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Black holes are not infinitely dense because physical infinities do no exist, you are correct that it doesn't make sense and anyone claiming this is wrong.

The foundation of quantum mechanics is that the universe is quantum, which is to say, it is made up of a finite quantity of discrete "bits." An area of one Planck Length by one Planck Length contains one bit of information, either it contains mass or it doesn't. The concept of infinite density violates this by claiming that an infinite amount of information can be stored in an area smaller than that, which contradicts our understanding of quantum mechanics on a very fundamental level. The highest density something can have is to contain one bit of mass in every Planck area.

The reality is that black holes come from stars, with finite mass, and they occur when the force of gravity becomes so great that it overpowers the nuclear forces holding an atom together, causing electrons to collapse into the nucleus, and even for the subatomic particles to break down. However, the mass still occupies a discrete area, incredibly small but not infintismally so.

Now having said that, black holes are currently a controversial and unresolved subject because no consensus has formed around a satisfying resolution to what's called the Black Hole Information Paradox, which pits several fundamental principles against each other.

Information, in the context of physics, cannot be destroyed, because the destruction of information would mean reducing the complexity of the universe, reducing entropy and violating the second law of thermodynamics. What this means is that it's hypothetically possible, given sufficient knowledge of the physical world, to reconstruct everything that has ever happened. Every event that occurs leaves behind evidence, and that evidence can be jumbled up and garbled beyond recognition, but never completely erased from history.

For this to be true in the context of black holes, it is necessary for there to be some method of extracting information from a black hole - but there isn't. Or at least, nobody has been able to explain how that would work. Some have speculated that the information is emitted in Hawking radiation, but this doesn't really make any sense because of what Hawking radiation is. Others have speculated that the information could be stored on the event horizon, but this has been rejected as the event horizon is just a mathematical abstraction (this is sometimes referred to as a question of whether black holes have "hair," with most physicists agreeing that "black holes have no hair"). Various other hypotheses have been proposed over the decades, but all of them seem to contradict some principle that we have very good reasons for believing in, and we can't overturn any of those principles without solid evidence, which we don't have.

To restate the problem: if black holes can consume physical things that have high entropy, and reduce them to a state of low physical complexity, then they are capable of reducing entropy in a closed system, and our understanding thermodynamics is fundamentally wrong. If they are capable of storing that entropy in an infintesmally small area, then our understanding of quantum mechanics is fundamentally wrong. If it's possible to retrieve information from a black hole, then our understanding of relativity is fundamentally wrong. So either some fundamental principle in physics is wrong, or someone will eventually come up with some very clever solution that reconciles these competing principles.

The hardest part to wrap your head around is the fact that the interior quite literally does not exist from your perspective, nor will it ever. From our perspective outside of a black hole, everything that has ever fallen into a black hole is not actually inside it yet, it is actually squashed up against the surface impossibly thin, asymptotically approaching the radius we observe as the event horizon. Due to time dilation, there is in fact no black hole yet; just a collapsing star paused in time approaching infinitely close to the moment where its mass is contained within its Schwarzschild radius. We can't observe these objects squished up against that boundary because the light they emit gets red shifted into infinity, but that's where they are and will remain forever.

But from the perspective of an object falling in, nothing special happens at the event horizon. It observes the flow of time to be as normal, and it simply becomes causally disconnected from the entire rest of the universe as it enters a space that does not exist for us.

I'm not exactly sure how black hole evaporation works into this, though. Just when you think you got something about physics understood, you realize you actually don't.

When I was a kid an astrophysicist in my family explained black holes to me using a simplified model like this.

Imagine the fabric of space-time as a literal fabric, a trampoline. When you put a heavy object like a planet, or a bowling ball onto that trampoline it warps and distorts the shape of the surface. You can see how other objects will be attracted to that heavy object, how two heavy objects might be attracted to each other, and to some degree how an object might orbit around that heavy object, like one of those spinning coin funnels. If we imagine that bowling ball being much heavier, much denser, we can imagine how it will distort the fabric more and more, making its effect on other objects more dramatic, until eventually it is so dense and heavy that it tears through the fabric of that trampoline. Unlike the trampoline though, the fabric of space-time won't spring back when the ball tears through, and that heavy hole in the middle of the trampoline will continue to very dramatically affect other objects.

So it isn't that it has infinite density, but that it doesn't matter if it's denser than the point at which it tears a hole, it's still going to tear the hole either way. So it's more like the "speed limit" of density.

I'm not sure exactly how accurate that is, but to an 8-year-old Emma it made sense and was a lot less scary than the perception that black holes were some sort of a "glitch in The Matrix"