1. Nov 4, 2015

### Gerinski

I have read that the Schwarzschild radius of a black hole with the mass-energy of the observable universe is roughly equal to the actual Hubble radius of 13.8 billion light years. And I have read that contrary to some popular esoteric interpretations such as "the universe is a black hole", "we are inside a black hole" etc, this simply means that the universe is spatially flat, or nearly so, that the equivalence of Hubble radius and Schwarzschild radius for a flat universe is derived from the Friedmann equations.

So far so good, but there are things I do not understand.

As I understand, this means that was the universe not expanding, it would collapse into a black hole, it has already the total average density of a black hole with an event horizon the size of it. But this seems highly counterintuitive, the density of the observable universe seems incredibly thin, it is by far mostly empty space. How can it have the same density as a black hole?

Alright, a big share of its energy contents is dark energy, but even so, how can we then make the equivalence to a black hole? Dark energy may contribute to the total energy density of the universe but it causes it to expand, so it goes against the tendency to collapse gravitationally. It can not be right to include dark energy in the mass-energy computation to say that the mass-energy of the observable universe is equal to that of a black hole the same size, is it?

Related question: in a black hole, the density at the singularity is infinite, whatever mass divided by zero volume. But if we express the density as the mass of the black hole divided by the volume of its Schwarzschild sphere, what would the density of a black hole be like?

Thanks

2. Nov 4, 2015

### bcrowell

Staff Emeritus
This is a non sequitur. There is no logical connection between the average density of a black hole and the density of the object that collapses to form it. The average density you're talking about (mass of the black hole divided by the volume inside its event horizon) is not especially interesting and has no special properties or logical status.

If the universe were not expanding at some moment in time, then the Friedmann equations predict collapse to a Big Crunch singularity, not a black hole singularity. This is what happens in a closed FRW cosmology with zero cosmological constant. No minimum density is required.

Black hole spacetimes have very little in common with cosmological spacetimes. Cosmological spacetimes are homogeneous. Black hole spacetimes aren't.

3. Nov 4, 2015

### Smattering

Can someone please comment on the question whether this is just a coincidence, or it results from some kind of law or principle?

Volume and mass increase with the third power of a sphere's radius, and the Schwarzschild radius is proportional to the mass. Thus, it seems to me that the sphere's Schwarzschild radius should grow faster than the radius of the sphere itself. So it seems at least possible for me that the Hubble radius and the Schwarzschild radius of the observable universe had quite different values in the past.

But then again, this over-simplistic model does not consider expansion of space and much other phenomenons that might play a role.

4. Nov 4, 2015

### bcrowell

Staff Emeritus
The Friedmann equations give $H_0\sim \sqrt{\rho}$ (where the $\sim$ means that we make simplifications such as ignoring like the contribution of pressure to the stress-energy). Since the Hubble radius is $1/H_0$, it follows that the mass inside the observable universe is on the order of the Hubble radius (in geometrical units, where $G=c=1$). This is not a coincidence.

The part about a black hole is irrelevant and misleading, however. It just happens that the radius of a black hole's event horizon is on the order of its mass in geometrical units. There is no physical analogy between the observable universe and a black hole.

5. Nov 5, 2015

### Smattering

O.k., this is very interesting. Ignoring the metric expansion of space, the observable universe's radius should grow by one lightyear per year, right? Assuming that mass is homogeneously distributed on cosmological scales, this implies that the mass should grow faster than the radius of the observable universe (volume of a sphere, etc. pp.).

So when the Friedmann equations imply that the mass stays in the same order as the radius, is this due to the metric expansion of space?

Yes, I am aware that the observable universe is not a black hole. This interpretation makes no sense when assuming that the universe continues homogeneously beyond our hubble volume. After all, what we denote as "obversable universe" or "our hubble volume" should not be different from any other hubble volume. So if our hubble volume was a black hole, then any other hubble volume would also have to be a black hole. And this just makes no sense.

6. Nov 5, 2015

### martinbn

One more thing to keep in mind is that the radius $r =2m$ is not the distance from the horizon to the centre. Nor is the black hole a ball. The spacetime inside the hole is very counter-intuitive, it is not even stationary.

7. Nov 5, 2015

### Smattering

What is the meaning of "stationary" with respect to spacetime?

8. Nov 5, 2015

### Gerinski

Thanks, but it must have some value. It has mass X and event horizon's sphere volume (as measured from outside) Y, so it must be possible to say what X/Y is like for typical black holes. I'm just intrigued if such a value would be counterintuitively small for laymen like me. I know this does not reflect any physical density. The space inside of the black hole's event horizon does not contain any stuff, all of its mass resides at the hypothetical singularity, not in the space enclosed within the event horizon.
But still we can divide X/Y and see what turns out. Perhaps the density of a black hole measured in that way would be (say for example) not much denser than lead? (just guessing).

9. Nov 5, 2015

### martinbn

10. Nov 5, 2015

### Smattering

O.k., I think now I understand what you were referring to by "not stationary".

11. Nov 5, 2015

### bcrowell

Staff Emeritus
That's an interesting paper. I find it easy to believe that we can't meaningfully assign a volume of $(4/3)\pi r^3$ to the interior, since obviously that's a Euclidean formula, and nothing here is Euclidean. What is less obvious to me is why they think the interior volume can and should be characterized as the maximal volume of a spherically symmetric spacelike surface. Even in the Minkowski case, this surprises me. I would have expected that one could have made the interior volume arbitrarily large or arbitrarily small, based on intuition from the fact that a spacelike geodesic neither maximizes nor minimizes length. I guess there is something subtle and (to me) non-obvious that happens when you increase the number of dimensions and require symmetry.

12. Nov 5, 2015

### bcrowell

Staff Emeritus
You can divide the charge of an electron by the S&P 500 index and add Barack Obama's year of birth, and it will have some value. That doesn't mean it's a meaningful thing to consider.

Why don't you just go ahead and calculate it? It's going to depend on the mass of the black hole, which can take on any value.

13. Nov 5, 2015

### Smattering

When they say "long spacelike 3d cylinder" - I guess they are referring to a cylinder with a 3d surface, right?

But if there is there is really so much space inside and the singularity is located at the other end of the cylinder - how long will it take for an infalling object to reach the singularity?

14. Nov 5, 2015

### Gerinski

Of course, and the mass defines its Schwarzschild radius as well. So I will try go ahead and calculate it, it does not seem too difficult a task. Thanks for the big help.

15. Nov 5, 2015

### bcrowell

Staff Emeritus
We can't say that there is "really" that much space inside. The paper makes a somewhat arbitrary definition of the volume. It's not the only possible definition.

The proper time to reach the singularity is quite short -- on the order of r/c, where r is the radius of the event horizon.

16. Nov 6, 2015

### martinbn

A space-like geodesic doesn't minimize or maximize the space-time interval, but if you restrict to a space-like hypersurface, which will have a definite metric, there should be a maximal one i.e. one with maximal length where by length one means the length from the induced metric. (Of course I am vague here. If you consider curves on a non-compact subset, say missing points, there need not be a maximal length.)

17. Nov 6, 2015

### Smattering

O.k. I understand that. But do you have an idea what the authors mean by the following quote:

Why does matter have "newer and newer space" to fall into? And how can it be that the time to reach the singularity stays in the order of r/c if matter has newer and newer space to fall into?

18. Nov 6, 2015

### martinbn

For the first one because space-time is dynamic so space changes with time. For the second, because the singularity is like a moment of time, no matter how much new space you get Sunday is coming in two days.

19. Nov 6, 2015

### Smattering

I can understand that in the sense that mass bends space-time and the curvature of space-time changes the movement of masses which results in yet a different curvature of space-time and so on ...

But on the other hand, I would have thought that a singularity has already maximum density so that there cannot be any further movement of masses--at least not in the Schwarzschild case where the singularity is located in a vacuum. And if the distribution of mass (or energy) stays constant, why does the curvature of space-time keep changing?

But isn't the singularity also like a location in space?

Best regards,
Robert

20. Nov 6, 2015

### Staff: Mentor

No. The singularity is spacelike, not timelike. A "location in space" would be described by a timelike curve in spacetime.

21. Nov 6, 2015

### Smattering

O.k., just to make sure that I am interpreting this correctly: Can you please tell me which of the following statements are correct respectively wrong?

1. The center of the BH is not a location, but rather it is a point in time.
2. There is exactly one point in time at the center of the BH.
3. All infalling objects will reach the center of the BH at exactly the same point in time regardless when they crossed the EH (because of 2).
4. The center of the BH can be considered the end of time.
5. The travel time from the EH to the singularity is proportional to the Schwarzschild radius.
6. The singularity is not a point, but rather it has a spatial extent.
7. The spatial extent of the singularity grows with the age of the BH.
8. Objects that cross the EH at different points in time will hit the singularity at different locations.

Thank you very much in advance.

Best regards,
Robert

22. Nov 6, 2015

### bcrowell

Staff Emeritus
These are all incorrect for the same reason. They all talk about the singularity as if it were a point or set of points. The singularity is not part of the spacetime manifold at all. Because the singularity is not a point or point-set, there is no obvious, simple, and correct way to define its dimensionality. Similar considerations apply to its spacelike or timelike character, but there is a standard (not simple, not obvious) definition of whether a singularity is timelike or spacelike: http://adsabs.harvard.edu/full/1974IAUS...64...82P . By this definition, a Schwarzschild black hole's singularity is spacelike.

#7 is also wrong because the singularity is spacelike, not timelike, so it doesn't make sense to talk about how it changes over time.

4 is correct. 5 is correct if "travel time" means proper time, and I think it should also probably refer to the maximum proper time for free-falling particles.

23. Nov 6, 2015

### Staff: Mentor

All of them are unsatisfactory, because you are trying to use ordinary language to describe something that really can't be described using ordinary language. Our ordinary language concepts of "location in space" and "point in time" were not developed to deal with black holes. So I won't sign up to saying any of these statements are "right", or even "wrong", because "wrong" implies that the terminology being used is applicable in the first place, and it isn't.

As I believe I've said before in this thread, a much more fruitful viewpoint is to drop all the ordinary language terms involving "space" and "time" altogether, and instead look at things geometrically. Here is a geometric description of two different black hole spacetime models:

(1) An "eternal" black hole formed by the gravitational collapse of a massive object like a star. "Eternal" here means we are using classical GR only and ignoring any quantum effects like Hawking radiation, and also assuming that nothing else falls into the hole once it forms. In that case the spacetime geometry has three regions:

- Region C: the region occupied by the collapsing matter. This region has the same geometry as a collapsing closed universe, i.e., it is homogeneous and isotropic as seen by "comoving" observers within the collapsing matter. Note that a portion of region C is outside the event horizon of the spacetime, but the other portion is inside it. Like a collapsing universe, Region C has a "scale factor" which can be thought of as the "size of the universe" as seen by comoving observers, and which each such observer sees as decreasing with time (here "time" means "proper time along the observer's worldline", which is well-defined), until the scale factor reaches zero at the endpoint of Region C (see below for more on this endpoint).

- Region I: the vacuum region outside the collapsing matter, and also outside the event horizon. This region has the exterior Schwarzschild geometry, which is a static spacetime region in which there is a well-defined notion of "observers at rest" (these are observers who "hover" at a constant altitude above the horizon), and a well-defined notion of "time dilation" for those observers, relative to an observer at rest at infinity.

- Region II: the vacuum region outside the collapsing matter, but inside the event horizon. This region has the interior Schwarzschild geometry. This geometry is not static; there is no well-defined notion of "observers at rest", and there is no well-defined notion of "time dilation".

The singularity in this spacetime is a spacelike hypersurface labeled $r = 0$ that has one endpoint at the future endpoint of Region C (the point at which, heuristically, the collapsing matter reaches infinite density, forms the singularity, and vanishes), and extends from there all the way to infinity in the other direction. This spacelike hypersurface has zero "radius", meaning it is more properly thought of as a spacelike line--i.e., it is composed of a series of points (a continuous infinity of them) lined up in a spacelike direction, rather than a series of 2-spheres lined up in a spacelike direction.

(Note, btw, that what I've said above does not contradict what bcrowell said about the singularity not being part of the spacetime. The statements I made above, to be strictly correct, should be reinterpreted as talking about calculated quantities, limits of various things as $r \rightarrow 0$, instead of "actual" quantities applying to an "actual" spacelike line at $r = 0$. Or, they should be interpreted as talking about an "extended" spacetime which has been mathematically enlarged to include the singularity, even though that extension doesn't describe a physically real part of the original spacetime.)

The geometry of Region II has a highly counterintuitive property, which is that there are spacelike hypersurfaces that start at the boundary of the collapsing matter, and extend to infinity in the other direction, just as the singularity does. But unlike the singularity, these hypersurfaces are composed of 2-spheres lined up in a spacelike direction--a continuous infinity of them. That means that the 3-volume of one of these hypersurfaces is infinite.

(2) A black hole formed by the gravitational collapse of a massive object, which then radiates Hawking radiation and eventually evaporates away. This geometry has four regions:

- Region C: the region occupied by the collapsing matter. This works the same as for the first model above.

- Region I: the vacuum region outside the collapsing matter, and outside the horizon, in which the black hole is measured to have a nonzero mass. This is similar in many ways to Region I in the first model above, but it has one key difference: it is not static, because the mass of the hole changes with time. Furthermore, the mass of the hole also depends on the radial coordinate in this region (this is implicit in the discussion we had earlier in the thread about how the orbital radius of the test object affects "when" the hole's mass is observed to increase--when Hawking radiation is emitted by the hole and flies outward, "when" the hole's mass is observed to decrease depends on the orbital radius of the test object in the same way). So, although we can still pick out observers who "hover" at some altitude above the horizon, the properties of spacetime measured by those observers are no longer constant; they change with time (for example, the proper acceleration required to hold station at a given altitude decreases), and these changes can be thought of as the mass of the hole, as measured by those observers, decreasing with time.

- Region II: the vacuum region outside the collapsing matter, and inside the horizon. This region is also similar in many ways to Region II in the first model, but it also has one key difference: the spacelike hypersurfaces (and the singularity itself, as a spacelike line) that extended to infinity in one direction in the first model, no longer do so. They now end in a finite "length" (more technically, after a finite affine parameter along any spacelike geodesic within them), and their endpoints are on the boundary between Region II and Region I, i.e., the event horizon. This happens because the horizon's radial coordinate is no longer constant in this model; it decreases from its maximum value, at the point where the horizon crosses the boundary of Region C, to zero at the point where the horizon meets the singularity at $r = 0$, which is the point at which the hole finally evaporates. In between those two points, spacelike hypersurfaces that would have remained inside Region II to infinite extent in the first model above, instead cross into Region I.

- Region F: the vacuum region in which the hole is observed to have completely evaporated, and which is therefore geometrically flat, with no mass or energy present. The boundary between this region and Region I is the outgoing null surface composed of light emitted radially outward from the point of the hole's final evaporation. The worldline of an observer "hovering" at a given altitude will intersect this boundary at some point (the observer perceives this as the light flash passing him on its way out), and after this happens ("after" according to the proper time along that observer's worldline), that observer will be in a flat spacetime region and will perceive the hole to be gone.

24. Nov 7, 2015

### Smattering

In what sense is the singularity "not part of the spacetime manifold"? Can you please try to explain what criteria something has to fulfill in order to be "part of the spacetime manifold"?]

25. Nov 7, 2015

### bcrowell

Staff Emeritus
A singularity represents a breakdown in the metric, which is the only apparatus we have for measurement. Without a metric, you can't tell the difference, for example, between one point and many points. Suppose I have a two-dimensional space with coordinates (u,v), and I ask you whether {(u,v)|v=0} is a point or a line. You'd probably say it was a line, and if the metric was $ds^2=du^2+dv^2$, you'd be right. On the other hand, if the metric was $ds^2=v^2du^2+dv^2$, it would be a point.

In my (u,v) space I gave an example where there are two possible metrics we could imagine. At a singularity, it's even worse. There is *no* possible metric that we can extend to the singularity.

So because we can say so little about any points or sets of points at a singularity, we choose not to call them points or sets of points at all. This isn't just a mathematical convention. In the case of a black hole singularity, it represents the fact that there is no "there" there, nothing that we can ever know through observation, even in principle. (In the case of a timelike singularity, we could in principle observe whatever popped out of it -- which in John Earman's famous phrase could be anything, including green slime or your lost socks. But we would still not be observing the singularity itself.)

Last edited: Nov 7, 2015