The acceleation would be greater on the side of the star closer to the black hole, so the star would be pulled apart into an accretion disk, which would become hotter due to friction and emit x-rays. To an outside observer the disk would slow down as it nears the event horizon, and become dramatically dimmer, whereas to the star it would get increasingly bright then cross the horizon. Two black holes can merge in finite time in the perspective of the black holes. A distant observer would not be able to see the black holes themselves anyway. 

You've asked an excellent question.  Indeed, one I've asked for about a decade from time to time, never getting a straight answer to it.  When I ask a physicist, he tells me to ask an astronomer or cosmologist.  When I ask one of them, they tell me to ask a physicist.  Surely there must be an expert in General Relativity around to respond!

But your main point is correct--to an outside observer, things take infinitely long to fall through an event horizon if they are moving through space and time toward it.  I state things that way because perhaps when black holes merge they are so distorting spacetime that their behavior may be less a case of moving through spacetime as going along with the wrapping of it while remaining in place. (I am only speculating--a fancy word for guessing.)  Such wraping can take place in less than infinite time.

It is also the case that as a moving object falls toward the event horizon, for its light to communicate that fact to an outside observer the light must climb out of the gravitational potential well around the hole.  This redshifts and dims the light to the point that at the horizon, you'd get no more information out.  So all those fantastic effects like pulsar jets and such take place further outside the horizon--far enough away for the light to "get out" and reach us. 

I groan, however, when I read about how some astronomer observed a black hole "swallowing" a star, etc., and agree with you: shouldn't it take forever to witness that?

One of the most important things to bear in mind in thinking about black holes, is the distinction between what happens at the event horizon, and what happens near the event horizon. Jim has given some examples of the effect of a black hole on the universe around it, as opposed to what happens as things cross the horizon.Now, the event horizon is a very narrow place; in fact it's thickness as seen by us is of sub-atomic dimensions. (It does not have zero thickness, because we seem to live in a quantum world, which in turn seems to mean that a mathematical zero thickness is not empirically meaningful.) Of course, as you and Jim obviously are well aware, the event horizon is not very special from the point of view of anyone crossing it, or contemplating it from within the black hole.

At the same time, from our point of view looking from outside, that event horizon is thin enough that for practical purposes we should regard things happening in its neighbourhood at no more than a sub atomic particle at a time. So when an electron falls in, we can in principle trace its history indefinitely, but when a planet falls in, things become complicated. Let's consider the simplest case we can manage. Suppose that a very small, relatively cold, non-rotating neutron star falls in a straight line towards a large (and therefore cool) non-rotating black hole. What we see happening is not what from our point of view happens to each particle as it takes forever to cross the event horizon; instead we see what happens at this side of the horizon, the tidal effects slurping our soft little stronger than steel neutron star towards the place where the event horizon is a reaching out to meet it. (There are some animations online illustrating such events, in particular colliding black holes, though offhand I do not know their addresses. It should be simple enough to surf and look for them.)

In practice of course, the frequency of straight inward trajectories, nonrotating black holes and neutron stars etc, is negligible. What we may expect in real life is far messier, with lots of splashing, radiation, jets and so on. By the time all that has died down the event horizon has moved anyway.

Exactly what that expansion of the event horizon does to our prediction of each terminal crossing of the horizon I am not prepared to say, any more than I am prepared to say that the foregoing is better than nonsense. But it does seem plain that the event horizon will move, and that it will do so in a good deal less time than infinity, and that any particle close enough to where it begins to take for ever to vanish is not going to reverse its progress just to stay outside the event horizon for our benefit.

In general I cannot help wondering whether in the light of the monotonic growth of the event horizon, the idea of the infinite time to cross the horizon is not empirically simplistic.

Ideas, anybody?

Cheers,

Joooooooooooooooooooooooo....

On rereading my last comment, it seems that I owe everybody, including myself, a brief and more lucid description of the operative point, which also to my mind is the operative answer to the original question.

Firstly, remember that at best the event horizon of a typical size of black hole never retreats, and in practice may advance quite dramatically in response to almost any major event. (For microscopic black holes, this is not generally so.)

Now, for a large object to fall into a black hole counts as a major event by most standards. Whatever else results, we know that there will be a surge in the diameter of the black hole, and in particular of the diameter of its event horizon.

As I remarked, and clearly as the other correspondents were fully aware, such an event would be a messy one, with jets, gravitational waves, electromagnetic field distortions and so on. However, let us imagine a nice, well conducted, quiet black hole, isolated in space, and with a single electron falling into it. All is peace, apart from the periodic emission of photons by which we track the progress, the eternal progress, of our electron towards the event horizon.

Bear in mind that nothing is going to look particularly interesting in the messages we retrieve until that electron is microscopically close to the event horizon. All of that 12th of never eternal slow down of the signals takes place in the final few micro-meters or so.

Whoops! It seems that I was sold a pup, a no-good, time-biding black hole! Behind it there was a neutron star about to fall in. It was quite a big neutron star. It caused our black hole to expand enormously, several millimetres in fact!

Now, in our monitoring of the electron, we had followed its progress down to within a micro-meter of the event horizon. We had sequenced and timed each returned photon, and had determined the near-as-dammit position from which each had been emitted. We also could predict when and from where we would receive each next photon for as long as we might be interested in following the electron's fascinating progress.

Now suddenly the entire volume of space in which our recent observations were taken has vanished behind an event horizon from which we now can receive no signals whatsoever. Nor will we receive any future signals, not from there anyway. Not up to, at, nor even after the 12th of never. We have been robbed of a potentially infinite number of tell-tale photons. Our infinite fall-in has been brutally curtailed.

Am I making sense?

If I had been making observations from the other side of the black hole, the side where the big, messy, neutron star fell in, all its huge volume of quarks and leptons would have caused an extremely rapid advance of the event horizon, rapidly swallowing the space in which the eternal falling-in of each successive particle otherwise would have proceeded for ever. They no longer should take forever; in fact, their fading squeaks would be quite abrupt, I should say. The volume of space from which we should be hearing them is not where we could detect anything, though no doubt they could still see our ever-blue-shifting, frustrated expressions.

Now am I making something more like sense?