[Paper Review] Equivalence principle in classical electrodynamics

Bozhidar Z. Iliev|ArXiv.org|Mar 1, 2003

Cosmology and Gravitation Theories18 references3 citations

TL;DR

This paper formulates the equivalence principle in classical electrodynamics by interpreting electromagnetic potentials as a linear connection in a one-dimensional vector bundle over spacetime. It demonstrates that normal (inertial) frames exist at any point or along non-self-intersecting paths, where potentials vanish, and the equivalence principle holds universally in these cases; however, it fails on higher-dimensional submanifolds unless the field is a pure gauge.

ABSTRACT

The principle of equivalence in gravitational physics and its mathematical base are reviewed. It is demonstrated how this principle can be realized in classical electrodynamis. In general, it is valid at any given single point or along a path without selfintersections unless the field considered satisfies some conditions.

Motivation & Objective

• To extend the equivalence principle—originally formulated in general relativity—to classical electrodynamics using differential geometry.
• To establish a rigorous mathematical framework for inertial and normal frames in the context of electromagnetic potentials.
• To clarify the conditions under which the equivalence principle holds in electrodynamics, particularly in relation to the geometry of spacetime and vector bundles.
• To connect the physical notion of inertial frames with the mathematical concept of normal frames for linear connections in vector bundles.
• To explore the implications of the equivalence principle for minimal coupling and the Aharonov-Bohm effect in classical field theory.

Proposed method

• Models the electromagnetic potential as a linear connection (specifically, a covariant derivative) in a one-dimensional complex vector bundle over spacetime.
• Applies the theory of linear transports along paths and normal frames for linear connections to characterize inertial frames in electrodynamics.
• Uses the condition that a frame is normal if the connection coefficients (Γ) vanish along paths, which corresponds to the vanishing of the electromagnetic potential in that frame.
• Derives the transformation law for potentials under frame changes, showing that a gauge transformation with λ = −f₀ can eliminate the potential in a neighborhood where F = 0.
• Applies theorems from differential geometry (e.g., from [bp-normalF-LTP]) to prove existence of normal frames at points and along injective paths.
• Distinguishes between normal frames and strong normal frames, showing that on open sets where the field is pure gauge, both types coincide.

Experimental results

Research questions

• RQ1 Under what conditions does the equivalence principle hold in classical electrodynamics?
• RQ2 How can the concept of inertial frames in electrodynamics be mathematically formalized using linear connections and normal frames?
• RQ3 What is the geometric role of electromagnetic potentials in the context of parallel transport and connection coefficients?
• RQ4 Why does the equivalence principle fail on submanifolds of dimension ≥2 unless the field is a pure gauge?
• RQ5 How does the Aharonov-Bohm effect relate to the equivalence principle and the existence of normal frames in electrodynamics?

Key findings

• The equivalence principle in electrodynamics is not a postulate but a theorem that holds at any single point or along any non-self-intersecting path.
• Normal frames for the electromagnetic connection exist at points and along injective paths, and in such frames, the electromagnetic potential vanishes.
• On open neighborhoods where the field strength F vanishes (i.e., pure gauge fields), the electromagnetic potential can be made identically zero via a gauge transformation.
• In such pure gauge regions, the existence of a normal frame implies the existence of a strong normal frame, meaning both the 2-index and 3-index connection coefficients vanish.
• The equivalence principle fails on submanifolds of dimension ≥2 unless the field is a pure gauge, as normal frames do not exist in general on such sets.
• The minimal coupling principle in interacting field theories can be justified via the equivalence principle when the electromagnetic field is represented as a linear connection.