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I understand that the typical density of the super massive black hole is close that of the water. It is also my understanding that this density is not true matter density because the volume used to calculate the density is measured from event horizon (so I suppose this is rather potential gravitational energy density?).

If it was the case that the universe were to collapse into a super massive black hole, can we still assume that it would be close to that of the water? If so, why, and if not, how would you calculate the density? Note that the equations using the Shwarzschild Radius don't work in this case.

Luis
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The density of a black hole is defined simply as the mass within the event horizon divided by the volume within the event horizon. This gives an average density, but doesn't imply that the density is uniform within the event horizon. So when you hear statements like the density of a supermassive black hole is the same as water don't take this too literally.

I suppose it has some meaning in that the average density can be related to the curvature at the event horizon, and therefore to the tidal forces you would experience if you jumped in. Even so I doubt anyone outside the popular science media would attach much significance to the density of a black hole.

If you're interested, the Schwarzschild radius is given by:

$$ r_s = \frac{2GM}{c^2} $$

and if you take the volume to be $4/3 \pi r_s^3$ that gives an average density of:

$$ \rho = \frac{3c^2}{8\pi Gr_s^2} $$

Set this equal to 1000 kg per cubic metre (density of water) and you get a radius of about $4 \times 10^{11}$ and this corresponds to a black hole mass of about $10^8$ solar masses, which I guess counts as supermassive. Note however that the average density is proportional to $r_s^{-2}$, which is proportional to $m^{-2}$, so if you took a larger mass like the whole observable universe the average density of the corresponding black hole would be lower than water - vastly lower in fact.

Note that I referred to the observable universe in my comments above. Your question just refers to the universe but the universe as a whole can't collapse into a black hole. In a black hole the event horizon divides the interior of the black hole from the rest of the universe. But the whole universe can't form a black hole because there is nothing outside it for the event horizon to divide it from.

The universe as a whole does have an average density, and you can calculate this using the FLRW metric. If you wind time back towards the Big Bang the average density of the universe increases and at some point it would be the same density as water. The universe seems unlike to recollapse in the future, so in the future its average density is just going to keep decreasing.

John Rennie
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