Relativistic one-dimensional billiards

In this article we study the dynamics of one-dimensional relativistic billiards containing particles with positive and negative energy. We study configurations with two identical positive masses and symmetric positions with two massless particles between them of negative energy and symmetric positions. We show that such systems have finitely many collisions in any finite time interval. This is due to a phenomenon we call $\mathit{\text{tachyonic}}\mathit{\text{collision}}$, which occur at small scales and produce changes in the sign of the energy of individual particles. We also show that depending on the initial parameters the solutions can be bounded with certain periodicity or unbounded while obeying an inverse square law at large distances.