The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Flores, Jose L.¹  Sanchez, Miguel² 

[1] Univ Malaga, Dept Algebra Geometria & Topol, Fac Ciencias, Campus Teatinos, E-29071 Malaga, Spain

[2] Univ Granada, Fac Ciencias, Dept Geometria & Topol, Campus Fuentenueva S-N, E-18071 Granada, Spain

关键词: Geodesic completeness;    Plane and pp-waves;    Ricci flat Lorentz manifolds;    Newtonian mechanics;    Trajectories under a potential;    Harmonic functions and polynomials;   

DOI  :  10.1016/j.jde.2019.11.061

来源: Elsevier

【 摘 要 】

The 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) is an element of R-2 x R, harmonic in z (i.e. source-free), the trajectories of its associated dynamical system (z) overdot (s) = -del 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 significant case that V is bounded polynomially in z for u in bounded intervals. The mathematical and physical implications of this polynomial EK conjecture, as well as the non-polynomial one, are discussed beyond their original scope. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free