Black Holes As Actual Infinities

[Agent Smith]

A little philosophy to get this party going ...
Aristotle (Greek philosopher), way back in the BC era, claimed that only potential infinities exist and that there are no actual infinities. This issue isn't resolved to date, because even though it seems so tempting to say that the set of naturals is infinite, we assert the existence of at least 1 infinity (the naturals) as an axiom and not a theorem.

Then someone I met mentioned black holes, those that are collapsed stars or primordial ones that were formed during The Big Bang itself. My questions:

1. Is a black hole an actual infinity?
2. If it is how may we enrich our mathematics?

 

[topsquark]

A little philosophy to get this party going ...
Aristotle (Greek philosopher), way back in the BC era, claimed that only potential infinities exist and that there are no actual infinities. This issue isn't resolved to date, because even though it seems so tempting to say that the set of naturals is infinite, we assert the existence of at least 1 infinity (the naturals) as an axiom and not a theorem.

Then someone I met mentioned black holes, those that are collapsed stars or primordial ones that were formed during The Big Bang itself. My questions:

1. Is a black hole an actual infinity?
2. If it is how may we enrich our mathematics?

A black hole theoretically contains a singularity: a point where the laws of Physics break down. Specifically: at R = 0, the metric is no longer a defined quantity. Some say this is a point of infinite mass density, but as we understand very little of the Physics that goes on inside of the Schwarzschild radius I don't believe that we can actually make that statement.

-Dan

 

[Agent Smith]

A black hole theoretically contains a singularity: a point where the laws of Physics break down. Specifically: at R = 0, the metric is no longer a defined quantity. Some say this is a point of infinite mass density, but as we understand very little of the Physics that goes on inside of the Schwarzschild radius I don't believe that we can actually make that statement.

-Dan

Gracias for the response.

So we're talking about density here. It's not the mass that's important, it's the "size" of the space within which it's contained? I don't know how that works; can you explain it to me? I vaguely recall reading about how the sun (with its current mass) if shrunk to the size of the earth would become a black hole.

 

[topsquark]

Gracias for the response.

So we're talking about density here. It's not the mass that's important, it's the "size" of the space within which it's contained? I don't know how that works; can you explain it to me? I vaguely recall reading about how the sun (with its current mass) if shrunk to the size of the earth would become a black hole.

It's a bit of both mass and density. Every object has a Schwarzschild radius. [imath]r_S = \dfrac{2GM}{c^2}[/imath]. If the object is compressed to a size below that radius, it becomes a black hole.

-Dan

 

[mmm4444bot]

Is a black hole an actual infinity

What concept are you referring to as an "actual infinity", specifically? Strength of gravity, infinitesimal size, number of possible shapes in higher dimensions…

Your question seems like metaphysics. Whatever it is, there's probably no definite answer. Science has not determined the meaning of math "blowing up" in relativity theory under certain circumstances at very small scales. "Singularity" is the name that physicists have assigned for that, and it's a sign that they don't know what such results in the math mean. You may need to wait for science to reconcile relativity theory with quantum theory in a way that doesn't make relativity even more pathological.
[imath]\;[/imath]

 

[Agent Smith]

It's a bit of both mass and density. Every object has a Schwarzschild radius. [imath]r_S = \dfrac{2GM}{c^2}[/imath]. If the object is compressed to a size below that radius, it becomes a black hole.

-Dan

So black holes have to be "perfectly" spherical, because if it were (say) an ellipse, we could have one radius below the Schwarzschild radius and the other one above it? I can see the denominator is \(\displaystyle c^2\) and so the Schwarzschild radius is going to be really small unless the mass is off the charts. Gracias.

What exactly happens once an object is shrunk to below the Schwarzschild radius? How does the mass density become \(\displaystyle \infty\)?

What concept are you referring to as an "actual infinity", specifically? Strength of gravity, infinitesimal size, number of possible shapes in higher dimensions…

Your question seems like metaphysics. Whatever it is, there's probably no definite answer. Science has not determined the meaning of math "blowing up" in relativity theory under certain circumstances at very small scales. "Singularity" is the name that physicists have assigned for that, and it's a sign that they don't know what such results in the math mean. You may need to wait for science to reconcile relativity theory with quantum theory in a way that doesn't make relativity even more pathological.
[imath]\;[/imath]

I'm referring to the referent of singularity. Most people seem to equate it to infinity. If black holes exist and there's something infinite about them then an actual infinity exists. Si, my topic is metaphysical to the extent infinity is metaphysical. I specifically mentioned axiom of infinity in math, which states that there exists at least 1 infinity, the naturals (if it were a theorem, we would have proof, but it seems we don't). So if black holes have some component that's infinity, we could update Aristotle's position from there are no actual infinities to there are. How would this then impact mathematics? Could we upgrade the axiom of infinity to a theorem of infinity?

 

[topsquark]

So black holes have to be "perfectly" spherical, because if it were (say) an ellipse, we could have one radius below the Schwarzschild radius and the other one above it? I can see the denominator is \(\displaystyle c^2\) and so the Schwarzschild radius is going to be really small unless the mass is off the charts. Gracias.

What exactly happens once an object is shrunk to below the Schwarzschild radius? How does the mass density become \(\displaystyle \infty\)?

Generally, once you compress a dimension of the object down to such a size, the event horizon will be spherical. The object need not be. From that point, there is nothing more to keep it from getting smaller and smaller, ie. more and more dense. The gravitational field strength just gets bigger and bigger which is why most would say that we have an infinite mass density at r = 0. However, as there is no way to "see" inside the event horizon, this is anyone's guess.

-Dan

 

[mmm4444bot]

my topic is metaphysical to the extent infinity is metaphysical

Oh, I think I now see what kind of party you're throwing: The question is of no matter.
[imath]\;[/imath]

 

[Agent Smith]

Generally, once you compress a dimension of the object down to such a size, the event horizon will be spherical. The object need not be. From that point, there is nothing more to keep it from getting smaller and smaller, ie. more and more dense. The gravitational field strength just gets bigger and bigger which is why most would say that we have an infinite mass density at r = 0. However, as there is no way to "see" inside the event horizon, this is anyone's guess.

-Dan

So Newton was, in a sense, wrong. Mass is not what matters, density is what does. In high school I was taught [imath]F = G \frac{m_1 m_2}{r^2}[/imath] as if to say that once 2 objects touch/are in contact, [imath]F = \infty[/imath]. Can you clear this up for me?

Oh, I think I now see what kind of party you're throwing: The question is of no matter.
[imath]\;[/imath]

Didn't mean to cause more confusion than there already is. Was on another forum, the topic was mathematical infinity and The Axiom Of Infinity came up midway; also Wiki briefly mentions Aristotle's objections against acutal infinities.

 

[topsquark]

So Newton was, in a sense, wrong. Mass is not what matters, density is what does. In high school I was taught [imath]F = G \frac{m_1 m_2}{r^2}[/imath] as if to say that once 2 objects touch/are in contact, [imath]F = \infty[/imath]. Can you clear this up for me?

Didn't mean to cause more confusion than there already is. Was on another forum, the topic was mathematical infinity and The Axiom Of Infinity came up midway; also Wiki briefly mentions Aristotle's objections against acutal infinities.

Gravity depends on mass... the Schwarzschild radius depends on density.

I can't clear it up, because it isn't correct. In Newtonian Physics the gravitational force between two touching objects is [imath]F = \dfrac{G m_1 m_2}{d^2}[/imath], where d is the distance between the center of mass of the two objects. I suppose if you are dealing with point particles what you say is true... then d = 0 when they touch.

-Dan

 

[Agent Smith]

Gravity depends on mass... the Schwarzschild radius depends on density.

I can't clear it up, because it isn't correct. In Newtonian Physics the gravitational force between two touching objects is [imath]F = \dfrac{G m_1 m_2}{d^2}[/imath], where d is the distance between the center of mass of the two objects. I suppose if you are dealing with point particles what you say is true... then d = 0 when they touch.

-Dan

So is that the reason why people expressed (feigned) concern about CERN creating mini black holes?

Why would the Schwarzschild radius depend on density? Black holes were hypothesized in the 1800s or earlier, Laplace being one of the scientists who did so, based simply on mass (stars so massive that its escape velocity is greater than the speed of light). Density didn't figure in these conjectures.
Perhaps a rubber sheet analogy is in order. How a given mass is distributed over space decides whether its black hole material or not, oui? I can imagine a heavy, but small ball bearing causing a huge depression in "space" but a basket ball not doing much in the way of warping space, even though its as heavy as the ball bearing. What sayest thou?

 

[Agent Smith]

A black hole theoretically contains a singularity: a point where the laws of Physics break down. Specifically: at R = 0, the metric is no longer a defined quantity. Some say this is a point of infinite mass density, but as we understand very little of the Physics that goes on inside of the Schwarzschild radius I don't believe that we can actually make that statement.

-Dan

"the metric is no longer a defined quantity" because ... division by 0?

 

[Otis]

"the metric is no longer a defined quantity" because ...

Measuring less than the Plank Length yields nonsensical results (uncertainty seems to grow with measurements taken within smaller and smaller volumes, eventually surpassing the volume itself). As topsquark said, the reasons have to do with regular physics (what we currently understand about quantum mechanics and gravity) breaking down at such scales. Measuring distance relies on regular physics and the associated, universal constants that come into play. Since we don't understand what lies beyond current knowledge, we need new physics (and likely, better technology, too).
[imath]\;[/imath]

 

[Agent Smith]

Measuring less than the Plank Length yields nonsensical results (uncertainty seems to grow with measurements taken within smaller and smaller volumes, eventually surpassing the volume itself). As topsquark said, the reasons have to do with regular physics (what we currently understand about quantum mechanics and gravity) breaking down at such scales. Measuring distance relies on regular physics and the associated, universal constants that come into play. Since we don't understand what lies beyond current knowledge, we need new physics (and likely, better technology, too).
[imath]\;[/imath]

Non liquet, but what does "breaking down at such scales" mean in mathematical terms?
Guessing ...
1. Infinity enters into the equations
2. Division by 0 occurs in the equations?
3. We get a contradiction in the equations?

Also would like to know if once we're at the Planck scale, we have to switch from classical to quantum physics?

 

[Otis]

what does "breaking down at such scales" mean in mathematical terms?

You mean something besides nonsensical results? I'm not sure what you're after.
[imath]\;[/imath]

 

[Agent Smith]

You mean something besides nonsensical results? I'm not sure what you're after.
[imath]\;[/imath]

I'm not sure either. What do you mean by "nonsensical results"?

 

[topsquark]

1. There was little "feigned" concern about CERN creating mini-black holes. It was a possibility, though a small one. Such black holes could be dangerous if they do not dissolve quickly enough.

2. Your comment about Laplace and black holes in the 1800s is full of holes. In 1800 stars still burned on coal and no one knew that c was the speed limit. Yes, I have heard of black holes mentioned back then, but they were not the objects that we now call black holes.

3. Two components of the Schwarzschild metric depend on [imath]\dfrac{GM}{R}[/imath]. When R goes to 0, the metric becomes undefined, ie. length and time scales become unmeasurable.

4. We have to switch to the Quantum scale long before we get to the Planck scale. Anything smaller than, say, [imath]10^{-8}[/imath] m starts showing suggestions of Quantum effects.

5. Non-sensical results: time and distance scales no longer existing. Everything is now composed of the most elementary particles, which we may not even understand. The Physics we understand about our Universe ends at the event horizon.

-Dan

 

[Otis]

I'm not sure either. What do you mean by "nonsensical results"?

Mathematical results that make no sense in the context for which the math was done.

If at some point you realize specifically what you're trying to ask, then please feel free to post your question again. Until then, people can only guess. I had previously guessed that you desire a specific example; that is why I posted the statements about uncertainty in the measurements. It makes no sense when the uncertainty (that is, the numerical quantification of measurement error) becomes larger than possible in the given context. That indicates a serious issue, and the concensus is that measurements below the Plank Scale are currently not humanly possible. I'm afraid that's something we all just have to get used to, until there is a major breakthrough.
[imath]\;[/imath]

 

[Agent Smith]

Mathematical results that make no sense in the context for which the math was done.

If at some point you realize specifically what you're trying to ask, then please feel free to post your question again. Until then, people can only guess. I had previously guessed that you desire a specific example; that is why I posted the statements about uncertainty in the measurements. It makes no sense when the uncertainty (that is, the numerical quantification of measurement error) becomes larger than possible in the given context. That indicates a serious issue, and the concensus is that measurements below the Plank Scale are currently not humanly possible. I'm afraid that's something we all just have to get used to, until there is a major breakthrough.
[imath]\;[/imath]

You mean like [imath]3 \text{ cm} \pm 10 \text{ cm}[/imath]? I remember posting something about a "joke" that the age of the earth is [imath]4 \text{ billion years} \pm 10 \text{ billion years}[/imath]. Could we discuss that, because though some have reasoned that the earth can't be [imath]-6 \text{billion years old}[/imath] and thus found the measurement ridiculous, my own take on this was that the earth will form after [imath]6 \text{ billion years}[/imath]. I couldn't . It's possible that what we think is earth isn't earth?

 

[Agent Smith]

Nonsensical mathematical results (as far as I can tell):
1. Division by 0 [undefined]
2. Infinity in the equations such that know methods of dealing with them don't work [no further computation possible]
3. Contradictions like 1 = 0 [breaking math]