3+1 Formalism in General Relativity

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3+1 formalism in general relativity
Eric Gourgoulhon
Laboratoire Univers et Th´eories (LUTH)
CNRS / Observatoire de Paris / Universit´e Paris Diderot
F-92190 Meudon, France
eric.gourgoulhon@obspm.fr
http://www.luth.obspm.fr/
∼
luthier/gourgoulhon/
2008 International Summer School on
Computational Methods in Gravitation and Astrophysics
Asia Pacific Center for Theoretical Physics, Pohang, Korea
28 July - 1 August 2008
Eric Gourgoulhon (LUTH) 3+1 formalism in general relativity APCTP School, 30 July 2008 1 / 34

Plan
1
The 3+1 foliation of spacetime
2
3+1 decomposition of Einstein equation
3
The Cauchy problem
4
Conformal decomposition
Eric Gourgoulhon (LUTH) 3+1 formalism in general relativity APCTP School, 30 July 2008 2 / 34

The 3+1 foliation of spacetime
Outline
1
The 3+1 foliation of spacetime
2
3+1 decomposition of Einstein equation
3
The Cauchy problem
4
Conformal decomposition
Eric Gourgoulhon (LUTH) 3+1 formalism in general relativity APCTP School, 30 July 2008 3 / 34

The 3+1 foliation of spacetime
Framework: globally hyperbolic spacetimes
4-dimensional spacetime (M , g) :
M : 4-dimensional smooth manifold
g: Lorentzian metric on M :
sign(g) = (−, +, +, +)
(M , g) is assumed to be time
orientable: the light cones of g can be
divided continuously over M in two sets
(past and future)
The spacetime (M , g) is assumed to be
globally hyperbolic: ∃ a foliation (or
slicing) of the spacetime manifold M by
a family of spacelike hypersurfaces Σ
t
:
M =
[
t∈R
Σ
t
hypersurface = submanifold of M of
dimension 3
Eric Gourgoulhon (LUTH) 3+1 formalism in general relativity APCTP School, 30 July 2008 4 / 34

The 3+1 foliation of spacetime
Framework: globally hyperbolic spacetimes
4-dimensional spacetime (M , g) :
M : 4-dimensional smooth manifold
g: Lorentzian metric on M :
sign(g) = (−, +, +, +)
(M , g) is assumed to be time
orientable: the light cones of g can be
divided continuously over M in two sets
(past and future)
The spacetime (M , g) is assumed to be
globally hyperbolic: ∃ a foliation (or
slicing) of the spacetime manifold M by
a family of spacelike hypersurfaces Σ
t
:
M =
[
t∈R
Σ
t
hypersurface = submanifold of M of
dimension 3
Eric Gourgoulhon (LUTH) 3+1 formalism in general relativity APCTP School, 30 July 2008 4 / 34