[Paper Review] Is the relativity principle consistent with classical electrodynamics? Towards a logico-empiricist reconstruction of a physical theory

Márton Gömöri, László E. Szabó|arXiv (Cornell University)|Dec 22, 2009

Relativity and Gravitational Theory16 references3 citations

TL;DR

This paper investigates whether the relativity principle (RP) is consistent with classical electrodynamics (ED) by analyzing operational definitions of electrodynamical quantities and applying Lorentzian pedagogy. It shows that while transformation rules derived from RP match those of covariance, the concept of a field 'moving' with a velocity at a point lacks a coherent physical meaning, undermining the foundation of RP in ED and leaving the principle's consistency unanswerable in a clear way.

ABSTRACT

The transformation rules for the basic electrodynamical quantities are routinely derived from the hypothesis that the relativity principle (RP) applies for Maxwell's electrodynamics. These derivations leave open several questions: (1) Is the RP a true law of nature for electrodynamical phenomena? (2) Are, at least, the transformation rules of the fundamental electrodynamical quantities, derived from the RP, true? (3) Is the RP consistent with the laws of electrodynamics in one single inertial frame of reference? (4) Are, at least, the derived transformation rules consistent with the laws of electrodynamics in one single frame of reference? (1) and (2) are empirical questions; we will investigate problems (3) and (4). First we will develop a formalism of the RP. In the second part, we will deal with the operational definitions of the fundamental quantities. In the third part of the paper we will show that the proper transformation rules are indeed identical with the ones obtained by presuming the covariance, and that the covariance is indeed satisfied. Problem (3) raises conceptual problems to which there seems no satisfactory solution in electrodynamics; thus, contrary to the widespread views, the question we asked in the title has no obvious answer.

Motivation & Objective

• To assess whether the relativity principle (RP) is consistent with classical electrodynamics (ED) in a single inertial frame.
• To investigate whether the transformation rules of electrodynamical quantities derived from RP are consistent with ED laws in a single frame.
• To clarify the operational meaning of 'field moving with velocity v' in ED, which is central to applying RP to moving systems.
• To challenge the widespread assumption that RP automatically holds for electrodynamical phenomena by exposing conceptual and semantic ambiguities in the notion of a moving system.

Proposed method

• Formalizing the relativity principle (RP) as a statement about the invariance of physical descriptions across inertial frames when expressed in terms of co-moving measurements.
• Applying Lorentzian pedagogy—where laws in a single frame account for all physical phenomena—to derive transformation rules consistent with ED covariance.
• Using operational definitions of electrodynamical quantities (E, B, J) in both stationary and moving frames to analyze measurement consistency.
• Constructing a family of solutions Eλ(r,t), Bλ(r,t) that converge to a given field configuration E(r,t), B(r,t) as λ→0, while preserving local dynamics.
• Analyzing the Jacobian matrix of the electric field to assess whether a local velocity field can be consistently defined at a spacetime point.
• Demonstrating that the condition ∂tE = D E v leads to contradictions unless the Jacobian is invertible, and showing that such invertibility cannot be generally guaranteed.

Experimental results

Research questions

• RQ1 Is the relativity principle consistent with the laws of classical electrodynamics in a single inertial frame of reference?
• RQ2 Are the textbook transformation rules for electrodynamical quantities consistent with ED laws in a single frame?
• RQ3 What is the physical meaning of an electromagnetic field 'moving with velocity v' at a point in space and time?
• RQ4 Can a consistent operational definition of a field's velocity be established in classical electrodynamics?
• RQ5 Does the covariance of Maxwell's equations in different frames imply the validity of the relativity principle for electrodynamical systems?

Key findings

• The transformation rules for electrodynamical quantities derived from the relativity principle are consistent with the covariance of Maxwell’s equations in different inertial frames.
• The concept of a field 'moving with velocity v' at a point cannot be consistently defined in classical electrodynamics, as it leads to contradictions when the Jacobian of the electric field is not invertible.
• Even when solutions converge to a given field configuration (Eλ → E, Bλ → B as λ→0), the local velocity field cannot be uniquely determined, undermining the idea of a field's motion.
• The relativity principle cannot be applied to moving systems in ED without a coherent definition of 'motion' of the field, which the paper shows is not available.
• The paper concludes that the question of whether RP holds in ED remains unanswerable not due to empirical failure, but due to conceptual incoherence in the notion of a moving electromagnetic system.