Black hole formation watched from a distance

[haushofer]

TL;DR Summary

How is black hole formation observed by a distant observer?

Dear all,

For a new book I'm writing I'm investigating some common misconceptions in physics. And of course, that means confronting myself with my own confusion. One thing I've never got clear in my head, and which I find hard to answer using google and my textbooks on GR, is the following: how exactly is black hole formation observed by an observer sitting at a distance? If we take a solution of, say, infalling matter or light, crushing itself into a black hole, how much coordinate time would an observer at a distance measure for the matter or light to form an event horizon? I'm sure this can be answered technically by looking at Penrose diagrams, but I've never felt completely comfortable with those. My intuitive answer would be that it would take a huge amount of coordinate time (an infinite amount?) according to our observer for the matter to collapse into a volume smaller than its Schwarzschild radius such that the event horizon actually forms. Am I right? And if so, can someone give some details? What would this mean concretely for the black holes we observe, especially the famous picture by the Event Horizon Telescope? In the end I want to translate this to something concrete in my book, like this picture.

As a follow up one could also ask the same question about black hole evaporation of course, but let's first focus on the black hole formation process. Any insights are more than welcome.

 

[martinbn]

Is there a reason why you expect that the answer shouldn't depend on what exacly the formation is, and on what time coordinate you use? Or do you have something more specific in mind?

 

[haushofer]

Is there a reason why you expect that the answer shouldn't depend on what exacly the formation is, and on what time coordinate you use? Or do you have something more specific in mind?

Well, let's compare to the particle which flies into the black hole from great distance: a distant observer measures an infinite coordinate time between the events "particle starts traveling" and "particle enters horizon", while the particle itself measures a finite proper time between the same two events. But when it comes to the observation of the actual formation of the black hole, I have intuitive doubts (I know, nature doesn't care about my intuition).

Does the event horizon form in a finite amount of coordinate time according to the distant observer? I.e., when we see the famous Event Horizon picture of the black hole, is it plausible that the formation of the black hole and its event horizon is in a much more advanced stadium than from our perspective?

 

[jbriggs444]

Does the event horizon form in a finite amount of coordinate time according to the distant observer?

Unlike the situation in the flat space-time of special relativity, specifying an observer is not enough to specify a global coordinate system.

If you want Schwarzschild coordinates, you have to say so.

 

[haushofer]

Unlike the situation in the flat space-time of special relativity, specifying an observer is not enough to specify a global coordinate system.

If you want Schwarzschild coordinates, you have to say so.

Yes, let's take Schwarzschild coordinates, and put Earth at r= far away from the black hole (at r=0; horizon at r=2M).

 

[martinbn]

What are Schwarzshild coordinates?

 

[vanhees71]

https://en.wikipedia.org/wiki/Schwarzschild_metric

 

[martinbn]

https://en.wikipedia.org/wiki/Schwarzschild_metric

Is this for my question? The thread is about black hole formation. The Schwarzschild is eternal it cannot be relevant.

 

[PAllen]

Well, the standard simple model of gravitations collapse to BH, rather than eternal case, is the Oppenheimer-Snyder solution. The following notes cover this collapse in a modern way, then discusses trapped surfaces and light escape during the process (also, generalizing to fluid ball rather than dust ball):

http://events.asiaa.sinica.edu.tw/school/20070129/talk/960201rezzolla.pdf

In any case, as to what you would visually 'see' from a distance during and idealized collapse (i.e. no remaining accretion disc), this was well covered way back circa 1970 in MTW - within a quite short amount of proper time for the distant observer, the collapsing star would be blacker than empty intergalactic space. Thus, the term black hole was coined. The light that will emanate forever from the near outside of the horizon is of such low frequency and intensity that it could never be observed in principle (imagine a detector for 'photons' with wavelength of 1 light year arriving once per month).

 

[PAllen]

Well, the standard simple model of gravitations collapse to BH, rather than eternal case, is the Oppenheimer-Snyder solution. The following notes cover this collapse in a modern way, then discusses trapped surfaces and light escape during the process (also, generalizing to fluid ball rather than dust ball):

http://events.asiaa.sinica.edu.tw/school/20070129/talk/960201rezzolla.pdf

In any case, as to what you would visually 'see' from a distance during and idealized collapse (i.e. no remaining accretion disc), this was well covered way back circa 1970 in MTW - within a quite short amount of proper time for the distant observer, the collapsing star would be blacker than empty intergalactic space. Thus, the term black hole was coined. The light that will emanate forever from the near outside of the horizon is of such low frequency and intensity that it could never be observed in principle (imagine a detector for 'photons' with wavelength of 1 light year arriving once per month).

Actually, adding some detail from MTW, when the quantum nature of emission is considered, about 10 milliseconds (of distant observers proper time) after a distant observer sees a collapsing 10 solar mass star begin to dim, the last photon they will ever see arrives. Note that Hawking radiation does not change this because for more that a 100 billion years, the BH is growing by absorbing CMB radiation, rather than shrinking by emitting Hawking radiation. The relevant discussion is pp. 872-3 of MTW.

 

[PAllen]

Actually, adding some detail from MTW, when the quantum nature of emission is considered, about 10 milliseconds after a distant observer sees a collapsing 10 solar mass star begin to dim, the last photon they will ever see arrives. Note that Hawking radiation does not change this because for more that a 100 billion years, the BH is growing by absorbing CMB radiation, rather than shrinking by emitting Hawking radiation. The relevant discussion is pp. 872-3 of MTW.

Note, it also follows, that a visible EH in the sense of the EH telescope forms within 10 milliseconds of the beginning of catastrophic collapse for a 10 solar mass star. Note, this is consistent with the time scales of BH merger events, where two BH of masses of 10s of solar masses merge within 2 seconds or less, once they get close, emitting 3 solar masses of energy as GW in much less than a second - all as observed from earth. The same fallacious 'takes forever' arguments would apply to BH mergers, except, of course, that they are fallacious.

 

[romsofia]

Well, you could go back to the original paper on the matter, "On continued gravitational contraction" by Snyder and Oppenheimer.

Which, I only know because of this lecture by Jacobson:
which might also be helpful.

 

[PAllen]

Well, you could go back to the original paper on the matter, "On continued gravitational contraction" by Snyder and Oppenheimer.

Which, I only know because of this lecture by Jacobson:
which might also be helpful.

Well, I am familiar with this paper, and could have linked to it. However, it has some issues for the purpose at hand, and from the modern point of view. It stops its analysis when the dust boundary reaches the horizon, noting that this happens in finite proper time for a boundary riding observer, but infinite coordinate time for a distant observer per the chosen coordinates. It does not examine at all what happen to the dust after horizon crossing. The paper I linked follows the dust all the way to the singularity, and distinguishes coordinate features from observations. Further, it generalized to the more plausible fluid case, noting differences in the formation of apparent and true horizons in this case. Thus, I made a deliberate, IMO, well founded, decision not to link this paper.

Does anyone insist on linking to Maxwell's original papers when discussing electromagnetism?

 

[romsofia]

Well, I am familiar with this paper, and could have linked to it. However, it has some issues for the purpose at hand, and from the modern point of view. It stops its analysis when the dust boundary reaches the horizon, noting that this happens in finite proper time for a boundary riding observer, but infinite coordinate time for a distant observer per the chosen coordinates. It does not examine at all what happen to the dust after horizon crossing. The paper I linked follows the dust all the way to the singularity, and distinguishes coordinate features from observations. Further, it generalized to the more plausible fluid case, noting differences in the formation of apparent and true horizons in this case. Thus, I made a deliberate, IMO, well founded, decision not to link this paper.

Does anyone insist on linking to Maxwell's original papers when discussing electromagnetism?

OP is writing a book, and I personally like books that talk about the history of the subject, while also discussing the physics, and subsequent progression. If this were just a post about the topic, your link would suffice since it's modern AND through.

However, the lecture notes you posted don't mention the original work, and I would be remiss if I didn't mention it given the context (and my bias for books :-) )

 

[PAllen]

OP is writing a book, and I personally like books that talk about the history of the subject, while also discussing the physics, and subsequent progression. If this were just a post about the topic, your link would suffice since it's modern AND through.

However, the lecture notes you posted don't mention the original work, and I would be remiss if I didn't mention it given the context (and my bias for books :-) )

Yes, definitely agreed. I hadn’t thought of that aspect. Here is the Oppenheimer Snyder original:

https://journals.aps.org/pr/pdf/10.1103/PhysRev.56.455

 

[vanhees71]

Is this for my question? The thread is about black hole formation. The Schwarzschild is eternal it cannot be relevant.

No that was to answer your question, what Schwarzschild coordinates are.

 

[martinbn]

No that was to answer your question, what Schwarzschild coordinates are.

But it doesn't, because my question was "what are Schwarzschild coordinates for a black hole?", not just for the Schwarzschild black hole.

 

[vanhees71]

I thought Schwarzschild coordinates refer to the Schwarzschild solution and thus also a Schwarzschild black hole. For other black holes (most usefully FAPP Kerr black holes) of course you use other coordinates.

 

[PeroK]

But it doesn't, because my question was "what are Schwarzschild coordinates for a black hole?", not just for the Schwarzschild black hole.

Do you mean the time-dependence of the star collapsing to within its previous surface? You have a dynamic collapsing solution between a static initial solution an approximately static end solution?

 

[martinbn]

Do you mean the time-dependence of the star collapsing to within its previous surface? You have a dynamic collapsing solution between a static initial solution an approximately static end solution?

I meant that I understand the question to be about a generic formation of a black hole.

 

[PeroK]

I meant that I understand the question to be about a generic formation of a black hole.

That leaves me none the wiser regarding your question, I'm sorry to say.

 

[pervect]

You might want to consider what case you want to understand, and what you want to observe. A couple of possible things I can think of observing - "hot" test particles emitting a known proper frequency, and of course gravitational waves.

What case you want to study is also important. The simple cases are unfortunately probably not representative of what actually happens. The issue as I understand it is stability. Perfect spherical symmetry is a natural assumption to make, but since it is felt that actual solutions diverge in an unstable manner from the perfectly symmetrical case, the perfectly spherical case is probably misleading. Unfortunately, the realistic non-spherical cases are hard.

Openheimer-Snyder solutions, which were previously mentioned, would represent the unstable perfectly spherical collapse.

This is a bit generic, so let's give something more specific. Consider what's called "mass inflation", and it's impact on the stability of horizons. Poisson and Israel have some papers on mass inflation, IIRC. Google finds https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.63.1663

Gravitational wave emission from binary inspirals have been extensively studied, though they require numerical simulations. I've read bits and pieces about these simulations, but I don't recall any of the details or authors. It isn't something I consdier that I really understand, and is something I'd like to understand much better than I do. Probably digging into the Ligo papers would give a starting place, at least.

There are some other papers on realistic, rotating, and non-spherical collapse, but from what I recall reading it's dificult and not fully understood. It may have been Andrew Hamilton that wrote some of what I read, but I don't really recall for sure.

 

[PAllen]

The OP asked mostly for, IMO, for qualitative statements about observation of BH formation by a distant observer. For this, I think the description in MTW and the lecture notes I provided is sufficient. However the author of the lecture notes I linked is an expert in numerical relativity analyses of realistic collapse. The following are two professional papers by this author on realistic collapse. Unfortunately, they don’t have much to say on the observations of distant observers:

https://arxiv.org/abs/gr-qc/0403029
https://arxiv.org/abs/gr-qc/0503016

 

[haushofer]

Note, it also follows, that a visible EH in the sense of the EH telescope forms within 10 milliseconds of the beginning of catastrophic collapse for a 10 solar mass star. Note, this is consistent with the time scales of BH merger events, where two BH of masses of 10s of solar masses merge within 2 seconds or less, once they get close, emitting 3 solar masses of energy as GW in much less than a second - all as observed from earth. The same fallacious 'takes forever' arguments would apply to BH mergers, except, of course, that they are fallacious.

Thanks! I took a quick glance at MTW, which I only have as a pdf, so I hadn't thought about consulting it. I also wasn't really familiar with the explicit Oppenheimer-Snyder solution, so I'll definitely take a look at that. Just to be sure: can we state that for us on Earth the formation of a black hole would take an infinite amount of time if we define "formation of the black hole" by "the star has shrunk to its gravitational radius" (as Oppenheimer and Snyder), but that the difference between this asymptotic state (as observed by us from earth) and the black hole as observed by a "cocollapsing observer" is for all practical purposes zero within a very small timescale?

I'll dive into the other reactions too, but due to a fusion at school things are a bit hectic. Do know that I read and appreciate every reaction!

By the way, it's historically quite curious that the Oppenheimer-Snyder paper was written when Europe was collapsed into World War 2. I'm sure this has been poetically expressed by other authors before ;)

 

[PAllen]

By the way, it's historically quite curious that the Oppenheimer-Snyder paper was written when Europe was collapsed into World War 2. I'm sure this has been poetically expressed by other authors before ;)

Also historically curious is how little this work by Oppenheimer was recognized at the time. Later, some historians of science have noted that this was easily Oppenheimer’s most Nobel worthy work (there were actually 2 related papers by Oppenheimer that are almost impossible to track down now), yet received almost no attention until after his death.

 

[PeterDonis]

my question was "what are Schwarzschild coordinates for a black hole?", not just for the Schwarzschild black hole.

If "Schwarzschild coordinates" means specifically the coordinate chart on Schwarzschild spacetime with that name, obviously there are no such coordinates for any black hole other than the Schwarzschild black hole.

If you have some other meaning of "Schwarzschild coordinates" in mind, you will need to clarify what it is, since I'm not aware of any other meaning than the one described above.

 

[PeterDonis]

The Schwarzschild is eternal it cannot be relevant.

The "eternal" Schwarzschild spacetime (more technically called "maximally extended Schwarzschild spacetime" is not the only possible application of the Schwarzschild spacetime geometry. That geometry describes any region of any spacetime that is vacuum and spherically symmetric (by Birkhoff's theorem). The most common application is the vacuum exterior region of a spherically symmetric configuration of matter, as in the Oppenheimer-Snyder paper that has been referred to, or in a model of a spherically symmetric static body like a planet or star.

 

[PeterDonis]

I understand the question to be about a generic formation of a black hole.

Since @haushofer has adopted Schwarzschild coordinates, that restricts discussion to a spherically symmetric black hole. That is also the only case where we have a closed form solution (the Oppenheimer-Snyder solution). There is no known closed form solution for the non-spherically symmetric case, although I believe it has been studied numerically.

 

[PAllen]

...Just to be sure: can we state that for us on Earth the formation of a black hole would take an infinite amount of time if we define "formation of the black hole" by "the star has shrunk to its gravitational radius" (as Oppenheimer and Snyder), but that the difference between this asymptotic state (as observed by us from earth) and the black hole as observed by a "cocollapsing observer" is for all practical purposes zero within a very small timescale?
...
I

I would express this as follows:

1) For a co-collapsing observer, reaching the horizon takes a fraction of a second for a stellar BH once catastrophic collapse has set in. For this observer, reaching the singularity takes a similar length of time.

2) A distant observer can never truly observe the horizon or anything reaching or crossing it. This is by definition of a horizon (no information from it reaches null infinity, which translates, in practice, to distant observers). In principle, the distant observer forever sees, at any given time, a state of the collapse from just before the outer collapse surface reaches the horizon. (But 'sees' is in a mathematical sense. See next bullet).

3) However, all signal from the collapse ends a very short time after the beginning of the collapse as seen by the distant observer. That is, instead of seeing some 'frozen image', what really happens is that all information from the collapsing body vanishes in a very short time, the result being far more black than intergalactic space (under the thermodynamic model suggested in MTW, ignoring Hawking radiation, no photons whatsoever ever reach the distant observer after a short time per the distant observer). Also, for any 'frozen star' model, if you wait long enough for a return signal, you can probe the surface with radiation. This is impossible, in principle for a BH horizon. Thus a BH horizon is observationally distinct from compact object, over sufficiently long time scales.

The event horizon telescope obviously can't directly observe a horizon. That would contradict the definition of a horizon. It's aim is to observe predicted features of a horizon generated just outside it, that distinguish it from as many compact object models as possible.

Note: In the above, I have deliberately refrained from ever mentioning coordinate time, or any notion of 'now at a distance'. I have only used locally observed time (proper time) and direct observations. Much of the 'never happens' language is based on using particular coordinate charts. However, an invariant statement can be made that distinguishes a horizon from the surface of a compact body. The former is never in the causal past of a distant observer (however, it does transition from being in the causal future to the 'possibly now' , i.e. no longer in the causal future of a distant observer). In contrast, any event on the surface of compact object is eventually in the causal past of a distant observer.

 

[haushofer]

I would express this as follows:

1) For a co-collapsing observer, reaching the horizon takes a fraction of a second for a stellar BH once catastrophic collapse has set in. For this observer, reaching the singularity takes a similar length of time.

2) A distant observer can never truly observe the horizon or anything reaching or crossing it. This is by definition of a horizon (no information from it reaches null infinity, which translates, in practice, to distant observers). In principle, the distant observer forever sees, at any given time, a state of the collapse from just before the outer collapse surface reaches the horizon. (But 'sees' is in a mathematical sense. See next bullet).

3) However, all signal from the collapse ends a very short time after the beginning of the collapse as seen by the distant observer. That is, instead of seeing some 'frozen image', what really happens is that all information from the collapsing body vanishes in a very short time, the result being far more black than intergalactic space (under the thermodynamic model suggested in MTW, ignoring Hawking radiation, no photons whatsoever ever reach the distant observer after a short time per the distant observer). Also, for any 'frozen star' model, if you wait long enough for a return signal, you can probe the surface with radiation. This is impossible, in principle for a BH horizon. Thus a BH horizon is observationally distinct from compact object, over sufficiently long time scales.

The event horizon telescope obviously can't directly observe a horizon. That would contradict the definition of a horizon. It's aim is to observe predicted features of a horizon generated just outside it, that distinguish it from as many compact object models as possible.

Note: In the above, I have deliberately refrained from ever mentioning coordinate time, or any notion of 'now at a distance'. I have only used locally observed time (proper time) and direct observations. Much of the 'never happens' language is based on using particular coordinate charts. However, an invariant statement can be made that distinguishes a horizon from the surface of a compact body. The former is never in the causal past of a distant observer (however, it does transition from being in the causal future to the 'possibly now' , i.e. no longer in the causal future of a distant observer). In contrast, any event on the surface of compact object is eventually in the causal past of a distant observer.

Many thanks for your extensive answer. I think I got it. I justed printed out MTW's relevant chapters and will dive into the Oppenheimer-Snyder solution. Whenever I have more questions, I'll let you know.

And when I've found a publisher for my book, of course. Who knows, maybe it will even turn into a nice Insight :P

 

[vanhees71]

Since @haushofer has adopted Schwarzschild coordinates, that restricts discussion to a spherically symmetric black hole. That is also the only case where we have a closed form solution (the Oppenheimer-Snyder solution). There is no known closed form solution for the non-spherically symmetric case, although I believe it has been studied numerically.

and also visualized nicely. Here's a collection of pictures as well as movies made by my astrophysics colleagues in Frankfurt:

https://relastro.uni-frankfurt.de/gallery/

 

[TeethWhitener]

Leonard Susskind's GR lecture series on Youtube covers this at a fairly easy-to-understand level:

The first 45 minutes or so are an introduction to Penrose diagrams, and then he proceeds to use them to introduce black hole formation.

Edit: he starts talking about black hole formation at around 44:40.

 

[pervect]

Just to be sure: can we state that for us on Earth the formation of a black hole would take an infinite amount of time if we define "formation of the black hole" by "the star has shrunk to its gravitational radius"

In order to answer this, we need to interpret the actual physical observations. These observations might be, for instance, observation of one or more beacons falling into the black hole with a "frequency vs Earth (TAI) time of reception" plot. If we have multiple beacons, we might want to include angle, as well, though it's questionable if we could resolve the angular differences with multiple beacons.

I'd rather talk about these actual physical observations than the interpretation thereof, as the interpretation involves concepts of simultaneity and coordinate time. These interpretations are generally recognized as requiring conventions. Perhaps some of the question is about these conventions, but I won't address that in this post.

Without doing any calculations, we can say that a signal from an infalling beacon, emitted at the event horizon, never reaches infinity. The beacon can be regarded as being located at the edge of a cloud of dust undergoing an idealized Openheimer-Snyder collapse. Also note, we are not taking into account the possible evaporation of the black hole, which complicates things considerably.

I have done more detailed calculations in the past, but I'm not sure exactly where - I could try and search more if there is some interest. But basically, the frequency of the beacon decreases (approximately) exponentially as one approaches the horizon.

The proper time of the beacon when it reaches the event horizon is finite, and no signal emitted from the beacon reaches the observer at infinity at or after the time at which the beacon reaches the event horizon.

 

[bob012345]

If it takes an infinite amount of time to form a black hole then wouldn't we have to say they don't actually exist, and any future black holes are only and always will be only in the process of being formed? If so, what then are those images of black holes all about?

 

[PAllen]

If it takes an infinite amount of time to form a black hole then wouldn't we have to say they don't actually exist, and any future black holes are only and always will be only in the process of being formed? If so, what then are those images of black holes all about?

Please read the thread. These questions have all been dealt with in the thread.