Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics (Q456745)

In this paper the authors study the 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 their approach is based on the introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in \(d\) colours, by which they 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 their study of what they call discriminant sets of tropical lines \(\Sigma^+\) and \(\Sigma^-\) and their singularities. All this allows them to get an analytic criterion describing all periodic billiard trajectories, including the light-like ones as those of a special interest.