A Conic Section Approach to the Relativistic Reflection Law

TL;DR

This paper derives the relativistic reflection law for a moving mirror using conic sections, showing that the effective surface of reflection forms a conic and applying Fermat's principle to confirm the law.

Contribution

It introduces a novel geometric approach using conic sections to derive the relativistic reflection law for both uniform and accelerating mirrors.

Findings

Effective surface of reflection is a conic for uniform motion.

The relativistic reflection law aligns with the bi-angular conic equation.

Fermat's principle confirms the derived reflection law.

Abstract

We consider the reflection of light, from a stationary source, off of a uniformly moving flat mirror, and derive the relativistic reflection law using well-known properties of conic sections. The effective surface of reflection (ESR) is defined as the loci of intersection of all beams, emanating from the source at a given time, with the moving mirror. Fermat principle of least time is then applied to ESR and it is shown that, assuming the independence of speed of light, the result is identical with the relativistic reflection law. For a uniformly moving mirror ESR is a conic and the reflection law becomes a case of bi-angular equation of the conic, with the incident and reflected beams coinciding with the focal rays of the conic. A short calculus-based proof for accelerating mirrors is also given.