Global Existence and Completeness of Classical Solutions in Higher Dimensional Einstein-Klein-Gordon System

Abstract

In this paper we study the global existence and completeness of classical solutions of gravity coupled a scalar field system called Einstein-Klein-Gordon system in higher dimensions. We introduce a new ansatz function to reduce the problem into a single first-order integro-differential equation. Then, we employ the contraction mapping in the appropriate Banach space. Using Banach fixed theorem, we show that there exists a unique fixed point, which is the solution of the theory. For a given initial data, we prove the existence of both local and global classical solutions. We also study the completeness properties of the spacetime. Here, we introduce a mass-like function for $D\ge 4$ in Bondi coordinates. The completeness of spacetime along the future directed timelike lines outward to a region which resembles the event horizon of the black hole.

arXiv comment35 pages, no figure, title slihgtly changed, subsection 1.1 extended, some typos corrected. Accepted in General Relativity and Gravitation 2024