Relative speeds in GR are also only unambiguously defined locally, and if you're at the event horizon you are necessarily falling in; it comes at you at the speed of light.
Light beams aimed directly outward from just outside the horizon don't escape to large distances until late values of t. For someone at a large distance from the black hole and approximately at rest with respect to it, the coordinate t does correspond well to proper time.
So if you, watching from a safe distance, attempt to witness my fall into the hole, you'll see me fall more and more slowly as the light delay increases. You'll never see me actually get to the event horizon.
My watch, to you, will tick more and more slowly, but will never reach the time that I see as I fall into the black hole. Notice that this is really an optical effect caused by the paths of the light rays. This is also true for the dying star itself. If you attempt to witness the black hole's formation, you'll see the star collapse more and more slowly, never precisely reaching the Schwarzschild radius.
Now, this led early on to an image of a black hole as a strange sort of suspended-animation object, a "frozen star" with immobilized falling debris and gedankenexperiment astronauts hanging above it in eternally slowing precipitation. This is, however, not what you'd see. The reason is that as things get closer to the event horizon, they also get dimmer. Light from them is redshifted and dimmed, and if one considers that light is actually made up of discrete photons, the time of escape of the last photon is actually finite, and not very large.
So things would wink out as they got close, including the dying star, and the name "black hole" is justified. As an example, take the eight-solar-mass black hole I mentioned before. If you start timing from the moment the you see the object half a Schwarzschild radius away from the event horizon, the light will dim exponentially from that point on with a characteristic time of about 0.
The times scale proportionally to the mass of the black hole. If I jump into a black hole, I don't remain visible for long. Also, if I jump in, I won't hit the surface of the "frozen star.
Some have pointed out that I really go through the event horizon a little earlier than a naive calculation would imply. The reason is that my addition to the black hole increases its mass, and therefore moves the event horizon out around me at finite Schwarzschild t coordinate. This really doesn't change the situation with regard to whether an external observer sees me go through, since the event horizon is still lightlike; light emitted at the event horizon or within it will never escape to large distances, and light emitted just outside it will take a long time to get to an observer, timed, say, from when the observer saw me pass the point half a Schwarzschild radius outside the hole.
All this is not to imply that the black hole can't also be used for temporal tricks much like the "twin paradox" mentioned elsewhere in this FAQ. Suppose that I don't fall into the black hole—instead, I stop and wait at a constant r value just outside the event horizon, burning tremendous amounts of rocket fuel and somehow withstanding the huge gravitational force that would result.
If I then return home, I'll have aged less than you. In this case, general relativity can say something about the difference in proper time experienced by the two of us, because our ages can be compared locally at the start and end of the journey. If an external observer sees me slow down asymptotically as I fall, it might seem reasonable that I'd see the universe speed up asymptotically—that I'd see the universe end in a spectacular flash as I went through the horizon. This isn't the case, though.
What an external observer sees depends on what light does after I emit it. What I see, however, depends on what light does before it gets to me. And there's no way that light from future events far away can get to me. Faraway events in the arbitrarily distant future never end up on my "past light-cone," the surface made of light rays that get to me at a given time.
It will be a rocky ride as your matter perturbs the environment and the spacetime undulates. As you fall toward the singularity, you are badly broken. The part of your body closest to the singularity is accelerated drastically faster than the part of your body farthest from the singularity, stretching you miserably. Simultaneously, your overall anatomy is forced to converge toward that point, crushing you.
In a microsecond, less time than it would take to blink your eye, you are simultaneously flayed, shredded, and pulverised to death. Your organic matter is then pummelled, tattered, and inevitably shattered into elementary constituents. Ultimately, your fundamental bits spray toward the cut in spacetime and cease to be. The rip leads to nowhere. The singularity is an end to space and time, an end to existence. Death by singularity is the paramount existential death — the death of your fundamental particles, the removal from reality of you and your constituent stuff.
Actual nonexistence. Singularities, as they involve malefic infinities, deserve to be treated with great suspicion. They are such an anathema to the entire paradigm of the scientific pursuit of reality that essentially all physicists suspect general relativity ceases to be the complete physical description of gravity at such dramatic scales, the singular core a false prophecy. Rephrased: the mathematics is telling us that the physical description offered by relativity is broken there.
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