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

110 Asymptotic flatness and global quantities
Example : Let us return to the example considered in Sec. 6.2.3, namely Schwarzschild spacetime
in isotropic coordinates (t,r,θ,ϕ) 1 . The conformal factor was found to be Ψ =
1 + m/(2r) [Eq. (6.25)] and the conformal metric to be ˜γ = f. Then Dj˜γij = 0 and only
the first term remains in the integral (7.45):
with
so that we get
MADM = − 1
2π lim
r→∞
∂Ψ
∂r
r=const
∂
= 1 +
∂r
m
2r
∂Ψ
∂r r2 sin θ dθ dϕ, (7.46)
= − m
2r 2,
(7.47)
MADM = m, (7.48)
i.e. we recover the result (7.31), which was obtained by means of different coordinates
(Schwarzschild coordinates).
7.3.3 Newtonian limit
To check that at the Newtonian limit, the ADM mass reduces to the usual definition of mass,
let us consider the weak field metric given by Eq. (5.14). We have found in Sec. 6.2.3 that the
corresponding conformal metric is ˜γ = f and the conformal factor Ψ = 1 − Φ/2 [Eq. (6.26)],
where Φ reduces to the gravitational potential at the Newtonian limit. Accordingly, D j ˜γij = 0
and DiΨ = − 1
2 DiΦ, so that Eq. (7.45) becomes
MADM = 1
4π lim
St→∞
St
s i DiΦ √ q d 2 y. (7.49)
To take Newtonian limit, we may assume that Σt has the topology of R 3 and transform the
above surface integral to a volume one by means of the Gauss-Ostrogradsky theorem:
MADM = 1
4π
DiD i Φ f d 3 x. (7.50)
Now, at the Newtonian limit, Φ is a solution of the Poisson equation
Σt
DiD i Φ = 4πρ, (7.51)
where ρ is the mass density (remember we are using units in which Newton’s gravitational
constant G is unity). Hence Eq. (7.50) becomes
MADM = ρ f d 3 x, (7.52)
Σt
which shows that at the Newtonian limit, the ADM mass is nothing but the total mass of the
considered system.
1 although we use the same symbol, the r used here is different from the Schwarzschild coordinate r of the
example in Sec. 7.3.1.