1 Introduction
References
2 Basic Differential Geometry
2.1 Introduction
2.2 Differentiable Manifolds
2.2.1 Notion of Manifold
2.2.2 Vectors on a Manifold
2.2.3 Linear Forms
2.2.4 Tensors
2.2.5 Fields on a Manifold
2.3 Pseudo-Riemannian Manifolds
2.3.1 Metric Tensor
2.3.2 Signature and Orthonormal Bases
2.3.3 Metric Duality
2.3.4 Levi-Civita Tensor
2.4 Covariant Derivative
2.4.1 Affine Connection on a Manifold
2.4.2 Levi-Civita Connection
2.4.3 Curvature
2.4.4 Weyl Tensor
2.5 Lie Derivative
2.5.1 Lie Derivative of a Vector Field
2.5.2 Generalization to Any Tensor Field
References
3 Geometry of Hypersurfaees
3.1 Introduction
3.2 Framework and Notations
3.3 Hypersurface Embedded in Spacetime
3.3.1 Definition
3.3.2 Normal Vector
3.3.3 Intrinsic Curvature
3.3.4 Extrinsic Curvature
3.3.5 Examples: Surfaces Embedded in the Euclidean Space R3
3.3.6 An Example in Minkowski Spacetime: The Hyperbolic Space H3
3.4 Spacelike Hypersurfaces
3.4.1 The Orthogonal Projector
3.4.2 Relation Between K and Vn
3.4.3 Links Between the ▽ and D Connections
3.5 Gauss-Codazzi Relations
3.5.1 Gauss Relation
3.5.2 Codazzi Relation
References
4 Geometry of Foliations
4.1 Introduction
4.2 Globally Hyperbolic Spacetimes and Foliations
4.2.1 Globally Hyperbolic Spacetimes
4.2.2 Definition of a Foliation
4.3 Foliation Kinematics
4.3.1 Lapse Function
4.3.2 Normal Evolution Vector
4.3.3 Eulerian Observers
4.3.4 Gradients of n and m
4.3.5 Evolution of the 3-Metric
4.3.6 Evolution of the Orthogonal Projector
4.4 Last Part of the 3+1 Decomposition of the Riemann Tensor.
4.4.1 Last Non Trivial Projection of the Spacetime Riemann Tensor
4.4.2 3+1 Expression of the Spacetime Scalar Curvature.
References
5 3+1 Decomposition of Einstein Equation
5.1 Einstein Equation in 3+1 form
5.1.1 The Einstein Equation
5.1.2 3+1 Decomposition of the Stress-Energy Tensor ..
5.1.3 Projection of the Einstein Equation
5.2 Coordinates Adapted to the Foliation
5.2.1 Definition
5.2.2 Shift Vector
5.2.3 3+1 Writing of the Metric Components
5.2.4 Choice of Coordinates via the Lapse and the Shift
5.3 3+1 Einstein Equation as a PDE System
5.3.1 Lie Derivatives Along m as Partial Derivatives
5.3.2 3+1 Einstein System
5.4 The Cauchy Problem
5.4.1 General Relativity as a Three-Dimensional Dynamical System
5.4.2 Analysis Within Gaussian Normal Coordinates
5.4.3 Constraint Equations
5.4.4 Existence and Uniqueness of Solutions to the Cauchy Problem
5.5 ADM Hamiltonian Formulation
5.5.1 3+1 form of the Hilbert Action
5.5.2 Hamiltonian Approach
References
6 3+1 Equations for Matter and Electromagnetic Field
6.1 Introduction
6.2 Energy and Momentum Conservation
6.2.1 3+1 Decomposition of the 4-Dimensional Equation
6.2.2 Energy Conservation
6.2.3 Newtonian Limit
6.2.4 Momentum Conservation
6.3 Perfect Fluid
6.3.1 Kinematics
6.3.2 Baryon Number Conservation
6.3.3 Dynamical Quantities
6.3.4 Energy Conservation Law
6.3.5 Relativistic Euler Equation
6.3.6 Flux-Conservative Form
6.3.7 Further Developments
6.4 Electromagnetism
6.4.1 Electromagnetic Field
6.4.2 3+1 Maxwell Equations
6.4.3 Electromagnetic Energy, Momentum and Stress...
6.5 3+1 Ideal Magnetohydrodynamics
6.5.1 Basic Settings
6.5.2 Maxwell Equations
6.5.3 Electromagnetic Energy, Momentum and Stress...
6.5.4 MHD-Euler Equation
6.5.5 MHD in Flux-Conservative Form
References
7 Conformal Decomposition
7.1 Introduction
7.2 Conformal Decomposition of the 3-Metric
7.2.1 Unit-Determinant Conformal "Metric"
7.2.2 Background Metric
7.2.3 Conformal Metric
7.2.4 Conformal Connection
7.3 Expression of the Ricci Tensor
7.3.1 General Formula Relating the Two Ricci Tensors
7.3.2 Expression in Terms of the Conformal Factor
7.3.3 Formula for the Scalar Curvature
7.4 Conformal Decomposition of the Extrinsic Curvature
7.4.1 Traceless Decomposition
7.4.2 Conformal Decomposition of the Traceless Part
7.5 Conformal Form of the 3+1 Einstein System
7.5.1 Dynamical Part of Einstein Equation
7.5.2 Hamiltonian Constraint
7.5.3 Momentum Constraint
7.5.4 Summary: Conformal 3+1 Einstein System
7.6 Isenberg-Wilson-Mathews Approximation to General Relativity
References
8 Asymptotic Flatness and Global Quantifies
8.1 Introduction
8.2 Asymptotic Flatness
8.2.1 Definition
8.2.2 Asymptotic Coordinate Freedom
8.3 ADM Mass
8.3.1 Definition from the Hamiltonian Formulation of GR
8.3.2 Expression in Terms of the Conformal Decomposition
8.3.3 Newtonian Limit
8.3.4 Positive Energy Theorem
8.3.5 Constancy of the ADM Mass
8.4 ADM Momentum
8.4.1 Definition
8.4.2 ADM 4-Momentum
8.5 Angular Momentum
8.5.1 The Supertranslation Ambiguity
8.5.2 The "Cure".
8.5.3 ADM Mass in the Quasi-Isotropic Gauge
8.6 Komar Mass and Angular Momentum
8.6.1 Komar Mass
8.6.2 3+1 Expression of the Komar Mass and Link with the ADM Mass
8.6.3 Komar Angular Momentum
References
9 The Initial Data Problem
9.1 Introduction
9.1.1 The Initial Data Problem
9.1.2 Conformal Decomposition of the Constraints
9.2 Conformal Transverse-Traceless Method
9.2.1 Longitudinal / Transverse Decomposition of A ij
9.2.2 Conformal Transverse-Traceless Form of the Constraints
9.2.3 Decoupling on Hypersurfaces of Constant Mean Curvature
9.2.4 Existence and Uniqueness of Solutions to Lichnerowicz Equation
9.2.5 Conformally Flat and Momentarily Static Initial Data
9.2.6 Bowen-York Initial Data
9.3 Conformal Thin Sandwich Method
9.3.1 The Original Conformal Thin Sandwich Method .
9.3.2 Extended Conformal Thin Sandwich Method
9.3.3 XCTS at Work: Static Black Hole Example
9.3.4 Uniqueness Issue
9.3.5 Comparing CTT, CTS and XCTS
9.4 Initial Data for Binary Systems
9.4.1 Helical Symmetry
9.4.2 Helical Symmetry and IWM Approximation
9.4.3 Initial Data for Orbiting Binary Black Holes
9.4.4 Initial Data for Orbiting Binary Neutron Stars
9.4.5 Initial Data for Black Hole: Neutron Star Binaries.
References
10 Choice of Foliation and Spatial Coordinates
10.1 Introduction
10.2 Choice of Foliation
10.2.1 Geodesic Slicing
10.2.2 Maximal Slicing
10.2.3 Harmonic Slicing
10.2.4 1+log Slicing
10.3 Evolution of Spatial Coordinates
10.3.1 Normal Coordinates
10.3.2 Minimal Distortion
10.3.3 Approximate Minimal Distortion
10.3.4 Gamma Freezing
10.3.5 Gamma Drivers
10.3.6 Other Dynamical Shift Gauges
10.4 Full Spatial Coordinate-Fixing Choices
10.4.1 Spatial Harmonic Coordinates
10.4.2 Dirac Gauge
References
11 Evolution schemes
11.1 Introduction
11.2 Constrained Schemes
11.3 Free Evolution Schemes
11.3.1 Definition and Framework
11.3.2 Propagation of the Constraints
11.3.3 Constraint-Violating Modes
11.3.4 Symmetric Hyperbolic Formulations
11.4 BSSN Scheme
11.4.1 Introduction
11.4.2 Expression of the Ricci Tensor of the Conformal Metric
11.4.3 Reducing the Ricci Tensor to a Laplace Operator
11.4.4 The Full Scheme
11.4.5 Applications
References
Appendix A: Conformal Killing Operator and Conformal Vector Laplacian
Appendix B: Sage Codes
Index
