3+1 formalism and bases of numerical relativity - LUTh ...

Chapter 4
3+1 decomposition of Einstein
equation
Contents
4.1 Einstein equation in 3+1 form . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Coordinates adapted to the foliation . . . . . . . . . . . . . . . . . . . 54
4.3 3+1 Einstein equation as a PDE system . . . . . . . . . . . . . . . . . 59
4.4 The Cauchy problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 ADM Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Einstein equation in 3+1 form
4.1.1 The Einstein equation
After the first two chapters devoted to the geometry of hypersurfaces and foliations, we are now
back to physics: we consider a spacetime (M,g) such that g obeys to the Einstein equation
(with zero cosmological constant):
4 14
R − R g = 8πT , (4.1)
2
where 4R is the Ricci tensor associated with g [cf. Eq. (2.16)], 4R the corresponding Ricci scalar,
and T is the matter stress-energy tensor.
We shall also use the equivalent form
4
R = 8π T − 1
T g , (4.2)
2
where T := g µν Tµν stands for the trace (with respect to g) of the stress-energy tensor T.