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Wikipedia lists the gravitational potential as $$V(\vec{x})=-\int_{\mathbb{R}^3} \frac{\rho(\vec{r})G}{|\vec{x}-\vec{r}|} dv(\vec{r}) $$ with $dv(\vec{r})$ the volume element, G the gravitational constant and $\rho$ the mass density. I wonder how and when the integral converges, in particular around $\vec{x}$ and towards infinity? What are required conditions for the mass density and how does this match a posssibly finite universe? In case of negative answers, does general relativity fare better?

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