The 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 R 2 with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potential V (z, u), (z, u) ∈ R 2 × R, harmonic in z (i.e. source-free), the trajectories of its associated dynamical systemz(s) = −∇zV (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 ∈ R. The mathematical and physical implications of this polynomial EK conjecture, as well as the non-polynomial one, are discussed beyond their original scope.

Kinetic energy and angular momentum of free particles in the gyratonic pp-waves space-times

“…(5), a positive variation of u corresponds to a negative variation of the time coordinate t. Therefore, if u grows from left to right in all Figures, the time coordinate t grows from right to left. Thus, we see that in Figure [4] the final Kinetic energy is smaller than the initial energy, whereas in Figure [5] the final Kinetic energy is larger than the initial energy. This is exactly the behaviour discussed in Ref.…”

“…Ehlers and Kundt [2,3] asserted that the vacuum pp-waves (parallelly propagated plane-fronted waves) are geodesicaly complete (see Ref. [4]), and also showed that one may add the amplitudes of two pp-waves running in the same direction, in which case the principle of linear superposition holds (section 2-5.3 of Ref. [2]).…”

On completeness and dynamics of compact Brinkmann spacetimes

“…The Ehlers-Kundt conjecture says that in the global case, with n = 2, i.e. H is defined globally on R × R 2 , if g is Ricci-flat, then g is complete exactly when H is quadratic on z, in which case g is said to be a plane wave (see [15] for the most recent results on the subject). Observe that Ricci-flatness is equivalent to harmonicity with respect to z ∈ R 2 of H. This fact together with the reduction of the geodesic equation to a mechanical system, that will be discussed below, show the post-Newtonian character of pp-waves, which applies also to Brinkmann spaces.…”

Exact Parallel Waves in General Relativity

“…Let us first recall the causal ladder for Lorentzian manifolds: from [48,Sec. 3] and [49]. Note that "stably causal" was first understood as the causality being a stable property under perturbations, but Hawking showed [50] that this is equivalent to the existence of a global time function.…”