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

64 3+1 decomposition of Einstein equation
(4.89), (4.90) and (4.91), only the first one involves second-order time derivatives. Moreover the
sub-system (4.89) contains the same numbers of equations than unknowns (six) and it is in a
form tractable as a Cauchy problem, namely one could search for a solution, given some initial
data. More precisely, the sub-system (4.89) being of second order and in the form
∂2
γij
= Fij
∂t2 γkl, ∂γkl ∂γkl
∂xm, ∂t , ∂2γkl ∂xm∂xn
, (4.92)
the Cauchy problem amounts to finding a solution γij for t > 0 given the knowledge of γij
and ∂γij/∂t at t = 0, i.e. the values of γij and ∂γij/∂t on the hypersurface Σ0. Since Fij
is a analytical function 1 , we can invoke the Cauchy-Kovalevskaya theorem (see e.g. [100]) to
guarantee the existence and uniqueness of a solution γij in a neighbourhood of Σ0, for any initial
data (γij,∂γij/∂t) on Σ0 that are analytical functions of the coordinates (x i ).
The complication arises because of the extra equations (4.90) and (4.91), which must be
fulfilled to ensure that the metric g reconstructed from γij via Eq. (4.77) is indeed a solution of
Einstein equation. Equations (4.90) and (4.91), which cannot be put in the form such that the
Cauchy-Kovalevskaya theorem applies, constitute constraints for the Cauchy problem (4.89).
In particular one has to make sure that the initial data (γij,∂γij/∂t) on Σ0 satisfies these
constraints. A natural question which arises is then: suppose that we prepare initial data
(γij,∂γij/∂t) which satisfy the constraints (4.90)-(4.91) and that we get a solution of the Cauchy
problem (4.89) from these initial data, are the constraints satisfied by the solution for t > 0 ?
The answer is yes, thanks to the Bianchi identities, as we shall see in Sec. 10.3.2.
4.4.3 Constraint equations
The main conclusions of the above discussion remain valid for the general 3+1 Einstein system
as given by Eqs. (4.63)-(4.66): Eqs. (4.63)-(4.64) constitute a time evolution system tractable
as a Cauchy problem, whereas Eqs. (4.65)-(4.66) constitute constraints. This partly justifies the
names Hamiltonian constraint and momentum constraint given respectively to Eq. (4.65) and
to Eq. (4.66).
The existence of constraints is not specific to general relativity. For instance the Maxwell
equations for the electromagnetic field can be treated as a Cauchy problem subject to the
constraints D · B = 0 and D · E = ρ/ǫ0 (see Ref. [171] or Sec. 2.3 of Ref. [44] for details of the
electromagnetic analogy).
4.4.4 Existence and uniqueness of solutions to the Cauchy problem
In the general case of arbitrary lapse and shift, the time derivative ˙γij introduced in Sec. 4.4.2 has
to be replaced by the extrinsic curvature Kij, so that the initial data on a given hypersurface
Σ0 is (γ,K). The couple (γ,K) has to satisfy the constraint equations (4.65)-(4.66) on Σ0.
One may then ask the question: given a set (Σ0,γ,K,E,p), where Σ0 is a three-dimensional
manifold, γ a Riemannian metric on Σ0, K a symmetric bilinear form field on Σ0, E a scalar
1 it is polynomial in the derivatives of γkl and involves at most rational fractions in γkl (to get the inverse
metric γ kl