Well-posed first-order reduction of the characteristic problem of the linearized Einstein equations

TL;DR

This paper introduces a specific set of first-order variables that reformulate the linearized Einstein equations' characteristic problem into a well-posed form, ensuring solutions can be reliably estimated from initial data.

Contribution

It provides a novel choice of variables that guarantees manifest well-posedness for the characteristic problem of linearized Einstein equations.

Findings

The reformulation yields a system with an a priori estimate of solutions.

The algebraic criterion for manifest well-posedness is established.

The approach parallels symmetric hyperbolicity in Cauchy problems.

Abstract

A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that there exists an \textit{a priori} estimate of the solution of the characteristic problem in terms of the data. The notion of manifest well-posedness consists of an algebraic criterion sufficient for the existence of the estimates, and is to characteristic problems as symmetric hyperbolicity is to Cauchy problems. Both notions have been made precise elsewere.