Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: rwjefferson on 14/11/2010 23:05:30

Is black hole mass infinitely dense?
Is this proved or dogma?
peace
rwjefferson
force is inertial differential

I look forward to an answer to this question, because my own belief is that it is not possible to measure infinite density (or infinite anything), so any such claim could be only theoretical.

I don't think anyone can definitively answer this question since our theories don't cover it, and we certainly can't look inside a black hole to see the answer. I think most people would guess that a black hole isn't infinitely dense, because infinities tend not to actually exist in nature (although we use them as good approximations to a lot of things when working out mathematical models).

It isn't proven as such. Penrose and Hawking singularities dictate that a black hole has a singular region which is infinitely dense. Infinitely dense in the sense matter is crushed to a certain point inside the black hole. By all practical means, it seems correct since information cannot escape from our universe.

The density of the smallest common black holes formed by the collapse of a star is quite easy to calculate, assuming the mass to be 1.989*10^30 *1.4 Kg= 2.785*10^30 Kg and the diameter (re Hawking) to be 8.272 Km the density works out at
1.3 *(1,392,000/8.272)^3=tons/M^3=6.1948*10^15 tons per cubic meter.
This of course the density of the blackhole as defined by the event horizon, the density of the so called central singularity is a matter for conjecture.

It is true that smaller black holes are very dense indeed but Infinity is a purely mathematical term and only applicable to mathematical and not physical matters. the ultimate or peak density of a collapsing black hole will be higher than the density needed to collapse it in the first place but I am sure it will NOT be infinite. The truth is, we don't know and cannot observe what happens but we know very well a lot of physical things that could allow us to model it.
The simplest model could be based on a non rotating Swartzchild black hole and Hawking radiation. As the black hole collapses the event horizon of which we are familiar prevents radiation from coming out to free space but it does not prevent radiation from escaping for a short distance before falling back into the hole. Each layer of this event horizon can radiate hawking radiation even though it will eventually fall back but this takes time and saps the energy of the hole as the matter continues to collapse towards a singular point this radiation gets more and more intense and continues to sap the hole until it reaches an equilibrium point. It is possible to calculate the size of this point (although I have not done it yet) but is most definitely not infinitely small and is probably larger than a Planck unit so this does not require quantum gravity. Just saying the collapse is to a mathematical singularity is really a copout used by early workers in the field before Hawking radiation was conceived that has been copied by most of the popular writers in the field.
Its about time this fallacy was put to bed by someone with authority. Because it causes endless questions like this and prevents people from turning their thoughts to more useful matters.
It is interesting to note that like white dwarf stars and neutron stars the more massive the hole is the smaller the final collapse will be because of the greater energy that will have to be generated. This is true even though the first event horizon of the hole becomes larger as the mass of the hole increases. This is a linear function of about one mile for every solar mass inside.

Infinity is a sign that there is something wrong with the theory  the physics needs refining. In order to know what happens in a black hole we need a theory of quantum gravity, which we do not have at present, but it is likely that at very small scales something happens to prevent us getting to infinity.

Piakia I agree that a theory of quantum gravity would be very helpful that is what everyone says but I do not believe that it is essential for any progress to be made. A great deal of work has been done for many years on the topic of string theory and quantum gravity and is still going on. This has produced a vast number of potential mathematical models with no link with reality and no way of selecting which one may be correct. I strongly believe that we have not used enough modelling based on what we DO know to help direct us to solutions of this problem. Notably the detailed modelling using standard classical and quantum theories of the processes of collapse inside the event horizon of a rotating black hole right up to the limit where our theories fail. I strongly believe that the devil is in the detail.
We know that the Kerr black hole collapses to a "ring" singularity. The equations have been solved. This is just the same as saying we know that Our universe will eventually die through the thermodynamic heat death. This does not mean that interesting things do not happen on the way, like us observing our universe!

There is ample evidence that stellar mass and super large blackholes exist but is there any evidence that small ones exist ?, the universe is not old enough or cool enough for stellar mass ones to have lost any of their mass so the only way small ones could exist is if they were created at the time of the 'bigbang'.
Have any been detected ?.

Infinity is a purely mathematical term and only applicable to mathematical and not physical matters.
Wow!!! Someone with (possibly infinitely) more knowledge of the subject than I, has actually said what I have frequently been castigated for saying, in another forum. I'm going to celebrate. Thanks, Soul Surfer. [;D]

Infinity is a purely mathematical term and only applicable to mathematical and not physical matters.
Wow!!! Someone with (possibly infinitely) more knowledge of the subject than I, has actually said what I have frequently been castigated for saying, in another forum. I'm going to celebrate. Thanks, Soul Surfer. [;D]
Well, it's a matter of opinion, or even choice of what kind of infinities we are dealing with. It isn't necesserily a true statement. For instance, infinities pop up in theory all the time. A good example is an infinite spacetime void. This is actually acceptable, and in every sense of the theory, is a physical infinity as well.

infinities pop up in theory all the time. A good example is an infinite spacetime void. This is actually acceptable, and in every sense of the theory, is a physical infinity as well.
For "infinite spacetime void" substitute "boundless spacetime void", it fits the theory just as well, but could involve a completely different physical reality.

A truly mathematically infinite universe requires there to be an infinite number of identical earths in which every person is doing the exact same thing as we are doing at this moment and an greater infinity of worlds like the earth where people are doing slightly different things and that is before we come to a range of other options like rearranging the stars.
This to my mind is completely stupid and that is why I much prefer the concept of unbounded or indefinite for the physical concept.

I much prefer the concept of unbounded or indefinite for the physical concept.
So do I !!!
I’m not expert enough where mathematical infinities are concerned too be able to make any profound comments about what any of the more complex forms might be like, but for a true physical infinity, I think you have not gone far enough. If you apply Cantor's "infinity of infinities" to the Universe you would, as you say, have infinite numbers of everything, and everything that happens, happening an infinite number of times. However, this would be just a mathematical infinity. A mathematical ‘reality’ means only logical selfconsistency and this does not require physical existence to validate it.
One often meets statements like: “In infinity everything that can happen will happen, an infinite number of times.” This statement contains at least two elements that are at variance with the concept of physical infinity: “…will happen”, because there can be no past or future in infinity; and “…number of times”, because there can be no numbers, or differentiated times in infinity. I said “at least two” because, strictly, “happen” has no place in infinity because it implies action, and action involves change.
How, then, can we reword the original statement so as to bring it into line with the concept of physical infinity? It would have to be something like: “Every thing that can be, is”.

A truly mathematically infinite universe requires there to be an infinite number of identical earths in which every person is doing the exact same thing as we are doing at this moment and an greater infinity of worlds like the earth where people are doing slightly different things and that is before we come to a range of other options like rearranging the stars.
This to my mind is completely stupid and that is why I much prefer the concept of unbounded or indefinite for the physical concept.
Mindboggling, yes, but not stupid; it is an inevitable consequence of an infinite universe. I find it far more plausible and easy to understand than a finite universe, with the weird geometry that it implies.

Mindboggling, yes, but not stupid; it is an inevitable consequence of an infinite universe. I find it far more plausible and easy to understand than a finite universe, with the weird geometry that it implies.
But an infinite (unbounded) universe has infinities in it! That's not at all a comfortable concept IMO. 'weird' geometry is walk in the park in comparison!

Quot homines tot sententiae; but as long as you discuss mathematical infinities as though they were synonymous with physical infinity you are likely to keep going round in circles.

Quot homines tot sententiae; but as long as you discuss mathematical infinities as though they were synonymous with physical infinity you are likely to keep going round in circles.
Or at least go round in a circle that after an infinite amount of time will bring you back to your starting place

A truly mathematically infinite universe requires there to be an infinite number of identical earths in which every person is doing the exact same thing as we are doing at this moment and an greater infinity of worlds like the earth where people are doing slightly different things and that is before we come to a range of other options like rearranging the stars.
This to my mind is completely stupid and that is why I much prefer the concept of unbounded or indefinite for the physical concept.
I find this far less weird than lots of things in quantum mechanics, which does appear to be real, so I don't think it's stupid at all.
The one thing I think everyone can agree on about whether infinity can be physically realized is that no one has proof one way or the other, just opinions.

The one thing I think everyone can agree on about whether infinity can be physically realized is that no one has proof one way or the other, just opinions.
Absolutely true (at least in my opinion) [;D]
The more I think about infinity, though, the more I find it difficult to escape various "opinions"; two of which are:
1. Nothing that is finite can become infinite.
2. Accepting a physical infinity is preferable to being stuck with infinite regression.

I like the way you say it Bill [:o)], but your first assumption is quite a revelation!!!

your first assumption is quite a revelation!!!
This means that either I have it wrong, or you need to do some more thinking. [:\]
Give me an example of something finite that becomes infinite, we can work from there.

I mean this assumption is right: nothing finite can become infinite.

nothing finite can become infinite
Great! Someone who is willing to put this in writing. Would you also agree about the next step: "Anything that is infinite must also be eternal"?

it sounds very logical to me, but some people might disagree about the properties of time...
And what about something eternal, does it have to be infinite?

And what about something eternal, does it have to be infinite?
Before I try to answer that one, let me ask you one more question, just to make sure we are on the same page.
What is the difference between infinite and eternal?

Infinite in space and eternal (infinite) in time.

We are all playing with words because we just do not know. But things in this universe are logical and black holes with huge negative masses are certainly not logical; and neither are worm holes to other universes.
There is an alternative electric option. What if super massive black holes are at the centre of every galaxy like a giant magnetic hub around which all the stars rotate in a set order. The magnetic field not only spins the stars into position [ which gravity cant] but can accept feedback to avoid collisions. If collisions occur then the tiny galaxy may turn into a vibrating quasar or a magnetar. What do you think?
CliveS

There is a lot of playing with words, it can be fun, but is usually not very productive. However, there are also lots of lines of discussion that are less productive than they could be, simply because different people ascribe different shades of meaning to particular words. Infinity is a concept that suffers greatly in this regard. For example; it is important to establish if we are talking about mathematical infinities or a physical infinity. It is also crucial to distinguish between infinite and boundless. That's before getting into some of the finer points. Take, for example, ArkAngel's response to my question:
Infinite in space and eternal (infinite) in time
Infinity is not infinite space, nor is eternity infinite time. We have to think of them in that way because we are unable to even imagine the possible dimensions of infinity. While there is nothing wrong with ArkAngle's answer in terms of our 4D thinking, it is important to remember that it is a sort of "model", perhaps the nearest we can come to understanding the real thing. Just think about a spider walking through "Flatland". [;)]
You raise an interesting point, Clive, but it will take me a little while to get my brain around it. [:\]

Singularities are not necessarily 0dimensional points; for instance they may well be circles, spheres & hyperspheres; so energy 'density' never becomes infinite. [:I]

Infinity is not infinite space, nor is eternity infinite time. We have to think of them in that way because we are unable to even imagine the possible dimensions of infinity. While there is nothing wrong with ArkAngle's answer in terms of our 4D thinking, it is important to remember that it is a sort of "model", perhaps the nearest we can come to understanding the real thing.
How can we even know there's a "real thing" to be modelled? Just because we can imagine invisible pink unicorns doesn't mean they exist. Similarly, just because we can grok the concept of infinity doesn't necessarily mean anything actually is infinite.

just because we can grok the concept of infinity doesn't necessarily mean anything actually is infinite
Perhaps you should start by asking yourself if there can ever have been a time when there was nothing. It's surprising where you can go from there. [8D]

Similarly, just because we can grok the concept of infinity doesn't necessarily mean anything actually is infinite.
A Series can be! [;)] (tongue firmly in cheek!)
No one can say for sure that anything physical has infinite dimension (gravity, distance, etc).
Re: singularities 
Wikipedia  that bastion of all knowledge [;)] says of singularities:
"A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the scalar invariant curvatures of spacetime, some of which are a measure of the density of matter."
It also says there are theories that specify that singularities can not exist. mmmm [???]

A Series can be! (tongue firmly in cheek!)
Very sensible; you knew that would set me off if you said it seriously, didn't you? [:X]
"A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the scalar invariant curvatures of spacetime, some of which are a measure of the density of matter."
OK. You quoted it, you translate it so the plebs can understand it. [:P]

Is it possible that the universe has appeared from nothing? If not, the universe is eternal (even if maybe discrete in some way). But it doesn't mean it is necessarily infinite in the other dimensions.
It is open to a more casual and funny discussion i hope...
By experiences, Physics tend to say any singularity is very improbable, to say the least...

I just feel that we all can ever imagine to do with black holes theorize.In fact a black whole is a tunnel between universes. Black hole attracts matter that does not collapse into a point, as predicted, but a black one out the other end of the "white hole" say, the theory goes.

Is it possible that the universe has appeared from nothing?
If there had ever been a time when there was nothing, there would be nothing now. Manifestly, there is something now, so something must be eternal.
But it doesn't mean it is necessarily infinite in the other dimensions.
I've been trying to get my head round this for some time. Does "eternal" automatically imply "infinite"? I feel it should, but have yet to formulate the necessary argument to my own satisfaction.
Welcome, joshrego. Hope to see lots of posts from you. Don't take all of us too seriously, though.
In fact a black whole is a tunnel between universes.
Is this an assumption, or do you have evidence that would make William of Ockham turn in his grave?

Perhaps you should start by asking yourself if there can ever have been a time when there was nothing. It's surprising where you can go from there. [8D]
A similar question would be whether there could be a place where there was no time.
In both cases, the answer depends on my assumptions. If I assume that nothing cannot become something, I answer "No, there could not be a time when there was nothing." If I assume that nothing can become something, I answer, "Yes, there could have been nothing at one point in time, and then something at a later point." I'm not even sure what it would mean to have a place without time. How long would it last? :)
However, when I think of spacetime as a four dimensional structure, I can imagine bounds in both time directions.
Of course, I also have to ask whether what I can imagine has anything to do with what is.

I've been trying to get my head round this for some time. Does "eternal" automatically imply "infinite"? I feel it should, but have yet to formulate the necessary argument to my own satisfaction.
I think "eternal" just means "outside time", not "infinite in duration". Something eternal is something that doesn't have extension in the temporal direction. Its value is the same at all points on the time axis. I don't think we'd have any way of observing such an object since our experience is so aligned to the temporal dimension.
Wiktionary offers the following definition for "eternal": "(philosophy) existing outside time; as opposed to sempiternal, existing within time but everlastingly"

Something eternal is something that doesn't have extension in the temporal direction. Its value is the same at all points on the time axis.
This captures a major factor in the problem. If it "doesn't have extension in the temporal direction", how can it be present "at all points on the time axis"?

Something eternal is something that doesn't have extension in the temporal direction. Its value is the same at all points on the time axis.
This captures a major factor in the problem. If it "doesn't have extension in the temporal direction", how can it be present "at all points on the time axis"?
1) Its value might be zero at all points on the time axis.
2) Its value might be some fixed nonzero value. If it doesn't change with time, how would we notice its presence?

We noticed gravity and understood it well  at a point in our history when gravitational attraction was observable as a fixed (in place and in time) nonzero quantity

a black hole is mathematically defined as a singularity
what does this say about a black holes density  or any other property for that matter???

Its value might be zero at all points on the time axis.
I suspect you are still thinking in terms of mathematical infinities. Eternity can neither have, nor be placed on, a time axis. It is not time.
We noticed gravity and understood it well  at a point in our history when gravitational attraction was observable as a fixed (in place and in time) nonzero quantity
Are you saying that gravity is eternal?

Is black hole mass infinitely dense?
Is this proved or dogma?
peace
rwjefferson
force is inertial differential
It isn't proven as such. Penrose and Hawking singularities dictate that a black hole has a singular region which is infinitely dense.
We are all playing with words because we just do not know.
I don't think anyone can definitively answer this question since our theories don't cover it...
In other words, your own personal theories are new or disproved.
Bonus Questions
According to the dictates of Penrose and Hawking; how much dirt is in a hole 1x1x1x1?
What is the relative density at the eye of a drain?
Is black hole mass closer to infinitely less than dense?
Fellow Geezers
Force as inertial differential is not a new theory.
Force as inertial differential is not a disproved theory.
Force as inertial differential is the law of physics.
peace
ron

You are not getting away as easily as that. [:P]
Its value might be zero at all points on the time axis.
What does this mean?
If it has zero value, is it really there?
How can eternity be accommodated at any point on a time line?

Fellow Geezers
Force as......
Oi! I'll have you know there is only one Geezer on this forum.

....a black hole is really just a very dense defusionised star
A singularity is a mathematical construct
never the twain shall meeteth

Infinity exists, matter accelerated to lightspeed has to be an 'infinite slope', as matter otherwise would break the light barrier at some point if accelerated enough. And the predecessor to a black hole is a finite object becoming a singularity. It's like 'distance' though, plasticity incorporated, but I'm sure it exist, even if we never will 'see it' practically.

Any particle has a maximum acceleration point. Beyond that acceleration, the particle will break and changed into photons going at the speed of light...

Okay CPT :)
That's one solution, but as seen from its own frame, for example accelerating at one G continuously, it would take about five years to reach 0.999934 % of lightspeed with the equation being "v/c = tanh (at/c). Since tanh(at/c) is always less than 1, you can never reach the speed of light."
And to you that would be noticed as one Earth Gravity.

yes, but you will need an increasing toward infinity force to accelerate at a constant rate until the speed of light is reached, which is not possible. My point was about increasing acceleration though...

Yeah, I guessed that you might mean it that way :)

Truth passes through three stages.
First, it is ridiculed.
Second, it is violently opposed.
Third, it is accepted as being selfevident.
You are not getting away as easily as that. [:P]
Kindly stick to the subject. Please feel free to launch other topics if you wish to propose a new theory. Thanks!
....a black hole is really just a very dense defusionised star...
We are all playing with words because we just do not know.
Please answer the questions.
1) How dense is the dirt within a hole 1'x1'x1'?
2) What is the relative density of the eye of a hurricane after another million tons of airmass is swallowed?
3) Does a black hole demonstrate the most basic law of physics or the actions of spooky warp?
ItS
peace
r~

It's not very good form to quote from a post in a different thread, particularly when it's entirely out of context.

1) How dense is the dirt within a hole 1'x1'x1'?
2) What is the relative density of the eye of a hurricane after another million tons of airmass is swallowed?
3) Does a black hole demonstrate the most basic law of physics or the actions of spooky warp?
Are you on drugs?

I think Mr Kip Thorn said that there is no matter inside a black hole, "that all the matter is converted to the energy in the form of curvature of spacetime" If that is correct density whould have no meaning right? since there is no matter to have a density? But maybe your stuck with the same problem anyway. infinite energy density?

These theories you are expousing miss an extremely important point  they all attempt to view a black hole from the outside. What does a black hole look like from the inside? Well, at one time, the answer was fairly simple  "look around you  you are in one!" That is, if the universe was expanding at a decreasing rate, and eventually would contract into a "big crunch" by definition, the universe would be a black hole.
Recent information indicates thhat the universe may not collapse. If so, this observation is meaningless. But that has not been positively determined yet. So, possibly, we all live inside a black hole.

... infinite energy density?
Density is a property of matter; energy is antidense.
... positive or negative infinite energy density?
What is the energy density within the eye of a hypersonic cyclone? Does infinite energy density tell which way the cosmic wind will blow?
in the spirit of
peace
ron~

Density isn't the word for it any more. We do not know what 'energy' is although we know of its transformations. The center of a black hole is 'energy' and nobody have ever put a 'size' to energy, neither have we a measured a 'density' for it. It's like all other ideas where we go out from what we know here to describe what we never ever will be able to measure.
We can't know the inside of a black hole, we can have theories though :) And I have to admit that I love theories too ::)) But before someone show me a parcel of pure energy and then prove that it also have a 'density' I will wait with trusting to those descriptions:)

What is the energy density within the eye of a hypersonic cyclone?
Depends on the cyclone, but it's a finite number.

the predecessor to a black hole is a finite object becoming a singularity
Within this definition, is a singularity infinite? If so, are you saying that something finite becomes infinite?

Yor_on might have something else in mind, but I think the explanation is that general relativity treats it as infinitely dense, but in reality it probably isn't. Once it gets small enough, general relativity probably isn't an accurate theory anymore, and the infinite density that it gives is probably a result of the theory breaking down.
That said, you can talk about the singularity, to mean the point at which the theory breaks down.

I cannot see why what exists beyond the event horizon needs to of quasi infinite density, the matereal of a Neutron star has nearly a high enough density to produce an event horizon so why does the contents of a black hole have to be vastly greater

I cannot see why what exists beyond the event horizon needs to of quasi infinite density, the matereal of a Neutron star has nearly a high enough density to produce an event horizon so why does the contents of a black hole have to be vastly greater
A neutron star doesn't have enough density to produce an event horizon. Assuming it does is due to mistakenly applying a solution to GR which is valid in empty space to the region within the neutron star as well.
An analogy to this would be the Shell theorem in Newtonian gravity. If you're outside the earth's surface, you can apply the shell theorem to predict the gravitational field anywhere by assuming all of the earth's mass is concentrated at its center. This isn't physically true, but it's a useful mathematical trick. Of course, if you start digging towards the center of the earth, this solution isn't valid anymore. You have to recalculate the force of gravity taking into account only the mass contained within a shell whose radius is equal to your distance from the earth's center.
In the same way, you can solve Einstein's field equations to get a solution for a nonrotating, spherically symmetric object. By making an analogous solution to the shell theorem, where all the object's mass is concentrated at it's center, you can come up with a solution to these equations. This solution has an event horizon, or Schwarzschild radius. Obviously if this radius lies within the object, it doesn't really exist, since this solution is only valid outside of the object. I suspect that's the event horizon you're referring to. Solve the equations within the object, you should see that no horizon actually exists.
That the Schwarzschild radius does tells us is that if you take all the mass in your object and squash it to within the Schwarzschild radius, you do suddenly get a physical event horizon appearing. Because any force keeping the object from collapsing further would have to be communicated faster than light, it has to keep collapsing at this point, forming a black hole with a singularity at its center.

Thanks JP :)
I think an infinite density is an equivalence to energy. And I don't think you can reach that density, as little as matter can reach the speed of light in a vacuum. Before that it have to become that scarlet pimpernel 'energy', but I don't know of course :)

Mass is equivalent to energy and mass density is equivalent to energy density. So you have infinite energy density, but finite energy. How can this happen? It's infinite density squashed into an infinitely small point, so the infinities "cancel" in a sense. In reality, things becoming infinitely large and small at that point is probably a sign that GR doesn't work for tiny things.

Huh, Are you trying to scare me?
"infinite energy density, but finite energy"
I just knew there was something :)
So a density can't be equivalent to a energy then?
==
Mass has then no defined density?
Ahem :)
(and how the he* will I be able to sleep after this?)

Mass density is mass/volume. Energy density is energy/volume. Mass or energy density can become infinite in two ways. Either mass or energy becomes infinite or volume decreases towards zero. In the case of a black hole it's mass/energy is finite, but the volume of the singularity decreases to a point, which is zero volume.
This isn't exactly the same as dividing mass/0, which is mathematically illegal. It's taking mass/x as x>0, which is mathematically legal. But this whole idea of dividing by volume that's going to zero is a sign that the black hole singularity probably isn't a physical thing.

Okay, that makes sense.
And yes, that's how I understand the center.
As not describable.
Very nice and concise explanation JP.
==
I think I can sleep now :)

the volume of the singularity decreases to a point, which is zero volume.
Can anything be smaller than the Planck mass and still exist?
How do you distinguish between having "zero volume" and not existing?

the volume of the singularity decreases to a point, which is zero volume.
Can anything be smaller than the Planck mass and still exist?
No one knows. We don't have high enough energy experiments to test what happens at such small scales, and our best tested theory (the standard model) doesn't give answers for things that small. It might be that smaller things do exist. It might also be that the fabric of space takes on a totally different character at such lengths, so size as we know it doesn't mean the same thing (I read that somewhere on a writeup of quantum loop gravity). But no one knows.
How do you distinguish between having "zero volume" and not existing?
Easy. On exists and one doesn't!
I get your point, though. Zero volume, as far as we know, is a mathematical construct that's probably accurate as a physical model up to a certain point, and then it has problems. We won't know what it really means in this case until we figure out a theory of quantum gravity.
If I model a black hole as a singularity, I get an answer that makes sense and appears to model black holes pretty well. If I model a black hole as not existing... well, I have no black hole.

On exists and one doesn't!
You say that "one exists", but then qualify that by saying that it is a "mathematical construct". However, as discussed with respect to mathematical infinities, something that can be said to exist mathematically does not necessarily exist in reality.
I suppose that causes no problems as long as we realise we are talking about models.

I think that in a way it's like JP pointed out to me before. Some things we can describe mathematically, like the idea of gluons, and find evidence for indirectly. But never being able to prove first handedly by direct observations. A black holes center just got to be one of those things :)
It falls back to how rigorously you expect mathematics to mimic 'reality', and that seems to have a lot to do with your definitions, and also conceptions, of reality. I avoid the word preconceptions because I think most people do want their mathematics to be as close to what is observed as possible. But in hindsight we've always had preconceptions, it's hard to avoid I guess, one can only build on what one know, and for me that changes constantly I'm afraid :)