Localized past stability of the subcritical Kasner-scalar field spacetimes

TL;DR

This paper proves the nonlinear stability of subcritical Kasner-scalar field solutions near the big bang singularity in four-dimensional spacetime, showing they asymptotically resemble Kasner solutions and develop quiescent singularities.

Contribution

It introduces a new stability proof for the entire subcritical Kasner-scalar field family, using a conformal representation and harmonic time slicing to analyze local and global stability.

Findings

Perturbed solutions are asymptotically Kasner near the singularity.

Solutions develop quiescent, crushing big bang singularities with curvature blow-up.

The stability results include both global and localized in-space theorems.

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 $\tau = \exp\bigl(\frac{2}{\sqrt{3}}\phi\bigr)$, where $\phi$ 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 $\tau=0$, 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 $M =\bigcup_{t\in (0,t_0]} \tau^{-1}(\{t\}) \cong (0,t_0] \times \mathbb{T}^{3}$. The second is a localised version where the perturbed spacetimes are given by $M=\bigcup_{t\in (0,t_0]}\tau^{-1}(\{t\})\cong \bigcup_{t\in (0,t_0]} \{t\}\times\mathbb{B}_{\rho(t)}$ with time-dependent radius function $\rho(t)=\rho_0+(1-\vartheta)\rho_0\bigl(\bigl(\frac{t}{t_0}\bigr)^{1-\epsilon}-1\bigr)$. Spatial localisation is achieved through our choice of zero-shift, harmonic time slicing that leads to hyperbolic evolution equations with a finite propagation speed.