[pdonis]

> Why not?

Because "volume" is a geometric concept, not a physical thing. It has a geometric center, but not a center of mass. What you appear to be thinking of when you talk about calculating the center is the geometric center, not the center of mass. (And note that, unless an actual physical system has a high degree of symmetry, the geometric center defined by its spatial volume will not be the same as its physical center of mass.)

> Is the expansion of the universe directly deducible from GR?

The fact that the spacetime describing the universe cannot be stationary (i.e., that it must be either expanding or contracting) is deducible from the original 1915 Einstein Field Equation (i.e., without a cosmological constant) plus the assumptions of homogeneity and isotropy--roughly speaking, that the universe looks the same at all spatial locations and in all directions. We then pick out the "expanding" option as the one describing our actual universe based on observations.

Einstein actually discovered this in 1917, and he was bothered by it, because he believed (as did most physicists and astronomers at that time) that the universe was static--that it did not change with time on large scales. So he added the cosmological constant to his field equation to allow it to have a static solution that could describe a homogeneous and isotropic universe. Then, about ten years later, when evidence began to mount for the expansion of the universe, Einstein called adding the cosmological constant "the biggest blunder of my life"--because if he had trusted his original field equation, he could have predicted the expansion of the universe a decade before it was discovered.

Today, we believe that there is in fact a nonzero cosmological constant (our best current value for it is small and positive), but we also understand, what Einstein did not explore very thoroughly, that the Einstein Static Universe is an unstable solution, like a pencil balanced on its point: any small perturbation will cause it to either expand forever or collapse to a Big Crunch. So this solution is not considered a viable candidate to describe our actual universe. And we also know that there are no other static solutions that describe a homogeneous and isotropic universe.

[wyager]

> What you appear to be thinking of when you talk about calculating the center is the geometric center, not the center of mass.

No, I mean something along the lines of $Integral_V x*m(x) dx / Integral_V m(x)dx$ where $m$ is the mass-energy density function. The usual way of finding the center-of-momentum frame of a system that people mean when they say "invariant mass".

[pdonis]

In cases where the integral you describe is well-defined and physically meaningful, yes, you are correct, it is a center of mass (or center of momentum) integral not a geometric center. But it's not the center of mass of the "volume" over which the integral is done, it's the center of mass of the stress-energy over which the integral is done. In order to obtain the function m(x), you need to look at the stress-energy tensor.

Also, the integral you describe will not, in general, be invariant; it will depend on your choice of coordinates, because you are integrating over a spacelike surface of constant coordinate time, and which surfaces those are depends on your choice of coordinates.

Your intuition about the "usual" invariant mass is based on the special cases where the integral you describe can be equated to one of the known invariants, the ADM mass, the Bondi mass, or the Komar mass. (Strictly speaking, even the Komar case is problematic, because the integral in question in a general stationary spacetime does not necessarily converge. In cases where it does converge, AFAIK the spacetime must be asymptotically flat and the Komar mass is equal to the ADM mass.) But an expanding universe is not one of those cases.