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

104 Asymptotic flatness and global quantities
7.2.1 Definition
We consider a globally hyperbolic spacetime (M,g) foliated by a family (Σt)t∈R of spacelike
hypersurfaces. Let γ and K be respectively the induced metric and extrinsic curvature of the
hypersurfaces Σt. One says that the spacetime is asymptotically flat iff there exists, on each
slice Σt, a Riemannian “background” metric f such that [276, 277, 251]
• f is flat (Riem(f) = 0), except possibly on a compact domain B of Σt (the “strong field
region”);
• there exists a coordinate system (x i ) = (x,y,z) on Σt such that outside B, the components
of f are fij = diag(1,1,1) (“Cartesian-type coordinates”) and the variable r :=
x 2 + y 2 + z 2 can take arbitrarily large values on Σt;
• when r → +∞, the components of γ with respect to the coordinates (x i ) satisfy
γij = fij + O(r −1 ), (7.1)
∂γij
∂x k = O(r−2 ); (7.2)
• when r → +∞, the components of K with respect to the coordinates (x i ) satisfy
The “region” r → +∞ is called spatial infinity and is denoted i 0 .
Kij = O(r −2 ), (7.3)
∂Kij
∂x k = O(r−3 ). (7.4)
Remark : There exist other definitions of asymptotic flatness which are not based on any coordinate
system nor background flat metric (see e.g. Ref. [24] or Chap. 11 in Wald’s textbook
[265]). In particular, the spatial infinity i 0 can be rigorously defined as a single point in
some “extended” spacetime ( ˆ M, ˆg) in which (M,g) can be embedded with g conformal to
ˆg. However the present definition is perfectly adequate for our purposes.
Remark : The requirement (7.2) excludes the presence of gravitational waves at spatial infinity.
Indeed for gravitational waves propagating in the radial direction:
This fulfills condition (7.1) but
γij = fij + Fij(t − r)
r
∂γij
∂xk = −F ′ ij(t − r)
r
x k
r − Fij(t − r)
r2 + O(r −2 ). (7.5)
x k
r + O(r−2 ) (7.6)
is O(r −1 ) since F ′ ij = 0 (otherwise Fij would be a constant function and there would be no
radiation). This violates condition (7.2). Notice that the absence of gravitational waves
at spatial infinity is not a serious physical restriction, since one may consider that any
isolated system has started to emit gravitational waves at a finite time “in the past” and
that these waves have not reached the spatial infinity yet.