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

52 3+1 decomposition of Einstein equation
Let us assume that the spacetime (M,g) is globally hyperbolic (cf. Sec. 3.2.1) and let be
(Σt)t∈R by a foliation of M by a family of spacelike hypersurfaces. The foundation of the 3+1
formalism amounts to projecting the Einstein equation (4.1) onto Σt and perpendicularly to Σt.
To this purpose let us first consider the 3+1 decomposition of the stress-energy tensor.
4.1.2 3+1 decomposition of the stress-energy tensor
From the very definition of a stress-energy tensor, the matter energy density as measured
by the Eulerian observer introduced in Sec. 3.3.3 is
E := T(n,n) . (4.3)
This follows from the fact that the 4-velocity of the Eulerian observer in the unit normal vector
n.
Similarly, also from the very definition of a stress-energy tensor, the matter momentum
density as measured by the Eulerian observer is the linear form
i.e. the linear form defined by
In components:
p := −T(n,γ(.)) , (4.4)
∀v ∈ Tp(M), 〈p,v〉 = −T(n,γ(v)). (4.5)
pα = −Tµν n µ γ ν α. (4.6)
Notice that, thanks to the projector γ, p is a linear form tangent to Σt.
Remark : The momentum density p is often denoted j. Here we reserve the latter for electric
current density.
Finally, still from the very definition of a stress-energy tensor, the matter stress tensor
as measured by the Eulerian observer is the bilinear form
or, in components,
S := γ ∗ T , (4.7)
Sαβ = Tµνγ µ αγ ν β
As for p, S is a tensor field tangent to Σt. Let us recall the physical interpretation of the stress
tensor S: given two spacelike unit vectors e and e ′ (possibly equal) in the rest frame of the
Eulerian observer (i.e. two unit vectors orthogonal to n), S(e,e ′ ) is the force in the direction e
acting on the unit surface whose normal is e ′ . Let us denote by S the trace of S with respect
to the metric γ (or equivalently with respect to the metric g):
(4.8)
S := γ ij Sij = g µν Sµν . (4.9)