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

7.2.2 Asymptotic coordinate freedom
7.3 ADM mass 105
Obviously the above definition of asymptotic flatness depends both on the foliation (Σt)t∈R and
on the coordinates (x i ) chosen on each leaf Σt. It is of course important to assess whether this
dependence is strong or not. In other words, we would like to determine the class of coordinate
changes (x α ) = (t,x i ) → (x ′α ) = (t ′ ,x ′i ) which preserve the asymptotic properties (7.1)-(7.4).
The answer is that the coordinates (x ′α ) must be related to the coordinates (x α ) by [157]
x ′α = Λ α µx µ + c α (θ,ϕ) + O(r −1 ) (7.7)
where Λ α β is a Lorentz matrix and the cα ’s are four functions of the angles (θ,ϕ) related to the
coordinates (x i ) = (x,y,z) by the standard formulæ:
x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ. (7.8)
The group of transformations generated by (7.7) is related to the Spi group (for Spatial infinity)
introduced by Ashtekar and Hansen [25, 24]. However the precise relation is not clear because
the definition of asymptotic flatness used by these authors is not expressed as decay conditions
for γij and Kij, as in Eqs. (7.1)-(7.4).
Notice that Poincaré transformations are contained in transformation group defined
by (7.7): they simply correspond to the case cα (θ,ϕ) = const. The transformations with
cα (θ,ϕ) = const and Λα β = δα β constitute “angle-dependent translations” and are called supertranslations.
involves a boost, the transformation (7.7) implies a
Note that if the Lorentz matrix Λα β
change of the 3+1 foliation (Σt)t∈R, whereas if Λα β corresponds only to some spatial rotation
and the cα ’s are constant, the transformation (7.7) describes some change of Cartesian-type
coordinates (xi ) (rotation + translation) within the same hypersurface Σt.
7.3 ADM mass
7.3.1 Definition from the Hamiltonian formulation of GR
In the short introduction to the Hamiltonian formulation of general relativity given in Sec. 4.5,
we have for simplicity discarded any boundary term in the action. However, because the gravitational
Lagrangian density (the scalar curvature 4R) contains second order derivatives of the
metric tensor (and not only first order ones, which is a particularity of general relativity with
respect to other field theories), the precise action should be [209, 205, 265, 157]
4 √ 4
S = R −g d x + 2 (Y − Y0) √ h d 3 y, (7.9)
V
where ∂V is the boundary of the domain V (∂V is assumed to be a timelike hypersurface), Y the
trace of the extrinsic curvature (i.e. three times the mean curvature) of ∂V embedded in (M,g)
and Y0 the trace of the extrinsic curvature of ∂V embedded in (M,η), where η is a Lorentzian
metric on M which is flat in the region of ∂V. Finally √ hd 3 y is the volume element induced
by g on the hypersurface ∂V, h being the induced metric on ∂V and h its determinant with
∂V