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Suppose one has a static and spherically symmetric spacetime with line element defined by: $$ds^{2}=-c^{2}e^{\nu(r)}dt^{2}+e^{-\nu(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$$ where $\nu(r)$ is the metric coefficient defined by the Einstein equations in terms of the density $\rho(r)$: $$\rho(r)=\dfrac{-c^{2}}{8\pi G r^{2}}\dfrac{d}{dr}\left(r(e^{\nu(r)}-1)\right).$$

From this density: How could one read the total mass enclosed in a sphere of radius $r$? By the usual formula $$M(r)=4\pi\int_{0}^{r}\rho(s)s^{2}ds\;?$$ Since there is a metric factor $e^{-\nu(r)}$ modifying the radial distance: Does it enter in some way into the mass-density relation?

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  • $\begingroup$ Komar mass formula $\endgroup$ – Slereah Oct 27 '18 at 23:41

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