Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes

TL;DR

This paper develops a mathematical framework for analyzing spherically symmetric Klein-Gordon fields coupled with gravity in asymptotically Anti-de-Sitter spacetimes, addressing well-posedness and setting the stage for stability results.

Contribution

It establishes local well-posedness for the Einstein-Klein-Gordon system with negative cosmological constant using a novel renormalized approach and weighted estimates.

Findings

Proved local existence of solutions depending on initial data norms

Introduced a renormalized system of equations for analysis

Provided estimates crucial for future stability studies

Abstract

We initiate the study of the spherically symmetric Einstein-Klein-Gordon system in the presence of a negative cosmological constant, a model appearing frequently in the context of high-energy physics. Due to the lack of global hyperbolicity of the solutions, the natural formulation of dynamics is that of an initial boundary value problem, with boundary conditions imposed at null infinity. We prove a local well-posedness statement for this system, with the time of existence of the solutions depending only on an invariant H^2-type norm measuring the size of the Klein-Gordon field on the initial data. The proof requires the introduction of a renormalized system of equations and relies crucially on r-weighted estimates for the wave equation on asymptotically AdS spacetimes. The results provide the basis for our companion paper establishing the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this system.