Global Existence and Long Time Behavior in Einstein-Belinski-Zakharov Soliton Spacetimes
Claudio Muñoz, Jessica Trespalacios
TL;DR
The paper establishes a rigorous global-in-time theory for vacuum Einstein equations under Belinski–Zakharov symmetry in the cosmological (timelike $\alpha$) sector, proving global existence for small data and deriving decay mechanisms via energy and virial methods. By combining the inverse-scattering–driven BZ structure with modern PDE techniques, the authors construct a cosmological energy $E[\Lambda,\phi;\alpha]$ and new virial functionals to describe decay inside the light cone, including a precise long-time decay result $\lim_{t\to\infty} \int_{|x-vt|\lesssim t(\log t)^{-2}} (\cdots)\,dx = 0$ for $|v|<1$. The framework yields a robust global theory (Theorem GLOBAL0) and a decay theory (Theorems MT20 and MT2) with explicit applications to gravitational solitons, notably generalized Kasner 1-solitons, as well as to Einstein–Rosen-type cylindrical waves. The results provide a bridge between integrable-systems techniques and nonlinear PDE methods, with implications for the long-time dynamics of cosmological spacetimes and the stability of solitonic GR geometries.
Abstract
We consider the vacuum Einstein field equations under the Belinski-Zakharov symmetry, which leaves the problem as a 1+1-dimensional quasilinear system of PDEs. Depending on the chosen signature of the metric, these spacetimes contain most of the well-known special solutions in General Relativity. In this paper, {\color{blue} we consider the case of cosmological metrics, in the Belinsky-Zakharov notation}, and prove global existence of small Belinski-Zakharov spacetimes under a natural nondegeneracy condition. We also construct new energies and virial functionals to provide a description of the energy decay of smooth global cosmological metrics inside the light cone. Finally, some applications are presented in the case {\color{blue} of the particular metrics called} generalized Kasner solitons.