Ehlers-Kundt Conjecture about Gravitational Waves and Dynamical Systems

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.

Final version with minor changes and some new references

References in corpus (8)

• A Brief History of Gravitational Waves
• Radiation and Boundary Conditions in the Theory of Gravitation
• Finsler pp-waves
• The causal boundary of wave-type spacetimes
• Geodesics in nonexpanding impulsive gravitational waves with $\Lambda$, Part I
• On the completeness of impulsive gravitational wave space-times
• Completeness of general pp-wave spacetimes and their impulsive limit
• Geodesics in nonexpanding impulsive gravitational waves with $\Lambda$, II

Cited by in corpus (1)

• Exact Parallel Waves in General Relativity