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

Mirda Prisma Wijayanto, Fiki Taufik Akbar, Bobby Eka Gunara

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 $D\geq 4$ 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.

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 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.
Paper Structure (15 sections, 9 theorems, 141 equations)

This paper contains 15 sections, 9 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Motivation and Challenges
  3. Structure of the Paper
  4. Higher Dimensional Einstein-Scalar System
  5. Einstein Equation
  6. Equation of Motion
  7. Existence of Classical Solution
  8. Local Existence
  9. Global Existence
  10. Completeness Properties of the Spacetime
  11. Appendix
  12. Estimate for (\ref{['fk']})
  13. Estimate for (\ref{['f1 higher']})
  14. Estimate for (\ref{['estimate psi2']})
  15. Estimate for (\ref{['estimate dhdu']})

Key Result

Lemma 1

Let $D\geq 4$. For any $\mathcal{L}_1(\hat{x})>\hat{d}$, there exists $\delta(\hat{x},\hat{d})>0$ such that if $u_0<\delta$, the map $\mathcal{F}$ defined in (Fcurl) is contained in the closed ball of radius $\hat{x}$ in the space $\hat{X}$.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 2
  • ...and 6 more