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I was thinking about energy-matter densities in General Relativity and I can't seem to find a reference that answers my question, so I was hoping someone could help me.

Imagine that I live in a space-time with the following geometry: $$ds^2=-c^2dt^2+A(r)dr^2+r^2(d\theta^2+\sin^2(\theta)d\varphi^2),$$ where $A(r)$ is some function of the coordinate $r$, and, in this space-time, I have a matter density, $\rho(r)$. I'm wondering: this matter density is measured by which observer? I mean, if, for example, $\rho(r)=\alpha e^{-r^2}$, this gaussian density would be measured by which observer? And if a person wanted to assembly an apparatus to have this matter density, what should it measure in it's proper frame of reference?

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The infinitesimal physical radial distance locally measured by an observer at $r=r_0$ is $dr_{physical}^2(r_0) = A(r_0)dr^2$. If $A(r) \rightarrow 1$ when $r\rightarrow +\infty$, this means that $dr$ is the infinitesimal radial distance as seen by an observer at infinity. –  Trimok Jun 30 '13 at 17:08
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