Nonlinear realization of unstable linearized Kasner perturbations

Determine whether there exist one-parameter families of four-dimensional vacuum Einstein metrics through a Kasner metric whose linearizations are the unstable solutions v_k, w_k, and z described in Theorem 2.3 and Theorem 2.4, respectively, including the inhomogeneous cases with k\geq 2.

Background

The paper constructs explicit unstable solutions of the linearized Einstein equation around a non-flat four-dimensional Kasner spacetime. The modes v_1 and w_1 are identified with linearizations of Taub’s explicit Bianchi II Kasner-transition family, but the corresponding nonlinear realization of the higher-order modes v_k and w_k for k\geq 2, and of the exceptional mode z, is not established.

The issue is complicated by the Killing fields of the Kasner background and by linearization-stability constraints. The subsequent discussion shows that any prospective family realizing v_k for k\geq 2 would necessarily be inhomogeneous, but it does not construct such families or prove that they exist.

References

The question that arises is whether there are one-parameter families of metrics s \mapsto g_s solving the Einstein vacuum equation, with linearization given by our unstable solutions v_k, w_k and z in Theorem~\ref{thm: self-similar solutions} and Theorem~\ref{thm: main linearized Einstein}.

— The linear instability of Kasner spacetimes  (2609.20809 - Petersen, 17 Sep 2026) in Section 1, subsection “The linear instability of Kasner spacetimes,” immediately before Remark “Non-linearized version of v_1 and w_1”