Einstein Equations: Local Energy, Self-Force, and Fields in General Relativity: Domoschool 2019

Preface
Contents
Part I Main Lectures
Introduction to the Wang–Yau Quasi-local Energy
Contents
1 Introduction
2 Isometric Embedding of Spheres into R3,1
3 Surface Hamiltonian and Its Properties
3.1 Surface Hamiltonian
3.2 Graphical Surfaces in R3,1
3.3 Some Related Inequalities
4 Construction of the Wang–Yau Energy
4.1 The Choice of { T, n }
4.2 A Variational Property
4.3 Expression of the Wang–Yau Quasi-local Energy
4.4 Relation to Other Quasi-local Energy
5 Positivity of EWY (, τ)
5.1 A Motivation to Jang's Equation in R3,1
5.2 Jang's Equation on a General (Ω, g, k)
5.3 Comparison with a Euclidean Domain
5.4 A Physical Interpretation of H̃ - "426830A Ỹ, ν̃"526930B
5.5 Some Comment on Ỹ and ν'
References
Gravitational Self-force in the Schwarzschild Spacetime
Contents
1 Introduction
2 The Various Steps of Gravitational Self-force Computations
2.1 Metric Perturbations: The Regge–Wheeler–Zerilli Formalism
2.2 Curvature Perturbations: The Teukolsky Formalism
3 Scalar Self-force: Computational Details
3.1 Scalar Wave Equation
3.2 Angular Part of the Perturbation
3.3 Radial Part of the Perturbation
3.4 Computing the Scalar Field Along the Source World Line
3.5 Solutions of the Radial Homogeneous Equation
3.5.1 Heun Confluent (Exact) Solutions
3.5.2 PN Solutions
3.5.3 MST or Hypergeometric-Like Solutions
3.6 Analytical Versus Numerical Results
4 So Far So Good: A List of SF Accomplishments
5 Concluding Remarks
References
Geometry and Analysis in Black Hole Spacetimes
Contents
1 Introduction
2 Background
2.1 Minkowski Space
2.2 Lorentzian Geometry and Causality
2.3 Conventions and Notation
2.4 Einstein Equation
2.5 The Cauchy Problem
2.6 Asymptotically Flat Data
2.7 Komar Integrals
3 Black Holes
3.1 The Schwarzschild Solution
3.1.1 Orbiting Null Geodesics
3.2 Black Hole Stability
3.3 The Kerr Metric
4 Spin Geometry
4.1 Spinors on Minkowski Space
4.2 Spinors on Spacetime
4.3 Fundamental Operators
4.4 Massless Spin-s Fields
4.5 Killing Spinors
4.6 Algebraically Special Spacetimes
4.6.1 Petrov Type D
4.7 Spacetimes Admitting a Killing Spinor
4.8 GHP Formalism
5 The Kerr Spacetime
5.1 Characterizations of Kerr
6 Monotonicity and Dispersion
6.1 Monotonicity for Null Geodesics
6.2 Dispersive Estimates for Fields
7 Symmetry Operators
7.1 Symmetry Operators for the Kerr Wave Equation
8 Outlook
8.1 Teukolsky
8.2 Stability for Linearized Gravity on Kerr
8.3 Nonlinear Stability for Schwarzschild
8.4 Nonlinear Stability for Kerr
References
Study of Fundamental Laws with Antimatter
Contents
1 Introduction
2 Testing CPT
3 Testing the Weak Equivalence Principle
4 Anti-hydrogen at CERN
4.1 Confinement
4.2 Anti-hydrogen Beams
4.3 The Beginning of Anti-hydrogen Spectroscopy
5 Positrons and Positronium
6 The Mu-Atom
7 Conclusion
References
Part II Proceedings
Quantum Ergosphere and Brick Wall Entropy
Contents
1 Introduction
2 Geometry of an Evaporating Black Hole
3 Mode Counting and Calculation of Entropy
4 Calculation of Luminosity
5 Conclusion
References
Geodesic Structure and Linear Instability of Some Wormholes
Contents
1 Introduction
2 Modelling the Metric of a Wormhole
2.1 An Embedding for the EBMT Wormhole
3 Deriving Wormholes from Einstein's Equation
4 An Embedding for the AdS Wormhole
5 The Timelike and Null Geodesics Motion
5.1 General Spherically Symmetric Case
5.2 The Case of the AdS Wormhole
6 Linear Instability of the EBMT Wormhole
6.1 Radial Perturbations of the EBMT Wormhole
6.2 A Master Equation for R
6.3 Linear Instability of the EBMT Wormhole
6.4 A Comparison with [2, 4, 8, 14]
References
New Trends in the General Relativistic Poynting–Robertson Effect Modeling
Contents
1 Introduction
2 General Relativistic 3D PR Effect Model
2.1 Critical Hypersurfaces
3 Stability of the Critical Hypersurfaces
4 Analytical form of the Rayleigh Dissipation Function
4.1 Discussions of the Results
5 Conclusions
References
Brief Overview of Numerical Relativity
Contents
1 ADM Formalism of Numerical Relativity
1.1 ADM Variables and Adapted Coordinates
1.2 ADM Evolution and Constraints
2 BSSN Formalism of Numerical Relativity
3 Further Considerations
3.1 Initial Data
3.2 Gauge Choice
3.3 Potential Application to Cosmology
References
Length-Contraction in Curved Spacetime
Contents
1 Introduction
2 Proper Frame Measurement
3 Length-Contraction and Measurement
3.1 Relative 3-Velocity
3.2 Length-Contraction Vector
3.3 Measurement
4 Volume Forms
4.1 Metric Volume
4.2 De-Contracted Volume
4.3 Contracted Volume
5 Examples
5.1 Lorentz Boost
5.2 Square Flux
5.3 Rotating Disc
5.4 Schwarzschild Spacetime
6 Discussion
7 DomoSchool Memories
References
Exact Solutions of Einstein–Maxwell(-Dilaton) Equations with Discrete Translational Symmetry
Contents
1 Introduction
1.1 Majumdar–Papapetrou Solution
2 Alternating Crystal
2.1 Constructing the Solution
2.2 Geometry
3 Uniform Crystal
3.1 Constructing the Solution
3.2 Geometry
4 Smooth Crystal
4.1 Constructing the Solution
4.2 Geometry
5 Uniform Reduced Crystal
5.1 Constructing the Solution
5.2 Geometry
6 Conclusions and Summary
References
Exact Solutions of the Einstein Equations for an Infinite Slab with Constant Energy Density
Contents
1 Introduction
2 Einstein Equations for Spacetimes with Spatial Geometry Possessing Plane Symmetry
3 Solution with Isotropic Pressure
4 Solution with Vanishing Tangential Pressure
5 Concluding Remarks
References
Emergence of Classicality from an Inhomogeneous Universe
Contents
1 Introduction
2 Preliminary Results
3 Quantum Dynamics and Classical Emergence
4 Discussion
Appendix: The Killing Tensors of the Lagrangian
References