[Mentors' note: split off from this thread]
Do all black holes have the same density?
I am not sure what you mean by "density". The following may or may not be relevant. It certainly is not rigourous..
Density is a derived quantity, not a fundamental one, equal to mass divided by volume. The volume of a singularity is not defined, so neither is the density. It just isn't a useful concept for black holes.
Short answer: No, as you can see just by looking at the relationship between the mass and the Schwarzschild radius. Doubling the mass doubles the "radius" so increases the "volume" eight-fold, commensurately reducing the "density".
Longer answer: Those scare-quotes are in the short answer for a reason. The volume of a black hole, and hence its density, aren't especially well defined. The first and most obvious problem is that the mass is not distributed evenly inside the black hole; as far as GR will tell us (and there's no reason to believe this particular prediction) it is concentrated in a point of zero volume at the singularity in the "center" so there's no physical significance to any density you might calculate. Second, the Schwarzschild radius is not the length of a line segment between the central singularity and the event horizon (in fact, there is no such thing) so you can't just plug it into the formula for the volume of a sphere.
Usually when someone speaks of the "density" of a black hole, they mean the ratio of the mass to the volume of a hypothetical sphere whose surface area is equal to the surface area of the event horizon. And if that's what you mean by the density, you can go with the short answer above: the bigger the black hole, the lower the density.
But the utility of George's calculation is, for example, suppose some process carried away most of the angular momentum of the stars within our galaxy, so they inevitably start collapsing. Then, the calculation show that stars are inside the event horizon before they are touching (and in some sense, before 'whatever' happens at the center, assuming it is not a singularity). In fact if there were a cloud of dust with the mass of the milky way, that had low angular momentum and was collapsing, its [average] density just before the event horizon formed would be less than that of air.
I always thought they're all made of "black hole matter" with constant average density, like some exotic black diamond. Patently false.
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