Infinite, Singularity, Existence

[420MuNkEy]

Einsteins equations predict black holes (although, he did not believe they could exist outside theory and even wrote an article defending this view), and at the 'heart' of the black hole is something that's become known as a singularity. Put simply, a singularity (in the context of black holes) is where the density becomes infinite and the volume becomes 0. Now, this obviously raises a bunch of questions, but one of the most interesting (imo) of all is, can ∞ really exist? If so, is the equation ∞=∞ always true?

Now for a thought experiment: Imagine two black holes equal to each other in every way perfectly overlap each other. Do they combine or remain separate? In either situation, are the forces at work complemented, negated, or unaffected by the other forces?

If black holes are able to combine and compliment each other, does this mean that ∞=∞ is not always true? If so, mathematics seems to be lacking a symbol to represent the concept properly.

Another thought experiment: Imagine that enough black holes form to bring every particle of matter past an event horizon, and suppose black holes can merge and compliment each others forces. Because all that are left are black holes, their gravity acts on each other and merges them all into a singularity. Would the force from the creation and/or existence of this singularity be enough to cause a 'Big Bang'? Would this mean that existence is merely an infinite loop of 'big bangs'?




I kinda went off on a crazy tangent and just jotted down some thoughts that I thought fellow INTP's might be interested in. I don't have enough knowledge of advanced mathematics to even begin trying to translate these thoughts to equations so as to be able to test them.

[nickgray]

can ∞ really exist

Sure. Black holes exist, even supermassive black holes exist. To further confuse you (mathematically) there can exist infinities that are bigger than other infinities :)

Imagine that enough black holes form to bring every particle of matter past an event horizon

Not possible.

Because all that are left are black holes, their gravity acts on each other and merges them all into a singularity.

Perhaps you're confusing something... the "trick" of the black holes is not their gravity, but rather their size. For example if Sun all of a sudden turned into a black hole the planets would still orbit it. However it'll get much smaller in size and you'll be able to get so close that you wouldn't be able to escape the gravity of it (event horizon).

Imagine two black holes equal to each other in every way perfectly overlap each other. Do they combine or remain separate?

Well, if they're close enough to interact gravitationally, sure they'll combine somehow. What matter is that they'll interact, it doesn't matter if they're "equal" to each other.

[420MuNkEy]

can ∞ really exist
Sure. Black holes exist, even supermassive black holes exist. To further confuse you (mathematically) there can exist infinities that are bigger than other infinities :)I am familiar with this concept, however I don't agree with it. ∞ represents meaning. You can't subtract 1 from ∞, however you can add one to both sides of the equation negating it. The terms meaning itself isn't changed by this. You can't reduce such an equation and retain the operations that are supposed to alter the infinity, the the term itself has static meaning. :confused:


Imagine that enough black holes form to bring every particle of matter past an event horizon Not possible.It's not a question of whether or not you think it's possible for the situation to arise, it's a hypothetical situation.


Because all that are left are black holes, their gravity acts on each other and merges them all into a singularity.Perhaps you're confusing something... the "trick" of the black holes is not their gravity, but rather their size. For example if Sun all of a sudden turned into a black hole the planets would still orbit it. However it'll get much smaller in size and you'll be able to get so close that you wouldn't be able to escape the gravity of it (event horizon).If there were no particles of matter (as explained earlier in the hypothetical), the black holes would be the only force in existence and would inevitably interact.


Imagine two black holes equal to each other in every way perfectly overlap each other. Do they combine or remain separate?Well, if they're close enough to interact gravitationally, sure they'll combine somehow. What matter is that they'll interact, it doesn't matter if they're "equal" to each other.Yes, just being specific so that no one argues variables that are irrelevant to the idea (one taking precedence over another or whatever).

[Artifice Orisit]

Imagine that enough black holes form to bring every particle of matter past an event horizon
Not possible.
For anyone who was wondering.
It's because particles are spontaneously created in a true vacuum thus all matter cannot become part of a black hole if there's still empty space around them.

Put simply, a singularity (in the context of black holes) is where the density becomes infinite and the volume becomes 0.
That messes with me too, it makes me wonder if a black hole in our universe is the creation of another universe and that if white holes exists than they're either active big-bangs and/or there's a white hole in our universe’s origin point.

I wonder if the origin point is like the Jerusalem of the universe, a place worshiped by alien physicist-priests across the universe because it's quite literally the gateway to a higher universe (a white hole?), one that only those whom are enlightened enough to transcend 4D space/time can use.
...
Whoa 0.0

[420MuNkEy]

For anyone who was wondering.
It's because particles are spontaneously created in a true vacuum thus all matter cannot become part of a black hole if there's still empty space around them.There wouldn't be empty space around them, that's kidna the base of the inevitable interaction mentioned later :p


That messes with me too, it makes me wonder if a black hole in our universe is the creation of another universe and that if white holes exists than they're either active big-bangs and/or there's a white hole in our universe’s origin point.This would be the situation in my mind if all the ∞'s (for lack of a better term) of singularities were equal and didn't compliment or negate each other.

[nickgray]

It's not a question of whether or not you think it's possible for the situation to arise, it's a hypothetical situation.

Yep, I realize that... it's just way too hypothetical :) Still, suppose it happens. So what? There's still space between them that prohibits a lot of them from interacting with each other. It wouldn't be much different than now, it's just that there will be black holes instead of galaxies.

the black holes would be the only force in existence and would inevitably interact.

Space itself expands. Particles can all of a sudden pop up an a vacuum (so there will be particles out there). Also, there's Hawking Radiation to consider.

I am familiar with this concept, however I don't agree with it

A set of natural numbers from 5 to infinity is bigger than a set of natural numbers from 10 to infinity. It's a very simple example though.

---
There wouldn't be empty space around them, that's kidna the base of the inevitable interaction mentioned later

Uhhhhh... how? Right now most of the matter is not in a "black hole state" and there are enormous amounts of "empty" space. You propose to create black holes out of that matter (obviously, the "size" of it will be much smaller afterwards) and expect to have less "empty" space around? It's something like this: suppose you're in an amphitheater and you hold a soccer ball in your hand. You then proceed to empty the soccer ball out of air. A crude explanation, but it'll do, I suppose.

[420MuNkEy]

Yep, I realize that... it's just way too hypothetical :)Nothing is too hypothetical :smoker:


A set of natural numbers from 5 to infinity is bigger than a set of natural numbers from 10 to infinity. It's a very simple example though.You are referring to the value of the set, not infinity. ∞-50<∞+50, but this is only because you can subtract ∞ from either side. The value of ∞ is static, and there seems to be a lack of a term to replace it in situations where it's used as if it's not.


Uhhhhh... how? Right now most of the matter is not in a "black hole state" and there are enormous amounts of "empty" space. You propose to create black holes out of that matter (obviously, the "size" of it will be much smaller afterwards) and expect to have less "empty" space around? It's something like this: suppose you're in an amphitheater and you hold a soccer ball in your hand. You then proceed to empty the soccer ball out of air. A crude explanation, but it'll do, I suppose.Again, it's a theoretical state. Try not to think of how it would get that way. Use it as the starting point for the thought. I'll try to make it a little clearer: Every point of space is within the event horizon of a black hole, what happens?

[nickgray]

Every point of space is within the event horizon of a black hole, what happens?

But the space expands. And black holes slowly evaporate via Hawking Radiation. You gotta have a non-expanding space and just enough matter to form such a black hole. Yet there's a much bigger problem for such a case - you need new physics for this new universe :)

[Nicholas A. A. E.]

can ∞ really exist?Yep.If so, is the equation ∞=∞ always true?If you define it so. But by common use, no. Also, if you define it the way most people do, you CAN subtract 1 from infinity. You still get infinity. The best generalization I've seen of all this is hyperreal numbers, where for example ∞-1<∞ in a meaningful way.

Now for a thought experiment: Imagine two black holes equal to each other in every way perfectly overlap each other. Do they combine or remain separate? In either situation, are the forces at work complemented, negated, or unaffected by the other forces?[Pretty sure the Pauli exclusion principle forbids this as a contradiction in terms. But in case I'm wrong, I'm pretty sure they combine and are thereby augmented. Complemented. That's what would happen if they were placed a nonzero distance away from each other, anyway.

If black holes are able to combine and compliment each other, does this mean that ∞=∞ is not always true? If so, mathematics seems to be lacking a symbol to represent the concept properly.I don't see how you can make that connection.

Another thought experiment: Imagine that enough black holes form to bring every particle of matter past an event horizon, and suppose black holes can merge and compliment each others forces. Because all that are left are black holes, their gravity acts on each other and merges them all into a singularity. Would the force from the creation and/or existence of this singularity be enough to cause a 'Big Bang'? Would this mean that existence is merely an infinite loop of 'big bangs'?
There's definitely not enough information to say whether. But I suspect no. I don't think the Big Band was merely an emergent phenomenon.

re. the part I bolded: this doesn't necessarily happen, depending on the curvature of the universe.

[420MuNkEy]

But the space expands. And black holes slowly evaporate via Hawking Radiation. You gotta have a non-expanding space and just enough matter to form such a black hole. Yet there's a much bigger problem for such a case - you need new physics for this new universe :)
You're still thinking of how it would end up this way. It's a hypothetical state frozen in time. For that moment frozen, it's encapsulating every single point in space and no black hole is evaporated whatsoever :storks:. Also, just to deter another post to question the origins of such a state, lets say that the expanding space from the next moment of time is already inside an event horizon.

[420MuNkEy]

I don't see how you can make that connection.If the density of a singularity is ∞ and it is combined with another singularity with a density of ∞, the and the sum is >∞, and can't be =∞. One could argue it's greater than or equal to ∞, then it becomes impossible to know how much more than ∞ it is or if it's just plain ol' ∞. The term looses meaning.

[nickgray]

You're still thinking of how it would end up this way.

Well, yeah. As I've said - such a situation would require different physics, it's definitely an impossible situation in our universe.

I've no idea what will happen :) Space expansion isn't really limited by the gravity and even if everything is "inside" the event horizon, space could expand and some parts wouldn't be in the event horizon anymore.

Another point I'd like to make is that you're considering the possibility of a "big crunch" happening in such a scenario. It's the expansion of space itself that will reverse and cause the big crunch. I've no idea if even such a big black hole may affect it. Ultimately, we've don't know what the hell dark energy is and what's happening inside the singularity. So even speculation is kinda pointless, since you've asked about black holes and expansion of space :)

[Vrecknidj]

A few passing comments while I go back to trying to digest the information here.

1) We have to be careful with words. For example, people use the word "energy" in everyday speak, to refer to things like feeling like they can get much accomplished in a day. Physicists will use the word "energy" in an entirely different way. Similarly here. Mathematicians use the word "infinity" in very specific kinds of ways. When we talk about physical senses of infinity (such as is being discussed here, with singularities) we might very well be discussing something other than the mathematical infinity.

2) Mathematically speaking, there are different "sizes" of infinity, and it's provable. Whether there are any physical analogues to this is entirely speculative. One of the more "ordinary" kinds of infinity in mathematics is the kind that doesn't get any bigger when you add a finite number to it (though what this means is open to further debate, I suppose). Black holes, can, of course, get bigger. So, thinking of them and thinking of infinities at the same time can lead to some serious confusion. (After all, the singularity being what it is, it seems that no matter how big the black hole gets, the singularity "inside" it shouldn't really get any bigger.)

Dave

[Nicholas A. A. E.]

If the density of a singularity is ∞ and it is combined with another singularity with a density of ∞, the and the sum is >∞, and can't be =∞. One could argue it's greater than or equal to ∞, then it becomes impossible to know how much more than ∞ it is or if it's just plain ol' ∞. The term looses meaning.
In the system of hyperreal numbers, ∞ is not a number and no operations are defined on it, but there is a set (infinitely large) of infinite numbers which have several operations defined on them.

If we call the density of one singularity H and say the other's is also H, then the density of the new black hole is 2H. 2H > H.

[420MuNkEy]

The confusion inherent in '∞' is what made me suggest that a new term needs to be created to be used in such situations. I know that many mathematicians have literally gone insane with ∞ on their mind over the course of history :p. (I think there's actually a documentary about it called 'Dangerous Knowledge')

[Agent Intellect]

The infinity that arises in calculating the density of a black hole comes from the density formula (d=m/v). The mass of the black hole is not infinite (the 'stuff' inside it) but the density would be infinite simply because you are dividing a number by zero.

The information of a black hole (http://en.wikipedia.org/wiki/Black_hole_thermodynamics)can be calculated by it's entropy on the two dimensional surface of the event horizon - when another black hole merges with a black hole the event horizon will increase in size equal to the amount of entropy in both black holes (a black hole is the largest amount of entropy there can be inside a certain amount of space). This lead some scientists to the holographic universe theory (http://community.livejournal.com/ref_sciam/1190.html).

Another thought experiment: Imagine that enough black holes form to bring every particle of matter past an event horizon, and suppose black holes can merge and compliment each others forces. Because all that are left are black holes, their gravity acts on each other and merges them all into a singularity. Would the force from the creation and/or existence of this singularity be enough to cause a 'Big Bang'? Would this mean that existence is merely an infinite loop of 'big bangs'?

Sounds like the Fecund Universes (http://en.wikipedia.org/wiki/Lee_Smolin#Fecund_universes) theory of Lee Smolin.

[420MuNkEy]

Sounds like the Fecund Universes (http://en.wikipedia.org/wiki/Lee_Smolin#Fecund_universes) theory of Lee Smolin.It's similar in the sense that a new universe is created from the collapse of a black hole, but from what I could tell (without reading his book), the universe in which the black hole was formed survives, leaving the possibility of multiple universes. I was thinking something more like the violent collapse of one universe being so forceful it would re-birth itself (insane, I know :p).

[Darby]

The infinity that arises in calculating the density of a black hole comes from the density formula (d=m/v). The mass of the black hole is not infinite (the 'stuff' inside it) but the density would be infinite simply because you are dividing a number by zero.

Thank you, I was hoping someone would bring this up, because much like the d=m/v equation, I always saw it as similar to the vertical line equation(I am more familiar with this is all) where slope is undefined or infinite, because slope is rise over run, blah blah blah and therefore when it travels 0 forward, the number for rise is automatically seen as infinite compared to the zero.

I never realized so few people saw it this way

[Vrecknidj]

The funny thing about the slopes example is this.

On the Cartesian coordinate system, any two lines that have perpendicular slopes have this trait: when you multiply their slopes, you always get -1.

There is one pair of perpendicular lines for which this may not be true. If you pick a horizontal and a vertical line, your slopes are 0 and undefined. And, for all I know, if you multiply zero by undefined, you may not get -1.

But, since coordinate axes can be rotated without losing any information, it strikes me that you can then rotate the vertical and horizontal lines.

In other words, I want to make this case. Since all* the perpendicular lines' slopes are such that their product is -1, then the product of the slopes of the horizontal and vertical lines is also -1. Therefore 0 * ∞ = -1 (where "∞" here simply means a finite number divided by zero).

Dave

[Darby]

The funny thing about the slopes example is this.

On the Cartesian coordinate system, any two lines that have perpendicular slopes have this trait: when you multiply their slopes, you always get -1.

There is one pair of perpendicular lines for which this may not be true. If you pick a horizontal and a vertical line, your slopes are 0 and undefined. And, for all I know, if you multiply zero by undefined, you may not get -1.

But, since coordinate axes can be rotated without losing any information, it strikes me that you can then rotate the vertical and horizontal lines.

In other words, I want to make this case. Since all* the perpendicular lines' slopes are such that their product is -1, then the product of the slopes of the horizontal and vertical lines is also -1. Therefore 0 * ∞ = -1 (where "∞" here simply means a finite number divided by zero).



it would look something like this:

∞/0 = Horizontal

0/∞ = Vertical

they are perpendicular slopes, and so:
(∞/0)*(0/∞) = -1
the infinites would cancel:
(1*0)/(0*1) = -1
which means:
0/0 = -1

EDIT:[and if they dont cancel: (∞*0)/(0*∞) = -1]

which personally doesn't make any sense to me, and is assuming that the infinites are equal (∞=∞) when in truth it is:

(y/0) = ∞
Horizontal as long as y≠0

and:

(0/x) = ∞
Vertical as long as x≠0

I think I got that right, maybe not.

EDIT: oops, forgot to say that that last part just means that we are dealing with two different infinites. A y infinite, and an x infinite, they are not necessarily equal

[Vrecknidj]

"the infinities would cancel"

Maybe. Good catch on the edit.

Your explanation is good, thanks.

Dave

[Latro]

You ask "is infinity equal to infinity?" Mathematically this isn't really a well-defined question. Infinity in calculus is a mere description of a behavior; the rigorous concept of a limit being infinite or of a limit as a variable approaches infinity is not actually formally defined using infinity. It's formally defined by allowing things to grow without bound.

In calculus, however, infinity need not be effectively "equal" to infinity. If this were true, we would expect "infinity - infinity" to be 0, which needn't be the case. For example:
lim x->infinity x^2-x
Should be expected to be infinity-infinity, at a glance. However, this is clearly an infinite quantity. Why? Factor it:
lim x->infinity x(x-1)
which is infinity*infinity, which is clearly infinity.

Formally using "equations" and "algebra" on infinity does not work in the usual way; doing things with transfinite numbers, hyperreals, etc. has very unexpected results that are counterintuitive, and thus must be investigated very formally after a buildup through other advanced material like real analysis.

Set theoretic infinity is an entirely different concept, involving the inability to take the natural numbers up to k and put them in one-to-one correspondence with the elements of a set for any natural number k. This makes a set "infinite." Inability to map the elements of the set to, for example, the set of natural numbers, creates a "larger" infinity in some...somewhat abstract sense. More confusing in set theory is that you can have a set that seems "larger" than another set (that is, that contains that set) with "the same number of elements" as that set. The naturals and the rationals are an example; the naturals and the pairs of naturals are another; the reals and the n-tuples of reals are another.

And nick: the set N-{1,2,3,4} has the same cardinality as N-{1,2,...,9}, assuming you were referring to cardinalities.

And Vreck: 0*infinity can be anything you want it to be. It's called an indeterminate form. If you do, for example:
lim x->0+ x*(1/x)
You get 1, since x*(1/x)=1, but individually those appear to approach 0 and infinity. (I used 0+ because it has to be from the right to be +infinity).

You can use this example and vary that 1 and get back any real number you want.

"Infinities cancelling" is also not well-defined.



Overall there is a lot of mathematics lacking here...people using too much intuition and not enough formal logic.

[Nicholas A. A. E.]

When I said "infinitely large" I meant that the number of infinite numbers is infinite, as the number of real numbers is infinite (but not necessarily of the same cardinality. that, I don't know).

And yeah, you guys should realize that operators like multiplication just aren't defined on infinite numbers, traditionally. You can't use induction to figure out how they would work when applied to infinity - you just have to arbitrarily assign a definition to the scenario.

As I've mentioned a couple times though, this entire discussion is 100% resolved by using hyperreal numbers. I guess no one else has heard of those though. Wiki does a good job of explaining them IIRC.

[Vrecknidj]

And Vreck: 0*infinity can be anything you want it to be. It's called an indeterminate form. If you do, for example:
lim x->0+ x*(1/x)
You get 1, since x*(1/x)=1, but individually those appear to approach 0 and infinity. (I used 0+ because it has to be from the right to be +infinity).Yep. The example I usually use in class is 0/0. When i discuss ratios to, say, precalculus students, and show that 24/4 = 6/1 and we talk about the means and the extremes, I then go on to show that 0/0 = x/y and we talk about just how frustrating this notion is when compared to all the other ratios.Overall there is a lot of mathematics lacking here...people using too much intuition and not enough formal logic.Well, maybe. Formal logic might be disambiguating, but I find that there are compelling discoveries to be made when I allow the ambiguities. In this regard, it's a little like how sometimes I prefer concise writing to poetry and sometimes the reverse, because, sometimes, the vagueness and ambiguities in poetry express things that I can't articulate as profoundly in concise writing.

Dave