Oppenheimer-Snyder metric: how to integrate it to recover the total mass-energy? Weinberg recaps dustball collapse in comoving coordinates, where the energy-momentum tensor is
$$T^{\mu\nu}(t) = \rho(t)\left(\begin{array}{l}
1&0&0&0\\
0&0&0&0\\
0&0&0&0\\
0&0&0&0
\end{array}\right)$$
from which he derives the metric
$$
d\tau^2 = dt^2-Q^2(t)\left(\frac{1}{1-kr^2}dr^2 + r^2d\Omega^2\right)
$$
$r$ ranges from $0$ to $a$ throughout time, as we have comoving coordinates. The initial, uniform proper density is $\rho_0$, and increases with time:
$\rho(t)=\rho_0/Q^3(t)$.
$Q(t)$ shrinks from $1$ to $0$.
Now, with spherical symmetry, we can integrate the mass-energy of shells to get the total mass-energy.
It seems the integral should be
$$
m(t)=\int_0^a\frac{\rho_0}{Q^3}4\pi (Qr)^2\sqrt{\frac{Q^2}{1-kr^2}}\,dr=\frac{\rho_0}{Q^3}V_\textrm{proper}
$$
When I do this, then m(t) is constant in time, but the resulting $m$ is somewhat larger than the mass-energy $M$ that an external observer would measure, the $M$ in the Schwarzschild metric. When the ratio $r_0/r_S$ (initial radius over Schwarzschild radius) is large, the ratio $m/M$ approaches 1 from above.
How to make sense of this?
How to get total mass-energy $M$ from some integral of either the metric or $T^{\mu\nu}$?
 A: You're dealing with a spacetime that is asymptotically flat but not stationary. Under these conditions, there are two common measures of mass, the Bondi mass and the ADM mass. These two can differ in general if there is gravitational radiation, but the symmetry of the O-S spacetime guarantees that there can't be any radiation, so they are the same in this case.
The Bondi/ADM mass gives a measure of how much mass the object has according to a distant observer. The explicit expression is given by Wald, p. 293, as the limit of a certain integral over a sphere, as the radius approaches infinity. But the exterior metric of the O-S spacetime is the Schwarzschild metric. Therefore the integral you would be evaluating would be the one for the Schwarzschild metric. The only issue would be verifying that the external and internal metrics match properly, which is exercise 32.4 in Misner, Thorne, and Wheeler.
The result should be that the ADM/Bondi mass is the same as the parameter $M$ appearing in the O-S solution. The discrepancy you report is because you're not using a valid method for finding the mass. Conceptually, your method doesn't work because you're just trying to integrate the rest mass of all the matter fields, but there is also mass-energy stored in the gravitational field itself. Note that if you applied your method to the Schwarzschild metric, which is a vacuum solution, you'd get zero.
Normally we expect these measures of mass in an asymptotically flat spacetime to be conserved. However, that is not necessarily the case when there is a timelike singularity, as there is in the O-S metric. Once the singularity forms, the spacetime is no longer globally hyperbolic, so we can't even really define Cauchy surfaces in order to talk about "when" we're measuring the mass.
