[Paper Review] Normal frames for linear connections in vector bundles and the equivalence principle in classical gauge theories

Bozhidar Z. Iliev|CERN Bulletin|Oct 21, 2001

Advanced Differential Geometry Research4 citations

TL;DR

This paper establishes the existence of normal frames for arbitrary linear connections in vector bundles, generalizing the concept of normal coordinates from Riemannian geometry to gauge theories. By defining inertial frames for gauge fields and formulating the equivalence principle in this context, it extends the principle of equivalence beyond gravity to include non-gravitational gauge interactions, showing that normal frames exist at points and along injective paths for any linear connection.

ABSTRACT

Frames normal for linear connections in vector bundles are defined and studied. In particular, such frames exist at every fixed point and/or along injective path. Inertial frames for gauge fields are introduced and on this ground the principle of equivalence for (system of) gauge fields is formulated.

Motivation & Objective

• To generalize the concept of normal frames and coordinates from symmetric linear connections on manifolds to arbitrary linear connections in vector bundles.
• To define inertial frames for gauge fields and formulate the principle of equivalence in classical gauge theories.
• To demonstrate that results on normal frames from prior work on derivations and transports along paths apply mutatis mutandis to linear connections in vector bundles.
• To unify the description of gravitational and non-gravitational gauge interactions through the framework of normal frames and the equivalence principle.
• To provide a geometric foundation for extending the equivalence principle to systems of gauge fields, including gravity.

Proposed method

• Defining linear connections in vector bundles via the standard axiomatic approach, ensuring compatibility with tensorial and module structures.
• Introducing linear transports along paths and establishing their equivalence to linear connections through a bijective correspondence.
• Deriving the fundamental equation for frames normal to a linear connection, which matches equations from prior work on derivations and transports.
• Constructing inertial frames for individual gauge fields and their direct sum, using the product of normal frames from each connection.
• Applying the minimal coupling principle to field Lagrangians by replacing partial derivatives with covariant derivatives associated with the total connection.
• Proving that normal frames for the direct sum connection correspond to inertial frames for the system of gauge fields, thereby realizing the equivalence principle.

Experimental results

Research questions

• RQ1 Can normal frames for arbitrary linear connections in vector bundles be systematically defined and constructed?
• RQ2 How can the principle of equivalence be generalized from gravity to non-gravitational gauge fields using the concept of inertial frames?
• RQ3 What conditions ensure the existence of normal frames at a point or along an injective path for a linear connection in a vector bundle?
• RQ4 How do normal frames for individual gauge fields combine to form an inertial frame for a system of interacting gauge fields?
• RQ5 To what extent can the geometric equivalence principle be extended to gauge theories with non-Abelian structure groups?

Key findings

• Any linear connection in a vector bundle admits frames normal at any given point in the base manifold.
• Normal frames also exist along any injective smooth path in the base manifold, generalizing the local existence result to curves.
• The existence and properties of normal frames for linear connections in vector bundles are governed by the same fundamental equation as in prior work on derivations and transports, allowing direct transfer of results.
• Inertial frames for a system of gauge fields can be constructed as the direct product of normal frames for each individual connection, provided the fields are independent.
• The principle of equivalence for a system of gauge fields reduces to the statement that inertial frames coincide with normal frames for the total connection, making the principle a tautological consequence of frame definition.
• The framework allows a geometric generalization of the equivalence principle to include gauge theories, where Lorentz invariants are replaced by invariants of the gauge group representation.