Undergrad Black Hole Diameter: Is It Finite or Infinite?

[Ibix]

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).

 

[Ibix]

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'.

 

[PeterDonis]

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.

 

[cmb]

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.

A vague description of what?

It was the answer to your question 'why do you think that' and this is an absolutely clear as a bell answer why I thought that.

At no point did you say 'can you please give me the basis for your reasoning about black holes?', so of course I never gave you an answer to that.

"I have seen..." is a good answer to "why do you think...?". You seem to be thinking that you asked a different question and deserve a different answer.

If you had asked "how can you justify that?" then I'd not have given that answer. I'd have said I can't and might then have directed you at the text which caused me to think that. (Frankly, pretty much every lay-person's text on the subject says it, find one that does not. Try wikipedia.)

You seem to strive on probing on very specific points, but don't ask very specific questions.

Do you think this has been a friendly sort of exchange to help me with my understanding of this sort of thing?

Don't worry, I am off this thread now. Nothing more to ask/say.

 

[PeterDonis]

A vague description of what?

Of the collapse of a massive object to form a black hole. I did not mean to imply that your answer to my question was vague; I agree that it wasn't.

You seem to strive on probing on very specific points, but don't ask very specific questions.

You seem to be getting very hung up on the particular words I used to ask one particular question, instead of reading my latest response, post #35, as a whole and thinking about what it says.

Do you think this has been a friendly sort of exchange to help me with my understanding of this sort of thing?

I think that, if you want to gain a better understanding of how black holes actually work, you should read my post #35 and think about what it says.