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

Chapter 8
The initial data problem
Contents
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
8.2 Conformal transverse-traceless method . . . . . . . . . . . . . . . . . 127
8.3 Conformal thin sandwich method . . . . . . . . . . . . . . . . . . . . . 139
8.4 Initial data for binary systems . . . . . . . . . . . . . . . . . . . . . . 145
8.1 Introduction
8.1.1 The initial data problem
We have seen in Chap. 4 that thanks to the 3+1 decomposition, the resolution of Einstein
equation amounts to solving a Cauchy problem, namely to evolve “forward in time” some initial
data. However this is a Cauchy problem with constraints. This makes the set up of initial data
a non trivial task, because these data must obey the constraints. Actually one may distinguish
two problems:
• The mathematical problem: given some hypersurface Σ0, find a Riemannian metric γ,
a symmetric bilinear form K and some matter distribution (E,p) on Σ0 such that the
Hamiltonian constraint (4.65) and the momentum constraint (4.66) are satisfied:
R + K 2 − KijK ij = 16πE (8.1)
DjK j
i − DiK = 8πpi . (8.2)
In addition, the matter distribution (E,p) may have some constraints from its own. We
shall not discuss them here.
• The astrophysical problem: make sure that the solution to the constraint equations has
something to do with the physical system that one wish to study.