Localized past stability of the subcritical Kasner-scalar field spacetimes
F. Beyer, T. A. Oliynyk, W. Zheng
TL;DR
The paper proves nonlinear past stability for the entire subcritical Kasner-scalar field family in four dimensions, establishing both global-in-space and localised past stability results. It develops a conformal tetrad formulation with zero shift and uses the time function $\tau=\exp(\tfrac{2}{\sqrt{3}}\,\phi)$ to synchronise the big bang, then applies a Fuchsian framework to obtain global past existence and decay. The main contributions are two stability theorems—one global-in-space on $M_{0,t_0}$ and one localised on truncated cones—together with AVTD behaviour and curvature blow-up at $\tau=0$, showing $C^2$-inextendibility for generic Kasner data. This localisation, via finite propagation speed, strengthens prior global results and demonstrates stability under spatially localised perturbations in four dimensions, with precise control of the renormalised variables and curvature invariants.
Abstract
We prove the nonlinear stability, in the contracting direction, of the entire subcritical family of Kasner-scalar field solutions to the Einstein-scalar field equations in four spacetime dimensions. Our proof relies on a zero-shift, orthonormal frame decomposition of a conformal representation of the Einstein-scalar field equations. To synchronise the big bang singularity, we use the time coordinate $τ= \exp\bigl(\frac{2}{\sqrt{3}}φ\bigr)$, where $φ$ is the scalar field, which coincides with a conformal harmonic time slicing. We show that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete to the past and terminate at quiescent, crushing big bang singularities located at $τ=0$, which are characterised by curvature blow up. Specifically, we establish two stability theorems. The first is a global in-space stability result where the perturbed spacetimes are of the form $M =\bigcup_{t\in (0,t_0]} τ^{-1}(\{t\}) \cong (0,t_0] \times \mathbb{T}^{3}$. The second is a localised version where the perturbed spacetimes are given by $M=\bigcup_{t\in (0,t_0]}τ^{-1}(\{t\})\cong \bigcup_{t\in (0,t_0]} \{t\}\times\mathbb{B}_{ρ(t)}$ with time-dependent radius function $ρ(t)=ρ_0+(1-\vartheta)ρ_0\bigl(\bigl(\frac{t}{t_0}\bigr)^{1-ε}-1\bigr)$. Spatial localisation is achieved through our choice of zero-shift, harmonic time slicing that leads to hyperbolic evolution equations with a finite propagation speed.