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)

  • $\begingroup$ He is clearly assuming the spatial slices of the manifold have an end, as is the standard assumption in cosmology. On a compact manifold you can do whatever you want, but the noncompact case is the interesting one in GR, generally speaking. $\endgroup$
    – Ryan Unger
    Commented May 25, 2017 at 1:40
  • $\begingroup$ @ocelouvskyopoulo7 Weinberg states: "This is why it was so important to identify the coordinate system as quasi-Minkowskian; if $g_\mu\nu$ approached the metric of spherical polar coordinates at infinity" it would diverge (for his particular example). Weinberg was speaking of convergence over an infinite slice not a finite one. But yes, I'm interested in finite spaces myself $\endgroup$
    – R. Rankin
    Commented May 25, 2017 at 2:39

1 Answer 1


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.

  • $\begingroup$ @Bob_Bee Thanks Bob, I'm only interested in the integral quantities from over the whole slice. Would this apply to a background space of constant curvature? that is what i'm interested in: an isolated system on a background of constant positive curvature. maybe it's expanding or contracting, but I'm only looking at one moment of cosmological time. Any thoughts? $\endgroup$
    – R. Rankin
    Commented May 25, 2017 at 2:47
  • $\begingroup$ @ R. Rankin. Not sure. Consider the spatial slices at one time in FLRW cosmology. Constant curvature. Maybe open, maybe closed, depends on k. But asymmetrically in space at spatial infinity it still has the same curvature. And certainly not Minkowki. The spacetime curvature R is not 0 at infinity, I'm pretty sure. Can't remember the Einstein universe, you mean the static with cosmological constant? Maybe you can look at some of the pseudotensors, one might make sense. That article has some applications for some of them. Sorry. If I think of anything else I'll come back $\endgroup$
    – Bob Bee
    Commented May 25, 2017 at 4:42
  • $\begingroup$ @Bob_Bee Again thank you; I would hope however that for constant R>0 there is no spatial infinity, since this would correspond to the 3-sphere. I definitely hadn't planned on stereographically projecing it to the plane. the finite region of integration was why I asked the question, it seems there would be no issues of convergence. As weinberg states, the pseudotensor is lorentze covariant, so I should be able to choose the comoving coordinates and go from there integrating over the three sphere? $\endgroup$
    – R. Rankin
    Commented May 25, 2017 at 9:10
  • $\begingroup$ @R.Rankin The Einstein universe is constant positive curvature, hyper spherical as you imply, and static. R is >0 but note that R is not the actual curvature, which is k. Still, both positive and constant, with k =1. In general R in FLRW includes a term for k but also other terms, and it's k that determines the curvature. Thing about the Einstein universe is that it is unstable to just about any perturbation, in $\rho$ or $\Lambda$. See en.m.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric. So may not be the best thing to try. More $\endgroup$
    – Bob Bee
    Commented May 26, 2017 at 3:06
  • $\begingroup$ And you might integrate the pseudotensor and have it converge to a value, but so what? The universe is static how would you show it means anything? If you perturbative off it it is unstable so it won't help even having gravitational waves on it. You might also note that there a a lot of results on pseudotensors and when they are equivalent or not, and in my view there's been very little clarity on it. Just google Weinberg pseudotensors and you'll see a variety of papers comparing many of the pseudotensors in different cases. Not sure if any were for closed spacetimes, worth checking. $\endgroup$
    – Bob Bee
    Commented May 26, 2017 at 3:32

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