Smearing of chaos in sandwich pp-waves

TL;DR

This paper investigates how chaos in geodesic motion within sandwich pp-waves diminishes as the wave duration decreases, ultimately becoming non-chaotic in impulsive limits, with explicit geodesic solutions illustrating focusing effects.

Contribution

It generalizes previous chaos results to non-homogeneous sandwich pp-waves and demonstrates how chaos smears with decreasing wave duration, providing explicit geodesic solutions and physical interpretation.

Findings

Chaos diminishes as wave duration decreases

Geodesics become non-chaotic in impulsive limit

Test particles form closed hypotrochoidal curves

Abstract

Recent results demonstrating the chaotic behavior of geodesics in non-homogeneous vacuum pp-wave solutions are generalized. Here we concentrate on motion in non-homogeneous sandwich pp-waves and show that chaos smears as the duration of these gravitational waves is reduced. As the number of radial bounces of any geodesic decreases, the outcome channels to infinity become fuzzy, and thus the fractal structure of the initial conditions characterizing chaos is cut at lower and lower levels. In the limit of impulsive waves, the motion is fully non-chaotic. This is proved by presenting the geodesics in a simple explicit form which permits a physical interpretation, and demonstrates the focusing effect. It is shown that a circle of test particles is deformed by the impulse into a family of closed hypotrochoidal curves in the transversal plane. These are deformed in the longitudinal direction in such a way that a specific closed caustic surface is formed.