general relativity - Using the gravitational pseudotensor on a finite space

I get that the gravitational pseudotensor is generally only used for asymptotically flat spaces (aka quasi-Minkowski). In these cases, conserved total energy momenta can often be found for some system (and the various pseudotensors out there tend to agree with one another).

I recently came across Weinberg (Gravitation and Cosmology p.167) who states the reason for choosing quasi-Minkowski coordinates is to ensure convergence of the integral for energy and momenta. If this is the sole reason, then can't I utilize the gravitational pseudotensor in a finite space? (The Einstein universe being the simplest example I can think of)

Answer

Even in a compact or finite space there is not a well defined and definitive pseudotensor that has no problems. That's known as the quasi-local mass issue, and it is not settled, from anything I've seen except that in special cases a number of proposed pseudo-tensors give the same answer. But not in all cases or in general.

See a relatively recent review of the status and different versions of that at http://link.springer.com/article/10.12942/lrr-2009-4 by Szabados. But it doesn't give any easy general conclusions.

The different mass (some people call it energy, but more accurately in papers they label mass what should be invariant, and sometimes distinguish with energy what should be part of a 4 vector) definitions like ADM and Bondi masses work ok in asymptotically flat spacetimes, but it's always the global mass, not something local or pseudo-local or a density. They do get conserved, and one can use it to compute masses, and energies, where the mass energy distribution is isolated (does not extend to infinity), and when you are far enough. So it works ok for the gravitational radiation from black holes, but not well at all for the mass (or mass energy) in an expanding universe, even for local regions (the redshift causes energy loss, the cosmological energy causes gains). The psueudopseudo-local masses like that of Hawking and others also have their problems.

And as @Rankin correctly stated in his comment, Weinberg was not talking about finite spaces, more about using Minkowski like coordinates at infinity, where instead if you use spherical coordinates you get infinity, so there's nothing covariant about them even in these known cases.