# Black holes

1. May 11, 2004

### shrumeo

Here's a non-physicist asking a question:

How can black holes have different masses and radii?

I understand that black holes are formed when a critical density is reached and poof, all that matter is condensed into a sigularity, with so-called infinite density. Then we talk about black holes being massive, so many times the mass of the sun, etc. But, if there is only one critical density, then how can black holes have different masses? Wouldn't all the other mass be at the event horizon, since you can never actually approach the singularity? How can the singularity have different masses if by definition it is infinitely dense? Is this just our approximation? (I can see the possibility of two collapsing stars of different mass becoming black holes and prehaps any mass "within" the event horizon at the time of collapse would become part of the singularity, so maybe the black holes would have different masses, but to me talking about the mass of a singularity sounds funny.)

If black holes are really singularities, then how can the radii of their event horizons differ?

This sounds like a paradox that will most likely be explained very handily by someone out there.

2. May 11, 2004

### mathman

The critical radius of an object is determined by its mass. If the density is great enough so that its entire content is inside this radius, it is a blackhole. This density is dependent on the radius, since the critical radius is linear with the mass, so that the density is proportional to r-2.

3. May 13, 2004

### Labguy

The "critical density" in terms related to total mass is ~ 3.2 solar masses. Any stellar remnant above this will collapse to a black hole.

But, 3.2 Ms is not an infinite amount of mass. Neither is the >106 Ms black hole at the center of the galaxy M-87. So, your statement above about a singularity being of infinite density is not correct as an infinite density would have to equate to infinite mass. The infinite density idea was generally dropped quite a few years ago.

4. May 13, 2004

### jcsd

Infinmote densitity doesn't necessarily equal infinite mass: $\frac{m}{V}\rightarrow\infty\mbox{ as }V\rightarrow0$

GR shows that there is a singularity of infinite density in a black hole as there is a finite mass occupying a single point.

The critical density (ignoring other factors)that mathman refers (in thi case he is slightly wrong as the density is proportional to r-3)to is:

$$\frac{3c^6}{32 \pi G^3M^2}$$

Last edited: May 13, 2004
5. May 13, 2004

### Phobos

Staff Emeritus
As jcsd said, the General Relativity model has a black hole as a singularity of infinite density. Infinite density is based on zero volume, not the mass. The original mass determines whether or not the object can collapse into a black hole or not (enough weight to crush everything down to a point). The newer String Theory, which is not well proven like GR, has a lower limit on the volume and therefore, density is not infinite (was this what you were referring to labguy?).

shrumeo - The mass at the event horizon does reach the singularity. But from our outside reference frame, it looks like it stops at the event horizon (time dilation). The event horizon is not a phyical thing, but just the distance around the singularity at which nothing can escape (i.e., the distance at which the escape velocity is the speed of light). The more mass crunched down into the singularity, the further away the event horizon (radius) reaches.

6. May 13, 2004

### Labguy

With a Kerr-Newman black hole (rotating and with charge) or even just Kerr, there is a ring singularity with a finite size larger than the defined mathematical point. The point works for the math described above, but how many believe there are static, non-rotating black holes? Not many I think.

Of course, the "point source" definition of a singularity would have infinite density even if there was but one gram of matter, much less >3.2 Ms.

Last edited: May 13, 2004
7. May 13, 2004

### jcsd

Actually a ring singularity has zero volume too, as the ring has a thickness of zero.

8. May 14, 2004

### Labguy

Kerr thinks it is a torus, i.e., donut, with a thickness depending on the rate of the BH rotation as per angular momentum. Might be new info or theory, but I hadn't heard the "thickness of zero" version yet. Or, new to me at least, I'm old.

9. May 14, 2004

### jcsd

I think you msay be a little confused, the ring singularity is the outer equator of a torus with some very interesting properties, but it itself has zero voulme.

10. May 14, 2004

### Labguy

Ok, I might agree with that, but the mass (and all matter of whatever form) is somewhere inside the event horizon and is of a finite quantity which equates to the measured mass of the BH.

I realize that no single web site is "all-authorative", but several even doubt the existance of a singularity at all and consider the mass of the BH to be just another exotic form of compressed matter like a "quark soup" or some condensate. The problem is with using the idea (term) of infinite density which could only apply in an area/point of zero volume. A lot of theorists have abandoned trying to deal with or acknowledge infinite density.

From just the site I posted before, it is mentioned several times:
"And the second is the apparition of a "peculiarity" under the shape of an infinity of density, that is physically unacceptable." and:

"Consequently, matter, all the matter, is projected toward the equator where it accumulates under the shape of a relativistic tore.
In the tore, and solely there, the centrifugal force is oriented inwards. (relativistic effects) We have therefore in the tore a confinement effect. Let's note that matter is then under shape of à very high-density plasma of particles. (Neutrons, protons, electrons, etc.) But one notices that this model doesn't imply that the density is infinite."

I realize that some of the discussion pertains to effects outside the event horizon, but some implicates the singularity. Also, see:
http://www.innerx.net/personal/tsmith/Sidharth.html
about 1/3 down the page where the graphic shows the ring singularity in purple color. To me, it is getting complicated. Complicated to the point where many believe (I agree) that no singularity at all is necessary for a BH to exist. Even Hawking, in 1987, wrote a whole page trying to explain why he had abandoned the idea of singularities.

11. May 14, 2004

### jcsd

I'll be careful what you read on the web, the degenenracy pressure in a quark-gluon soup isn't strong enough to stop graviational collapse in most cases though it could lead to a so-called strange star which would have a narrow band of possible masses inbetween a neutron star and a black hole.

It's not the infinite density that's important to theorists it's the infinte curvature that's more worrying.

Some people think that a singularity will be prevented in quantum gravity, but the concept is far from abandoned as singularity at the centre of a black hole is an inevitable result of GR. As far as I'm aware Stephen Hawking was arguing that the big bang singularity is a removable singularity just like the event horizon of a black hole.

Again be careful, the site you quoted is hardly 100% reliable as it's (or at least appears very much to be one) an anti-big bang site, also a torus accreted matter does form, but that's not the singularity.

12. May 14, 2004

### Labguy

I'll be as careful as I can, but too many respectable theorists out there see no need for a singularity, as popularly defined, at the center of black holes, so no need for infinite density anywhere. The 1987 Hawking argument against singularities was not the origin of the BB, it was singularities in black holes. I'll try to dig around and find the sources I'm remembering.

13. May 14, 2004

### jcsd

There's a slight distinction as there's no solid theory behind black holes not having a singularity (Roger Penrose proved that black holes necessarily form singularites in 1964), so it's more corrcet to say that some believe that a singularity may be avoided.

14. May 14, 2004

### Labguy

Ok.

I don't know which of the 23 versions of inflation theory is correct, but Guth probably doesn't either. I suppose they get together sometime and brainstorm.

15. May 14, 2004

### jcsd

What's inflationary theory got to do with it black hole singularties? suffice to sya there's only one theory that can deal solidly with the interior of a black hole: GR; and singularities are an inevitiable result of the gravitational collapse of stars in GR.

16. May 14, 2004

### meteor

Has the diameter of the ring singularity of a Kerr or Kerr-Newman BH been calculated by somebody? Is the diameter of the ring a function of the mass of the black hole?

17. May 14, 2004

### Labguy

The point was that there is more than one viable theory as to most anything in cosmology, including BH geometry, around today, and no single theory has been "resolved" to the satisfaction of all.

Hawking, then Penrose dumped the 1964 "proof" that singularities have to exist in black holes. If you are sticking to that premis as the absolute gospel, you'll find a lot who will disagree. That was the point. Change the subject. A log-pile of physicists will argue string theory vs. not, Higgs field vs. not, ad infinitum, and it won't be settled in our lifetime. Gee, even GR and quantum gravity don't fit together yet so let's wait and see. You should consider other ideas as at least possibilities, or buy more books.

18. May 15, 2004

### Stingray

?... Black holes have nothing to do with cosmology. Inflation is a completely unrelated topic.

There is no disagreement that GR is the correct theory except AT the singularity. There are various singularity theorems proven by Hawking, Penrose, and others which were never dumped. These things are theorems, though. They say if so and so is true, then something happens. The "so and so" part isn't necessarily satisfied in the real universe (but its probably close enough). One could reasonably argue whether enough of the implications of GR are understood, but you're not going to find anyone claiming that the theorems are actually wrong.

meteor,
Diameter doesn't have meaning there, but the geometry is understood. It is related to the mass and angular momentum of the hole.

19. May 15, 2004

### jcsd

but inflationary theories irrelevant, and the only solid theory that is relevant predicts singularities.

You have to make the distinction between the belief that the singularity may or should be avoided (which is no founded on any particular theory just the idea that singularities can't be physically real and so is an attempt to second-guess a future theory) and having the theory to say that the singularity is avoided.