The discussion centers on the nature of black holes, specifically addressing the question of whether their diameter is finite or infinite. Participants clarify that a black hole does not have a well-defined diameter or volume due to the singularity at its core, which is not a spatial point but a moment in time. The classical General Relativity (GR) model, while mathematically consistent, does not provide a complete understanding of black holes, as it excludes the singularity from the manifold. The conversation emphasizes the necessity of mathematical comprehension to grasp the complexities of black holes and their properties.
PREREQUISITESAstronomy students, physicists, and anyone interested in advanced concepts of black holes and spacetime geometry will benefit from this discussion.
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 said: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.
I never claimed it was the volume of a black hole.Ibix said: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.
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.cmb said: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).
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.cmb said:I'm interested to hear why that is not a helpful calculation to make,
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 said:only a mistake to misinterpret the answer says something it doesn't, (which I didn't).
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.cmb said:I have seen explanations of black holes that describe it (a given mass) as collapsing to a singularity with 'infinite' gravitational characteristics.
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.cmb said:I understood that to mean with no volume. Otherwise, what does 'collapsing to a singularity' mean?
That is the only way to determine the hole's mass.cmb said:If one calculates the mass of a black hole from the gravitational fields around it
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.cmb said:one then calculates the geometric volume within the event horizon
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.cmb said:the event horizon (which is a 3D volume).
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 said:it seems questions can be wrong
A vague description of what?PeterDonis said: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.
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.cmb said:A vague description of what?
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.cmb said:You seem to strive on probing on very specific points, but don't ask very specific questions.
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.cmb said:Do you think this has been a friendly sort of exchange to help me with my understanding of this sort of thing?