Numerical Relativity

FSU Jena, SS 2023

Lecture period: 03.04.2023 - 07.07.2023

Last updated: 10.02.2023

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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

• T.W.Baumgarte & S.L.Shapiro Numerical Relativity and Compact Binaries Phys. Rep.376, 41 (2003)
• E.Gourgoulhon 3+1 Formalism and Bases of Numerical Relativity Lecture Notes in Physics (Springer) (2012)
• E.Gourgoulhon Construction of initial data for 3+1 numerical relativity J. Phys.: Conf. Ser. 91 012001 (2007)
• E.Gourgoulhon An introduction to relativistic hydrodynamics
• E.Gourgoulhon An introduction to the theory of rotating relativistic stars
• E.Arnowitt, S.Deser, C.W.Misner The Dynamics of General Relativity
• D.Hilditch An Introduction to Well-posedness and Free-evolution

General

• M.Blau Lecture Notes on General Relativity
• S.M.Carrol Lecture Notes on General Relativity
• G.J.Galloway Notes on Lorentzian causality
• P.T.Chru'sciel Lectures on Energy in General Relativity

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

• A.Carlotto The general relativistic constraint equations
• Y.Choquet-Bruhat & J.W.York "The Cauchy Problem, in General Relativity and Gravitation, one hundred Years after the Birth of Albert Einstein", Vol. 1, edited by A.Held, Plenum Press, New York (1980)
• Y.Choquet-Bruhat Beginnings of the Cauchy problem
• R.Bartnik & J.Isenberg The Constraint Equations
• G.B.Cook Initial Data for Numerical Relativity
• J.A.Font Numerical Hydrodynamics and Magnetohydrodynamics in General Relativity
• P.Grandcleent & J.Novak Spectral Methods for Numerical Relativity
• L.Lehner & F.Pretorious Numerical Relativity and Astrophysics
• J.L.Jaramillo & E.Gourgoulhon Mass and Angular Momentum in General Relativity
• J.M.Marti & E.Mueller Numerical Hydrodynamics in Special Relativity
• O.Rinne Numerical and analytical methods for asymptotically flat spacetimes
• U.Sperhake The numerical relativity breakthrough for binary black holes
• W.Tichy The initial value problem as it relates to numerical relativity

Numerical methods

Books and lectures notes:

• J.P.Boyd Chebyshev and Fourier Spectral Methods
• M.W.Choptuik Relativistic Astrophysics and Numerical Relativity, Numerical Analysis for Numerical Relativists
• B.Gustafsson, H.O.Kreiss and J.Oliger, Time-dependent problems and difference methods, John Wiley,New York 1995.
• C.W.Shu Numerical Methods for Hyperbolic Conservation Laws

Papers:

• M.J.Berger & J.Oliger Adaptive mesh refinement for hyperbolic partial differential equations
• M.J.Berger & P.Colella Local adaptive mesh refinement for shock hydrodynamics

Web resources and Software

• AMR AMR is a package of Fortran routines for the numerical solution of hyperbolic conservation laws in 2 and 3 space dimensions
• Baumgarte's NR: Starting from Scratch
• Cactus and Einstein toolkit a community project for numerical relativity
• Choptuik and Pretorius webpages for lectures, tutorial, and examples
• Cococubed.com F.X.Timmes (ASU) web resources
• git and go a quick intro on git
  
• GRChombo open-source code for numerical general relativity simulations based on Chombo
• GRwiki yes, a wikipage!
• GRHypS 1+1 GR code for evolutions with puncture gauge
• GR1D 1+1 GR and neutrino transport code for supernova expolosion
• nuLib Basic standard set of neutrino matter interaction routines that can be readily incorporated in radiation-hydrodynamics codes
• PARAMESH A parallel adaptive mesh refinement community toolkit. See also this paper
• PETSc a suite of data structures and routines for the scalable (parallel) solution of scientific applications modeled by PDEs.
• PySpinSph Spin Weighted Spherical Harmonics for Python/NumPy
• SageMath and SageManifold. See also these lecture notes and this paper. Online notebooks of interest (Thanks to E.Gourgoulhon):
  • Kruskal diagram ipynb (cell 44)
  • Kruskal diagram ipynb (cell 95)
  • Kruskal diagram ipynb (cell 19)
  • 3+1 split
  • 3+1 split ipynb
• xAct and contributed examples and these lectures
• pyg a simple python3 program to plot & animates 1D data
• PIRSA online lectures on NR (Lehner and Pretorius)

Lectures

The following are semi-private notes. If you decide to use them, please send feedbacks, cite the source and let me know.

• LaTeX notes PDF (2021, v0.1)

Exercises

Tutor: B.Daszuta

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