Criterion for testing the covariance of physical laws and Gordon optical metric

TL;DR

This paper establishes a criterion for testing the covariance of physical laws based on three rules, revealing that the Gordon metric does not satisfy these rules and discussing covariance issues in relativistic quantum mechanics.

Contribution

It introduces a new criterion for covariance of physical laws and demonstrates that the Gordon metric fails this criterion, clarifying foundational issues in relativity and optics.

Findings

Gordon metric does not satisfy the covariance criterion

Covariance of Dirac equation conflicts with mass-energy equivalence

A new test for physical law covariance is proposed

Abstract

The problem of covariance of physical quantities has not been solved fundamentally in the theory of relativity, which has caused a lot of confusion in the community; a typical example is the Gordon metric tensor, which was developed almost a century ago, and has been widely used to describe the equivalent gravitational effect of moving media on light propagation, predicting a novel physics of optical black hole. In this paper, it is shown that under Lorentz transformation, a covariant tensor satisfies three rules: (1) the tensor keeps invariant in mathematical form in all inertial frames; (2) all elements of the tensor have the same physical definitions in all frames; (3) the tensor expression in one inertial frame does not include any physical quantities defined in other frames. The three rules constitute a criterion for testing the covariance of physical laws, required by Einstein's principle of relativity. Gordon metric does not satisfy Rule (3), and its covariance cannot be identified before a general refractive index is defined. Finally, it is also shown as well that in the relativistic quantum mechanics, the Lorentz covariance of Dirac wave equation is not compatible with Einstein's mass-energy equivalence.