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

54 3+1 decomposition of Einstein equation
(2) Full projection perpendicular to Σt
This amounts to applying the Einstein equation (4.1), which is an identity between bilinear
forms, to the couple (n,n); we get, since g(n,n) = −1,
4 14
R(n,n) + R = 8πT(n,n). (4.18)
2
Using the scalar Gauss equation (2.95), and noticing that T(n,n) = E [Eq. (4.3)] yields
R + K 2 − KijK ij = 16πE . (4.19)
This equation is called the Hamiltonian constraint. The word ‘constraint’ will be justified
in Sec. 4.4.3 and the qualifier ‘Hamiltonian’ in Sec. 4.5.2.
(3) Mixed projection
Finally, let us project the Einstein equation (4.1) once onto Σt and once along the normal n:
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R(n, γ(.)) − R g(n,γ(.)) = 8πT(n, γ(.)). (4.20)
2
=0
By means of the contracted Codazzi equation (2.103) and T(n,γ(.)) = −p [Eq. (4.4)], we get
or, in components,
D · K − DK = 8πp , (4.21)
DjK j
i − DiK = 8πpi . (4.22)
This equation is called the momentum constraint. Again, the word ‘constraint’ will be
justified in Sec. 4.4.
Summary
The Einstein equation is equivalent to the system of three equations: (4.15), (4.19) and (4.21).
Equation (4.15) is a rank 2 tensorial (bilinear forms) equation within Σt, involving only symmetric
tensors: it has therefore 6 independent components. Equation (4.19) is a scalar equation
and Eq. (4.21) is a rank 1 tensorial (linear forms) within Σt: it has therefore 3 independent
components. The total number of independent components is thus 6 + 1 + 3 = 10, i.e. the same
as the original Einstein equation (4.1).
4.2 Coordinates adapted to the foliation
4.2.1 Definition of the adapted coordinates
The system (4.15)+(4.19)+(4.21) is a system of tensorial equations. In order to transform it
into a system of partial differential equations (PDE), one must introduce coordinates on the