Stability of Minkowski Space-time with a translation space-like Killing field

TL;DR

This paper proves the nonlinear stability of Minkowski space-time with a translation space-like Killing field by reducing the Einstein equations to a 2+1 wave system with a scalar field and overcoming decay challenges.

Contribution

It introduces a generalized wave gauge with a cubic weak null structure to establish global existence in a setting with weaker decay properties.

Findings

Proves nonlinear stability of Minkowski space with a translation Killing field.

Develops a generalized wave gauge with null structure to handle decay issues.

Establishes global existence for the reduced Einstein equations in this setting.

Abstract

In this paper we prove the nonlinear stability of Minkowski space-time with a translation Killing field. In the presence of such a symmetry, the 3 + 1 vacuum Einstein equations reduce to the 2 + 1 Einstein equations with a scalar field. We work in generalised wave coordinates. In this gauge Einstein equations can be written as a system of quasilinear quadratic wave equations. The main difficulty in this paper is due to the weaker decay of free solutions to the wave equation in 2 dimensions, compared to 3 dimensions. This weak decay seems to be a deterrent for proving a stability result in the usual wave coordinates. In this paper we construct a suitable generalized wave gauge in which our system has a "cubic weak null structure", which allows for the proof of global existence.