Relativity and Black Hole Question

[PeterDonis]

oh, yes, I forgot to say we need the Schwarzschild metric outside the matter region. Although, for the general case of varying density, I guess we can just use one region. But the metric would not be nice and simple.

Not if there's vacuum outside the collapsing matter; then you still need two regions, a matter region and a vacuum region. You're correct that the metric in both regions would be more complicated than in the simple idealized model we've been discussing; but it would still be a *different* metric, a different geometry, in the matter region and in the vacuum region.

If there isn't vacuum outside the collapsing matter, then you're modeling a collapsing universe, not a collapse to a black hole. See further comments below.

I thought the event horizon forms inside the matter region described by the FRW metric. And as matter falls in through the event horizon, the event horizon gets bigger, until eventually it ends up outside the matter region.

This is true as long as there *is* a vacuum region outside the matter region. If there isn't, then the term "event horizon" has no meaning, because there's no null infinity in the spacetime.

Let me describe this another way, to make it clearer what I'm saying. Consider a small piece of matter at the exact center of the spherically symmetric body of matter that is collapsing. At some point on the worldline of this piece of matter, an outgoing null ray will be emitted that just happens to reach the surface of the collapsing matter at the exact instant that that surface is at $r = 2M$, i.e., at the horizon radius for the collapsing matter (as given by its total mass). That outgoing null ray then stays at $r = 2M$ forever; i.e., it marks the event horizon.

If we are looking at the spacetime as a whole, then yes, the entire path of that null ray, starting from the piece of matter at the exact center, marks the event horizon, so in that sense, yes, there is an event horizon inside the matter region. But the only reason it *is* an event horizon is that that null ray never escapes to infinity; it gets trapped at $r = 2M$, even though it has exited the collapsing matter and is in the vacuum region. But for the idea of "escape to infinity" to make sense in the first place, there has to *be* the vacuum region exterior to the collapsing matter; if there isn't, then as I said above, you're describing a collapsing universe, with finite spatial volume, so there is no "infinity" and therefore no meaningful concept of "escape to infinity". So there is no meaningful concept of "event horizon" that applies to the FRW metric by itself; the concept only applies if the FRW metric is joined to the exterior vacuum (Schwarzschild) metric.

(Also, if the collapsing FRW metric describes the entire spacetime, then there is no "center" and no "surface"--the matter occupies the entire spacetime and the spacetime is homogeneous--so the description I gave above of the null ray being emitted from the center and reaching the surface at $r = 2M$ is meaningless. That description only makes sense if the collapsing FRW metric is joined to an exterior vacuum metric.)

 

[yuiop]

If the infalling matter is negligible in mass compared to the black hole, the Schwarzschild solution is not *exactly* correct, but it's a very, very good approximation, certainly good enough that if it did in fact predict that matter could not fall through the horizon, that would be a problem. There is no problem because it does *not* predict that.

Could you elaborate on why you think there "would be a problem"?

 

[PeterDonis]

Could you elaborate on why you think there "would be a problem"?

"Would be a problem" in the sense that that model wouldn't work if it didn't match observations, so we would have to find a different model. To some extent this is circular, since we interpret observations in terms of theory; but our observations of highly compact, massive objects, such as the various stellar black hole candidates and the supermassive object at the center of our galaxy, would be, IMO, very hard to reconcile with any theory that did *not* predict that there were event horizons and matter could fall into them.

(Technically, the above is true of a classical theory, i.e., one that doesn't include quantum effects on black holes, such as Hawking radiation or the more speculative suggestions about quantum effects preventing a true event horizon from forming at all. But even in the speculative quantum models where there isn't actually an event horizon, in that any information that goes into a "black hole" region eventually comes back out, there is still a long period of time where there is a compact region containing trapped surfaces that looks like a standard classical black hole. The only difference is in the far future--"far" meaning times many, many orders of magnitude longer than the current age of the universe. So for practical purposes, such models are the same as the standard classical black hole model, since in that model, you can't locally distinguish between a true event horizon and a local trapped surface anyway; you have to know the entire future of the spacetime to do that.)