Localized past stability of the subcritical Kasner-scalar field spacetimes

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 $\ddot{I}„=\exp (\frac{2}{\sqrt{3}}\ddot{I}•)$, where $\ddot{I}•$ 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 $\ddot{I}„=0$, 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 $M={\bigcup }_{t\in (0,t_0]}\ddot{I}{„}^{-1}(\{t\})\cong (0,t_0]\times {\mathbb{T}}^3$. The second is a localised version where the perturbed spacetimes are given by $M={\bigcup }_{t\in (0,t_0]}\ddot{I}{„}^{-1}(\{t\})\cong {\bigcup }_{t\in (0,t_0]}\{t\}\times {\mathbb{B}}_{\ddot{I}(t)}$ with time-dependent radius function $\ddot{I}(t)=\ddot{I}{}_0+(1-\vartheta )\ddot{I}{}_0\left(\right(\frac{t}{t_0}{)}^{1-\hat{I}\mu }-1)$. Spatial localisation is achieved through our choice of zero-shift, harmonic time slicing that leads to hyperbolic evolution equations with a finite propagation speed.

Typos corrected and the statement of Lemma 4.2 separated into two cases