Future stability of the $1+3$ Milne model for Einstein-Klein-Gordon system

TL;DR

This paper proves the nonlinear future stability of the 1+3 Milne model for the Einstein-Klein-Gordon system, demonstrating that small perturbations lead to geodesically complete spacetimes in a rigorous mathematical framework.

Contribution

It establishes the first nonlinear stability result for the Einstein-Klein-Gordon system around the Milne model using a novel energy scheme in the CMC gauge.

Findings

Perturbed spacetimes are future causally geodesically complete.

The stability proof is achieved within the constant mean curvature gauge.

The energy scheme only involves spatial derivatives, not normal derivatives.

Abstract

We study the small perturbations of the $1+3$-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the $1+3$ splitting of the Bianchi-Klein-Gordon equations. Moreover, we treat the Bianchi-Klein-Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi-Klein-Gordon equations with the spatially covariant derivatives while the normal derivative is not allowed.