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I am not exactly sure if this is the right place to post this, so do comment if not.

Consider a universe similar to ours, filled with only a gas, distributed uniformly throughout it.

What is the lowest possible density $d$ of the gas, such that for any point $p$ in the universe, there is sufficient mass within a radius $r$, for $r$ to be (so to speak) $p$'s Schwarzschild radius?

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  • $\begingroup$ What's the formula for the Schwarzschild radius? What's the mass within a radius $r$? $\endgroup$
    – Kevin
    Commented Apr 4, 2018 at 0:26

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The answer is zero. Really.

The Schwarzschild radius of a mass $M$ black holes is $R=2GM/c^2$. So black holes have "density" (scare quotes because volume acts up a bit in curved spacetime, but we will ignore that right now) $$\rho = \frac{M}{4\pi R^3/3}=\frac{3 c^6}{32\pi G^3 M^2}.$$ So the bigger the mass, the lower the density of the black hole. So for any density you can name, I will be able to name a black hole mass or radius that give me a lower density.

The stability of the gas filled universe will depend on other things too, such as the expansion. The expansion factor does create an adjustment to the Schwartzchild radius for very large black holes.

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  • $\begingroup$ What prevents our universe from "being" a black hole, then? The fact that it is expanding due to dark energy? $\endgroup$
    – Ola
    Commented Apr 4, 2018 at 12:18
  • $\begingroup$ Well, maybe it is. If the density of the Universe is greater than the critical density (which is very close to the density described in the answer), then the Big Crunch will happen, and that means, no matter how fast we move in any direction, it is still on our future world line. Kinda sounds like life inside an event horizon. $\endgroup$
    – JEB
    Commented Apr 4, 2018 at 13:37
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    $\begingroup$ The cosmological constant further complicates things: it allows universes with more mass that yet do not collapse. The quick answer to Ola's comment is that black holes are concentrations of mass surrounded by largely empty space while cosmological models have the same density of stuff everywhere but have some global curvature. The kinds of singularities and overall topology are different (black holes: space is $R^3$ minus a point, times a time dimension; cosmology space is $S^3$ or $R^3$ times a time dimension, with at least a past spacelike singularity as a past & maybe a future boundary). $\endgroup$
    – Anders Sandberg
    Commented Apr 4, 2018 at 13:49

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