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

4.5 ADM Hamiltonian formulation 65
field on Σ0 and p a vector field on Σ0, which obeys the constraint equations (4.65)-(4.66):
R + K 2 − KijK ij = 16πE (4.93)
DjK j
i − DiK = 8πpi, (4.94)
does there exist a spacetime (M,g,T) such that (g,T) fulfills the Einstein equation and Σ0 can
be embedded as an hypersurface of M with induced metric γ and extrinsic curvature K ?
Darmois (1927) [105] and Lichnerowicz (1939) [176] have shown that the answer is yes for
the vacuum case (E = 0 and pi = 0), when the initial data (γ,K) are analytical functions
of the coordinates (x i ) on Σ0. Their analysis is based on the Cauchy-Kovalevskaya theorem
mentioned in Sec. 4.4.2 (cf. Chap. 10 of Wald’s textbook [265] for details). However, on physical
grounds, the analytical case is too restricted. One would like to deal instead with smooth (i.e.
differentiable) initial data. There are at least two reasons for this:
• The smooth manifold structure of M imposes only that the change of coordinates are
differentiable, not necessarily analytical. Consequently if (γ,K) are analytical functions
of the coordinates, they might not be analytical functions of another coordinate system
(x ′i ).
• An analytical function is fully determined by its value and those of all its derivatives at a
single point. Equivalently an analytical function is fully determined by its value in some
small open domain D. This fits badly with causality requirements, because a small change
to the initial data, localized in a small region, should not change the whole solution at all
points of M. The change should take place only in the so-called domain of dependence of
D.
This is why the major breakthrough in the Cauchy problem of general relativity has been
achieved by Choquet-Bruhat in 1952 [127] when she showed existence and uniqueness of the
solution in a small neighbourhood of Σ0 for smooth (at least C 5 ) initial data (γ,K). We shall
not give any sketch on the proof (beside the original publication [127], see the review articles
[39] and [88]) but simply mentioned that it is based on harmonic coordinates.
A major improvement has been then the global existence and uniqueness theorem by Choquet-
Bruhat and Geroch (1969) [87]. The latter tells that among all the spacetimes (M,g) solution
of the Einstein equation and such that (Σ0,γ,K) is an embedded Cauchy surface, there exists
a maximal spacetime (M ∗ ,g ∗ ) and it is unique. Maximal means that any spacetime (M,g) solution
of the Cauchy problem is isometric to a subpart of (M ∗ ,g ∗ ). For more details about the
existence and uniqueness of solutions to the Cauchy problem, see the reviews by Choquet-Bruhat
and York [88], Klainerman and Nicolò [169], Andersson [15] and Rendall [212].
4.5 ADM Hamiltonian formulation
Further insight in the 3+1 Einstein equations is provided by the Hamiltonian formulation of
general relativity. Indeed the latter makes use of the 3+1 formalism, since any Hamiltonian
approach involves the concept of a physical state “at a certain time”, which is translated in
general relativity by the state on a spacelike hypersurface Σt. The Hamiltonian formulation