Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the "Double Null Foliation Gauge"

TL;DR

This paper demonstrates that analytic solutions to certain characteristic problems in Einstein vacuum equations can exist in larger regions than previously known, leveraging hyperbolicity and a geometric gauge, with implications for global solutions under small initial data.

Contribution

It introduces a geometric gauge for Einstein equations, extends the existence region of analytic solutions, and compares the approach with the simpler Burger equation case.

Findings

Existence region for solutions is larger than Cauchy-Kowalevski predicts.

Small initial data lead to global analytic solutions.

Geometric formulation facilitates extension of solutions.

Abstract

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region larger than the one provided by the Cauchy-Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the apriori estimates for the Weyl equations, associated to the "Bel-Robinson norms". In particular if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a "geometric" way the Einstein equations, then we show how the Cauchy-Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.