Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics

Abstract

We study the geometry of confocal quadrics in pseudo-Euclidean spaces of 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 the introduction of a new discrete combinatorial geometric structure associated with a confocal pencil of quadrics, a colouring in d colours. This is used to decompose quadrics of d+1 geometric types of a pencil into new relativistic quadrics of d relativistic types. A study of what we term discriminant sets of tropical lines Σ+ and Σ− and their singularities provides insight into the related geometry and combinatorics. This yields an analytic criterion describing all periodic billiard trajectories, including light-like trajectories as a case of special interest.