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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.

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    $\begingroup$ Why do you think there is a well defined mass? $\endgroup$ – MBN Aug 19 '14 at 11:15
  • $\begingroup$ @MBN Thank you, MBN. Yes. I mean the mass of the matter content, not that of the gravitational field. $\endgroup$ – David Aug 19 '14 at 11:53
  • $\begingroup$ 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. $\endgroup$ – Void Aug 19 '14 at 12:38
  • $\begingroup$ @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. $\endgroup$ – Void Aug 19 '14 at 14:09
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The question is what do we need the matter content of the universe for. As I understand it, in the usual case we want to find the conserved quantity associated with a certain conserved current gained by the projection of the energy-momentum tensor into a Killing vector, as for example in the paper by Abott and Deser.

The requirement of asymptotical behaviour seems to be connected only with the elimination of boundary terms, i.e. of the flux of the spatial part of the 4-vector current "through infinity". The integration of the divergence of the four-vector over the whole space-slice then gives us a time conservation of the space-slice sum of the zeroth component of the 4-vector.

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.

But by definition, there cannot be a purely matter-based conserved energy and thus and ADM-sense matter content. I also believe that the main problem with this kind of generalization is also the fact that the ADM mass is designed to portray the energy content of an "isolated" space-time or at least a space-time with a certain dominating background. A cosmological situation defines such a dominant background and thus the notion of ADM mass should not be expected to hold up.

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  • $\begingroup$ Great! Thank you very very very much for the detailed explaination! It is very helpful for me. $\endgroup$ – David Aug 19 '14 at 14:16

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