# Black Holes-to-be, Frozen in Time?

1. Sep 15, 2006

### Logex

I've been confused about something ever since I first learned about black holes. Perhaps someone can help clear this up for me.

Suppose we (sitting here on earth) observe a clock sitting on the surface of an imploding stellar core during a supernova, destined to become a black hole. (By observe, I mean we have a crystal ball, so no confusion about light traveling to us). As the core approaches the point of forming an event horizon, before a black hole has actually formed, we observe the clock slows down tremendously. Now a millisecond on the clock takes a month of our time. Closer to forming the event horizon, now an attosecond takes a billion years of our time.

From this perspective, it seems more proper to me to say that the stellar core is frozen just before forming an event horizon, from our standpoint here on earth. All black holes are in fact "asymptotically" black holes. All hypothetical discussion about "the interior of a black hole", if it is relevant to us as observers here on earth, should really be about the physical state of the collapsing core, frozen in time, frozen at the exact point of almost forming an event horizon.

Yet astrophysicists and cosmologists routinely discuss the interior of a black hole in a manner that seems to shrug over this point. I've discussed this with a well-known astrophysicist, and the best he could come up with was that Penrose diagrams are the best way to view this. Can anyone do better, or explain how Penrose diagrams help here?

Last edited: Sep 15, 2006
2. Sep 15, 2006

### michael879

Im pretty sure that the limit as t -> infinity isnt the creation of an event horizon. time doesnt have to have completely stopped inside the black hole for the event horizon to appear. Im no expert though, the last time I read anything reliable about black holes was a few years ago.

3. Sep 15, 2006

### MeJennifer

The theory is not for nothing called relativity (despite Einstein's initial reservations on calling it so).
What appears to us as frozen is locally inside the black hole (with the exception of the singularity) not frozen at all.
This seeming dichotomy is actually a consequence of the principle of relativity.

Last edited: Sep 16, 2006
4. Sep 16, 2006

### pervect

Staff Emeritus
"Crystal balls" are not particularly physical, unfortunately. Take a look at Geoffry Landis's very old newsgroup post about what would happen with a wormhole involving the twin paradox for a rough idea of the sort of thing that would happen. Basically, wormholes (and crystal balls) that communnicate instantaneously will act like time machines.

5. Sep 16, 2006

### cesiumfrog

That would be why the original term was "frozen stars". MWT has a detailed and quantitative explanation of how, as that clock slows down, we receive photons less and less often (even though in its own frame the stellar material continues radiating brightly), so within perhaps a second of our time the brightness will be utterly undetectable.. hence the term "black hole" is quite appropriate.

Of course the central core of the star is now already within a radius from which light would take infinite time to be received.. But sure, if we have enough time, we should always be able to verify that a given chunk from near the star's outer surface hasn't yet crossed that horizon..

The static black hole is quite a well understood exact solution to the Einstein field equation, whereas it's harder to discuss collapsing objects (for which no such solution has been found).

6. Sep 16, 2006

### WhyIsItSo

This confuses me.

How old is the universe presumed to be? Is it 15 Billion years?

Doesn't really matter. Since our perception of a Black Hole will be time slowing towards "frozen" asymptoticly, wouldn't that imply we shuold not be able to detect any Black Holes at all?

It seems logical that any Black Holes forming since the Big Bang would be a phenomenon we literally could never see be completed due to time dilation.

7. Sep 16, 2006

### pervect

Staff Emeritus
Just to clarify my last post a bit - if you had a crystal ball, it would probably act just like a wormhole, as described by the post I quoted. The reason for this is that this "wormhole" model is one of the very few models for "crystal balls" that is compatibile with relativity.

This means that you'd always see the person age at 1 second / second through the crystal ball, while you would see the usual relativistic time delays when you observed them directly.

Thus if you go on a journey FTL, or close to the event horizon of a black hole, the results would be much the same. When you go back to meet the person, you'd see a very old person "in person", ,and a younger person through the crystal ball - i.e. the crystal ball would then be a time machine, which would be viewing the past.

8. Sep 16, 2006

### Logex

Thanks for the historical perspective. This mostly confirms my understanding. From the rest-frame of an external observer, general relativity tells us that a clock on (or inside) an imploding stellar core runs more and more slowly as the mass density approaches the point where a given quantity of mass is enclosed within its Schwarzschild radius. This is independent of whether such a clock can be observed.

This is where I'm not sure I understand correctly. As I see it, the core of the star never shrinks below its Schwarzchild radius, from the standpoint of the external observer. It may still be that hypothetical light from inside would take an infinite amount of time to be received, because the core is still collapsing and space-time curvature is still increasing. Is this what you mean by saying that the core of the star is already within a radius from which light would take an infinite time to be received?

The discussions here about crystal balls and FTL travel are very interesting and important for complete understanding. However, it does seem to me that there are issues which are independent of the notion of observation, inasmuch as the equations of relativy do give answers about the temporal slowing inside a steep gravitational field relative to "external observers", and this is independent of the physics of observation.

Another way to express my dissatisfaction with the commonly used parlance about black holes is to say that although we can not receive information from inside a black hole (barring evaporation etc.), we can in fact describe the physical state of this region. It is the physical state of the collapsing core at the density achieved when it is nearly enclosed by its Schwarzchild radius, frozen in time (from our perspective!)

9. Sep 16, 2006

### JesseM

I think past a certain point, it would be impossible for observers starting from a given distance to "catch up" with a given chunk before it crossed the event horizon, even if they sent a signal moving at the speed of light to catch up with it and bounce off it so they could see it was still outside the horizon. Of course, this point of last possible verification would presumably depend on the observer's distance from the horizon. And even if you couldn't verify the chunk's existence by sending a signal towards it to bounce off it, if the chunk itself sent you a signal before crossing the horizon, you could in principle detect it no matter how close it had gotten to the horizon (although this would probably change with quantum gravity, it presumably wouldn't be meaningful to talk about a signal sent out less than one planck time before it reached the horizon)

10. Sep 17, 2006

### Thrice

Mmm according to that logic, how a black hole even form if it takes infinitely long (from the perspective of an outside observer) for anything to reach the event horizon?

It's not a foolish question. Scientists were stuck on this for quite a while after the schwarzschild metric was discovered. I won't give away the solution..

11. Sep 17, 2006

### Sojourner01

This sounds similar to something I was pondering a while back.

Suppose that you have a gradually accreting mass which is approaching critical mass to form a black hole. Then, suppose that it gets to the point where only the mass of a single electron is required to push it 'over the edge'. That electron wanders past, approaches the event horizon - and of course from our perspective, never crosses it because of time dilation. Therefore, we observe a single electron sitting on the event horizon.

An electron is a quantum object. Therefore, it has an uncertainty in position that may or may not place it inside the event horizon of the black hole. If the electron is inside the event horizon, the black hole has critical mass and is thus a 'complete' black hole. If the electron is outside the event horizon, the black hole never quite forms. Because of the electron's uncertainty, both these states exist in superposition.

So, since the black hole's state (hypothetically) relies entirely on the position of the electron, is this black hole in fact a macroscopic quantum object? In other words, is it an actual Schrodinger's Cat?

12. Sep 17, 2006

### Logex

Well, yes, exactly, that is the question... if you have an answer, please share! I have asked that exact question to a well-known astrophysicist, and he was unable to give me a succinct answer (he said that Penrose diagrams were the way to understand this).

So far, it looks like this involves a question of semantics, about what exactly is meant by the term "black hole" and "event horizon". That's largely what I'm after... what is a more correct way to talk about black holes? It's easy to find popular science descriptions of black holes that say things like "after the event horizon has formed, the core continues to contract to a point of infinite density". It seems that this is grossly misleading at best.

This is remarkably similar to something I heard Richard Feynman say over 20 years ago. I once had the good fortune to be able to ask him a related question, as an undergraduate student. I asked how Hawking radiation could work, if the particle destined for the black hole never actually makes it to the event horizon due to time dilation. He said he'd have to think about it. I had the opportunity to ask him again later, and he said that he thought it maybe had to do with the quantum uncertainty of the position of the particle and maybe something like quantum tunneling. (He also said that he didn't really know, and that this was a guess.)

The rest of Sojourner01's suggestion of course goes well beyond this, but it's in a similar vein.

13. Sep 17, 2006

### Thrice

Heh so it is. I skimmed the post the first time through.

Anyway the key is it's just a coordinate singularity. Switch to a different one & it disappears. Much of relativity is like this; the description of "what is really happening" is observer dependent. Although admittedly this is a rather exotic example.

Last edited: Sep 17, 2006
14. Nov 27, 2006

### Chris Hillman

Some common misconceptions about black holes

Hi Logex, I am coming into this thread very late, so bear with me.

The basic confusion here concerns one of the most common of all the many misconceptions due to inadequate verbal descriptions.

Before I say anything else let me enthusiastically recommend a rare example of a readable "math-free" popular book on gtr by an EXPERT on gtr, General Relativity from A to B, by Robert Geroch. In this book, Geroch attempts to provide clear and correct intuition for the black hole concept using only pictures, and I would hope that readers with a strong geometric imagination will find it invaluable.

OK, so that is this misconception? It's where you say "time slows down near the horizon". Of course, you mean the misnomer "gravitational time dilation effect". But of course gtr doesn't say "time slows down" anywhere anywhen--- that would be nonsense! What would time be "slowing down" with respect to, if not with respect to itself, time? What this really refers to, from the viewpoint of calculus, is rate of change of one variable with respect to another. But in the spirit of Geroch's book, I prefer a more geometric explanation:

According to the theory of curved surfaces introduced by Gauss, one of the basic effects of spacetime curvature is that two initially parallel geodesics can converge (positive curvature) or diverge (negative curvature). In Riemannian and Lorentzian manifolds, this idea is carried over with little change and is the topic of the "geodesic deviation formula".
Now consider two static observers in the exterior region of the Schwarzschild vacuum, radially separated and using their rocket engines to hover at, respectively $$r_1, \, r_2$$ where $$2m < r_1 < r_2 < \infty$$. If the lower observer sends time signals toward the upper observer, at a rate of one signal per second, the world line of these signals can be modeled as null geodesics which diverge (negative curvature in the appropropriate two dimensional section through our spacetime model), and thus the upper observer measures time between reception of two time signals to be greater than one second. The point is, both observers use idealized clocks which are completely equivalent. Time is certainly not "slowing down" for the lower observer; rather, signals he sends to the upper observer follow diverging null geodesics.

Well, he certainly did not mislead you, and it was not inappropriate for him to refer you to the library since so many fine serious physics books which discuss black holes are now available (as well as, unfortunately, some truly dreadful popular books).

Penrose diagrams, aka block diagrams or Carter-Penrose diagrams, are indeed the very best way to understand the causal structure of a spacetime. See for example the textbook by D'Inverno for a very clear exposition of the basic ideas, and then see
http://www.math.ucr.edu/home/baez/RelWWW/history.html for many examples (and I could give many more).

We don't detect black holes directly (the event horizon is not even a physical surface so there is nothing there to see if that even made sense, which it does not). MTW has a nice discussion of what a static observer hovering outside a collapsing star (treated using the highly idealized OS model) would literally "see"; the point is that distant observers certainly do not see the surface "frozen" just before it vanishes under the horizon; as you would expect from the naive notion of energy conservation, the luminosity exponentially decays, and the image reds out and winks out in a very short time.

I certainly hope that I have cleared up this misconception!

No clocks "run more slowly" (well, real ones might, but ideal ones by definition withstand any punishment and always run at the same rate). Rather, light signals from the surface of a collapsing object have world lines which can be modeled as null geodesics, and radial null geodesics diverge due to the curvature of spacetime.

The "frozen star" metaphor was known to be incorrect even BEFORE it was published, but unfortunately in the early days of gtr (say before about 1960) very few physicists really understood the role of curvature or the global geometry of the Schwarzschild vacuum.

It depends on what you mean by the "standpoint". Confusion of this kind is almost always clarified by setting up and analyzing a precise thought experiment.

Some troublesome phrases here:

1. "hypothetical light from inside": if you mean light traveling outwards through the horizon, that is forbidden by definition in gtr, and it makes little sense to try to postulate in theory T something which theory T forbids.

2. "infinite amount of time": measured by which observer, located where, in what state of motion with respect to the hole? Also, you presumably mean the time which elapses between two events on the world line of this observer.

3. "received": by the observer who measures said elapsed time?

4. "the core is still collapsing": according to the physical experience of an observer riding on the surface? If so, you need to explain how his experience is relayed to the other observer.

I hope you appreciate that I am NOT carping, just trying to point out some issues to think about.

Well, depending upon what you mean, I might disagree as per the above comments.

I don't know what you mean. I hope that "state" doesn't refer to a quantum state, since of course QFT lies outside the scope of gtr, which is a classical field theory.

More troublesome phrases:

1. "critical mass to form a black hole": you might mean Chandrasekhar mass, but that concept is not really relevant to the discussion.

2. "from our perspective": in the sense of optical imaging? in the sense of inferences from reception of light signals emitted by another observer? which one?

3. "never": according to the ideal clock carried by what observer? located where? in what state of motion wrt the hole?

4. "we observe a single electron sitting on the event horizon": it would have a timelike world line, but the event horizon consists of null world lines.

Let me suggest a better thought experiment, using a Vaidya null dust exterior. Imagine that we have a perfect fluid ball which is almost at the Buchdahl limit. (Buchdahl's theorem says that according to gtr, a static spherically symmetric ball of perfect fluid must have a surface radius larger than $$r_0 = 9/4 m$$, which is a bit larger than the Schwarzschild radius.) Now we imagine a spherically contracting "sandwich wave" of incoherent massless radiation (the "null dust") which has just enough mass-energy so that with the added mass, the star would violate the limit and thus will begin to collapse.

You shouldn't expect a succinct answer. (At least, if we reject misleading and flippant ones such as are offered in most popular books.) And he was right about the value of Carter-Penrose block diagrams, which are about as succinct as you are going to find (but the trouble lies in helping you to interpret the diagrams).

Well, "event horizon" and "black hole" are two of the most popularly misunderstood concepts in physics. However, it is true that the density of the collapsing fluid ball diverges in finite proper time, as measured by an observer riding with the fluid.

Well... instead of relying upon your memory of decades old conversation, I hope you will look up the book by Geroch which I recommended. Sorry, by the way, if I was incorrect in guessing that you are not au courant with mathematical physics generally! However, even if you studied physics at Cal Tech (good place to study physics!) I think you can learn alot from this particular popular book, and I hope reading it will encourage you to follow up by studying MTW (the T is Kip Thorne, another Cal Tech legend).

Chris Hillman

Last edited: Nov 27, 2006
15. Dec 1, 2006

### kesh

so as far as i understand it, a distant observer (ie us on earth) will not see a remnant star become a black hole, as in an object with an event horizon, only approuch one asymptotically.

forget what happens to an observer near the black hole, or the black hole's own view of it's own existence, we here on earth can't see a black hole form, and winding our own clocks back (so long as we remain a distant observer, which naturally won't remain the case) could never have seen one form

16. Dec 1, 2006

### Chris Hillman

Observing a black hole form?

Hi, kesh,

Well, if you mean "see" in the sense of telescopic observations, as I recently mentioned somewhere in this forum, a secondary problem with the "frozen star" picture is that gtr actually predicts that the luminosity of the collapsing surface decays exponentially, so that it would be more correct to think of "watching" the collapsing surface "red out/wink out" than "freeze". See for example Exc. 32.2 in Misner, Thorne, Wheeler, Gravitation, for the point I was trying to explain.

I should stress that we are both discussing, I think, highly idealized models here--- realistic models of black holes would probably include lots of highly visible stuff. Indeed, one of the attractions of observing black hole formation with gravitational rather than electromagnetic radiation is that the former should help astronomical instruments avoid being "distracted" by all the other stuff going on.

Again, it depends upon what you mean by "see". Astronomers confidently expect to be able to observe characteristic gravitational wave signals when black holes form, and no doubt the press release will scream "black hole formation seen!" (or possibly, following Kip Thorne's metaphor suggesting how different gravitational radiation is from EM radiation, "black hole formation heard!" According to gtr, this radiation should mostly come, roughly speaking, from the immediate exterior of the horizon, so I guess that few astronomers would be reluctant to speak of "direct observation of black hole formation".

Chris Hillman

17. Dec 1, 2006

### kesh

thanks, but i'm not satisfied. i meant see in the strongest sense as in any information reaching us that would indicate the black hole had formed. from my basic knowledge of gr the remnant star black hole forms in (creates itself) an inaccesible part of spacetime, not just an inaccesible part of space, and we can have no causal relationship with it, including observing its creation, except the assymptotic approuch to the same

Last edited: Dec 1, 2006
18. Dec 1, 2006

### Chris Hillman

No causal connection with a black hole?

Hi, kesh,

I suspect there is some confusion here about what we mean by "observe", "formation" and "black hole". Passing over the tricky "teleological" nature of event horizons, the technical definition of "black hole" used by gravitation theorists is rather different from what those who haven't studied this subject are likely to imagine, but I will go out on a limb and guess that you have in mind what would usually be loosely termed "the" [sic] "interior region". If so, according to gtr (as a relativistic classical field theory of gravitation, i.e. specifically excluding any considerations from quantum field theory) observers in "the" [sic] "exterior region" cannot receive signals from the interior, but observers in the interior certainly CAN receive signals from the exterior!

Also, according to gtr, the interior of a black hole is certainly NOT "inaccessible" in the sense in which most physicists would use that word, because an observer in the exterior can fall into (or rocket into) the interior. You might be thinking of the fact that once he does so, he cannot rocket back out.

Does this help clarify the situation?

By the way, I would highly recommend General Relativity from A to B by Robert Geroch, a nontechnical book which does an outstanding job of conveying via pictures accurate intuition for what we mean by "black hole" in the context of gtr.

Chris Hillman

19. Dec 1, 2006

### kesh

by observe i mean receive information from, any information, by any means. by formation i mean observing effects attributable to the existence of an event horizon (from the point of view of a distant observer) where there wasn't one before. black hole i mean an event horizon observed from the frame of a distant observer produced by a massive body.

but i am particularly interested in remnant star black holes as these posit an evolution from a region of spacetime where relativistic effects are not strong (the star) to one where they are so strong as to be an event horizon (the black hole). from what i understand this evolution cannot be observed to completion by a distant observer because of the time dilation that such an evolution produces. this is the kind of inaccesibility i am interested in

20. Dec 1, 2006

### Chris Hillman

Hi, kesh,

It is still not clear from me what theoretical context, if any, you have in mind. It is important that we understand whether you are asking about how one can describe the predictions of gtr, or more properly, how one can geometrically characterize and physically interpret models of black holes in gtr, or something else, such as specific astronomical observations. (Such observations always require some theoretical context for interpretation, but sometimes this context may be more general than gtr.)

Well, what about observationsof SagA, which have been described in various popsci venues, e.g. http://unisci.com/stories/20013/0906011.htm? Is that the kind of thing you are asking about?

I'll go out on a limb and assume that by "frame" you do NOT mean "frame field", but a more elementary notion which doesn't play very well with curved spacetime.

Well, event horizon vs. no event horizon is not really a strong field versus weak field issue, at least not according to gtr. However, if you restrict attention to stellar-mass black holes, then the region near an event horizon would be strongly curved. Perhaps you would be interested in Buchdahl's theorem? This gives an upper limit on how "dense" a perfect fluid ball can be, in gtr, before it must collapse to form a black hole, namely $$r = 9/4 m$$; see the textbook by Schutz.

Did you see my recent post in which I tried to offer a better explanation of "gravitational time dilation"?

You should probably avoid using the term "inaccessible" to avoid confusion. You might be interested in learning about the "absolute future" and "absolute past" of an event in a curved spacetime (say an FRW model) and the notion of Cauchy horizon. Try the textbook by D'Inverno.

I think it would facilitate this discussion if we have all studied at least some of the same textbooks.

Chris Hillman