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

4.4 The Cauchy problem 61
PDE system for the unknowns (γij,Kij,N,β i ). It has been first derived by Darmois, as early as
1927 [105], in the special case N = 1 and β = 0 (Gaussian normal coordinates, to be discussed
in Sec. 4.4.2). The case N = 1, but still with β = 0, has been obtained by Lichnerowicz in
1939 [176, 177] and the general case (arbitrary lapse and shift) by Choquet-Bruhat in 1948
[126, 128]. A slightly different form, with Kij replaced by the “momentum conjugate to γij”,
namely π ij := √ γ(Kγ ij − K ij ), has been derived by Arnowitt, Deser and Misner (1962) [23]
from their Hamiltonian formulation of general relativity (to be discussed in Sec. 4.5).
Remark : In the numerical relativity literature, the 3+1 Einstein equations (4.63)-(4.66) are
sometimes called the “ADM equations”, in reference of the above mentioned work by
Arnowitt, Deser and Misner [23]. However, the major contribution of ADM is an Hamiltonian
formulation of general relativity (which we will discuss succinctly in Sec. 4.5). This
Hamiltonian approach is not used in numerical relativity, which proceeds by integrating the
system (4.63)-(4.66). The latter was known before ADM work. In particular, the recognition
of the extrinsic curvature K as a fundamental 3+1 variable was already achieved
by Darmois in 1927 [105]. Moreoever, as stressed by York [279] (see also Ref. [12]),
Eq. (4.64) is the spatial projection of the spacetime Ricci tensor [i.e. is derived from the
Einstein equation in the form (4.2), cf. Sec. 4.1.3] whereas the dynamical equation in the
ADM work [23] is instead the spatial projection of the Einstein tensor [i.e. is derived from
the Einstein equation in the form (4.1)].
4.4 The Cauchy problem
4.4.1 General relativity as a three-dimensional dynamical system
The system (4.63)-(4.74) involves only three-dimensional quantities, i.e. tensor fields defined
on the hypersurface Σt, and their time derivatives. Consequently one may forget about the
four-dimensional origin of the system and consider that (4.63)-(4.74) describes time evolving
tensor fields on a single three-dimensional manifold Σ, without any reference to some ambient
four-dimensional spacetime. This constitutes the geometrodynamics point of view developed by
Wheeler [267] (see also Fischer and Marsden [122, 123] for a more formal treatment).
It is to be noticed that the system (4.63)-(4.74) does not contain any time derivative of
the lapse function N nor of the shift vector β. This means that N and β are not dynamical
variables. This should not be surprising if one remembers that they are associated with the
choice of coordinates (t,x i ) (cf. Sec. 4.2.4). Actually the coordinate freedom of general relativity
implies that we may choose the lapse and shift freely, without changing the physical solution
g of the Einstein equation. The only things to avoid are coordinate singularities, to which a
arbitrary choice of lapse and shift might lead.
4.4.2 Analysis within Gaussian normal coordinates
To gain some insight in the nature of the system (4.63)-(4.74), let us simplify it by using the
freedom in the choice of lapse and shift: we set
N = 1 (4.75)