# Black hole diameter

• I
Johan0001
TL;DR Summary
consequences of space time curvature
Hi All
I'm sure this question has been covered previously , but when searching I do not find a definitive answers.
I recently watch some talks given by Kip Thorne that had me thinking about black holes and their densities.

So my deduction is as follows .

Using General relativity, and the notion that space time converges into some king of singularity inside the event horizon.
Which to me is not palatable.

This would imply that space inside the event horizon stretches toward this singularity
If I could walk around the event horizon I would get some finite value for its circumference.
But assuming I could traverse the black hole and walk through it to measure the diameter . I would walk forever into the singularity.
What I mean by this is that the Diameter is infinitely large, or undefined.

Am I correct in this assumption?

If so it would mean that the volume of the black hole (assuming we use the sphere behind the event horizon) would be infinite as well?

since r^3 is infinitely large?

Am I correct in this assumption.

If this is so then the Density would be infinitely small or zero?

Density =Mass/Volume

Am I correct in this assumption.

But how can it be that there is infinitely large amount of space inside the black hole .
Surely the space inside the black hole must be finite, and cannot go on forever.

Any comments to direct me into the right path?

## Answers and Replies

Mentor
black holes and their densities
A black hole does not have a well-defined density, because it does not have a well-defined volume. It is not an ordinary object.

the notion that space time converges into some king of singularity inside the event horizon.
Which to me is not palatable
If by "not palatable" you mean that you don't think it's physically reasonable, many physicists agree with you; that's one of the main things that most physicists hope a theory of quantum gravity, if and when we find one, will fix.

However, as a mathematical model, the classical GR model with a singularity inside a black hole is perfectly consistent.

This would imply that space inside the event horizon stretches toward this singularity
No, it doesn't. Nor is this true.

If I could walk around the event horizon
You can't.

I would get some finite value for its circumference.
It is true that a black hole's event horizon, or more precisely a single 2-sphere on the horizon, has a finite circumference; but there is no way to directly measure it by walking around it.

assuming I could traverse the black hole and walk through it to measure the diameter
You can't. The space inside a black hole is not an ordinary "interior" of a 2-sphere. There is no spatial center. The singularity at ##r = 0## is not a point in space at the center; it is a moment of time which is to the future of all events inside the horizon.

I would walk forever into the singularity.
No, you wouldn't. Once you are inside the horizon, you will reach the singularity in a finite time. However, that doesn't mean what you think it means. As above, the singularity is a moment of time in your future, not a place in space. You can't avoid it once inside the horizon, and will reach it in a finite time, for the same reason you can't avoid reaching tomorrow and will reach it in a finite time.

the Diameter is infinitely large, or undefined
It is undefined. However, that does not mean your argument is valid. See above.

If so it would mean that the volume of the black hole (assuming we use the sphere behind the event horizon) would be infinite as well?
No, the volume is also undefined. More precisely, you can cut spacelike slices through the interior of the black hole that have any volume you want, from zero to infinity.

how can it be that there is infinitely large amount of space inside the black hole
Because the spacetime geometry inside it is nothing like the ordinary space you are used to.

pinball1970, PeroK, Ibix and 1 other person
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A black hole does not have a well-defined density, because it does not have a well-defined volume. It is not an ordinary object.

If by "not palatable" you mean that you don't think it's physically reasonable, many physicists agree with you; that's one of the main things that most physicists hope a theory of quantum gravity, if and when we find one, will fix.

However, as a mathematical model, the classical GR model with a singularity inside a black hole is perfectly consistent.

No, it doesn't. Nor is this true.

You can't.

It is true that a black hole's event horizon, or more precisely a single 2-sphere on the horizon, has a finite circumference; but there is no way to directly measure it by walking around it.

You can't. The space inside a black hole is not an ordinary "interior" of a 2-sphere. There is no spatial center. The singularity at ##r = 0## is not a point in space at the center; it is a moment of time which is to the future of all events inside the horizon.

No, you wouldn't. Once you are inside the horizon, you will reach the singularity in a finite time. However, that doesn't mean what you think it means. As above, the singularity is a moment of time in your future, not a place in space. You can't avoid it once inside the horizon, and will reach it in a finite time, for the same reason you can't avoid reaching tomorrow and will reach it in a finite time.

It is undefined. However, that does not mean your argument is valid. See above.

No, the volume is also undefined. More precisely, you can cut spacelike slices through the interior of the black hole that have any volume you want, from zero to infinity.

Because the spacetime geometry inside it is nothing like the ordinary space you are used to.
+1

This saved me 15 minutes of typing. A very complete answer.

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Any comments to direct me into the right path?
Learn the maths is, unfortunately, about the only solution there is to understanding black holes properly. The problems with your post, I think, can be summarised by noting that neither the event horizon nor the singularity are places in the ordinary sense of the word (the former is a null surface and the latter is more like a moment in time)n so walking around or through them makes no sense. Also spacetime cannot be stretched or not stretched - it does not have mechanical properties like that.

Depending on your background, Sean Carroll's lecture notes on GR are a good source.

Johan0001
thank you guys for your comments

Johan0001 said: This would imply that space inside the event horizon stretches toward this singularity No, it doesn't. Nor is this true.

Peterdonis can you elaborate on this , my analogy of space time curvature close to the black hole was a stretch like effect , similar to the rubber band surface with a mass in the middle that creates a depression which "stretches " space time or the rubber surface around it. Obviously this is not correct.

How can one envision such a curvature or warping of the spacetime?

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similar to the rubber band surface with a mass in the middle that creates a depression which "stretches " space time or the rubber surface around it. Obviously this is not correct.
The Earth's surface is curved, but not stretched. So it is with spacetime.

The problem with the usual "rubber sheet" model is that it only describes spatial curvature - and it misses out curvature in the plane perpendicular to that which is actually more important. Furthermore there is no unique definition of "space" inside the event horizon and you can't sensibly draw that diagram.

Kruskal diagrams are my favourite depiction of a black hole. They show a slice through spacetime in the radial and time-like direction and they cover everything from infinity to singularity (and a couple of extra features like a white hole and a second exterior that are a result of this being an idealised model).

pinball1970
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my analogy of space time curvature close to the black hole was a stretch like effect , similar to the rubber band surface with a mass in the middle that creates a depression which "stretches " space time or the rubber surface around it.
Aside from the limitations that @Ibix described, there is one very important point to understand about the "rubber sheet" model: it only applies to the region outside the hole's horizon. The "rubber sheet" model does not apply to the region of spacetime inside the horizon at all.

How can one envision such a curvature or warping of the spacetime?
Spacetime curvature is the same thing as tidal gravity. So the best way to envision the curvature of spacetime inside the horizon is to imagine tidal gravity getting stronger and stronger as you get closer to the singularity. The "shape" of the tidal gravity is best described by its effects on objects: it either stretches or squeezes them, or some combination of the two. A black hole's tidal gravity is the same as the tidal gravity around any massive body: it stretches objects along the radial direction and squeezes them along the tangential directions. The stretching and squeezing get stronger and stronger as you approach the singularity.

Johan0001
Thanks guys , a lot of food for thought , fascinating stuff.
Need to think about this a lot more and get some good references to study.
I usually watch talks given by Penrose, Susskind, Brian Greene, Carrol to keep me on track.

The math is my problem though.

Regards

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The math is my problem though.
Then that is what you need to work on if you want understanding. Verbal descriptions are fine if you want a few facts to impress people with at parties, but if you want to be able to reason about them then you need the maths. Verbal descriptions are limited and error prone, and are open to wrong interpretation even when a competent scientist is being as clear as they can be.

If you want to learn and you haven't studied special relativity you'll need to do that first, since general relativity builds on those concepts. It ought to be in your reach with just high school maths. I like Spacetime Physics by Taylor and Wheeler, now a free download from Taylor's website, but there are plenty of other options. Do the exercises, and ask us if you have any questions or you get stuck.

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However, as a mathematical model, the classical GR model with a singularity inside a black hole is perfectly consistent.
Sort of. The singularity itself is excluded from the manifold.

As for diameter, the diameter of a black hole is not well-defined. The only well-defined size parameter for a black hole is the area of its horizon.

Mentor
Sort of. The singularity itself is excluded from the manifold.
The singularity being excluded from the manifold does not make the manifold itself inconsistent. A more precise way of stating what I was trying to state is that the classical GR model of a black hole with spacetime curvature increasing without bound as ##r \rightarrow 0## (where ##r## here is the "areal radius", i.e., based on the invariant geometric areas of 2-spheres, not a coordinate) is perfectly self-consistent. But, as you note, this model does not include the locus ##r = 0##.

Mentor
the diameter of a black hole is not well-defined
Yes, agreed.

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The singularity being excluded from the manifold does not make the manifold itself inconsistent. A more precise way of stating what I was trying to state is that the classical GR model of a black hole with spacetime curvature increasing without bound as ##r \rightarrow 0## (where ##r## here is the "areal radius", i.e., based on the invariant geometric areas of 2-spheres, not a coordinate) is perfectly self-consistent. But, as you note, this model does not include the locus ##r = 0##.
Yes, it is consistent. But it's also incomplete by construction. Which is what I meant by "sort of".

Mentor
it's also incomplete by construction.
It's geodesically incomplete. But that's not the same as being mathematically incomplete or inconsistent. Geodesic completeness is an expectation of physics, not math.

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It's geodesically incomplete. But that's not the same as being mathematically incomplete or inconsistent. Geodesic completeness is an expectation of physics, not math.
Right. I don't quite know what point you're trying to make here.

The point I'm trying to make is to hint at an interesting feature of General Relativity: it predicts that there are parts of the universe that the theory simply cannot describe, indicating that the theory is incomplete as a physical theory.

Mentor
I don't quite know what point you're trying to make here.
I'm trying to make sure the OP does not get confused.

The point I'm trying to make is to hint at an interesting feature of General Relativity: it predicts that there are parts of the universe that the theory simply cannot describe, indicating that the theory is incomplete as a physical theory.
I agree that this is a common opinion among physicists, and is one of the key issues driving the search for a theory of quantum gravity. I made a similar point in post #2, which was the original post of mine that you responded to.

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Right. I don't quite know what point you're trying to make here.

The point I'm trying to make is to hint at an interesting feature of General Relativity: it predicts that there are parts of the universe that the theory simply cannot describe, indicating that the theory is incomplete as a physical theory.
Which parts can the theory not describe?

cmb
I thought the radius of a black hole was precisely zero, but it's the space around it that is not well defined?

Are we into the realm of 'what do words mean' rather than physics? I mean, as you approach the thing, what 'is' the black hole and what is not? Could it be defined as a 'pure gravity field' in which it exists outside the event horizon too with such a definition (two orbiting black holes could therefore co-exist in a given space), or is there real physical mass in there somewhere (whatever it is we call a black hole)?

I think the answers to black holes often make as little sense as the questions!

Mentor
I thought the radius of a black hole was precisely zero
Why would you think that? It's not correct. The radius of a black hole is not well defined. But the area of its horizon is, and it is not zero.

it's the space around it that is not well defined?
Why would you think that?

Are we into the realm of 'what do words mean' rather than physics?
No. I have already referred to a perfectly consistent mathematical model in this thread that describes an idealized black hole. It is not a matter of words.

as you approach the thing, what 'is' the black hole and what is not?
The event horizon is the boundary of the spacetime region that is the black hole. It is a perfectly well-defined boundary. It is not locally detectable, but that doesn't make it any less well-defined.

Could it be defined as a 'pure gravity field' in which it exists outside the event horizon too
The spacetime of a black hole is curved everywhere, not just inside the horizon, so effects that could be called a "gravity field" are not limited to inside the hole. From far away, the hole's gravitational effects are no different from those of any other object with the same mass.

is there real physical mass in there somewhere
For an idealized "eternal" black hole, which is a mathematically valid solution but is not physically reasonable, no, there is no "physical mass" (nonzero stress-energy) anywhere; the solution is vacuum everywhere.

For a real black hole that forms from the gravitational collapse of a massive star, there is a region of spacetime inside the hole that is not vacuum, but is occupied by the collapsing matter that formed the hole. But unless you fall into the hole very soon after it forms, you will not be able to reach that matter region; all you will encounter inside the hole is vacuum.

In any case, the mass of the black hole itself is best thought of, not as a property of the matter that collapsed to form it, but as a property of the spacetime geometry as a whole.

Mentor
I think the answers to black holes often make as little sense as the questions!
Answers that don't involve math always run that risk, yes.

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I thought the radius of a black hole was precisely zero, but it's the space around it that is not well defined?
The black hole is usually defined to be everything inside the event horizon. That has a well defined radius, although it has to be defined formally in terms of the area of the horizon since you can't lay a ruler through the middle even in principle. Popsci sources often talk about the singularity as if it were a zero-dimensional point, but this is (a) not the same as the black hole, and (b) wrong.

Mentor
That has a well defined radius, although it has to be defined formally in terms of the area of the horizon
What you are calling the "radius" is more precisely called the "areal radius", i.e., ##\sqrt{A / 4 \pi}##, where ##A## is the area, but that name is misleading since it suggests that there is some actual spatial radius involved, which, as you note, there isn't.

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What you are calling the "radius" is more precisely called the "areal radius"
Indeed - but we often talk about the Schwarzschild radius as the radius of the event horizon. Formally we mean that the Schwarzschild radius is the areal radius of the horizon, but we often don't say so.

Mentor
Indeed - but we often talk about the Schwarzschild radius as the radius of the event horizon. Formally we mean that the Schwarzschild radius is the areal radius of the horizon, but we often don't say so.
Yes, but for this particular thread, given its title, I think it is important not to gloss over the issue like that.

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Yes, but for this particular thread, given its title, I think it is important not to gloss over the issue like that.
I agree, and you were correct to point out the distinction. The point I was trying to make in my last post is that we normally don't bother to specify "areal" radius for a black hole because we can't mean anything else because it's not definable. So if @cmb is doing any further reading, he/she shouldn't take the fact that (almost) everywhere talks about the "radius" of an event horizon to contradict the point that it only has an areal radius.

cmb
Why would you think that? It's not correct. The radius of a black hole is not well defined. But the area of its horizon is, and it is not zero.

Why would you think that?
Because I have seen explanations of black holes that describe it (a given mass) as collapsing to a singularity with 'infinite' gravitational characteristics.

I understood that to mean with no volume. Otherwise, what does 'collapsing to a singularity' mean?

Please explain it further to me courteously if you can, it feels like there is some sort of vexation element to your phrasing because I don't already understand this.

Let me ask a different question then, which might have been the question the OP wanted to ask;-

If one calculates the mass of a black hole from the gravitational fields around it, and one then calculates the geometric volume within the event horizon, what result (range of results) does/would one get dividing the latter into the former?

cmb
I've put in numbers from wiki for Cygnus X-1 (that is, assumed 21.2 solar masses), and using the Schwarzschild radius equation.

I get that the 'averaged density' of all 'stuff' inside the event horizon is ~4.2E16 kg/m^3.

This is curious as the density of a neutron star is given an order of magnitude higher. Does that mean the centre of a neutron star could be a black hole, and somehow a neutron star has formed a shell around it?

I will accept I might have made an error putting numbers into the calculator, but did it twice for the same result.

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I understood that to mean with no volume.
You often see this written, but it's wrong. The singularity turns out to be more like a time than a place, because the direction of decreasing ##r## is spacelike outside the horizon but timelike inside. That's one explanation for why you can't escape once you are inside the horizon - the singularity is literally in your future.
Otherwise, what does 'collapsing to a singularity' mean?
It means ordinary language isn't very good at this, I'm afraid.
If one calculates the mass of a black hole from the gravitational fields around it, and one then calculates the geometric volume within the event horizon, what result (range of results) does/would one get dividing the latter into the former?
The volume inside a black hole isn't well defined since there's no unique notion of space. Surfaces of constant Schwarzschild ##r## is probably as close as you can come, and they are infinite.
I get that the 'averaged density' of all 'stuff' inside the event horizon is ~4.2E16 kg/m^3.
I presume you did this by calculating ##M/V## where ##V=\frac 43\pi R_S^3##. Here you are assuming a Euclidean geometry to calculate the volume, which isn't anything like the Kerr geometry of a rotating black hole. So the result you get is meaningless - the Euclidean formula for volume is inappropriate and there isn't really an appropriate way to define the interior volume. Certainly not one that's comparable to a Euclidean volume. That's why your comparison to a neutron star is giving you weird results.

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PeroK
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I get that the 'averaged density' of all 'stuff' inside the event horizon is ~4.2E16 kg/m^3.
The mistake you are making is loosely analogous to deciding to paint the interior of a trumpet (for some good reason...), measuring the circumference of the rim of the bell, dividing by ##2\pi## to get the radius, calculating ##\pi r^2##, and buying paint to cover that area. You're going to run out of paint. Kerr and Schwarzschild geometries are somewhat more complicated and there's no singularity in a trumpet, but it's a loose analogy.

cmb
You often see this written, but it's wrong.
Yes, I do often see that, and I put it as answer the question I was asked; why I thought that.

The mistake you are making is loosely analogous to deciding to paint the interior of a trumpet (for some good reason...), measuring the circumference of the rim of the bell, dividing by ##2\pi## to get the radius, calculating ##\pi r^2##, and buying paint to cover that area. You're going to run out of paint. Kerr and Schwarzschild geometries are somewhat more complicated and there's no singularity in a trumpet, but it's a loose analogy.
Not sure what 'mistake' means. I calculated the answer to a well-defined question, but your point is that the answer doesn't tell us anything.

However, what it seems to say is that the volume inside a black hole is actually smaller than the volume the event horizon occupies as a sphere.

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Not sure what 'mistake' means. I calculated the answer to a well-defined question, but your point is that the answer doesn't tell us anything.
The mistake is thinking that ##\frac 43\pi R_S^3## has anything to do with the interior volume of a black hole. In fact, as I said, there isn't a sensible way to define the interior volume of a black hole. So your calculation has nothing to do with a density, which is why your comparison to neutron star densities gives you strange results.

cmb
The mistake is thinking that ##\frac 43\pi R_S^3## has anything to do with the interior volume of a black hole. In fact, as I said, there isn't a sensible way to define the interior volume of a black hole. So your calculation has nothing to do with a density, which is why your comparison to neutron star densities gives you strange results.
I never claimed it was the volume of a black hole.

It cannot be a 'mistake' because I asked what is a typical ratio of mass to volume of the 2-sphere defining the event horizon (which is a 3D volume).

Please note I did not call nor define this as a density.

I'm interested to hear why that is not a helpful calculation to make, and you are saying is that the 3D volume of the event horizon does not correlate with the physical reality of what is within the event horizon. I do understand that from what has been put previously. But it cannot be a 'mistake' to ask that question, only a mistake to misinterpret the answer says something it doesn't (which I didn't).

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It cannot be a 'mistake' because I asked what is a typical ratio of mass to volume of the 2-sphere defining the event horizon (which is a 3D volume).
You actually asked about "the geometric volume within the event horizon". As I pointed out, the geometry here is not Euclidean, and using the Euclidean formula for the volume of a sphere does not give you the interior volume of the 2-sphere. The volume of the interior of this particular 2-sphere has no unique definition.
I'm interested to hear why that is not a helpful calculation to make,
Because what you are doing is like trying to calculate the surface area of the northern hemisphere of the Earth by measuring the equator, deriving a radius, and calculating ##\pi r^2##. Euclidean geometry is the wrong tool for the job.
only a mistake to misinterpret the answer says something it doesn't, (which I didn't).
You immediately started speculating (in #27) about neutron star interiors based on interpreting your number as a density. But, as I pointed out, it isn't a density which is why you get weird results when you try to compare it to a density.

cmb
OK, yes I considered the density of neutron stars and figured there was something anomalous there. I withdraw my question of whether such a thing is possible, because it seems questions can be wrong and that was one of those 'wrong questions'.

Mentor
I have seen explanations of black holes that describe it (a given mass) as collapsing to a singularity with 'infinite' gravitational characteristics.
This is a very vague description (and you should give specific references instead of just saying "I have seen") and is not a good basis for reasoning about black holes.

I understood that to mean with no volume. Otherwise, what does 'collapsing to a singularity' mean?
The collapsing matter does end up at zero volume (at least, in the idealized classical model we are discussing). But that does not mean the black hole has zero volume. The black hole is not the collapsing matter. It is a configuration of spacetime geometry that is left behind by the collapsing matter.

If one calculates the mass of a black hole from the gravitational fields around it
That is the only way to determine the hole's mass.

one then calculates the geometric volume within the event horizon
One can't; there is no such thing. As has already been said earlier in this thread, the "volume" of a black hole is not well-defined.

the event horizon (which is a 3D volume).
No, it isn't. The event horizon is a 2-sphere. More precisely, it's an infinite series of 2-spheres connected along a family of outgoing null geodesics. But we can just look at anyone of the 2-spheres since they all have the same area.

The key thing to understand about the event horizon as a 2-sphere is that it does not enclose an ordinary 3D volume. The geometry of spacetime inside the event horizon is not the ordinary 3D geometry you are used to. It is something very different. So you cannot reason about it the way you would reason about an ordinary 3D volume enclosed by a 2-sphere surface. You just can't.

it seems questions can be wrong
It's not quite that the questions you are asking are wrong, it's that you don't yet appear to have grasped that the issue is not the particular questions you're asking, but the whole underlying conceptual scheme you are using to ask them. That whole underlying conceptual scheme is what you need to discard when you are talking about black holes.