Eric
GOURGOULHON (LUTH, CNRS / Observatoire de Paris)
3+1
Formalism and Numerical Relativity
6
lectures: 19, 23, 24, 25, 26, 27 October 2006
Lecture notes (also available as arxiv.org/abs/gr-qc/0703035)
Lecture 1 :
Thursday 19 October, 10 am - 1 pm :
1. Geometry
of hypersurfaces
Hypersurface embedded in spacetime; push-forward and pull-back mappings
First fundamental form, intrinsic curvature
Weingarten map, second fundamental form, extrinsic curvature
Examples: surfaces in R^{3}
Spacelike hypersurfaces: orthogonal projector, link between the induced and the ambient connections
3+1 decomposition of the Riemann tensor: Gauss-Codazzi relations
2. Geometry of foliations
Globally hyperbolic spacetimes and foliations by spacelike hypersurfaces;
Foliation kinematics: lapse function, normal evolution vector, Eulerian observers
Last part of the 3+1 decomposition of the Riemann tensor
Lecture 2 :
Monday 23 October, 10 am - 1 pm :
3. 3+1
decomposition of Einstein equation
Einstein equation in 3+1 form: constraint equations and dynamical equations
Coordinates adapted to the foliation; the shift vector
Rewriting of the 3+1 equations as a PDE system
The Cauchy problem; known existence and uniqueness theorems
Link with the ADM formulation
4. 3+1 form of the matter equations
3+1 writing of the Maxwell equations
3+1 writing of the magneto-hydrodynamical equations
Lecture 3 :
Tuesday 24 October, 10 am - 1 pm :
5.
Global quantities
ADM mass and momentum
Angular momentum
Komar mass for stationary spacetimes
Komar angular momentum for axisymmetric spacetimes
Quasi-local definitions of mass and energy: Hawking mass, Brown-York energy, etc...
6. Conformal decomposition
York-Lichnerowicz conformal decomposition
Conformal writing of the 3+1 system
Aymptotically isotropic gauges
Expression of the ADM mass and angular momentum
Lecture 4 :
Wednesday 25 October, 10 am - 1 pm :
7.
Initial data problem
Overview of methods for solving the constraint equations
Conformal transverse traceless method
Conformal thin sandwich method
Compact binaries in quasi-circular orbits
Lecture 5 :
Thursday 26 October, 10 am - 1 pm :
8.
Choice of coordinates
Constant mean curvature and maximal slicing
Minimal distortion gauge
Dirac gauge
Hyperbolic gauges
9. Treatment of black holes
Excision method
Boundary conditions from the dynamical horizon formalism
Puncture method
Lecture 6 :
Friday 27 October, 10 am - 1 pm :
10.
Numerical techniques to solve the 3+1 equations
Finite differences
Spectral methods
11. Review of numerical results
Gravitational collapse
Binary neutron star merger
Binary black hole merger