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To hook in on your Space-Time curvature, that's a nice example of different observers observing different timelines. I you would be sucked in a black hole time would be running normally for you, but for an observer at some distance, you seem to go slower and slower if you get sucked in further, eventually 'stopping' to move when you approach the event horizon.
Something I don't think modern physics does is distinctuate between an observer timeline and the actual, physical timeline. A lot of the theories I see seem to say if you were to enter a black hole, time would move faster for you, and someone observing you time for you would slow down. Really, you are still experiencing the same amount of time, but you would be observing that time at a faster rate.I also think our perception is linked to how we perceive it. For instance, we perceive it primarily through light. Light being the fastest/most accurate form of measuring it, the closer we get to its speed, the slower time seems to become. Really, I think you will still experience the same amount of time, but your perception of it is affected.