Yes, maybe, no? In the classic model, stars collapse until they hit a zero volume because no other force can resist the gravity. Singularities have zero volume, and so have infinite density, but they don’t have to be a single point – rotating black holes are thought to have a ring shape. However! We can’t see past the event horizon, so everything we know has to be found out through theory or testing indirectly what we see outside the event horizon. Some string theorists also think that the black hole isn’t a singularity but is instead a “fuzzball” due to quantum effects. There’s still a lot to learn about physics, for sure.
“Singularity” really is a mathematical concept to describe what we see and expect in the universe. The same with “infinity”. Actually, some infinities can be larger than other infinities. In some sense, singularity means that we label a given phenomenon as something that we cannot fully understand. So, yes, every black hole has singularities mathematically speaking but whether this is really how nature works or just the limits of our present-day understanding, that remains to be seen.
Yes, according to our current mathematical understanding of them. But that doesn’t mean that it’s for certain. As we make more observations as scientists we update our theories. Because the singularity is something we can’t physically observe, we can only infer things from what we can see (the impact of the black hole on its surroundings outside the event horizon).
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