Singularities, Density, and the Planck Length

[TeslaTrevor]

It is generally accepted that a star of sufficient mass collapsing in on itself will form a black hole (singularity) where density is infinite. I see a few problems arising with this, and I would like to have them clarified.

1.) Density=mass/area
If the mass of any star is finite, how can an infinite density for any real area be generated? If it is true that a black hole is infintesimally small, then that would break the assumption that nothing can be smaller than the Planck length which I have been hearing is a limit.

2.) If nothing can escape the event horizon of a black hole, how, and what is the method that causes Hawking radiation that spews from the black hole?

 

[Bill_K]

TeslaTrevor, A black hole is generally regarded as having a finite size, namely a sphere with radius 2GM/c² known as the Schwarzschild radius. The surface of this sphere is also referred to as the event horizon. It is generally stated that inside the hole, infalling matter continues to collapse to an infinitely dense singularity. However this is a classical approximation to a situation in which quantum mechanics must be important. No one knows what really replaces this apparent singularity.

Hawking radiation does not come from inside the hole, it arises from just outside the event horizon.

 

[TeslaTrevor]

Well it cannot collapse into a singularity at all because the smallest length that any object can have is the Planck length. This restriction also prevents infinite density.

 

[Dale]

The Planck length is not important in GR. This is one of the reasons that physicists are looking for a quantum theory of gravity.

 

[TeslaTrevor]

Whether or not the Planck length is relevant to any particular theory is irrelevant; I'm describing here how the Planck length is relevant to reality: in particular how the Planck length restriction prevents the possibility of singularities or infinite densities

 

[Dale]

Whether or not the Planck length is relevant to any particular theory is irrelevant; I'm describing here how the Planck length is relevant to reality: in particular how the Planck length restriction prevents the possibility of singularities or infinite densities

what makes you think that the Planck length prevents the possibility of singularities or infinite density?

 

[TeslaTrevor]

Since the Planck length is the smallest possible length that an object can have and all measurements of length are relative to the Planck length (is is the smallest distance at which Nature itself can distinguish two objects as being discrete). Since density is equal to mass/area, and neither mass nor area can be finite (the star collapsing has a finite mass and length or area is restricted by the Planck length) density must then always be finite as well. And if we have a function that graphs the size of a star versus time when a star collapses we find that it is continuous by the mean value theorem (like all functions accurately representing nature). So even as the star's mass approaches the Planck length (or if it can get smaller, the infintesimally small area) it can never reach it in a finite time, otherwise there would be a discontinuity along the graph of the function.

 

[Dale]

Since the Planck length is the smallest possible length that an object can have and all measurements of length are relative to the Planck length (is is the smallest distance at which Nature itself can distinguish two objects as being discrete).

Why do you think that? We have no experimental data on the Planck scale, so what would lead you to have this belief?

 

[WannabeNewton]

Classical GR isn't concerned with the Planck scale.

 

[TeslaTrevor]

Classical GR isn't concerned with the Planck scale.

Whether or not the Planck length is relevant to any particular theory is irrelevant; I'm describing here how the Planck length is relevant to reality: in particular how the Planck length restriction prevents the possibility of singularities or infinite densities

 

[JesseM]

Whether or not the Planck length is relevant to any particular theory is irrelevant; I'm describing here how the Planck length is relevant to reality: in particular how the Planck length restriction prevents the possibility of singularities or infinite densities

Well, in reality few physicists expect singularities of infinite density to exist, they are just a prediction of general relativity that will almost certainly change in a more accurate theory of quantum gravity.

 

[John232]

Density is not the only thing becomes infinite in the mathematics of the theory. When it pushes the limits of c all values are affected to the brinks of infinity. This is because the value of sqrt(1-v^2/c^2) approuches infinity when v approuches the same value of c. This is then divided by another value that then becomes undifined and the theory breaks down. It is also funny how the value of time decreases along with length that deal with this same value inversely.

 

[Dmitry67]

Since the Planck length is the smallest possible length that an object can have and all measurements of length are relative to the Planck length (is is the smallest distance at which Nature itself can distinguish two objects as being discrete). Since density is equal to mass/area, and neither mass nor area can be finite (the star collapsing has a finite mass and length or area is restricted by the Planck length) density must then always be finite as well. And if we have a function that graphs the size of a star versus time when a star collapses we find that it is continuous by the mean value theorem (like all functions accurately representing nature). So even as the star's mass approaches the Planck length (or if it can get smaller, the infintesimally small area) it can never reach it in a finite time, otherwise there would be a discontinuity along the graph of the function.

density=mass/volume is applicable to relatively flat spacetimes only. When we know that something MUST happen close to the Planks length, we don't know what exactly will happen. It is expected that close to such densities/energies, geometry will be very different from what we know, and the difference between matter and spacetime would dissapear. So very likely the "density" (in a classical sense) of such object won't be defined.

What you are describing is an extension of classical and semi-classcal concepts to regimes where they won't make any sense.

 

[Dale]

Whether or not the Planck length is relevant to any particular theory is irrelevant; I'm describing here how the Planck length is relevant to reality: in particular how the Planck length restriction prevents the possibility of singularities or infinite densities

You have made this comment several times now. I am trying to get you to understand that there is no experimental evidence whatsoever that "the Planck length is relevant to reality" nor that it poses any restriction which "prevents the possibility of singularities or infinite densities". Appeal to "reality" does not support your claims at all since we don't have any data. The significance of the Planck length is purely a theoretical prediction at this point.

The Planck length is relevant to QM, it is not relevant to GR, it is a theoretical conflict between the two most fundamental theories we have. So as I and others have said, this is one reason why physicists are trying to formulate a quantum theory of gravity. From philosophical considerations we suspect that the GR prediction of a singularity is wrong, but there is nothing in theory nor in evidence that confirms it.

 

[Naty1]

Tesla: first, those are good questions...but not with any precise answers.

One general comment: gravity, which is the weakest of all the classical forces in everyday situations we observe, dominates at singularities, changing space and time and likely mass as well. They all appear to lose their classical meaning.

I agree with you in general about density, but I don't think there is any theoretical nor experimental evidence that density IS infinite in a black hole. Nor at the big bang: We simply don't have any theory for the singularity at the big bang nor the singularity of a black hole. Nobody knows what happens at those entities.

That's one reason most physicsts believe we need to go beyond general relativity and quantum mechanics to some kind of quantum gravity...something that "unifies" those disparate theories and can describe those singularities.

One of the concepts arguing against "infinite" anything in a black hole is the entropy (information) of a black hole horizon...It's finite...search Holographic principle for some interesting reading. It's not even related to the VOLUME of the black hole but the AREA!

such as http://en.wikipedia.org/wiki/Holographic_principle

Planck length is an example of the discord between GR and QM: it plays no part as far as anyone knows in GR and yet is a cornerstone of QM, analogous to Heisenberg uncertantinty which is also absent from GR.

We've had some good discussions about Planck length in these forums...a search and some reading might be worthwhile. Check out "double special relativity" for some discussion on Planck length and special relativity length contraction. Does Planck length "contract" when observed at near light speed?

Hawking radiation only "spews" when the universe cools as LOT more...Nobody can yet observe Hawking radiation. If you search Hawking radiation you'll find some descriptions in these forums. An intuitive description which Hawking used was that two virtual particles created near the horizon might separate and one with positive energy might escape...the other disappearing behind the horizon...It's related to Unruh effect where particles "materialize" to an observer undergoing acceleration.

Also, density = mass/area may need a lot of work...nobody knows exactly what either mass of length (area) is...So I would not be surprised that all three concepts might take on new form at singularities. But all we know for sure is that our theories (GR and QM) so far don't fit. One perspective arises from degeneracy pressure,,,electron and then neutron..."mass" make take on new perspectives when these conditions arise.

One of the most interesting books on black holes I've come across is Leonard Susskind's THE BLACK HOLE WAR about his long running disagreements with Hawking over black holes...Barely more than HS math, but loads of fascinating concepts. Lee Smolin's THREE ROADS TO QUANTUM GRAVITY is also very good but not so closely related specifically to black holes.