Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

TL;DR

This paper explores the geometry and billiard dynamics within confocal quadrics in pseudo-Euclidean spaces, introducing a novel combinatorial structure and providing a comprehensive criterion for periodic trajectories, including light-like cases.

Contribution

It introduces a new discrete geometric structure and analytical criteria for periodic billiard trajectories in pseudo-Euclidean spaces, expanding understanding of relativistic quadrics.

Findings

Complete description of periodic billiard trajectories in ellipsoids.

Introduction of a coloring-based decomposition of confocal quadrics.

Analytic criterion for all types of billiard trajectories, including light-like.

Abstract

We study geometry of confocal quadrics in pseudo-Euclidean spaces of an arbitrary dimension $d$ and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of our approach is based on introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in $d$ colours, by which we decompose quadrics of $d+1$ geometric types of a pencil into new relativistic quadrics of $d$ relativistic types. Deep insight of related geometry and combinatorics comes from our study of what we call discriminat sets of tropical lines $\Sigma^+$ and $\Sigma^-$ and their singularities. All of that enable usto get an analytic criterion describing all periodic billiard trajectories, including the light-like ones as those of a special interest.