# Mass-Density relation in General Relativity

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?

• Komar mass formula – Slereah Oct 27 '18 at 23:41