# The FRW universe is NOT asymptotically flat? Its mass?

The Friedman-Robertson-Walker (FRW) metric in the comoving coordinates $(t,r,\theta,\varphi)$ which describes a homogeneous and isotropic universe is $$ds^2\,= -dt^2+\frac{a(t)^2}{1-kr^2}\,dr^2 + a(t)^2 r^2\,\Big( d\theta^2+\sin^2 \!\theta \,d\varphi^2 \Big)$$ where $k$ is the curvature normalized into $\{-1\,,0\,,+1\}$ which refers to a closed, flat and open universe, respectively; and $a(t)$ is the scale factor.

My question is, this FRW metric is NOT asymptotically flat at spatial infinity $r\to+\infty$, isn't it? Thus, we can not calculate the so-called ADM mass (Arnowitt-Deser-Misner), right? If so, how to get the mass of the matter content from the metric?

Note: I do not mean the trivial $m=\rho V$, I mean the mass obtained from the FRW metric.

The matter/material content determines geometry/metric, and reversely the metric reflects the matter content. So I'm trying to recover the material mass (not including the gravitational energy) from the FRW metric.

• Why do you think there is a well defined mass? – MBN Aug 19 '14 at 11:15
• @MBN Thank you, MBN. Yes. I mean the mass of the matter content, not that of the gravitational field. – David Aug 19 '14 at 11:53
• Oh, I read your question again and had to delete my answer since it gave only the $\rho V$ answer. I think there is no good definition of what you would like. It would probably be possible only for one type of source, i.e. for radiation a larger universe would be more energetic, but for dark energy it would be less energetic. Etc. – Void Aug 19 '14 at 12:38
• @David I tried to think about the whole thing a bit more and decided to rewrite and undelete the answer. It is now as close to a complete answer as I could get. – Void Aug 19 '14 at 14:09

In the FLRW models, there is however no time-like Killing vector and thus no kind of conserved Killing energy or matter content. The only possibility of a preferred time-like vector is through the orthogonal vector to all the three space-like Killing vectors. This is the four-vector used for the definition of matter density $\rho$ and the trivial (using Einstein equations) $$m=\rho V = \int \frac{3}{8 \pi} \frac{\dot{a}^2 + k}{a^2} dV_{sp} \,.$$ Where $dV_{sp}$ is just the spatial part of the volume element.