Numerical Study of Spacetimes - NTMF107

Course in 2021 (in English)

1. Introduction. Relativistic stellar models. TOV equations ([1] page 15, [2]). Rotating isentropic stars (equations and methods [3], examples [4-6]).
2. Grid and spectral representation of functions.
   Example 1: TOV equations solved using the pseudospectral method as a boundary value problem in Mathematica (also as pdf file).
   Example 2: TOV equations solved as ODE problem in Mathematica (also as pdf file) and in Python (Jupyter) (open in browser or view a pdf file)
3. Model of spherical collapse by Oppenheimer and Snyder [1]. Illustration and Problem #1
4. Method of lines -- Problem set #2 Wave equation in 1+1, boundary conditions
5. Wave equation in characteristic coordinate, gravitational collapse of scalar field -- see Paper [7] and notes in Czech. Mathematica notebook.
6. Einstein equations in 3+1 -- Textbooks: [8],[9], nice course on youtube Play list
7. Spacetime foliations (geodesic, maximal, harmonic and 1+log slicing) -- Textbooks: [8],[9]. Problem set #2: maximal slices and stationary limit of 1+log slicing in Schwarzschild spacetime pdf file (Jan 4 2021)

Additional informations

• Students are eligible for free licence of Wolfram Mathematica -- details.
• Free jupyter environment is available e.g. at colab.research.google.com or cocalc.com

References

1. Baumgarte, Thomas W. ; Shapiro, Stuart L.: Numerical Relativity: Solving Einstein's Equations on the Computer, Cambridge University Press, 2010. ISBN: 9780521514071
2. Tolman, R.C., Static solutions of Einstein's field equations for spheres of fluid, Phy. Rev. 55 (1939) 364
3. Nikolaos Stergioulas, Rotating Stars in Relativity LivingRev.Rel.6:3,2003
4. Butterworth, E. M. ; Ipser, J. R., On the structure and stability of rapidly rotating fluid bodies in general relativity. I. The numerical method for computing structure and its application to uniformly rotating homogeneous bodies. Astrophysical Journal, vol. 204 (1976) pt. 1, p. 200
5. Friedman, J. L. ; Ipser, J. R. ; Parker, L. Rapidly Rotating Neutron Star Models, Astrophysical Journal v.304, p.115
6. Marcus Ansorg, David Petroff, Black Holes Surrounded by Uniformly Rotating Rings, Phys.Rev. D72 (2005) 024019.
7. Michael Pürrer, Sascha Husa, Peter C. Aichelburg, News from Critical Collapse: Bondi Mass, Tails and Quasinormal Modes, Phys. Rev. D71 (2005) 104005. arxiv, doi:10.1103/PhysRevD.71.104005
8. Thomas Baumgarte, Numerical Relativity: Solving Einstein's Equations on the ComputerNumerical Relativity: Solving Einstein's Equations on the Computer, Cambridge University Press, 2010.
9. Eric Gourgoulhon , 3+1 Formalism and Bases of Numerical Relativity, Springer 2012. arxiv

Materiály k přednášce

Course in 2020 (in English)

1. TOV eqaution of the stellar structure. Nonobligatory problem: Mathematica notebook or Jupyter notebook (open in browser).
2. Oppenheimer-Snyder model of spherical collapse. Nonobligatory problem: pdf file
3. Method of lines -- Problem set #1 MOLwave1.ipynb , MOLwave1.py (While e.g. cocalc s OK, the animation does not work in windows version of Spyder and google.colab; use static version there)
4. 3+1 decomposition. So-called ADM equations. (Basic facts, derivation is here in Prague taught in the second course on General relativity. Apart from Chapter 4 in Gourgoulhon's book, you can watch T. Baumagarte's nice lectures here and here)
5. Spacetime foliations -- Problem set #2: maximal slices and stationary limit of 1+log slicing in Schwarzschild spacetime pdf file (Jan 4 2021)
6. Asymptotic flatness, ADM Mass
7. The initial data problem, various slaces in Schwarzschild spacetime, punctures. Brill initial data for gravitatonal wave. Using spectral methods for elliptic initial data PDEs.
8. BSSN evolution scheme. Problem set #3: Strong hyperbolicity of the BSSN system pdf file (Dec 21 2020)
9. Finding the event and apparent horizons

2018 (in Czech)

• Obyčejné a parciální dif. rovnice (ukázka principu metody konečných diferencí a pseudospektrální metody a jejich chování pro neanalytická řešení pro metodu konečných diferencí a pseudospektrální).
• Poznámky s odvozením separovaných rovic pro testovací pole "se spinem" s=0 a s=1. Principy Cauchyovy úlohy. CFL faktor. (pdf).
• Metoda čar pro vlnovou rovnici (RK3+FD2 (pdf,), oblast stability RK1-RK4 (pdf)).
• Vývoj perturbace pro Regge-Whelerovu rovnici (RK3+FD2 (pdf)).
• Příklad statického a dynamického prostoročasu: Schwarzschilova černá díra a Oppenheimerův-Snyderům model gravitačního kolapsu. (Kromě učebnic (např. MTW) jsou "obrázky" těchto prostoročasů v bakalářských pracech J. Haláčka a L. Honsy).
• Další výklad podle Eric Gourgoulhon: 3+1 Formalism and Bases of Numerical Relativity kapitoly:
  3. Geometry of foliations,
  4. 3+1 decomposition of Einstein equation,
  7. Asymptotic flatness and global quantities,
  8. The initial data problem,
  9. Choice of foliation and spatial coordinates,
  10. Evolution schemes.