Disproof of a widely-accepted mathematical conjecture

TL;DR

This paper discredits a widely-accepted mathematical conjecture related to Lorentz tensors, demonstrating it is flawed with a counterexample, and clarifies its implications for the Abraham-Minkowski controversy in light momentum.

Contribution

It provides the first counterexample to the conjecture, revealing its flaw and clarifying its limited applicability in relativistic electromagnetic theory.

Findings

The conjecture is mathematically invalid due to a counterexample.

The flawed conjecture has been widely accepted in textbooks for decades.

Møller's version of von Laue's theorem only yields a trivial zero 4-vector.

Abstract

A mathematical conjecture is successfully identified, which is used for relativistic analysis of dielectric Einstein-box thought experiment in a Letter (Ramos, Rubilar, and Obukhov, Phys. Lett. A 375, 1703 (2011)), where the authors conjecture (without any citations) that, the symmetry and divergence-less property of a Lorentz 4-tensor is a sufficient condition for the time-column space integrals to constitute a Lorentz 4-vector. This mathematical conjecture has been thought to be "a mathematical fact the validity of which was shown well" in textbooks. However in this paper, we indicate that this conjecture has never been proved mathematically. By enumerating a counterexample, we find that this mathematical conjecture is flawed, and it is not persuasive to use a flawed mathematical conjecture as a starting point to resolve Abraham-Minkowski controversy over light momentum in a dielectric medium. We also indicate that this flawed mathematical conjecture is actually a widely-accepted conjecture in the dynamics of relativity in textbooks for many decades. To eliminate a misunderstanding of this flawed conjecture in the community, we provide a detailed elucidation of why M{\o}ller's mathematical statement, also called "M{\o}ller's version of von Laue's theorem", only defines a trivial zero 4-vector for an electromagnetic stress-energy Lorentz 4-tensor.