NUMERICAL RELATIVITY - relativitywp.abdphysics.org

Preliminaries for numerical relativity
- Brief introduction of general relativity
  - Einstein’s equation
  - Nature of Einstein’s equation
- Gravitational waves
  - Linearized Einstein’s equation
  - Propagation of gravitational waves
  - Generation of gravitationalwaves
  - Gravitational-wave luminosity
  - Gravitational waves froma binary
  - Gravitational-wave detectors
- Black holes
  - Four dimensional black holes
  - Properties of four-dimensional black holes
  - Higher-dimensional black holes
- Neutron stars
  - Formation of neutron stars
  - Basic properties of neutron stars
  - Hydrostatic equations for cold neutron stars
  - Cold neutron-star equations of state
  - Structure and sequences of neutron stars
  - Supramassive and hypermassive neutron stars
  - Finite-temperature equations of state for high-density matter
  - Binary neutron stars
- Sources of gravitationalwaves
  - Inspiral of binary compact objects
  - Merger of binary neutron stars
  - Merger of binary black holes
  - Merger of black hole-neutron star binaries
  - Gravitational collapse and core collapse supernova
- Matched filtering techniques for gravitational-wave data analysis
Formulation for initial-value problems of general relativity
- Formulations based on spacetime foliation
- Gauge conditions
- Formulations in numerical relativity
- Formulations in axisymmetric spacetime
- Cartoon method
  - Cartoon method for axisymmetric spacetime
  - Cartoon method for higher-dimensional spacetime: Modified cartoonmethod
- Formulations for asymptotically de Sitter and Friedmann spacetime
  - Asymptotically de Sitter spacetime
  - Asymptotically Friedmann spacetime
Numerical methods for a solution of Einstein’s evolution equation
- Solving hyperbolic equations
- Handling advection terms
- Adaptive mesh refinement
  - Why necessary?
  - Spatial interpolation in the buffer zone
  - Time integration scheme in the buffer zone
  - Restriction
  - Kreiss–Oliger dissipation
- Testing numerical-relativity code: Vacuum spacetime
  - Examining convergence property
  - Propagation of linear gravitational waves
  - Evolution of black holes
Matter equations in general relativity
- Scalar fields
- Collisionless particles
- 3+1 form of stress-energy conservation: ∇aT a b =0
- Hydrodynamics
  - Basic equations
  - Properties of hydrodynamics equations
- Hydrodynamics with microphysics
- Electromagnetohydrodynamics
  - Definitions
  - Ideal magnetohydrodynamics
  - Properties of the ideal magnetohydrodynamics equations
  - Force-free electromagnetodynamics
- Radiation transfer and radiation hydrodynamics
  - Boltzmann’s equation
  - Moment formalism
  - Leakage scheme
- Numerical methods for hydrodynamics and magnetohydrodynamics: Handling transport term
  - Monotonicity preserving
  - Godunov’s theorem
  - Circumventing Godunov’s theorem .
  - Total variation diminishing
  - Reconstruction of numerical flux at cell interfaces
  - Approximate Riemann solvers
- Other ingredients in numerical hydrodynamics and magnetohydrodynamics
- Testing hydrodynamics and magnetohydrodynamics codes
- Testing a numerical-relativity code with matter
  - Stability of neutron stars
  - Oscillation of neutron stars
  - Collapse of unstable neutron stars
Formulations for initial data, equilibrium, and quasi-equilibrium
- Properties of initial-data equations
- York–Lichnerowicz formulation
- Mass, linear momentum, and angular momentum
  - ADM mass, linear momentum, and angular momentum in the Hamiltonian formulation
  - Komar mass and angular momentum
  - Virial relation
  - Irreduciblemass
- Initial data for pure gravitational waves
- Initial data for black holes
  - Time symmetric case
  - Time asymmetric case
Extracting gravitational waves
Finding black holes
Coalescence of binary compact objects
Gravitational collapse to a black hole
Non-radial instability and magnetohydrodynamics instability
Higher-dimensional simulations
Appendix A Killing vector and Frobenius’ theorem
Appendix B Numerical relativity in spherical symmetry
Appendix C Decomposition by spherical harmonics
Appendix D Lagrangian and Hamiltonian formulations of general relativity
Appendix E Solutions of Riemann problems in special relativistic hydrodynamics
Appendix F Landau – Lifshitz pseudo tensor
Appendix G Laws of black hole and apparent horizon
Appendix H Post-Newtonian results for coalescing compact binaries