Global existence of classical static solutions of four dimensional Einstein-Klein-Gordon system

TL;DR

This paper proves the global existence of static, spherically symmetric solutions in the four-dimensional Einstein-Klein-Gordon system by reducing the equations to a single integro-differential form and applying contraction mapping techniques.

Contribution

It establishes the global existence of classical static solutions for the Einstein-Klein-Gordon system under spherical symmetry, a novel analytical result.

Findings

Decay estimates of solutions obtained

Global existence proved using contraction mapping

Reduction to a single integro-differential equation

Abstract

In this paper we prove the global existence of classical static solutions of Einstein gravitational theory coupled to a real scalar field where the spacetime admits spherically symmetry. The equations of motions can then be reduced into a single first-order integro-differential equation. First, we obtain the decay estimates of the solutions. Then, in order to prove the global existence, we use the contraction mapping theorem in the appropriate function spaces.