What does it mean for something to have an infinite curvature (like a black hole?)?
Black holes do not necessarily have to have infinite curvature, but enough mass to prevent light from escaping its gravitational pull. Infinite curvature would require an infinitely massive object; it's only a theoretical idea and is probably not possible. Einstein didn't believe that space time could rip, but only bend.
If it could exist, it would mean that the space-time continuum was bent to infinity. In case you don't know, the space time continuum can be thought of as a soft surface (such as a mattress), that curves when a mass is on it, and that gives us the impression of gravitational acceleration. It’s a pretty “sketchy” way of thinking about it, but it works.
I checked for the definition of curvature, thinking I could come up with a good answer. I found many definitions of curvature! So I am leaving this one alone
In GR, curvature referes to one of several "curvatuare tensors". The most fundamental of these is the Riemann curvature tensor - if you know the values of the components of the Riemann curvature tensor, you can calculate the values of the other tensors (most notably the Ricci, Einstein, and perhaps the Weyl).
A tensor is not just a single number, but an "array" of related numbers.
"Infinite" means that the value of at least one of the components of said tensor (remember, a tensor is not just a single number) increases without bound as one approaches a point, such as the singularity of a black hole. This is usually taken to imply that the theory itself breaks down and is not valid at that singular point.
Unfortunatly, there isn't any really super-simple way to describe what the Riemann curvature measures, though this is discussed in, for instance:
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