More black hole questions

This has started to bother me. From numerous sources, I read/hear that to an outside observer, we would never see our brave/stupid/suicidal astronaut friend cross the event horizon of a black hole because of time dilation. This seems incorrect to me.

As a mythical starship approaches the speed of light, we see it going faster. Time dilation only affects the difference in our frame of reference relative to that of the crazy-ass pilot who keeps opening his throttle wider. Gravitational time dilation appears to be identical to time dilation due to acceleration, so we would not see the astronaut who is falling into a black hole fall ever more slowly, we would see him (or her) accelerate toward the speed of light, which is the speed at which they would seem to cross the event horizon.

Why would we, the amateur public, be given this misinformation? Is there a communication problem that even astrophysicists like Tyson cannot cope with? Or am I completely wrong?

And how do we figure on a singularity? Beyond the event horizon, we have no way to know what the internal dynamics would look like. Is it possible that a very massive object could simply have an event horizon without forming a singularity?

Or even have a gradient so extreme that we could not realistically discern that it is a full event horizon? Something only a little larger that its Schwarzschild Radius would look, for all intents and purposes, like a black hole.

From what I read, Sagittarius A*, by its mass and volume estimates, has an average density of about 2 atmospheres of Earth air, so it could easily contain an object that is simply very large, rather than a singularity. The math that we use for black holes suggests a singularity, but is that math valid beyond the event horizon?

To me, it seems likely that if matter cannot reach or exceed the speed of light, the notion that it could bend spacetime infinitely with its mass is equally implausible. We observe what appear to be black holes, but can we be sure that we are seeing genuine event horizons? If a mass can bend spacetime infinitesimally close to having an event horizon, with some kind of Zeno’s Paradox preventing, shall we say, full closure, would it matter from our point of view (being not inclined to dive in to find out)?

I guess I can’t quite follow your logic here, but: if we observe the clocks on a mythical spaceship traveling near the speed of light, they will be “running slow” relative to ours. In other words, for every hour (say) that passes on our clock, 15 minutes (say) will pass on ours. The closer you are to the speed of light, the greater the slow-down factor is; and as the velocity of the mythical spaceship approaches the speed of light, no time appears to elapse on the spaceship and everything appears to be frozen in place. Just like someone falling into a black hole.

Sure, it’s possible. But the laws of physics as we understand them don’t allow for anything but a singularity. As I and others pointed out in a recent thread, this probably just means that there must be laws of physics (currently unknown to us) that keep this sort of thing from occurring.

Note, though, that Sag A* is mind-bogglingly huge. For a more reasonably-sized black hole, such as that formed from a collapsing giant star, the density would have to be much larger — over 10[sup]12[/sup] kg per cubic centimeter of volume. The “density” of a black hole scales inversely proportionally to its mass, so smaller black holes are denser.

Again, we don’t know for sure. There are some proposals (stemming from string theory) in which the physics looks radically different just inside the event horizon of a black hole. But that’s a fairly large modification to the laws of physics as we understand them, because so far as we can tell, the event horizon isn’t that special “locally”. Under our current understanding, you wouldn’t notice anything in particular as you crossed the event horizon of a black hole; the curvature of spacetime isn’t particularly strong there or anything.