# Ehlers-Kundt Conjecture about Gravitational Waves and Dynamical Systems

## TL;DR

This paper investigates the Ehlers-Kundt conjecture relating gravitational plane waves to polynomial potentials in classical mechanics, proving the conjecture in cases where the potential is polynomially bounded in the spatial variable.

## Contribution

The paper proves the polynomial Ehlers-Kundt conjecture for bounded polynomial potentials, linking gravitational wave models to polynomial potentials in dynamical systems.

## Key findings

- The conjecture holds for potentials bounded polynomially in space.
- Trajectories are complete if and only if the potential is quadratic or lower degree.
- Implications extend to physical models of gravitational waves.

## Abstract

Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane ${\mathbb R}^2$ with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potential $V(z,u)$, $(z,u)\in {\mathbb R}^2\times {\mathbb R}$, harmonic in $z$ (i.e. source-free), the trajectories of its associated dynamical system $\ddot{z}(s)=-\nabla_z V(z(s),s)$ are complete (they live eternally) if and only if $V(z,u)$ is a polynomial in $z$ of degree at most $2$ (so that $V$ is a standard mathematical idealization of vacuum). Here, the conjecture is solved in the significative case that $V$ is bounded polynomially in $z$ for finite values of $u\in {\mathbb R}$. The mathematical and physical implications of this {\em polynomial EK conjecture}, as well as the non-polynomial one, are discussed beyond their original scope.