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

106 Asymptotic flatness and global quantities
respect to the coordinates (y i ) used on ∂V. The boundary term in (7.9) guarantees that the
variation of S with the values of g (and not its derivatives) held fixed at ∂V leads to the Einstein
equation. Otherwise, from the volume term alone (Hilbert action), one has to held fixed g and
all its derivatives at ∂V.
Let
St := ∂V ∩ Σt. (7.10)
We assume that St has the topology of a sphere. The gravitational Hamiltonian which can be
derived from the action (7.9) (see [205] for details) contains an additional boundary term with
respect to the Hamiltonian (4.112) obtained in Sec. 4.5 :
H = −
Σ int
t
NC0 − 2β i √ 3
Ci γd x − 2
St
N(κ − κ0) + βi(Kij − Kγij)s j √ q d 2 y, (7.11)
where Σint t is the part of Σt bounded by St, κ is the trace of the extrinsic curvature of St
embedded in (Σt,γ), and κ0 the trace of the extrinsic curvature of St embedded in (Σt,f) (f
being the metric introduced in Sec. 7.2), s is the unit normal to St in Σt, oriented towards the
asymptotic region, and √ q d2y denotes the surface element induced by the spacetime metric on
St, q being the induced metric, ya = (y1 ,y2 ) some coordinates on St [for instance ya = (θ,ϕ)]
and q := det(qab).
For solutions of Einstein equation, the constraints are satisfied: C0 = 0 and Ci = 0, so that
the value of the Hamiltonian reduces to
Hsolution = −2 N(κ − κ0) + β i (Kij − Kγij)s j √ 2
q d y. (7.12)
St
The total energy contained in the Σt is then defined as the numerical value of the Hamiltonian
for solutions, taken on a surface St at spatial infinity (i.e. for r → +∞) and for coordinates
(t,xi ) that could be associated with some asymptotically inertial observer, i.e. such that N = 1
and β = 0. From Eq. (7.12), we get (after restoration of some (16π) −1 factor)
MADM := − 1
8π lim
St→∞
(κ − κ0) √ q d 2 y . (7.13)
This energy is called the ADM mass of the slice Σt. By evaluating the extrinsic curvature
traces κ and κ0, it can be shown that Eq. (7.13) can be written
D j γij − Di(f kl
γkl) s i√ q d 2 y , (7.14)
MADM = 1
16π lim
St→∞
St
where D stands for the connection associated with the metric f and, as above, s i stands for the
components of unit normal to St within Σt and oriented towards the exterior of St. In particular,
if one uses the Cartesian-type coordinates (x i ) involved in the definition of asymptotic flatness
(Sec. 7.2), then Di = ∂/∂x i and f kl = δ kl and the above formula becomes
MADM = 1
16π lim
St→∞
St
St
∂γij ∂γjj
−
∂xj ∂xi
s i√ q d 2 y. (7.15)
Notice that thanks to the asymptotic flatness requirement (7.2), this integral takes a finite value:
the O(r 2 ) part of √ q d 2 y is compensated by the O(r −2 ) parts of ∂γij/∂x j and ∂γjj/∂x i .