What is the causal structure of a black hole collapse?

[wangyi]

Hi, I have a question on can the black hole be formed?

As we know,
First, in the Swartzchild solution of GR, an observer outside the black hole will find it takes infinity time for any object to fall into the black hole.
Second, a black hole is formed by a star falling to its center.

Now the question arises: when a star is falling, and nearly to form a black hole, say, if one little piece of matter falls onto the star, it forms a black hole. At this time, the time for a observer to see the last matter falling onto the star is nearly infinate, that means no matter how long a period of time T given, we will find a piece of matter small enough which falls onto the star taking longer time than T according to the observer outside. Then how can a black hole form?

I have been confused about it for long. Thank you.

 

[Phobos]

http://cosmology.berkeley.edu/Education/BHfaq.html#q4

 

[Zanket]

wangyi, what the distant observer sees and what an in-falling observer sees are different things. The distant observer never sees the hole form. An in-falling observer finds that there is a black hole there. Suppose the last little piece of matter is an in-falling observer, who crosses the nascent horizon at 5pm on his or her watch. The distant observer sees that time on the in-falling observer's watch ever more slowly approaches but never reaches 5pm.

 

[wangyi]

The distant observer never sees the hole form.

thank you, this is what i want to say, we are the distant observer, and are there real black holes in our point of view?

If no, the rediation from black hole can never be observed because outside the heavy object, the black hole is never formed. Then the information problem of black hole maybe questionable.

I am just confused with these.

regards
wangyi

 

[Zanket]

From our point of view as a distant observer, black holes do not form. From our point of view the term "frozen star" is more appropriate, where the surface of the collapsed star remains above the event horizon but is redshifted almost to blackness.

 

[pervect]

Hi, I have a question on can the black hole be formed?

As we know,
First, in the Swartzchild solution of GR, an observer outside the black hole will find it takes infinity time for any object to fall into the black hole.
Second, a black hole is formed by a star falling to its center.

Now the question arises: when a star is falling, and nearly to form a black hole, say, if one little piece of matter falls onto the star, it forms a black hole. At this time, the time for a observer to see the last matter falling onto the star is nearly infinate, that means no matter how long a period of time T given, we will find a piece of matter small enough which falls onto the star taking longer time than T according to the observer outside. Then how can a black hole form?

I have been confused about it for long. Thank you.

After a certain amount of time has elapsed, you cannot send a message to someone who is falling into a non-rotating black hole (or someone who is part of a newly forming non-rotating black hole) that will reach the person before he, she, or it reaches the singularity.

At this point, I regard the black hole as "having formed", or the person or object as "having fallen in", because there is nothing you can do that will influence or reach them - they have reached the singularity and been torn apart by the tidal forces, and you cannot send a message that will reach them before this happens.

However, you will never SEE anything that happens to anyone or anything that goes past the event horizon (unless you take a one way-trip through the event horizon yourself).

So I regard a non-rotating black hole as having "already happened" after the amount of time I mentioned in part 1 has passed by, it's just that you can't see it without taking a one-way trip yourself.

Rotating black holes are trickier.

[re-write for clarity]
It is possible in some theoretical solutions for an observer falling into a rotating black hole to see the entire future of the universe - this also means that there is no upper time limit where I can't send him a signal. So my argument that the black hole has "already formed" appears to fail in these cases, at first glance.

However, these theoretical solutions are currently believed to be unphysical, as they are not actually stable. The symmetrical solutions to the black hole (The Schwarzschild and Kerr-Newmann solutions) are stable for the region outside the event horizon, but inside the event horizon, very small assymetries grow, instead of shrinking, making the symmetrical solutions highly suspect / unphysical.

 

[wangyi]

Does that mean we can never see the Hawking radiation, and we can never see the black hole vaporize even when the background radiation temperature is low enough and the time is long enough, because in our point of view, the black hole never formed.
Is it right?
thank you.

 

[Ich]

At this point, I regard the black hole as "having formed", or the person or object as "having fallen in", because there is nothing you can do that will influence or reach them - they have reached the singularity and been torn apart by the tidal forces, and you cannot send a message that will reach them before this happens.

That´s exactly the point I have reached here.
But then they told me that I still can´t be sure that the person really has fallen in - he/she still could return.

 

[pervect]

Does that mean we can never see the Hawking radiation, and we can never see the black hole vaporize even when the background radiation temperature is low enough and the time is long enough, because in our point of view, the black hole never formed.
Is it right?
thank you.

If you take quantum mechanics into account, you'll eventually see the black hole explode, and you will also see the trapped light at or before the time of the explosion. My previous post did not take quantum mechanics into account (the problem is complex enough without it, IMO, and quantum mechanics doesn't really change the answer).

See for instance "Ted Bunn's Black Hole FAQ"

http://cosmology.berkeley.edu/Education/BHfaq.html#q9

which states that you will see the trapped light exactly at the same time as you see the black hole explode. (I haven't seen the equations for how this timing is worked out).

It's definitely wrong to argue that the black hole never formed and so could not explode.

[re-write]

Think of it this way. By controlling the flux of incoming particles, one can control whehter or not a black hole is "allwed" to evaporate or not. Thus, if I keep "feeding" a black hole, it will never evaporate, it will only grow. Imagine I have a simple switch which I can throw - one that says either "feed the black hole" or "do not feed the black hole".

Black holes evaporate slowly - so the space-time geometry of the hole will not change very much in the rather short ciritical time that I mentioned, the time during which external signals can reach the infalling observer.

Imagine that I chose not to throw the switch to decide the fate of the black hole until after this critical amount of time has elapsed.

Thus the logical interpretation in terms of causal chains is that an infalling observer reaches the singularity, first, before any information about whether or not I chose to let the black hole evaporate (by throwing my switch) can possibly reach him. The event of my throwing the switch is in his future - thus he reaches the singularity first, before he can know whether or not the hole will ever be allowed to evaporate, or whether it will keep growing.

 

[JesseM]

That´s exactly the point I have reached here.
But then they told me that I still can´t be sure that the person really has fallen in - he/she still could return.

Yes, although there is a time past which you cannot do anything to prevent them falling in, if the ship has sufficiently powerful rockets you can never be sure they didn't rocket away at the last moment before crossing the event horizon.

 

[wangyi]

It's definitely wrong to argue that the black hole never formed and so could not explode.

I am so glad to see so definite an answer:) but why? for the reason i gave at the beginning, has the real black hole(i.e. having singularities in our coordinate) form? if we can't wait until it formed, and it explode only after it formed, why we can wait until it explode?

thank you
wangyi

 

[pervect]

I'm guessing you didn't read the FAQ entry I quoted. I will therefore take the liberty of cutting and pasting it (it seems slightly more polite than just telling you to go read it :-)) If there is something that still puzzles you after reading the FAQ entry, ask away. I think the combination of the FAQ entry, below, and my example with the switch, pretty clearly answers the question, though.

Won't the black hole have evaporated out from under me before I reach it?
---------------------------------------------------------------------
We've observed that, from the point of view of your friend Penelope who remains safely outside of the black hole, it takes you an infinite amount of time to cross the horizon. We've also observed that black holes evaporate via Hawking radiation in a finite amount of time. So by the time you reach the horizon, the black hole will be gone, right?

Wrong. When we said that Penelope would see it take forever for you to cross the horizon, we were imagining a non-evaporating black hole. If the black hole is evaporating, that changes things. Your friend will see you cross the horizon at the exact same moment she sees the black hole evaporate. Let me try to describe why this is true.

Remember what we said before: Penelope is the victim of an optical illusion. The light that you emit when you're very near the horizon (but still on the outside) takes a very long time to climb out and reach her. If the black hole lasts forever, then the light may take arbitrarily long to get out, and that's why she doesn't see you cross the horizon for a very long (even an infinite) time. But once the black hole has evaporated, there's nothing to stop the light that carries the news that you're about to cross the horizon from reaching her. In fact, it reaches her at the same moment as that last burst of Hawking radiation. Of course, none of that will matter to you: you've long since crossed the horizon and been crushed at the singularity. Sorry about that, but you should have thought about it before you jumped in.

 

[chronon]

The event of my throwing the switch is in his future

No. The only reasonable meaning to give to this statement would be that the event was in the future light cone of the infalling observer, which is manifestly not the case.

 

[wangyi]

I'm guessing you didn't read the FAQ entry I quoted. I will therefore take the liberty of cutting and pasting it (it seems slightly more polite than just telling you to go read it :-)) If there is something that still puzzles you after reading the FAQ entry, ask away. I think the combination of the FAQ entry, below, and my example with the switch, pretty clearly answers the question, though.

Thank you very much, the idea in your FAQ is striking for me, while sorry for only on the second reading i came to understand your idea.

Do you mean we can only see a real black hole when it is actually evaporated and gave out the light it trapped?

but i still have a question (maybe my misunderstanding) that since the evaporation of the black hole happens after the background temperature is low enough, and the things happens near the black hole becomes slower in our view point, is it mean that now we can only see the real black hole whose
temperature is higher than 2.3K, i.e. it is very small, and we can see big black holes only long long afterwards?

 

[pervect]

The term "real" always bothers me a bit, but I think you've given a a reasonable summary. If a black hole does not evaporate, the event horizon prevents one from actually seeing what's inside the black hole. However, I regard this as being an optical illusion, rather than a philosphical statement - i.e. I don't regard the stuff inside as being non-existent in any sort of philosphical sense (anymore than do I regard the world as dissappearing when I close my eyes). Instead, I just regard the stuff inside a black hole as "existing" in some philosophical sense, but not being able to be seen because the light can't leave the (non-evaporating) black hole.

What I really try to focus on, though, is not philosphy so much as the causal structure near a black hole. By "causal structure" I mean which events can cause which others. As I've already argued, it's this causal structure which allows one to argue that the black hole can't evaporate before an infalling observer gets there. Causal structure is determined by the paths that light can take - because an event can cause another only inside the light cone of the first event.

I also don't really think about the quantum case very often - probably because I need more math to fully appreciate it. So I'm "biased" towards thinking of the non-evaporating BH rather than the quantum mechanical evaporating one.

There's a lot to say about the causal structure of black holes, too much for a short post. But I'm going to ramble on a bit, even though I can't really do the topic full justice.

Using Kruskal coordinates where the geodesics of light beams are always 45 degree angled lines is the easiest way to talk about the causal structure of a BH . As effects are limited to the light-cones of causes, by making the light-cones look like the light cones in flat space-time, Kruskal coordinates makes the job of determining causal structure a lot easier.

One thing to beware of is that the causal structure of an actual, physical collapse is significantly different than the causal structure of the Schwarzschild metric. (The causal structure of the Schwarzschild metric turns out to be that of a dynamic wormhole - this is very interesting, but not particularly physical).

The important elements of the actual Kruskal coordinates are often abstracted into a Penrose diagram, where some of the more complicated curves in the Kruskal diagram are "morphed" into simple straight lines.

Online, the only source I know of that talks about this stuff (causal structure) is
http://casa.colorado.edu/~ajsh/schww.html

and it may not really be clear enough to follow if it's one's only source of information.