Something I Don't Understand about Black Hole density.

[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?

 

[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".

 

[AndromedaRXJ]

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

 

[HallsofIvy]

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!

 

[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!

 

[AndromedaRXJ]

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!

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.

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!

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?

 

[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.

 

[elfmotat]

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.

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

 

[WannabeNewton]

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

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.

See here for further readings:
http://arxiv.org/pdf/0905.2391v2.pdf
http://arxiv.org/pdf/0907.0412v1.pdf
http://arxiv.org/pdf/0907.0414v1.pdf

 

[maxverywell]

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?

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...

 

[mpv_plate]

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.

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.

 

[AndromedaRXJ]

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

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 ≠ ∞.

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...

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

 

[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,

 

[maxverywell]

 

[WannabeNewton]

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,

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:
See here for a further reading: https://www.physicsforums.com/showthread.php?t=28672&page=2

 

[maxverywell]

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.

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)

 

[maxverywell]

EDIT:
See here for a further reading: https://www.physicsforums.com/showthread.php?t=28672&page=2

Post by chroot:

None of my relativity textbooks bother to define either the density of a black hole or the volume of a black hole. You can certainly measure the radius event horizon, of course, using test photons or test particles. At the very least you could measure the radius of the photonsphere and infer the radius of the event horizon. I don't see any reason why you could not simply define a black hole's (average) density to be its mass divided by the volume enclosed by its event horizon. Spacetime does not break down there, and neither do our models of it. The Schwarzschild solution is valid all the way down to the singularity (or, shall we say, to the suspected very tiny quantum-mechanical object!), at least in Eddington-Finkelstein coordinates.

- Warren

 

[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.

 

[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...).

 

[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!

 

[Chronos]

An article that might be of interest: http://curious.astro.cornell.edu/question.php?number=219 .

 

[WannabeNewton]

That's interesting Chronos but how are they calculating the volume i.e. what formula are they using?

 

[HallsofIvy]

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.

No. Saying something does NOT exist does NOT mean it is 0. 0 exists!

 

[pbuk]

A black hole is NOT an object that is "infinitely dense", a (stationery) black hole is simply a object whose Schwarzschild radius is greater than its physical radius. A neutron star with a mass of, say, 4 solar masses would be a black hole: although there is no way of knowing, there is no reason to assume that the density of such an object would be any different from the density of any other neutron star.

 

[Redbelly98]

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.

People can use the phrase "does not have X" to mean either that property X is undefined, or that X is zero. Whichever one of those is meant is supposed to be clear from the context, but those two interpretations have very different meanings from each other.

In this case, "does not have density" means it is undefined, not that it is zero.

EDIT: Aaaargh, sorry for necroposting. At least it was only weeks ago, not months or years.

 

[Chronos]

The customary way of defining the 'volume' of a black hole is the volume occupied by its event horizon. That is a highly unrealistic assumption.