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Relativistic one-dimensional billiards

Alfonso Artigue

Abstract

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 \textit{tachyonic 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.

Relativistic one-dimensional billiards

Abstract

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 \textit{tachyonic 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.

Paper Structure

This paper contains 14 sections, 8 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Free particle
  3. Definition of the dynamics
  4. Collisions
  5. Tachyonic collisions
  6. Example of a zero-energy tachyonic collision
  7. Mirror systems
  8. General properties of mirror systems
  9. Reduction to a one-parameter system
  10. A constant of motion
  11. Non-negative discriminant
  12. Negative discriminant
  13. Tachyonic collisions
  14. An estimation for tachyonic collisions

Key Result

Lemma 6

If $sr\neq 0$ then the system of equations ecuSudoku has just one other solution: which is different from the original if and only if the collision condition ecuCondiciónChoque holds.

Theorems & Definitions (27)

  • Remark 1
  • Remark 2
  • Remark 3: Interpretation of negative energy
  • Remark 4
  • Remark 5: Collision condition
  • Lemma 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Remark 10
  • ...and 17 more