Global Existence and Long Time Behavior in Einstein-Belinski-Zakharov Soliton Spacetimes

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.