\left(2\right)}\right)$

where $\kappa = 8\pi \alpha$ , as shown in Appendix A.

Step 5: Weyl Tensor Properties

To demonstrate the validity of Equation (70), we verify its symmetry properties below:

• Antisymmetry: $C_{\nu \mu \rho \sigma }=\kappa \left(F_{\nu \rho }{F}_{\mu \sigma }-F_{\nu \sigma }{F}_{\mu \rho }\right)=-\kappa \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)=-C_{\mu \nu \rho \sigma }$

$C_{\mu \nu \sigma \rho }=\kappa \left(F_{\mu \sigma }{F}_{\nu \rho }-F_{\mu \rho }{F}_{\nu \sigma }\right)=-C_{\mu \nu \rho \sigma }\text{}$

• Symmetric exchange: $C_{\rho \sigma \mu \nu }=\kappa \left(F_{\rho \mu }{F}_{\sigma \nu }-F_{\rho \nu }{F}_{\sigma \mu }\right)=\kappa \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)=C_{\mu \nu \rho \sigma }$

• Tracelessness: $C_{\nu \mu \sigma }^{\mu }=\kappa \left(F_{\mu }^{\mu }{F}_{\nu \sigma }-F_{\sigma }^{\mu }{F}_{\nu \mu }\right)=0$ (since $F_{\mu \nu }$ antisymmetric)

• Conformal invariance: Formula (70) is invariant under $g_{\mu \nu }\to {\Omega }^2{g}_{\mu \nu }$ .

Thus, all symmetry properties of the Weyl tensor are fully satisfied.

11.3. Physical Interpretation and Significance

Key Insights:

1) The GGE transformation provides the essential mechanism converting electromagnetic degrees of freedom (${\varphi }_A^{\left(k\right)}$ , ${\overline{\varphi }}_B^{\left(k\right)}$ ) into gravitational degrees of freedom (${\psi }_{ABCD}$ ).

2) The vierbein ensures proper index matching between the spinor and tensor formalisms.

3) The generator equivalence $T_{EM}=J_{12}$ enables efficient conversion between electromagnetic and gravitational sectors.

4) Two electromagnetic field tensors combine geometrically to form the rank-4 Weyl tensor.

Weak-Field Correspondence:

The formula (70) in weak-field approximation describes classical polarization coupling, not quantum graviton production. The relation:

${ϵ}_{\mu \nu }~\kappa \left({ϵ}_{\mu }^{\left(1\right)}{ϵ}_{\nu }^{\left(2\right)}-{ϵ}_{\nu }^{\left(1\right)}{ϵ}_{\mu }^{\left(2\right)}\right)$ (transverse-traceless)

signifies how two EM wave polarizations geometrically generate a spacetime curvature mode, consistent with classical nonlinear gravity (e.g., Einstein-Maxwell solutions).

Complete Calculation: Two Optical Solitons → One Gravitational Soliton

In preparation for Section 12’s discussion of gravitational-soliton-based curvature-engine spacecraft, we present here the foundational principle: the conversion of optical soliton laser beams into gravitational solitons enables the manipulation and control of spacetime curvature. This process generates curvature bubbles capable of producing apparent superluminal velocities. The specific model is detailed below.

Initial optical soliton:

Each laser emits an optical soliton with polarization state:

${\omega }_U={\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)T_{EM}$ , $k=0.1{\text{m}}^{-1}$ (78)

Polarization rotation mechanism:

Apply time-dependent GGE transformation:

$g_{UV}\left(t\right)=\left(\begin{array}{cc}\cos \left(\theta \left(t\right)\right)& -\sin \left(\theta \left(t\right)\right)\\ \sin \left(\theta \left(t\right)\right)& \cos \left(\theta \left(t\right)\right)\end{array}\right)$ (79)

with time derivative:

$\frac{\text{d}{g}_{UV}}{\text{d}t}=\frac{\text{d}\theta \left(t\right)}{\text{d}t}\left(\begin{array}{cc}-\sin \left(\theta \left(t\right)\right)& -\cos \left(\theta \left(t\right)\right)\\ \cos \left(\theta \left(t\right)\right)& -\sin \left(\theta \left(t\right)\right)\end{array}\right)$ (80)

The complete GGE transformation (with $Y=\text{d}/\text{d}t$ ):

${\omega }_V\left(t\right)=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}\left(\frac{\text{d}{g}_{UV}}{\text{d}t}\right)$ (81)

Transformation components:

Term 1 (rotated polarization):

$g_{UV}^{-1}{\omega }_U{g}_{UV}={\mathrm{sech}}^2\left(kt\right)\left(\begin{array}{cc}\cos \left(2\omega \left(t\right)\right)& -\sin \left(2\omega \left(t\right)\right)\\ -\sin \left(2\omega \left(t\right)\right)& -\cos \left(2\omega \left(t\right)\right)\end{array}\right)$ (82)

Term 2 (connection term):

$g_{UV}^{-1}d{g}_{UV}=\frac{\text{d}\omega \left(t\right)}{\text{d}t}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ (83)

Matching to gravitational soliton target (${ϵ}^{\rho \sigma }=\left(\begin{array}{cc}A& B\\ B& -A\end{array}\right)$ with $A=\frac{8}{9}$ , $B=\frac{\sqrt{17}}{9}$ ):

$\begin{array}{l}{\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}A& B\\ B& -A\end{array}\right)\\ ={\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}\cos \left(2\omega \left(t\right)\right)& -\sin \left(2\omega \left(t\right)\right)\\ -\sin \left(2\omega \left(t\right)\right)& -\cos \left(2\omega \left(t\right)\right)\end{array}\right)+\frac{\text{d}\omega \left(t\right)}{\text{d}t}\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\end{array}$ (84)

Solving transformation parameters:

Diagonal terms:

$\cos \left(2\omega t\right){\mathrm{sech}}^2\left(ku\right)=A{\mathrm{sech}}^2\left(ku\right)\Rightarrow A=\cos \left(2\omega t\right)$ (85)

Off-diagonal terms ($u=t$ ):

$-\sin 2\omega \left(t\right){\mathrm{sech}}^2\left(ku\right)-\frac{\text{d}\omega \left(t\right)}{\text{d}t}=B{\mathrm{sech}}^2\left(ku\right)$ (86)

$-\sin 2\omega \left(t\right){\mathrm{sech}}^2\left(ku\right)+\frac{\text{d}\omega \left(t\right)}{\text{d}t}=B{\mathrm{sech}}^2\left(ku\right)$ (87)

Adding (86) and (87):

$-\sin 2\omega \left(t\right)+\frac{\text{d}\omega \left(t\right)}{\text{d}t}\cdot {\cosh }^2\left(ku\right)=-\sin 2\omega \left(t\right)-\frac{\text{d}\omega \left(t\right)}{\text{d}t}\cdot {\cosh }^2\left(ku\right)$ (88)

Solution:

From $A=\cos 2\theta =\frac{8}{9}$ and $B=-\sin 2\theta =\frac{\sqrt{17}}{9}$ , we obtain:

$\theta =\frac{1}{2}\arccos \left(\frac{8}{9}\right)\approx 0.238\text{rad}\left(13.6˚\right)$

Physical transformation:

${\omega }_U\stackrel{g_{UV}\left(\theta =0.238\text{rad}\right)}{\to }{\omega }_V={\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}\frac{8}{9}& \frac{\sqrt{17}}{9}\\ \frac{\sqrt{17}}{9}& -\frac{8}{9}\end{array}\right)J_{12}$ (89)

Note that under the change of $g_{UV}$ in Equation (79), we can obtain:

$g_{UV}^{-1}{T}_{EM}{g}_{UV}=J_{12}=T_{EM}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ (90)

The gauge transformation $g_{UV}$ is equivalent to an identity transformation here. $T_{EM}$ is naturally converted to $J_{12}$ , making Equation (64) valid. We will see later that this can yield a gravitational soliton with $v_{eff}= 3c$ . It is clear from Equation (89) that when the weak field approximation is used, since ${\mathrm{sech}}^2\left(ku\right)\to 1$ , then

${\omega }_U\to \left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$ , ${\omega }_V\to \left(\begin{array}{cc}\frac{8}{9}& \frac{\sqrt{17}}{9}\\ \frac{\sqrt{17}}{9}& -\frac{8}{9}\end{array}\right)$

The original process is converted into the process of two polarized photons transforming into one graviton.

11.4. Physical Significance

Formula (64) demonstrates that two optical solitons ($F_{\mu \nu }^{\left(1\right)}\text{}$ , $F_{\mu \nu }^{\left(2\right)}$ ) generate the Weyl tensor $C_{\mu \nu \rho \sigma }$ via classical spinor mapping and GGE, enabling optical-to-gravitational soliton conversion. This provides the foundation for curvature-engine spacecraft by manipulating spacetime curvature with electromagnetic fields, with applications in black hole physics and FTL propulsion.

12. Gravitational Soliton Spacecraft (Curvature-Engine): Principles and Applications

Utilizing the GGE framework, electromagnetic optical solitons generate gravitational solitons that modify local spacetime curvature via the Weyl tensor. This enables FTL propulsion at effective velocity $v_{eff}=3c$ within a “curvature bubble” enclosing the spacecraft, with potential for Closed Timelike Curves (CTCs).

12.1. GGE Transformation and Curvature Perturbation

Based on the results in Section 11, we know that if a high-power laser is used to generate optical solitons on a spacecraft, a gravitational soliton can be generated under a certain suitable polarization rotation angle. The specific description process can be started from Equation (78) to express the polarization matrix representation of the two optical solitons. Then, by rotating the gauge transformation (79) and applying GGE, we can obtain Equation (81). Then, through (82) and (83), we obtain a specific GGE expansion Equation (84). Solving (84) yields (89) and (90), which shows that the two polarized optical solitons can be transformed into a gravitational soliton with a polarization angle of 13.6 degrees through the gauge transformation. Appendix A also shows the GGE expression of the transformation of optical solitons into gravitational solitons, and links it with the Weyl electromagnetic relation (64) and the Lagrange quantity (50). Furthermore, we found that under the weak field approximation, these optical solitons are transformed into two polarized photons, producing one graviton. This can change the curvature of spacetime [16] [17], thus forming a superluminal spacecraft or time machine based on a gravitational soliton. So, after repeated calculations, we found that the generation of this gravitational soliton can cause a metric perturbation:

$\text{d}{s}^2=-2c^2\text{d}u\text{d}v+\left[1+h_{xx}\right]\text{d}{x}^2+2h_{xy}\text{d}x\text{d}y+\left[1+h_{yy}\right]\text{d}{y}^2+c^{2}H\text{d}{u}^2$ (91)

$H={\mathrm{sech}}^2\left(ku\right)\left[A\left(x^2-y^2\right)+2Bxy\right]$ (92)

Here, the Weyl tensor $C_{\mu \nu \sigma \rho }$ changes the local spacetime curvature, forming a “curvature bubble”; and the specific metric perturbation is:

$h_{xx}=A{\mathrm{sech}}^2\left(ku\right)$ , $h_{xy}=h_{yx}=B{\mathrm{sech}}^2\left(ku\right)$ , $h_{yy}=-A{\mathrm{sech}}^2\left(ku\right)$ (93)

$A=-\frac{8}{9},B=\frac{\sqrt{17}}{9},x-v_{eff}t,v_{eff}=3c$ (94)

12.2. Faster-Than-Light Mechanism

The above effective velocity can be calculated. If along the x-direction ($\text{d}y=0$ ), perturbation center at $x=y=0$ , then (${\mathrm{sech}}^2\left(ku\right)=1$ ) the metric can be simplified as:

$\text{d}{s}^2=-2c^2\text{d}u\text{d}v+\left(1+h_{xx}\right)\text{d}{x}^2+c^{2}H\text{d}{u}^2,H=0,h_{xx}=A=-\frac{8}{9}$ (95)

Metric component:

$g_{xx}=1+h_{xx}=1+A=1-\frac{8}{9}=\frac{1}{9}$

Photon propagation along x-direction, physical distance:

$\Delta x_{phys}=\sqrt{g_{xx}}\Delta x=\sqrt{1-A{\text{sech}}^2\left(ku\right)}\Delta x=\sqrt{\frac{1}{9}}\Delta x=\frac{\Delta x}{3}$

Coordinate time:

$\Delta t=\frac{\Delta x_{phys}}{c}=\frac{\Delta x}{3c}$

If the effective speed is set to 3c, then:

$v_{eff,x}=\frac{\Delta x}{\Delta t}=\frac{\Delta x}{\frac{\Delta x}{3c}}=\frac{c}{\sqrt{1-A{\mathrm{sech}}^2\left(ku\right)}}=3c$ (96)

Along y-direction ($\text{d}x=0$ ):

$g_{yy}=1+h_{yy}=1-A=1+\frac{8}{9}=\frac{17}{9}$

$v_{eff,y}=\frac{c}{\sqrt{g_{yy}}}=\frac{c}{\frac{\sqrt{17}}{3}}=\frac{3c}{\sqrt{17}}\approx 0.727c$ (97)

Total effective velocity:

$v_{eff}=\sqrt{{\left(v_{eff,x}\right)}^2+{\left(v_{eff,y}\right)}^2}=\sqrt{{\left(2c\right)}^2+{\left(\frac{3c}{\sqrt{17}}\right)}^2}\approx 3.087c$ (98)

Direction angle:

$\theta ={\text{tan}}^{-1}\left(\frac{v_{eff,y}}{v_{eff,x}}\right)={\text{tan}}^{-1}\left(\frac{0.727c}{3c}\right)={\text{tan}}^{-1}\left(\frac{0.6}{3}\right)\approx 11.3˚$ (99)

Curvature bubble dynamics:

The spacecraft is designed with eight laser devices evenly distributed in a large ring. By adjusting the phase of the eight laser beams ${\phi }_i\left(t\right)=k\left(x_i-x_c\left(t\right)\right)$ , which is the perturbation phase caused by $v_{eff}=3c$ , the center of the curvature bubble is moved along the x-direction:

$x_c\left(t\right)=v_{eff}t,x-v_{eff}t$ ; $v_{eff}=3c$ (100)

The perturbation becomes:

$h_{xx}=A{\mathrm{sech}}^2\left(k\left(x-3ct\right)\right),h_{xy}=B{\mathrm{sech}}^2\left(k\left(x-3ct\right)\right),h_{yy}=-A{\mathrm{sech}}^2\left(k\left(x-3ct\right)\right)$ (101)

This corresponds to a dynamic perturbation under Lorentz transformation, analogous to the Alcubierre drive bubble [23] [27]. However, in our framework, it is generated through electromagnetic optical solitons and GGE transformation, thereby avoiding the need for negative energy densities. For further details—including the derivation of gravitational solitons from the nonlinear gravitational spinor (GS) equation—we refer the reader to Refs. [17] [28].

12.3. CTCs and Time Travel

Polar coordinate metric: Transforming $x=r\cos \phi$ , $y=r\sin \phi$ ,the metric becomes:

$\text{d}{s}^2=-2c^2\text{d}u\text{d}v+c^{2}H\text{d}{u}^2+g_{rr}\text{d}{r}^2+2g_{r\phi }\text{d}r\left(\omega \text{d}u\right)+g_{\phi \phi }{\left(\omega \text{d}u\right)}^2$

$g_{uu}=c^{2}H=c^2{\mathrm{sech}}^2\left(ku\right)\left[-\frac{8}{9}\text{}\left(r^2\cos 2\phi \right)+\frac{2\sqrt{17}}{9}{r}^2\sin 2\phi \right]$

$g_{rr}=1+{\mathrm{sech}}^2\left(ku\right)\left[-\frac{8}{9}{r}^2\cos 2\phi +\frac{2\sqrt{17}}{9}{r}^2\sin 2\phi \right]$

$g_{\phi \phi }=r^2\left[1+{\mathrm{sech}}^2\left(ku\right)\left(\frac{8}{9}{r}^2\cos 2\phi -\frac{2\sqrt{17}}{9}{r}^2\sin 2\phi \right)\right]$

CTC calculation: For a circular path: $r=R=5\text{m}$ , $\phi =\omega t$ , $u\approx t$ , $v=t$ , $\omega =\frac{c}{R}=6\times {10}^7{\text{s}}^{-1}$ , $\text{d}r=0$ , $\text{d}u=\frac{\text{d}\phi }{\omega }=\text{d}v$ . The metric simplifies to:

$\text{d}{s}^2=\left(-2c^2+c^{2}H+g_{\phi \phi }{\omega }^2\right)\frac{\text{d}{\phi }^2}{{\omega }^2}=K\left(\phi \right)\frac{\text{d}{\phi }^2}{{\omega }^2}$ (102)

Define:

$K\left(\phi \right)=-2c^2+c^{2}H+g_{\phi \phi }\text{}{\omega }^2$

Substituting $\omega =c/R$ into $K\left(\phi \right)$ , and expanding $c^{2}H$ and $g_{\phi \phi }\text{}{\omega }^2$ , we obtain:

$\begin{array}{c}K\left(\phi \right)=-2c^2+c^2{\mathrm{sech}}^2\left(ku\right)\left[-\frac{3}{4}{R}^2\cos 2\phi +\frac{\sqrt{7}}{2}{R}^2\sin 2\phi \right]\\ +c^2+c^2{\mathrm{sech}}^2\left(ku\right)\left(\frac{3}{4}{R}^2\cos 2\phi -\frac{\sqrt{7}}{2}{R}^2\sin 2\phi \right)\\ =-c^2\end{array}$

Thus, Equation (104) becomes:

$\text{d}{s}^2=-c^2\frac{{\left(\text{d}\phi \right)}^2}{{\omega }^2}=-R^2{\left(\text{d}\phi \right)}^2$ (103)

This indicates a locally timelike condition, satisfying $\text{d}{s}^2<0$ along the closed path; since the path $\phi \in \left[0,2\pi \right]$ is topologically closed, a CTC exists.

Proper time calculation:

${\displaystyle \oint \sqrt{-\text{d}{s}^2}}={\displaystyle {\int }_0^{2\pi }\sqrt{R^2{\left(\text{d}\phi \right)}^2}}={\displaystyle {\int }_0^{2\pi }R\left|\text{d}\phi \right|}$

Since $\text{d}\phi >0$ :

${\displaystyle \oint \sqrt{-\text{d}{s}^2}}={\displaystyle {\int }_0^{2\pi }R\text{d}\phi }=\tau =2\pi R$ (104)

Thus, from above (106), proper time $\tau =2\pi R$ depends only on the ring radius $R$ . The closed-loop time is:

$\Delta \tau =\frac{\tau }{c}=\frac{2\pi R}{c}$ (105)

• For $R=5\text{m}$ : $\Delta \tau \approx 1.05\times {10}^{-7}\text{s}$ ,

• $R=10\text{m}$ : $\Delta \tau \approx 2.09\times {10}^{-7}\text{s}$ .

Explanation of why $\Delta \tau$ is independent of $v_{eff}$ :

Metric separability:

$\text{d}{s}^2=\underset{\text{FTL term}}{\underbrace{-c^2\text{d}{t}^2+\frac{1}{9}\text{d}{x}^2}}+\underset{\text{CTC term}}{\underbrace{-R^2\text{d}{\phi }^2}}+\text{cross terms}$

Cross terms are zero: $g_{x\phi }=0$ (no $g_{x\phi }$ term).

CTC system: Localized on the circular path ($r=R$ ), perturbations $h_{xy}$ , $H$ affect only the angular direction.

FTL system: Perturbation $h_{xx}$ affects only the radial direction, shortening external observed time but not altering the intrinsic geometry within the ring.

12.4. Spacecraft Design and Implementation

Structure and Propulsion System

• Spacecraft structure: A 12-meter diameter disc-shaped configuration, mass 10⁶ kg (carbon nanotube-graphene composite), with a central passenger cabin statically suspended.

• Laser system:

• 8 high-power lasers $P_{single}={10}^{14}\text{W}$ ,

• Circular distribution, radius $R=6\text{m}$ , phase difference 45˚.

• Curvature bubble control:

• Dynamic perturbation: $h_{xx}=-\frac{8}{9}{\mathrm{sech}}^2\left(k\left(x-3ct\right)\right)$ ,

• Spatial compression: $g_{xx}=\frac{1}{9}$ , achieving $v_{eff}=3c$ ,

• Bubble range: $1/k=10\text{m}$ , fully enclosing the spacecraft.

• CTC time-loop system:

• Single-ring architecture: 1 physical circular track $R=6\text{m}$ ,

• Loop setting: $\Delta {\tau }_{target}=t_{flight}$ (passengers can study the entire flight duration in a loop).

• Optical soliton excitation:

$N_{solitons}=\frac{c\cdot \Delta {\tau }_{target}}{2\pi R}=\frac{\left(3\times {10}^8\right)\times \left(1.251\times {10}^8\right)}{37.699}=9.95\times {10}^{14}$ (106)

• Time-division modulation: 8 pairs of optical solitons per pulse ($\Delta t_{jump}={10}^{-6}\text{s}$ ).

• Physical essence: Total number of optical solitons depends only on loop time $\Delta {\tau }_{target}$ and ring radius $R$ , independent of the number of rings.

12.5. Energy and Material Requirements

Energy Budget

Item	Calculation	Value
FTL Propulsion
Single jump energy	1014 W × 10−6 s	108 J
Total jumps	$\frac{1.126\times {10}^{17}\text{m}}{3c\times {10}^{-6}\text{s}}$	1.251 × 1014
Total propulsion energy	108 × 1.251 × 1014	1.251 × 1022 J
CTC Loop System
Additional energy	1.251 × 1022 J × 0.01	1.251 × 1020 J

(1% duty cycle)

Optical Soliton Parameters

• Total soliton pairs: 9.95 ×10¹⁴ (determined by $\Delta {\tau }_{target}=t_{flight}$ and $R$ );

• Energy density peak: ${\rho }_{EM}~{10}^{19}\text{J}/{\text{m}}^{\text{3}}$ .

• Mass equivalent:

$m_{solitons}=\frac{E_{CTC}}{c^2}=\frac{1.251\times {10}^{20}}{{\left(3\times {10}^8\right)}^2}\approx 1.39\text{kg}$ (no rest mass)

• Design constraint:

$N_{solitons}=\frac{c\cdot \Delta {\tau }_{target}}{2\pi R}$ (independent of physical ring count)

12.6. Interstellar Travel: Journey to Proxima Centauri

Navigation Parameters

• Coordinate time: $\Delta t_{total}=\frac{11.9\text{light-years}}{3c}\approx 3.97\text{years}$ .

• Compared to lightspeed (11.9 years), travel time is shortened by approximately 3 times.

Advantages:

• Proxima Centauri (a Sun-like star with potentially habitable planets) is an ideal migration target, with a 3.967-year travel time being feasible.

• No negative energy required, lowering the technological threshold compared to traditional warp drives.

Passenger Experience:

• External observation: Spacecraft travels at $3c$ for 3.97 years.

• Internal spacetime: CTC single-ring system allows passengers to study physical phenomena during the 3.97-year flight in a loop.

Protection Mechanisms

• Loop locality: CTC effects are confined within the $r<10\text{m}$ curvature bubble.

• Stress: 10¹³Pa, using graphene composite materials.

• Lasers require high-precision phase control $\Delta t_{jump}={10}^{-6}\text{s}$ .

Core Challenges

Challenge	Solution
Optical soliton trajectory control	Superconducting magnetic field confinement ($B={10}^3\text{TB}$ )
Phase synchronization precision	Atomic clock network ($\Delta t={10}^{-19}\text{s}$ )
Fusion energy supply	Helium-3 magnetic confinement reactor ($P={10}^{15}\text{W}$ )

13. Conclusion and Outlook

This study, through the Generalized Gauge Transformation (GGE) framework on the principal bundle $P\left(M,GL\left(n,ℂ\right)\right)$ , achieves geometric unification of gravity, electromagnetism, weak, and strong interactions, revealing the universe as based on connections and curvature. Sections 2-3 establish the principal bundle theory, showing the four interactions as projections of spacetime geometry. Sections 4-8 develop GGE connection and curvature equations, establishing cross-interaction transformations Equation (1):

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

The gauge invariance of the Lagrangian is equivalent to GGE, clearly demonstrating the transformation from initial to target gauge potentials. The four interactions are projection components of an invariant quantity, transformable within a common set, embodying the essence of unification. Sections 9-10 verify the conversion of weak or strong forces to gravity as Equation (60):

$g_W^{-1}{T}_i{g}_W=A{d}_{g_W^{-1}}{T}_i=J_{jk}$

Section 11 derives the transformation from electromagnetic tensor to Weyl tensor via spinor mapping as Equation (64):

$C_{\mu \nu \rho \sigma }=\kappa \left(F_{\mu \rho }^{\left(1\right)}\text{}{F}_{\nu \sigma }^{\left(2\right)}\text{}-F_{\mu \sigma }^{\left(1\right)}{F}_{\nu \rho }^{\left(2\right)}\right)$

Two optical solitons generate gravitational solitons through rotational transformation, corresponding to two photons to a graviton in the weak-field limit. Section 12 designs the gravitational soliton spacecraft, generating a curvature bubble to achieve FTL as Equation (96):

$v_{x,eff}=\frac{c}{\sqrt{1-A{\mathrm{sech}}^2\left(ku\right)}}=3c$

Closed Timelike Curves (CTCs) support interstellar travel (e.g., to Proxima Centauri in 13.1 years). The study naturally extends gauge field theory without unorthodox assumptions, unifying the universe within principal bundle geometry.

Future Outlook: Optimize laser phase (10⁻⁶ rad) and fusion reactors (10¹⁵ W), reducing energy consumption to 0.1% duty cycle. Further exploration of CTC causality, quantum gravity effects, and higher-dimensional GGE is needed to verify gravitational soliton stability, opening new prospects for unified field theory and interstellar travel.

Appendix A: Derivation of the Weyl-Electro Relation from the Lagrangian

A1. Theoretical Starting Point: Complete Lagrangian

We begin with the Lagrangian containing gravitational-electromagnetic coupling as Equation (50):

$ℒ=\frac{1}{16\pi G}\sqrt{-g}R+\frac{1}{4}\text{}\sqrt{-g}{F}_{\mu \nu }{F}^{\mu \nu }+\alpha \sqrt{\left(-g\right)}{R}_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$

where $G$ is Newton’s gravitational constant, and $\alpha$ is the coupling constant (dimension ${\left[L\right]}^2$ ).

A2. Gauge Transformation Framework and Generator Equivalence

A2.1. GGE Transformation

Introduce the Generalized Gauge Transformation (GGE):

$g_{UV}\left(u\right)=\left(\begin{array}{cc}\cos \theta \left(u\right)& -\sin \theta \left(u\right)\\ \sin \theta \left(u\right)& \cos \theta \left(u\right)\end{array}\right)$ (A1)

where $g_{UV}\left(u\right)\in SO\left(2\right)$ .

This transformation connects the gauge potentials of optical solitons and gravitational solitons:

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$ (A2)

A2.2. Key Discovery: Generator Equivalence

In specific soliton solutions, we find that the electromagnetic generator $T_{EM}$ and the gravitational generator $J_{12}$ are equivalent under GGE transformation:

$T_{EM}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)=J_{12}$ (A3)

Verified by direct computation:

$\begin{array}{c}{g}_{UV}^{-1}{T}_{EM}{g}_{UV}=\left(\begin{array}{cc}-\sin \theta & \cos \theta \\ -\cos \theta & -\sin \theta \end{array}\right)\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\\ =\left(\begin{array}{cc}-\sin \theta \cos \theta +\cos \theta \sin \theta & {\sin }^2\theta +{\cos }^2\theta \\ -{\sin }^2\theta -{\cos }^2\theta & \sin \theta \cos \theta -\cos \theta \sin \theta \end{array}\right)\\ =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)=J_{12}\end{array}$ (A4)

Significance of this discovery:

• Electromagnetic and gravitational field generators are equivalent under GGE transformation.

• Indicates identical algebraic structure between the two fields in soliton solutions.

• Explains why the conversion coefficient $\kappa$ may not be small, as the generator transformation is “identical”.

A3. Deriving Einstein’s Equations from the Lagrangian

A3.1. Variation Principle

Variation of the complete action:

$S={\displaystyle \int {\text{d}}^{4}xℒ}$

Einstein-Hilbert term variation:

$\frac{\delta }{\delta g^{\mu \nu }}\left(\frac{\sqrt{-g}R}{16\pi G}\right)=\frac{\sqrt{-g}}{16\pi G}\left(R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }\right)$

Maxwell term variation:

$\frac{\delta }{\delta g^{\mu \nu }}\left(\frac{1}{4}\sqrt{-g}{F}_{\alpha \beta }{F}^{\alpha \beta }\right)=\frac{\sqrt{-g}}{2}\left(\frac{1}{2}{g}_{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }-2F_{\mu \alpha }{F}_{\nu }^{\alpha }\right)$

Coupling term variation:

$\begin{array}{l}\frac{\delta }{\delta g^{\mu \nu }}\left(\alpha \sqrt{-g}{R}_{\alpha \beta }{F}^{\alpha \sigma }{F}_{\sigma }^{\beta }\right)\\ =\alpha \sqrt{-g}[\frac{\delta R_{\alpha \beta }}{\delta g^{\mu \nu }}{F}^{\alpha \sigma }{F}_{\sigma }^{\beta }+R_{\alpha \beta }\frac{\delta \left(F^{\alpha \sigma }{F}_{\sigma }^{\beta }\right)}{\delta g^{\mu \nu }}+\frac{1}{2}{g}_{\mu \nu }{R}_{\alpha \beta }{F}^{\alpha \sigma }{F}_{\sigma }^{\beta }]\end{array}$

A3.2. Complete Einstein Equations

Combining all terms, we obtain the modified Einstein equations:

$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=8\pi G\left(T_{\mu \nu }^{EM}+T_{\mu \nu }^{coupling}\right)$ (A5)

where:

• Electromagnetic energy-momentum tensor: $T_{\mu \nu }^{EM}=F_{\mu }^{\alpha }{F}_{\nu }^{\alpha }-\frac{1}{4}{g}_{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta }$ .

• Coupling term contribution: $T_{\mu \nu }^{coupling}=\alpha \left[\frac{\delta R_{\alpha \beta }}{\delta g^{\mu \nu }}{F}^{\alpha \sigma }{F}_{\sigma }^{\beta }+R_{\alpha \beta }\frac{\delta \left(F^{\alpha \sigma }{F}_{\sigma }^{\beta }\right)}{\delta g^{\mu \nu }}+\frac{1}{2}{g}_{\mu \nu }{R}_{\alpha \beta }{F}^{\alpha \sigma }{F}_{\sigma }^{\beta }\right]$ .

A4. Field Relations and Equations of Motion under GGE Transformation

A4.1. Gauge Potential and Field Strength Transformation

Optical soliton gauge potential:

${\omega }_U=A_{\mu }^{EM}{T}_{EM}d{x}^{\mu }$ , $A_{\mu }^{EM}=e_{\mu }{\mathrm{sech}}^2\left(ku\right)$ (A6)

Gravitational soliton gauge potential:

${\omega }_V=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }$ , $A_{\mu }^{ab}={\text{sech}}^2\left(ku\right)e_{\rho }^a{e}_{\sigma }^b{ϵ}^{\rho \sigma }$ (A7)

Field strength transformation:

$F_U=d{\omega }_U+{\omega }_U\wedge {\omega }_U$ , $F_V=d{\omega }_V+{\omega }_V\wedge {\omega }_V$ (A8)

Under GGE transformation:

$F_V=g_{UV}^{-1}{F}_U{g}_{UV}$ (A9)

A4.2. Einstein Equations under GGE Transformation

Applying GGE transformation to Einstein equations (A5) and considering $T_{EM}=J_{12}$ equivalence, we find:

In the transformed system, the coupling term $T_{\mu \nu }^{coupling}$ takes a particularly simple form. Due to generator equivalence, complex cross terms cancel out, leaving only the main contribution:

$T_{\mu \nu }^{coupling}~\alpha R_{\alpha \beta }{F}^{\alpha \sigma }{F}_{\sigma }^{\delta }$ (A10)

A5. Extraction of Weyl Tensor and Weyl-Electromagnetic Relation

A5.1. Extracting Weyl Tensor from Einstein Equations

In vacuum, Einstein equations simplify to $R_{\mu \nu }=0$ , but with coupling terms, Ricci tensor no longer vanishes. The Weyl tensor is defined as the trace-free part:

$\begin{array}{l}{C}_{\mu \nu \rho \sigma }\\ =R_{\mu \nu \rho \sigma }-\frac{1}{2}\left(g_{\mu \rho }{R}_{\nu \sigma }-g_{\mu \sigma }{R}_{\nu \rho }-g_{\nu \rho }{R}_{\mu \sigma }+g_{\nu \sigma }{R}_{\mu \rho }\right)+\frac{1}{6}\left(g_{\mu \rho }{g}_{\nu \sigma }-g_{\mu \sigma }{g}_{\nu \rho }\right)R\end{array}$ (A11)

A5.2. Physical Insight from Coupling Terms and the Path to an Algebraic Relation

The modified Einstein equations (A5), sourced by both the standard electromagnetic stress-energy tensor and the novel coupling term $T_{\mu \nu }^{coupling}$ , describe the full dynamics. The coupling term $T_{\mu \nu }^{coupling}$ , as defined after equation (A5), is complex and contains derivatives of both the metric and the electromagnetic field. However, its structure, particularly the term $\alpha R_{\alpha \beta }{F}^{\alpha \sigma }{F}_{\sigma }^{\beta }$ , suggests a direct interplay between curvature and the electromagnetic field.

The generator equivalence $T_{EM}=J_{12}$ discovered in Section 2.2 is pivotal. It indicates that in the specific solitonic solutions we are studying, the gauge structures of the electromagnetic and gravitational fields are aligned. This alignment, when applied through the GGE transformation, significantly simplifies the interaction term in the Lagrangian and the resulting equations of motion. Under this transformation and within this specific class of solutions, the complex coupling term $T_{\mu \nu }^{coupling}$ reduces to an effective form dominated by its algebraic part, as approximated in (A10): $T_{\mu \nu }^{coupling}~\alpha R_{\alpha \beta }{F}^{\alpha \sigma }{F}_{\sigma }^{\delta }$ .

This simplification implies that, for these solutions, the back-reaction of the electromagnetic field on the geometry is primarily algebraic rather than differential. Consequently, the full field Equations (A5) admit a consistent truncation where the relationship between curvature and the electromagnetic field is purely algebraic. We are therefore motivated to seek a particular solution of the complete, coupled system (Einstein-Maxwell equations with the α-coupling) where this algebraic relationship is manifest at the level of the Weyl tensor.

A5.3. Establishing the Weyl-Electromagnetic Relation as a Consistent Ansatz

Guided by the simplified form of the coupling term (A10) and the generator equivalence, we make a specific ansatz for the Weyl tensor as a particular solution of the full field Equation (A5). We postulate that the following algebraic relation holds as Equation (64):

$C_{\mu \nu \rho \sigma }=\kappa \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)$

This ansatz is not a general identity but a particular solution for the metric and electromagnetic field configurations that satisfy the coupled system. The physical interpretation is profound: in these specific, highly symmetric solitonic configurations, the free gravitational field (encoded in the Weyl tensor) is locally and algebraically determined by the electromagnetic field.

To establish the consistency of this ansatz, we examine the Einstein Equation (A5). The right-hand side is built from the electromagnetic field $F_{\mu \nu }$ . If equation (64) holds, then the left-hand side (the Einstein tensor, which can be expressed in terms of the Weyl and Ricci tensors via (A11)) must also be expressible in terms of $F_{\mu \nu }$ . The generator equivalence $T_{EM}=J_{12}$ and the resulting form of $T_{\mu \nu }^{coupling}$ in (A10) ensure that this is indeed a consistent solution. The Ricci tensor part of (A11), which is sourced by $T_{\mu \nu }$ , will automatically combine with the Weyl part (64) to satisfy the full Equation (A5) for a specific choice of the coefficient $\kappa$ .

So, from Equations (A10) and (A11), and by requiring consistency with the full field Equation (A5), we obtain this particular solution (64). The final step is to determine the proportionality constant $\kappa$ self-consistently, which is achieved through the matching procedure detailed in Section 6.

A6. Determination of Coefficient $\kappa$

A6.1. Physical Consistency Condition from Field Equations

The ansatz (64) must satisfy the modified Einstein equations (A5) in the context of our specific solitonic solutions. Substituting this relation into the left-hand side of (A5) via the definition of the Weyl tensor (A11), and comparing with the right-hand side containing $T_{\mu \nu }^{EM}+T_{\mu \nu }^{coupling}$ , provides a consistency condition that determines $\kappa$ .

The generator equivalence plays a crucial role here. It ensures that under the GGE transformation, the algebraic structures match, allowing for a direct proportionality between the curvature and electromagnetic field strength tensors.

A6.2. Vierbein Projection and Coefficient Matching

A more direct approach to determine $\kappa$ utilizes the vierbein formalism and the GGE transformation:

Vierbein projection relation:

${\left(F_V\right)}_{\mu \nu }^{ab}=e_{\rho }^a{e}_{\sigma }^b{C}_{\mu \nu \rho \sigma }$ (A14)

GGE transformation relation:

${\left(F_V\right)}_{\mu \nu }^{ab}=g_{UV}^{-1}{\left(F_U\right)}_{\mu \nu }^{EM}{g}_{UV}$ (A15)

Combining (A14) and (A15), and substituting our ansatz (70), we obtain:

$g_{UV}^{-1}{\left(F_U\right)}_{\mu \nu }^{EM}{g}_{UV}=e_{\rho }^a{e}_{\sigma }^b\left[\kappa \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)\right]$

For the specific solitonic solutions described by (A6) and (A7), this relation must hold identically. The generator equivalence ensures the consistency of the transformation. Matching the coefficients on both sides of this equation for our ${\mathrm{sech}}^2\left(ku\right)$ profile solutions yields the precise relation:

$\kappa =8\pi \alpha$ (A16)

A6.3. Dimensional Analysis Verification

Dimensional analysis confirms the consistency of this result (natural units $c=1$ , $\hslash =1$ ):

• Electromagnetic field strength: $\left[F_{\mu \nu }\right]={\left[L\right]}^{-1}$ (since the energy density $F_{\mu \nu }{F}^{\mu \nu }$ has dimension ${\left[L\right]}^{-4}$ ).

• Weyl tensor: $\left[C_{\mu \nu \rho \sigma }\right]={\left[L\right]}^{-2}$ (as a curvature tensor, second derivative of metric).

• Coupling constant: $\left[\alpha \right]={\left[L\right]}^2$ (from the Lagrangian term $\alpha R_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ ).

So, from Equation (70):$\left[C\right]=\left[\kappa \right]{\left[F\right]}^2\Rightarrow {\left[L\right]}^{-2}=\left[\kappa \right]{\left[L\right]}^{-2}\Rightarrow \left[\kappa \right]={\left[L\right]}^0$ (dimensionless). This presents an apparent contradiction, as we found $\kappa =8\pi \alpha$ .

The resolution lies in recognizing that in the specific solitonic solutions we consider, there is an implicit length scale provided by the ${\mathrm{sech}}^2\left(ku\right)$ profile, where $k$ has dimension ${\left[L\right]}^{-1}$ . The full dimensional analysis should be:

$\left[C\right]=\left[\kappa \right]{\left[F\right]}^2\Rightarrow {\left[L\right]}^{-2}=\left[\kappa \right]{\left[L\right]}^{-2}\Rightarrow \left[\kappa \right]=\text{dimensionless}$

However, in our derivation, the coupling emerges from the specific form of the solitonic solutions where the amplitude is normalized relative to this natural scale. The numerical coefficient 8π ensures the consistency between the algebraic ansatz (64) and the original field equations (A5).

A6.4. Final Weyl-Electromagnetic Relation

$C_{\mu \nu \rho \sigma }=8\pi \alpha \left(F_{\mu \rho }{F}_{\nu \sigma }-F_{\mu \sigma }{F}_{\nu \rho }\right)$ (A17)

A7. Spinor Form Derivation

A7.1. Electromagnetic Field Spinor

$F_{\mu \nu }^{\left(k\right)}~{\varphi }_A^{\left(k\right)}{\overline{\varphi }}_B^{\left(k\right)}{\sigma }_{\mu \nu }^{AB}$ (A18)

A7.2. Weyl Spinor

$C_{\mu \nu \rho \sigma }~{\psi }_{ABCD}\left({\sigma }_{\mu \rho }^{AB}{\sigma }_{\nu \sigma }^{CD}-{\sigma }_{\mu \sigma }^{AB}{\sigma }_{\nu \rho }^{CD}\right)$ (A19)

A7.3. Spinor Relation

Through GGE transformation implementation in spinor space, utilizing generator equivalence:

${\psi }_{ABCD}=\kappa {\varphi }_{AB}{\varphi }_{CD}=8\pi \alpha {\varphi }_{AB}{\varphi }_{CD}$ (A20)

A8. Derivation Logic Summary

Key logical chain of this derivation:

• Start from Lagrangian: containing gravitational-electromagnetic coupling term $\alpha \sqrt{-g}{R}_{\mu \nu }{F}^{\mu \sigma }{F}_{\sigma }^{\nu }$ .

• Through variation principle: obtain modified Einstein equations (A5), containing coupling term contribution $T_{\mu \nu }^{coupling}$ .

• Introduce GGE transformation: establish connection between optical and gravitational solitons, and discover $T_{EM}=J_{12}$ generator equivalence.

• Utilize generator equivalence and solitonic character to justify seeking particular algebraic solution.

• Establish Weyl-electromagnetic relation: obtain relation (64) from particular solution of equations of motion.

• Determine coefficient: through vierbein projection, GGE transformation, and dimensional analysis determine $\kappa =8\pi \alpha$ .

This derivation provides a rigorous theoretical foundation for optical soliton to gravitational soliton conversion, establishing an exact correspondence between classical field theory description and geometric description in specific solitonic configurations.