I've learned from PBS Space Time videos that space rushes into a black hole's event horizon at the speed of light. I'm trying to square this with the analogy of stretched rubber sheet everyone uses; in that picture, the rubber doesn't move laterally. I'm wondering if maybe the rubber sheet actually represents spacetime, which doesn't "move" because time is just another dimension. If spacetime is curved in the time direction, maybe that appears to us as space moving over time.
Am I on the right track here, or way off base?
Thanks for the link to the question "Can gravity be interpreted as the acceleration of spacetime towards an object?" However, I believe that is different. Spacetime includes time, and therefore cannot be said to evolve over time. I'm not asking if spacetime accelerates; I'm asking about whether the space has velocity due to the curvature of spacetime.