Abelian decomposition for Einstein's theory and gauge theoretic approach to classical solutions of Einstein's equation

다국어 초록 (Multilingual Abstract)

In this report, we present some implications of Abelian decomposition of Einstein’s theory. Two fundamental solutions for A2- and B2-restricted gravity will be discussed. Indeed A2-decomposition is adapted to describe the topological property of spacetime and B2-decomposition is appropriate to handle all solutions
of gravitational waves. And we will propose a new method to solve Einstein’s equation in a gauge theoretical way. Using this method some well known solutions will be derived.

목차 (Table of Contents)

• 1 Introduction
• 2 Abelian decomposition of SU(2): A review
• 3 Abelian Decomposition of Gravitational Connection
• 3.1 Basic settings and notations
• 3.2 Two different Abelian Decompositions
• 3.3 A2 (Non Light-like) Isometry
• 3.4 B2 (Light-like) Isometry
• 3.5 Abelian Decomposition of Einstein’s Theory
• 3.5.1 A2 (Non Light-like) Decomposition
• 3.5.2 B2 (Light-like) Decomposition
• 3.6 Restricted Gravity
• 3.6.1 A2 Gravity
• 3.6.2 B2 Gravity
• 4 Fundamental Solutions of Restricted Gravity
• 4.1 A fundamental solution for A2-gravity : Cosmic String
• 4.1.1 Exterior solution for the cosmic string
• 4.1.2 Interior solution for the cosmic string
• 4.1.3 Gauge potential of the interior solution of cosmic string as a restricted potential
• 4.2 A fundamental solution for B2-gravity :
• Einstein-Rosen-Bondi wave solution
• 4.2.1 Plane wave solution of Bondi’s spacetime
• 4.2.2 Polarized wave solution of Bondi’s spacetime
• 4.3 The pp-wave spacetime
• 5 Gauge Theoretical Methods to Solve Einstein’s Equation
• 5.1 Most general vacuum potential of Einstein’s spacetime
• 5.2 Vacuum decomposition
• 5.3 Geodesic equation in an orthonormal frame
• 5.4 Potential of gauge potential
• 5.5 Purely electric potential
• 5.6 Static and spherically symmetric vacuumsolutions
• 5.7 StationaryWeyl Class
• 5.8 StaticWeyl Class
• 5.9 Levi-Civita Solution
• 6 Discussion