# Something I Don't Understand about Black Hole density.

1. Jun 13, 2013

### AndromedaRXJ

So I always hear that the density of a black hole is infinite, because it's volume is zero.

How can this be if, in mathematics, n/0 ≠ ∞? Would it be more correct to say a black hole's density is undefined or indeterminate?

I also hear that some people say that a black hole's volume is "infinitely small" and they distinguish it from zero, if I'm not mistaking. What exactly would this mean? Is it's volume something more like 0.∞(point infinity)? Does that make mathematical sense?

If not, then is there anything we can divide by to get infinity?

2. Jun 13, 2013

### phinds

Mathematically, the logical conclusion of the math models of a black hole leads to it having a zero volume and therefore infinite density at the singularity, BUT ... "singularity" really just means "the place where the model breaks down and we're not sure WHAT is going on".

3. Jun 13, 2013

### AndromedaRXJ

But how is that if division by zero doesn't result in infinity? Electrons don't have infinity density, do they?

4. Jun 13, 2013

### HallsofIvy

Staff Emeritus
AndromedaRXJ, you were right to begin with. What is MATHEMATICALLY correct is that a black hole does not have a "density". But those physicists just won't listen!

5. Jun 13, 2013

### WannabeNewton

When it comes to stationary black holes in general relativity we are interested in three properties of the stationary black holes: the charge, the spin, and the mass. What do I mean by stationary? A black hole is formed out of some form of gravitational collapse and after its turbulent formation it can also be "eating up" matter around it. Once the black hole has settled to some stationary state, such that the exterior region of the black hole is vacuum except for possible electromagnetic fields, we call it a stationary black hole. It can be shown that there exists only a single three parameter family of stationary black hole solutions to the Einstein field equations. The three parameters are called the spin, the charge, and the mass of the solutions and they completely characterize the solutions. Quantities which are mathematically nonsensical, things like infinities, aren't used in general relativity to physically characterize stationary black holes as far as I know, but I could easily be wrong!

6. Jun 13, 2013

### AndromedaRXJ

Really? lol

Okay then. But isn't it also incorrect to say it "does not have density"? That's basically saying zero density. And division by zero ≠ zero or infinity.

So can we just say it's density is undefined? Or is the volume just not zero, but rather infinitely(or infinitesimally) small? It seems like, in order to really have infinite density, we'd need infinite mass over finite volume.

Forgive me, but what does this have to do with a BH's density and volumn?

Edit: Okay, I guess what you're saying is, they don't have infinite density?

Last edited: Jun 13, 2013
7. Jun 13, 2013

### WannabeNewton

Yes I'm agreeing with Halls and Phinds that it is not a mathematically meaningful quantity to characterize a black hole by in classical GR, as far as I know.

8. Jun 13, 2013

### elfmotat

Why not? Point particle density is easily and usefully characterized by a delta function all the time in classical physics.

9. Jun 13, 2013

### WannabeNewton

The notion of a "point particle" is not trivial in general relativity. One must very carefully consider well defined limits; one cannot naively use "point particles" in the Newtonian sense. How would one incorporate delta function sources into the stress-energy tensor and still solve the non-linear PDEs making up the Einstein field equations? This is a very complicated subject in the theory of distributions and PDEs.

http://arxiv.org/pdf/0905.2391v2.pdf
http://arxiv.org/pdf/0907.0412v1.pdf
http://arxiv.org/pdf/0907.0414v1.pdf

10. Jun 14, 2013

### maxverywell

The volume of BH is actually not zero. BH is not only its physical singularity, it's the whole spacetime region from the singularity to the event horizon.

Think for example a black hole with the mass of our Sun, which has ~3km radius. For us, as outside observers, the volume of this BH is the volume of the space inside the event horizon, which is finite, and its mass is M_sun, which is also finite. Thus the density of BH is finite and (relatively) huge, but not infinite.

Outside of the event horizon, the spacetime is described by the same metric, no matter how the mass is distributed inside the event horizon. So, for an observer outside of the BH, there is no problem with infinities.

But inside the BH, we have problem at the physical singularity because everything that crosses the event horizon, inevitably moves towards it, which is a point in space (in the case of non-rotating BH). This means that all the infalling mass concentrates somehow at a mathematical point. But we actually don't know what exactly happens there, the general theory of relativity doesn't work there.

By the way, I think that by definition, a space region of zero volume has zero mass -- no such thing as infinite density. There is no space to put the mass, no matter how little, loosely speaking...

Last edited: Jun 14, 2013
11. Jun 14, 2013

### mpv_plate

Yes. And in this sense the supermassive black holes in the centers of galaxies have actually pretty low "density". Even lower than the density of air on Earth.

12. Jun 14, 2013

### AndromedaRXJ

I agree with this. I've always preferred to include the event horizon as part of a black hole since it's practically the reason why we call it black. Meaning inside the event horizon is inside the black hole, rather then, only inside the singularity is inside the black hole. But some people just include only the singularity. Wanted to avoid confusion but didn't know how to go about it initially.

Perhaps I should have asked "is the mass of a black hole(the singularity) infinitely dense because it occupies zero space?"

I would say no, because again, n/0 ≠ ∞.

So I guess I've gotten my answer. Once we get to the singularity, it's all a big mystery, and no one should say with confidence that there's infinite density anywhere.

However, regarding space-time it self, isn't that getting smaller as you approach the singularity?

Meaning if mass is occupying the singularity, then from it's "point of view", wouldn't it not seem so small after all, since the space-time around it, in comparison is also probably around the same size?

Sorry if that didn't make much sense. I should be in bed. It's 2:20 AM here. lol

13. Jun 14, 2013

### Grombely

Sorry, I would like to say that to the event horizon shouldn't be measured as part of a black hole's density as this is an 'effect' not acctual substance- sorry do not have time to talk more- though a black hole would be a definite volume & density,

14. Jun 14, 2013

### maxverywell

Last edited: Jun 14, 2013
15. Jun 14, 2013

### WannabeNewton

It is a null hypersurface and the black hole is the set theoretic difference of the space-time manifold with the causal past of future null infinity, nothing more; it is topologically closed in space-time and hence contains the event horizon. It also has an area which is dynamical according to the laws of black hole mechanics; this is one of the main geometric features of the event horizon along with the fact that its surface gravity is constant. You are right however that it is not an "actual substance" in the way many people think.

The problem with talking about things like the mass density of a black hole is that we don't know how matter content is distributed inside a black hole; general relativity cannot provide insight into this. Look up the "No-hair" theorem; this is why we characterize a black hole by its mass, charge, and spin as I said earlier because these are meaningful external observables. The mass density of the black hole will depend on how matter content inside is distributed and we don't know how as per general relativity; this is why you have to be careful because unless you can bring quantum gravity into this, you're just speculating.

EDIT:

Last edited: Jun 14, 2013
16. Jun 14, 2013

### maxverywell

Isn't all the matter located at the singularity and the region 0<r<2GM empty?
(I am considering a Schwartzchild BH formed after a complete collapse of a star to the singularity)

Last edited: Jun 14, 2013
17. Jun 14, 2013

### maxverywell

Post by chroot:

18. Jun 14, 2013

### WannabeNewton

As he said, no relativity textbook defines it (mass density). If you have a reference to a paper or text that actually does, feel free to share it. Again, the singularity theorems of GR only prove the existence of singularities-they don't provide insight into the nature of these singularities nor into the inextendability of time-like and/or null geodesics that terminate there, beyond the intrinsic topological properties of the manifold.

Last edited: Jun 14, 2013
19. Jun 14, 2013

### maxverywell

No I don't have any paper. I agree with you that the notion of the mass density of a BH is quite meaningless and cannot be well defined. Perhaps we could define the average density by dividing the mass of BH by $r_s^3$, but I don't know how useful it would be, because $r_s=2M$, so this average density is a function of M (no hair theorem...).

20. Jun 14, 2013

### WannabeNewton

I'm not averse to such a possibility max my friend but I've never seen a textbook that does it so I wouldn't know how valid/useful it would be as you say. On the other hand there may be papers that do indeed construct a meaningful notion of the mass density and if someone knows of one that would be a great share!