What is the definition of a black hole?

[JoeyJoystick]

TL;DR Summary

Discussing and challenging the definition of a black hole

Hi All,

This is my first post, so please bare with me and if I am going all wrong about, please let me know.

The definition of a black hole according NASA; 'A black hole is a place in space where gravity pulls so much that even light can not get out'. Now I am not challenging this at all. However, very often a black hole is considered an object with a high density as well. The reasons are obvious, but is this actually correct?

Why am I asking this question? Well, it's simple. A small black hole has a higher density than a large black hole. The larger the black hole, the lower the density. And what do we really know about the black hole other than that light can not escape? Not that much right?

For all I know, without claiming that this is the case, the mass inside a black hole could be compressed into a small nuclei with a diameter of several mm while the rest of the black hole is empty. I mean, after all there is a very high gravitational force right? Or it could be evenly spread across the available space inside of the back hole. But we never know this for sure right?

What I am essentially saying, thinking and hoping is that you prove me wrong here, because if it is true than I have a whole lot of other questions for the next thread. lol.

Joey

 

[Ibix]

Density isn't a well-defined concept for a black hole. Its internal volume is infinite by many measures. You can define a sphere enclosing a black hole and calculate the volume that sphere would contain in a Euclidean space (this is what you are probably doing), but that ignores the fact that spacetime is curved around a black hole and the Euclidean volume is inappropriate. If you do it anyway, you do find the density decreasing with mass, yes, but it's a meaningless statement.

 

[berkeman]

Summary:: Discussing and challenging the definition of a black hole

A black hole is a place in space where gravity pulls so much that even light can not get out

OMG, I saw that quote and thought, there's no way that NASA published that. It sounds like something from a kid's book or something. So I did a Google search on the quote, and sure enough it's from NASA:

https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-a-black-hole-k4.html

But if you look closely at the quote, you see it is targeted for K-4th grade kids. LOL, I was right!

@JoeyJoystick -- Welcome to the PF. The intro article at Wikipedia is probably a better place to start learning about Black Holes, and you can follow some of the references and links at the end of that article to get more information. We also have lots of good threads discussing the subtleties of Black Holes here at PhysicsForums -- a website search should turn up lots of interesting reading for you.

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

 

[sbrothy]

In alll fairness the wiki entry leads with:

"A black hole is a region of spacetime where gravity is so strong that nothing—no particles or even electromagnetic radiation such as light—can escape from it."

:)

 

[Ibix]

In alll fairness the wiki entry leads with:

"A black hole is a region of spacetime where gravity is so strong that nothing—no particles or even electromagnetic radiation such as light—can escape from it."

But it doesn't say "pull", which is problematic in GR since gravity is spacetime curvature, not a force, so it doesn't pull. Furthermore it says "region of spacetime" not "place in space". Describing a black hole as a "place in space" is somewhat dubious since the event horizon is a null surface, which isn't a place, and the singularity is a spacelike line, which isn't a place either. You can only really describe it as a place in space as long as you stay away from it and don't try to study it very much.

So despite differing in only a couple of words, the Wikipedia entry is much better.

 

[JoeyJoystick]

OMG, I saw that quote and thought, there's no way that NASA published that. It sounds like something from a kid's book or something. So I did a Google search on the quote, and sure enough it's from NASA:

https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-a-black-hole-k4.html

View attachment 267819
But if you look closely at the quote, you see it is targeted for K-4th grade kids. LOL, I was right!

@JoeyJoystick -- Welcome to the PF. The intro article at Wikipedia is probably a better place to start learning about Black Holes, and you can follow some of the references and links at the end of that article to get more information. We also have lots of good threads discussing the subtleties of Black Holes here at PhysicsForums -- a website search should turn up lots of interesting reading for you.

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

You are right. That was the link. It was actually the first link that showed. A rather simplistic presentation of a black hole compared to Wikipedia, but it got the point across. lol.

Anyway, thanks for the guidance. Will start reading shortly.

Joey

 

[JoeyJoystick]

Density isn't a well-defined concept for a black hole. Its internal volume is infinite by many measures. You can define a sphere enclosing a black hole and calculate the volume that sphere would contain in a Euclidean space (this is what you are probably doing), but that ignores the fact that spacetime is curved around a black hole and the Euclidean volume is inappropriate. If you do it anyway, you do find the density decreasing with mass, yes, but it's a meaningless statement.

And there you say it is a meaningless statement. Can you motivate that? Because this is actually the point I was trying to make. That as long as you have enough mass, even at a low density you can still have a black hole. As long as the escape velocity is higher than the speed of light. Now that may be meaningless, but I figured it demonstrates that a black hole does not necessarily need to have a high density.

And why am I going this way? Well it is a bit of a story, but never mind. Here I go.

Some 25 years ago I read a book from Isaac Asimov. I forgotten the title but it was about many different ways how we could meet our end. One of them was a collision with a black hole. Not exactly the kind of thing we need to worry about, but still interesting. Something he said triggered me in thinking that it is possible that the entire universe was a black hole. Now in 1996 internet was not widespread and not available to me at the time. So I plundered my father's library and went to the local library to pick a load of physics books. Now calculating this for me was not easy and yes you are correct that I did not include space time into this. I did find formulas and numbers. Among the numbers I found were the diameters of both the sun and the Earth if they would be a black hole. Which if I remember correctly was about 2km for the sun and about 2cm for planet earth. And I calculated this using basic physics from high school. This did not include Space Time, Einstein or anything more complicated than Newton. Being fully aware that this may be a problem I used these formulas to calculate the diameter for both the sun and the Earth and I got the same values that I found as a reference in the books I read. So now I thought I should start calculating this for the universe. For this I needed the mass of the universe. And this was an issue because the numbers I found varied a lot. By many magnitudes. However, I did see a line in there. The Line I saw was that the numbers for the mass of the universe went up was later publications. I did calculate the escape velocity of the universe based on 13billion light years old. And except for the oldest number, which was the lowest estimated density for the universe, all of them showed that the universe is a black hole.

Ok I was a lot younger back than. And the common knowledge was also more limited back than. I thought of a big bang from a singularity. Now we are talking about hyper inflation, string, multiple dimensions and god knows what else. Also, back than I had never even heard of the observable universe.

So now looking back, I do understand that my idea is majorly flawed. I did later find out that I was not the only person who has opted the idea and even Isaac Asimov had opted the idea in one of his publications if I remember correctly. And the people I shared my idea with mostly said that I was nuts. Fine. But no one has tried to explain why this is nuts and why this is not possible.

Please do not catch me on the numbers. It's all from the top of my head and I did not recalculate or verify the numbers. However, it is the concept I am talking about of course.

Joey

 

[JoeyJoystick]

But it doesn't say "pull", which is problematic in GR since gravity is spacetime curvature, not a force, so it doesn't pull. Furthermore it says "region of spacetime" not "place in space". Describing a black hole as a "place in space" is somewhat dubious since the event horizon is a null surface, which isn't a place, and the singularity is a spacelike line, which isn't a place either. You can only really describe it as a place in space as long as you stay away from it and don't try to study it very much.

So despite differing in only a couple of words, the Wikipedia entry is much better.

Agreed, but it was just to show that definitions do not include the term 'high density'.

Joey

 

[Ibix]

And there you say it is a meaningless statement. Can you motivate that?

I'm not sure what you mean by "motivate" in this context. That black holes do not have a well defined interior volume is simply a statement of fact. I can't prove that to you without you first learning general relativity, though.

Reading the rest of your post, though, you don't actually seem to be talking about black holes, but rather about the matter that eventually forms them.

That as long as you have enough mass, even at a low density you can still have a black hole. As long as the escape velocity is higher than the speed of light. Now that may be meaningless, but I figured it demonstrates that a black hole does not necessarily need to have a high density.

I'd quibble about black holes having an escape velocity, since the whole point is that not even light can escape them. But apart from that what you write here is true as far as it goes. In light of the rest of your post, it misses some important points.

And except for the oldest number, which was the lowest estimated density for the universe, all of them showed that the universe is a black hole [snip] But no one has tried to explain why this is nuts and why this is not possible.

I don't know where you are getting numbers for the mass of the universe from - our current understanding is that it's infinite in size and mass. Perhaps you are talking about the observable universe, which is the patch of the universe that we can see. That does have a finite mass.

However, the fundamental problem with your idea is that density isn't the controlling factor for whether something forms a black hole. There are at least two other conditions. First, you require that the density of your matter be greater than the density in the surrounding space. So if you have an patch of empty space with a chunk of matter in it, that matter will typically collapse towards its center. But if the universe is uniformly filled with matter there is no center for the matter to collapse towards. Or, to put it another way, at any point there is the same amount of matter on either side so the gravitational attractions are equal and no collapse can occur. Furthermore you can't talk about escape velocity in this context - escape to where from where? Everywhere's the same.

Secondly, above I was implicitly assuming that the matter I was talking about wasn't expanding. But galaxies are flying apart, not collapsing. This is a very different initial condition to a static chunk of matter that might undergo collapse, and mass isn't the only thing that affects gravity in general relativity. The dynamics of the matter, the way it moves and its internal stresses, all affect gravity. None of this looks like a black hole or a precursor to one.

Fundamentally, of course, the answer lies in the maths. This is a book length topic, which I shan't try to write here. The bottom line is that the solution to Einstein's field equations when we feed in "more or less uniform matter everywhere" looks very like our universe, and has none of the properties of a black hole spacetime. And a black hole spacetime doesn't look anything like our universe.

Among the numbers I found were the diameters of both the sun and the Earth if they would be a black hole. Which if I remember correctly was about 2km for the sun and about 2cm for planet earth.

The Schwarzschild radius for a mass $M$ is $2GM/c^2$, which works out to be 3km and 0.9cm for Sun and Earth, for the record.

 

[sbrothy]

But it doesn't say "pull", which is problematic in GR since gravity is spacetime curvature, not a force, so it doesn't pull. Furthermore it says "region of spacetime" not "place in space". Describing a black hole as a "place in space" is somewhat dubious since the event horizon is a null surface, which isn't a place, and the singularity is a spacelike line, which isn't a place either. You can only really describe it as a place in space as long as you stay away from it and don't try to study it very much.

So despite differing in only a couple of words, the Wikipedia entry is much better.

You're certainly right. The wording just seemed very similar. At first sight.

 

[JoeyJoystick]

I'm not sure what you mean by "motivate" in this context. That black holes do not have a well defined interior volume is simply a statement of fact. I can't prove that to you without you first learning general relativity, though.

Reading the rest of your post, though, you don't actually seem to be talking about black holes, but rather about the matter that eventually forms them.

I'd quibble about black holes having an escape velocity, since the whole point is that not even light can escape them. But apart from that what you write here is true as far as it goes. In light of the rest of your post, it misses some important points.I don't know where you are getting numbers for the mass of the universe from - our current understanding is that it's infinite in size and mass. Perhaps you are talking about the observable universe, which is the patch of the universe that we can see. That does have a finite mass.

However, the fundamental problem with your idea is that density isn't the controlling factor for whether something forms a black hole. There are at least two other conditions. First, you require that the density of your matter be greater than the density in the surrounding space. So if you have an patch of empty space with a chunk of matter in it, that matter will typically collapse towards its center. But if the universe is uniformly filled with matter there is no center for the matter to collapse towards. Or, to put it another way, at any point there is the same amount of matter on either side so the gravitational attractions are equal and no collapse can occur. Furthermore you can't talk about escape velocity in this context - escape to where from where? Everywhere's the same.

Secondly, above I was implicitly assuming that the matter I was talking about wasn't expanding. But galaxies are flying apart, not collapsing. This is a very different initial condition to a static chunk of matter that might undergo collapse, and mass isn't the only thing that affects gravity in general relativity. The dynamics of the matter, the way it moves and its internal stresses, all affect gravity. None of this looks like a black hole or a precursor to one.

Fundamentally, of course, the answer lies in the maths. This is a book length topic, which I shan't try to write here. The bottom line is that the solution to Einstein's field equations when we feed in "more or less uniform matter everywhere" looks very like our universe, and has none of the properties of a black hole spacetime. And a black hole spacetime doesn't look anything like our universe.

The Schwarzschild radius for a mass $M$ is $2GM/c^2$, which works out to be 3km and 0.9cm for Sun and Earth, for the record.

Hi Ibix,

Thanks for your elaborate response. It helps to better understand it.

And than you say; 'Fundamentally, of course, the answer lies in the maths.'

Has someone ever calculated this?

I struggle with imagining and visualizing all this, which, in 3D, I'm actually quiet good at. But I can obviously not apply this here.

So I am going to do more of the earlier suggested reading first... lol.

Joey

 

[Ibix]

Has someone ever calculated this?

Calculated what? An isotropic matter filled spacetime? Sure, Friedmann, Robertson, Lemaitre and Walker - FLRW spacetime (or sometimes FRW since Lemaitre published in French and was frequently overlooked) is the basis of cosmology. They predicted an expanding universe (prompting Einstein to try to force a steady-state solution by fine-tuning a cosmological constant - what he later called his greatest mistake) before Edwin Hubble observed cosmological redshift.

 

[George Keeling]

This is a book length topic, which I shan't try to write here.

So I am going to do more of the earlier suggested reading first... lol.

Hi Joey , hope the reading is going well. I've been at it for 2½ years. It's a great retirement pastime and I can recommend "Spacetime and Geometry : An Introduction to General Relativity" – by Sean M Carroll. One thing is that he has a good sense of humour. It's quite mathematical at the beginning and I struggled to remember how to differentiate and thought that contravariant vectors must be like column matrices and covariant vectors like row matrices. A great error . I joined Physics Forums and the people here are brilliant and very patient.
Good luck!

 

[George Keeling]

I don't know where you are getting numbers for the mass of the universe from - our current understanding is that it's infinite in size and mass.

Universe infinite in size and mass? That's extraordinary. Where can I read more about it, please?

 

[Ibix]

Universe infinite in size and mass? That's extraordinary. Where can I read more about it, please?

I would think Carroll covers it - chapter 8 of his lecture notes certainly does. The flat and negative curvature FLRW metrics are infinite in extent and have finite density matter everywhere. Modern cosmological models are a bit more complicated, but retain those features.

 

[OCR]

please bare with me

Well, thank you but. . . I think I'll politely decline that invitation ! .
Lol. . . carry on. .

.

 

[shjacks45]

A black hole is created when gravitational force exceeds nuclear forces, hence the original body implodes to an infinitesimal point whatever the mass was originally. The diameter of the "black hole" would be where the escape velocity exceeds C, which would be mass dependent. I would presume the space between would be mostly empty.

 

[Ibix]

A black hole is created when gravitational force exceeds nuclear forces

A stellar black hole forms that way, more or less, but that's not quite the point under discussion. The point is that the required density goes down as the available mass rises and why this does not lead to the universe being a black hole.

the original body implodes to an infinitesimal point

Absolutely not. The singularity is more like a point in time than a place in space.

I would presume the space between would be mostly empty.

Since neither the horizon nor the singularity are "places in space" by the usual meanings, this statement doesn't really make sense. However, the interior of a more-or-less realistic black hole would be expected to be vacuum unless you enter very shortly after horizon formation, yes.

 

[Nugatory]

The diameter of the "black hole" would be where the escape velocity exceeds C

That's sort of true, but you have to be careful about it. "Escape velocity exceeding $c$" does not mean that nothing can escape, it just means that nothing can escape on a ballastic trajectory. But in the case of a black hole, it really is the case that nothing can escape, even under continuous acceleration on a non-ballastic trajectory.

And of course as others have pointed out in this thread, the "diameter" is a problematic notion here.

I would presume the space between would be mostly empty.

That's not just presumption, it's a simple fact. The Schwarzschild solution is a vacuum solution with the stress-energy tensor zero everywhere; there is no point in space that is not a vacuum.

 

[shjacks45]

That's sort of true, but you have to be careful about it. "Escape velocity exceeding $c$" does not mean that nothing can escape, it just means that nothing can escape on a ballastic trajectory. But in the case of a black hole, it really is the case that nothing can escape, even under continuous acceleration on a non-ballastic trajectory.

And of course as others have pointed out in this thread, the "diameter" is a problematic notion here.
That's not just presumption, it's a simple fact. The Schwarzschild solution is a vacuum solution with the stress-energy tensor zero everywhere; there is no point in space that is not a vacuum.

I gave the most basic definition in keeping with Hawking's explanation.
The collapse of the star's mass after the force of gravity exeeds nuclear forces has no lower limit so the mass could be considered an infinitesimal point source.
"Mostly empty" because I assume infalling debris still takes a finite time from the black hole's action radius to the central gravity source. The debris will accelerate but slower speed than C in the gravity well.

 

[Ibix]

the mass could be considered an infinitesimal point source

As I said the last time you said that:

Absolutely not. The singularity is more like a point in time than a place in space.

"Mostly empty" because I assume infalling debris still takes a finite time from the black hole's action radius to the central gravity source.

Presuming that by "action radius" you mean "event horizon", then as I said the last time you said this:

Since neither the horizon nor the singularity are "places in space" by the usual meanings, this statement doesn't really make sense. However, the interior of a more-or-less realistic black hole would be expected to be vacuum unless you enter very shortly after horizon formation, yes.

The debris will accelerate but slower speed than C in the gravity well.

The usual meaning of "gravity well" is related to gravitational potential, which isn't defined inside a black hole. And it's worth noting that you need a lot of care with the word "accelerate" in GR. In the physically meaningful sense of accelerate ("proper acceleration"), nothing accelerates while falling into a black hole. In the other sense ("coordinate acceleration") you need to specify your coordinate system, since coordinate acceleration can be zero or non-zero depending on the trajectory and coordinate choice.

 

[sbrothy]

As I said the last time you said that:Presuming that by "action radius" you mean "event horizon", then as I said the last time you said this:The usual meaning of "gravity well" is related to gravitational potential, which isn't defined inside a black hole. And it's worth noting that you need a lot of care with the word "accelerate" in GR. In the physically meaningful sense of accelerate ("proper acceleration"), nothing accelerates while falling into a black hole. In the other sense ("coordinate acceleration") you need to specify your coordinate system, since coordinate acceleration can be zero or non-zero depending on the trajectory and coordinate choice.

A measure of BH's counter-intuitive-ness is how they turn all our notions upside - down, or rather inside-out :)

Space and time exchanges roles so to speak. Whatever that really "means".

 

[Ibix]

Space and time exchanges roles so to speak. Whatever that really "means".

It means nothing - it's another popsci misconception. The correct statement is that the tt and rr metric coefficients in exterior Schwarzschild coordinates have the same functional form as the zz and tt coefficients, respectively, in interior Schwarzschild coordinates. But that won't sell books. The claim is related to noting that as you walk towards the north pole north is forward and south is backward, and after you pass through the pole forward is south and backward is north. But you'd laugh at me if I wrote a book saying that north and south swap meaning if you cross the north pole.

 

[JoeyJoystick]

That's sort of true, but you have to be careful about it. "Escape velocity exceeding $c$" does not mean that nothing can escape, it just means that nothing can escape on a ballastic trajectory. But in the case of a black hole, it really is the case that nothing can escape, even under continuous acceleration on a non-ballastic trajectory.

And of course as others have pointed out in this thread, the "diameter" is a problematic notion here.
That's not just presumption, it's a simple fact. The Schwarzschild solution is a vacuum solution with the stress-energy tensor zero everywhere; there is no point in space that is not a vacuum.

Hi Nugatory,

Why is it that scientists are so certain about the state of the inside of a black hole? If we can not look inside and nothing escapes than how can we possibly know what it is like inside? Maybe all mass is collapsed into a single point, maybe it is spread evenly throughout the black hole, or maybe it is an entire universe with completely different space-time fabric. Ok, you can see that I am obviously not an expert hence my terminology, but I would really like to understand that better.

Nothing comes out of a black hole on a ballistic trajectory you said. Here you allow for other means of exiting a black hole. Is that what makes a black hole 'visible'? I have heard of jets coming out of black holes. Now I am not sure if they actually depart from the black hole or if this happens at the surface where a violent reaction of some kind forces half the mass to go in the black hole while the other half is ejected at tremendous speed to allow an actual escape. Where you hinting at something like this?

Joey

 

[Nugatory]

Nothing comes out of a black hole on a ballistic trajectory you said.

I said nothing comes out on any trajectory, whether ballastic (so that the concept of escape velocity applies) or not.

have heard of jets coming out of black holes. Now I am not sure if they actually depart from the black hole or if this happens at the surface

Neither. The jets are formed in the space outside the event horizon.

 

[Ibix]

Why is it that scientists are so certain about the state of the inside of a black hole?

We're certain about what general relativity says happens inside a black hole. You are correct that we don't have any direct experimental evidence, so "what the models say" is the best we can do. But we have a fairly high degree of confidence in the model (it's never failed an experimental test we can do), and proposing that it fails abruptly at the event horizon is abit like proposing that the Sun will rise in the west tomorrow. Sure, we can't rule it out (predictions can be wrong), but it would require an abrupt change to the way the world works and our experience is that that sort of thing doesn't happen. When models fail, they fail fairly gently.

That said, we know general relativity goes wrong somewhere inside a black hole. There are serious scientists who say it goes wrong at the event horizon and there are serious scientists who say it goes wrong far further on. There isn't any agreement yet and we don't have any experimental evidence for which of the many successor theories of gravity is correct, so GR's predictions are the best we can do.

Nothing comes out of a black hole on a ballistic trajectory you said.

That's not what he said. He said that "an escape velocity of $c$" would mean that ballistic objects could not leave but powered objects could. But that is not how black holes operate - nothing can escape, full stop. The point is that escape velocity isn't a meaningful concept at or below the event horizon.

 

[Alex Ford]

I have a couple of simple questions: Do black holes always exert their gravity in a swirl that corresponds, in observance from our vantage point, left to right? Are there right to left.

On Earth apparently above the equator (currently) water goes down the drain counter-clockwise. But then I've heard that below the equator, water reverses direction. I presume that this is to do with the gravity waves that start positively at Polar North and are negatively started at the South Pole. Since nature repeats itself at all levels, does that mean that there are black holes where the material of a feeding star are pulled in by gravity in a swirl that is in the opposite direction? What would this say about the gravitational force in terms of its polarity.

The answer to these questions would provide much more information with regard to black holes and black hole physics (physics that apply to a singularity that may or may not be different to quantum mechanics and other forms of physics. Alex Ford, Sep 13, 2020.

 

[Ibix]

I have a couple of simple questions

You seem to have a lot of confused concepts about rotating black holes. You can't describe rotation as "left to right" or "right to left". You can describe it as clockwise or counter-clockwise, but the same object viewed from opposite sides of its equator spin opposite ways - just think of the hands of a clock viewed from behind it.

The swirling motions of large storms are not relevant here (incidentally, that drain water swirls one way in the northern hemisphere and the other way in the southern is a myth - the Coriolis forces are far too tiny to have an effect on that scale). That is an interaction between the Earth's spin and moving air, which is not a relevant concept for black holes. It has absolutely nothing to do with gravitational waves (nor gravity waves, which are a kind of surface wave on water).

Since nature repeats itself at all levels, does that mean that there are black holes where the material of a feeding star are pulled in by gravity in a swirl that is in the opposite direction?

Except for very close to a rotating black hole, matter may move either way around a black hole. Very close, inside what is called the "ergosphere", it is impossible for matter to move in a sense opposite to the black hole's spin.

What would this say about the gravitational force in terms of its polarity.

Gravity isn't a force, and doesn't have a polarity in any meaningful sense.