Snellius meets Schwarzschild - Refraction of brachistochrones and time-like geodesics

TL;DR

This paper explores how a refraction law analogous to Snellius' law can be applied to solve variational problems in General Relativity, specifically for free fall in Schwarzschild spacetime, linking optics principles to gravitational trajectories.

Contribution

It introduces a refraction law analogy for time-like geodesics in Schwarzschild spacetime, extending variational calculus methods in General Relativity.

Findings

Derived a refraction law for free fall in Schwarzschild spacetime.

Connected optical refraction principles with gravitational geodesics.

Provided a new approach to solving variational problems in General Relativity.

Abstract

The brachistochrone problem can be solved either by variational calculus or by a skillful application of the Snellius' law of refraction. This suggests the question whether also other variational problems can be solved by an analogue of the refraction law. In this paper we investigate the physically interesting case of free fall in General Relativity that can be formulated as a variational problem w. r. t. proper time. We state and discuss the corresponding refraction law for a special class of spacetime metrics including the Schwarzschild metric.