Numerical Relativity
FSU Jena
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Topics
- 3+1 Geometry
- Cauchy surface, globally hyperbolic spacetime
- Hypersurfaces, normal vector and Eulerian observers
- Induced 3-metric and projections
- 3-Connection and intrinsic curvature
- Extrinsic curvature
- Acceleration of Eulerian observers
- Kinematical relations for the 3-metric
- Gauss relation and its contractions
- Codazzi equation
- Normal evolution vector and Lapse function
- Ricci equation
- 3+1 decomposition of Einstein equations
- Hamiltonian constraint
- Momentum constraint
- Adapted coordinates
- Evolution of the extrinsic curvature
- Adapted coordinates and PDEs
- Transport property of the constraints
- Historical remarks on 3+1 and Arnowitt, Deser and Misner (ADM) formulations
- Discussion on possible approaches for the solution
- Conformal decomposition
- 3-metric, conformal metric and background metric
- Expressions relating the Ricci tensors
- Traceless and conformal decomposition of the extrinsic curvature
- Time-evolution scaling
- Momentum scaling
- Conformal 3+1 GR
- Isenberg-Wilson-Mathews approximation of GR
- Asyptotic flatness and Global quantities
- Asymptotically flat spacetime
- Coordinates transformation at i0
- ADM mass
- Schwarzschild spacetime and weak field limit
- ADM momentum
- Quasi-isotropic gauge
- Komar mass and relation to ADM
- Komar angular momentum
- Initial data problem
- Conformal transverse-traceless decomposition
- Moment of time symmetry initial data for black holes
- Bowen-York punctures
- Brill waves
- Conformal thin-sandwich
- Extended Conformal thin-sandwich
- Helical Killing vector method for binaries in quasi-circular orbits
- Helical Killing vector and Kepler's third law
- Hyperbolic equations and well-posedness
- Hyperbolic equations in GR
- Wave and advection as models of hyperbolic equations
- Well-posed Cauchy problem and examples
- Concept of hyperbolicity for first order systems
- Symmetrizer, energy and characteristic fields
- 3+1 Wave equation analysis
- GR formulation for the Cauchy problem
- Harmonic gauge and generalized formulations
- Link to Z4 formulation and constraints
- From "ADM" to BSSNOK
- Slicing conditions
- Harmonic slicing
- Bona-Masso slicing
- Singularity avoding property
- Spatial gauge
- Minimal distortion
- Dirac gauge
- Gamma-freezing and Gamma-drivers
- Single puncture evolution, gravitational collapse and moving punctures
- Horizons
- Event horizon
- Apparent horizon
- Relativistic hydrodynamics
- 3+1 decomposition of matter fields
- Perfect fluids
- Relativistic Euler equation and properties
- Simple fluids and equation of state
- Stationary fluids and relativistic Bernoulli equation
- Irrotational fluids
- Conservative formulation, flat spacetime and and Newtonian limit
- Relativistic magnetohydrodynamics
- 3+1 decomposition of electromagnetism
- Conservative formulation and coupling with hydrodynamics
- Ideal magnetohydrodynamics
- Force-free limit
- Resistive magnetohydrodynamics
- Radiation hydrodynamcs
- Radiation fields and relativistic Boltzmann equation
- Newtonian limit
- Moment formalism
Sample of exercises solved during the tutorial sessions
- Derivation of Gauss-Codazzi-Ricci equations
- 3+1 decomposition of the Z4 system and constraints evolution equations
- Conformal decomposition with Mathematica
- Stokes theorem
- Numerical solution of TOV equations (ODEs integration)
- Numerical solution of geodesics (ODEs integration)
- Numerical solution of elliptic equations in 2D
- Numerical solution of the Regge-Wheeler equation (wave equation in 1+1D)
- Constraint equations in the generalized harmonic formulation
- Schwarzschild spacetime: maximal slicing and 1+log slices
- Apparent horizon for Schwarzschild: analytical and numerical computations
Final projects
- Apparent horizon finder in 3D
- Derivation of BSSN/Z4c equations in spherical coordinates
- RWZ equation and black hole quasi normal modes
- Brill waves initial data
- Bowen-York punctures initial data
- 1+1 ADM and black hole evolution in spherical symmetry
- 1+1 scalar field collapse
- 1+1 Special relativistic hydrodynamics
Literature
Books
Specialistic
- M.Alcubierre “Introduction to 3+1 Numerical Relativity”, Oxford Science Publications (2008)
- T.W.Baumgarte & S.L. Shapiro “Numerical Relativity”, Cambridge University Press (2010)
- L.Rezzolla & O.Zanotti "Relativistic Hydrodynamics” Oxford University Press (2013)
- M.Shibata “Numerical Relativity” World Scientific Publishing (2016)
- T.W.Baumgarte & S.L. Shapiro “Numerical Relativity, Starting from Scratch”, Cambridge University Press (2021)
General
- A.M.Anile "Relativistic Fluids and Magneto-fluids" Cambridge University Press (1990)
- R.Wald "General Relativity" University of Chicago Press (1984)
- Y.Choquet-Bruhat "General Relativity and the Einstein Equations" Oxford Mathematical Monographs (2009)
Lecture notes
Specialistic
General
Suggested articles
Original
- Hahn & Lindquist (1964) First supercomputer numerical relativity simulation of colliding black holes
- Smarr et al (1976), Anninos et al (1993), Anninos et al (1994) Head-on collision of black holes
- Stark & Piran (1985) Gravitational-Wave Emission from Rotating Gravitational Collapse
- Nakamura et al (1987) Gravitational collapse and gravitational waves from black holes
- Choptuik (1993), Abrahams & Evans (1993) Critical collapse
- Bonazzola et al (1993) Axisymmetric rotating relativistic bodies
- Bernstein (1993), Bona et al (1994) Anninos et al.(1995) 1+log slicing
- Balakrishna et al (1996) Alcubierre et al (2000) (2002) Gamma-drivers
- Anninos et al (1994) Apparent and event horizons
- Wilson & Mathews (1995) and (2000), Oechslin et al (2001) Neutron star mergers with IWM
- Shibata & Nakamura (1995), Baumgarte & Shapiro (1998); H.Beyer & O.Sarbach Gundlach & Martin-Garcia (2004) BSSNOK formulation and hyperbolicity analysis
- Bruegmann (1997) First 3D evolution of a short orbit
- Bowen & York (1980), Brandt & Bruegmann (1997) Puncture initial data
- Gourgoulhon (1998) on Bonazzola et al (1997) Shibata (1998) Teukolsky (1998) Irrotational binary neutron star configurations
- Font et al (1998} and (2002) Several tesbeds for general relativistic hydrodynamics
- Shibata & Uryu (1999) First binary neutron star merger in GR
- Alcubierre et al (2000) Grazing black hole collision from a full 3D numerical evolution and waveforms.
- Gourgoulhon et al (2000) Initial data for binary neutron stars in quasi-circular orbits
- Granclement et al (2001) CTS initial data for binary black hole in quasi-circular orbits
- Alcubierre et al (2002) Gauge conditions for black holes evolutions without excision ("It is possible to have the punctures move across the grid because of a non- vanishing shift.")
- Bona et al (2003) Gundlach et al (2005) Z4 formulation and constraint damping
- Bonazzola et al (2003) Garfinkle et al (2007) Brown (2009) Baumgarte et al Background metric and non-Cartesian coordinates
- Bonazzola et al (2003) Constrained scheme for GR (alternative to free evolution schemes)
- Bruegmann et al (2003) First one-orbit binary black hole simulation
- Alcubierre et al (2003), Babiuc et al and Cao&Hilditch Apple with Apple tests (testbeds for NR)
- Lindblom et al (2005) Generalized Harmonic formulation in first order form
- Pretorius (2005), Baker et al (2005), Campanelli et al (2005) Black hole mergers breaktrough
- Hannam et al (2006), Brown (2007), Hannam et al (2008), Thierfelder et al (2010) On the meaning of the moving puncture gauge
- Baker et al (2006), Campanelli et al (2007), Campanelli et al (2007), Herrmann et al (2007), Gonzalo et al (2007) Black hole kicks
- Palenzuela+ (2008) Palenzuela (2012) GR resistive magnetohydrodynamics
- Tichy (2011), Tichy (2012), Moldenhauer, et al (2014) Binary neutron star initial data with spin and eccentricity
- Le Tiec (2012) Helical Killing vector and Kepler's law
- Thorne (1981), Shibata et al (2011), Cardall et al (2013) GR relativistic radiation hydrodynamics, moment formalism and 3+1 decompositions
- Radice (2017) Shibata&Kiuchi (2017) GR viscous hydrodynamics simulations
- Foucart et al (2016) Radice et al (2021) Momentum scheme for neutrino radiation
Reviews
Numerical methods
Books and lectures notes:
Papers:
Web resources and Software
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