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<p>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, we consider the case of cosmological metrics, in the Belinski–Zakharov notation, and prove global existence of small Belinski–Zakharov spacetimes under a natural non-degeneracy 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 of the particular metrics called generalized Kasner solitons.</p>

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

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Claudio Muñoz,
  • Jessica Trespalacios

摘要

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, we consider the case of cosmological metrics, in the Belinski–Zakharov notation, and prove global existence of small Belinski–Zakharov spacetimes under a natural non-degeneracy 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 of the particular metrics called generalized Kasner solitons.