In general relativity (GR), does time stop at the event horizon or in the central singularity of a black hole?

I was reading through this question on time and big bang, and @John Rennie's answer surprised me. In the immediate environment of a black hole, where does time stop ticking if one were to follow a 'watch' falling into a black hole?

1. 
   
   

   At the event horizon?

   
   


2. 
   

   In the central singularity?

   
   

If time stops at the event horizon, does the watch get stuck there, or does it keep falling in all the way to the singularity. Guess I know less than I thought.

Answer

If you're sitting outside the event horizon watching a clock fall in, you will never see the clock reach the event horizon. You will see the clock slow as it approaches the horizon and you'll see it running slower and slower. However there is no sense in which time stops at the event horizon. You can wait as long as you want, and you'll see the clock creep closer and closer, but time will continue for both you and the clock.

Now suppose you're holding the clock. Assuming you can survive the tidal forces you'll cross the point where the external observer thinks the event horizon is (you would see no horizon there) and you would hit the singularity in a finite time. The problem is that at the singularity the spacetime curvature becomes infinite and there is no way to calculate your path in spacetime past this point. This is known as geodesic incompleteness (annoyingly Wikipedia has no good article on this but Google "geodesic incompleteness" for lots of info on the subject). It's because there is no way to calculate your trajectory past the singularity that it is said (but not by me!) that spacetime stops there.