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

7.3.4 Positive energy theorem
7.3 ADM mass 111
Since the ADM mass represents the total energy of a gravitational system, it is important to show
that it is always positive, at least for “reasonable” models of matter (take ρ < 0 in Eq. (7.52)
and you will get MADM < 0 ...). If negative values of the energy would be possible, then a
gravitational system could decay to lower and lower values and thereby emit an unbounded
energy via gravitational radiation.
The positivity of the ADM mass has been hard to establish. The complete proof was eventually
given in 1981 by Schoen and Yau [220]. A simplified proof has been found shortly after
by Witten [270]. More precisely, Schoen, Yau and Witten have shown that if the matter content
of spacetime obeys the dominant energy condition, then MADM ≥ 0. Furthermore, MADM = 0
if and only if Σt is a hypersurface of Minkowski spacetime.
The dominant energy condition is the following requirement on the matter stress-energy
tensor T: for any timelike and future-directed vector v, the vector − T(v) defined by Eq. (2.11)
2 must be a future-directed timelike or null vector. If v is the 4-velocity of some observer, − T(v)
is the energy-momentum density 4-vector as measured by the observer and the dominant energy
condition means that this vector must be causal. In particular, the dominant energy condition
implies the weak energy condition, namely that for any timelike and future-directed vector v,
T(v,v) ≥ 0. If again v is the 4-velocity of some observer, the quantity T(v,v) is nothing but the
energy density as measured by that observer [cf. Eq. (4.3)], and the the weak energy condition
simply stipulates that this energy density must be non-negative. In short, the dominant energy
condition means that the matter energy must be positive and that it must not travel faster than
light.
The dominant energy condition is easily expressible in terms of the matter energy density E
and momentum density p, both measured by the Eulerian observer and introduced in Sec. 4.1.2.
Indeed, from the 3+1 split (4.10) of T, the energy-momentum density 4-vector relative to the
Eulerian observer is found to be
J := − T(n) = En + p. (7.53)
Then, since n · p = 0, J · J = −E 2 + p · p. Requiring that J is timelike or null means J · J ≤ 0
and that it is future-oriented amounts to E ≥ 0 (since n is itself future-oriented). Hence the
dominant energy condition is equivalent to the two conditions E 2 ≥ p · p and E ≥ 0. Since p is
always a spacelike vector, these two conditions are actually equivalent to the single requirement
This justifies the term dominant energy condition.
7.3.5 Constancy of the ADM mass
E ≥ p · p . (7.54)
Since the Hamiltonian H given by Eq. (7.11) depends on the configuration variables (γij,N,β i )
and their conjugate momenta (π ij ,π N = 0,π β = 0), but not explicitly on the time t, the
2 in index notation, − T (v) is the vector −T α µv µ