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.

This post imported from StackExchange Physics at 2015-03-23 09:10 (UTC), posted by SE-user David