If a clock could survive in a black hole, would time stand still inside the black hole? What would space-time be like?

The answer is kind of boring since we do not have physics to describe the inside of black holes, the official answer is “we don’t know”. We can speculate given what happens right at the edge of the black hole though.

The first thing to state is: who (i.e., which observer) is asking? If the person asking is standing outside the black hole looking at a clock falling towards the event horizon, then the clock appears to move slower as it approaches the black hole… and actually, it would never fall inside the black hole because time gets infinitely slow at the event horizon. If the person asking is attached to the clock, then everything looks normal, time runs normally. Actually, general relativity says that the event horizon can be crossed and time would keep running exactly in the same way.

General relativity also says that the density of the black hole is infinite because its volume is formally zero – it is a singularity. For a static black hole, the singularity is a point, while for a rotating black hole, it is a ring, but still with zero volume. The thing is, quantum physics does not allow such precise numbers, so there must be something else actually happening there, but we still don’t have the physics to describe such a state.