Refractive index for the mechanical refraction of a relativistic particle

TL;DR

This paper analytically derives the refractive index for the mechanical refraction of relativistic particles across all speeds, linking classical metaphysics and modern physics through the optical-mechanical analogy.

Contribution

It extends the concept of refractive index to relativistic particles and unifies classical and relativistic results within a comprehensive analytical framework.

Findings

Refractive index matches Fermat's law at ultra-relativistic speeds.

Refractive index aligns with Descartes' metaphysical pseudo-Snell law at non-relativistic speeds.

Provides a unified analytical expression for relativistic mechanical refraction.

Abstract

We have analytically determined the refractive index for the mechanical refraction of a relativistic particle for its all possible speeds. We have critically analysed the importance of Descartes' metaphysical theory and extended it in this regard. We have considered the conservation of the tangential component of the relativistic momentum and the relativistic energy of the particle in the process of the mechanical refraction within the optical-mechanical analogy. Our result for the mechanical refractive index exactly matches with the forms of both the Fermat's result on Snell's law of optical refraction at the ultra-relativistic limit and the Descartes' metaphysical result on the pseudo-Snell law of optical refraction at the non-relativistic limit.