Universe Hubble radius equal to Schwarzschild radius

[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

 

[bcrowell]

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.

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.

 

[Smattering]

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.

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.

 

[bcrowell]

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.

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

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.

 

[Smattering]

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.

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?

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.

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.

 

[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.

 

[Smattering]

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

 

[Gerinski]

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.

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).

 

[martinbn]

What is the volume of a black hole? That's not as easy as the naive guess would suggest.

http://xxx.lanl.gov/abs/1411.2854

 

[Smattering]

What is the volume of a black hole? That's not as easy as the naive guess would suggest.

http://xxx.lanl.gov/abs/1411.2854

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

 

[bcrowell]

What is the volume of a black hole? That's not as easy as the naive guess would suggest.

http://xxx.lanl.gov/abs/1411.2854

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.

 

[bcrowell]

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.

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.

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).

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.

 

[Smattering]

What about this quote from the article?

That is: Inside the hole there is a long spacelike 3d cylinder with slowly varying radius, which grows longer with time.

This is a surprising result, because the volume is large. For instance, the black hole Sagitarius A has radius ~ 10^6 km and age ~ 10^9 years. Inside it, there is space for 10^34 km^3, enough to fit a million Solar Systems!

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?

 

[Gerinski]

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.

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.

 

[bcrowell]

But if there is there is really so much space inside[...]

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.

[...] 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?

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.

 

[martinbn]

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.

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.)

 

[Smattering]

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..

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

We find that the volume V(v) inside the sphere S_v grows with v. This makes sense: even if its surface-area remains constant, the horizon is still an outgoing null surface and the interior volume keeps growing with time. Matter, so to say, has newer and newer space where to fall into.

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?

 

[martinbn]

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?

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.

 

[Smattering]

For the first one because space-time is dynamic so space changes with time.

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?

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.

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

 

[PeterDonis]

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

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

 

[Smattering]

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

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.Robert

 

[bcrowell]

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).
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.

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. 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.

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.

 

[PeterDonis]

Can you please tell me which of the following statements are correct respectively wrong?

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.

 

[Smattering]

Thanks for your reply, bcrowell.

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.

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"?]

 

[bcrowell]

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"?]

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.)

 

[Smattering]

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.

O.k., I can certainly accept that, but I would like to understand *which* terms and concepts are not applicable, *when* exactly they are not applicable, and *why* they are not applicable.

bcrowell has already answered some of those questions in his last post.

(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:

Yes, let's ignore any quantum effects here.

- Region C: the region occupied by the collapsing matter.

When you say "region" here, you are referring to spacetime region not a purely spatial region, right?
By the way, why is this region called "C" whereas the other two are called "I" and "II"?

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.

Well, if all of the collapsing matter was located within the event horizon of the *spacetime*, then it would be something like a primordial black hole, wouldn't it?

- 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.

Is there a fundamental difference between this region and the region around any other massive spherical object that is not a black hole? I mean a difference other than that this one borders on the event horizon ...

- 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".

How do I distinguish a "static" geometry from one that is "not static"?

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.

Hm ... counterintuitive seems to be quite an understatement in this case. I will come back to this once I have finished thinking it through.

 

[PeterDonis]

I can certainly accept that, but I would like to understand *which* terms and concepts are not applicable, *when* exactly they are not applicable, and *why* they are not applicable.

You're looking at it backwards. Instead of starting from your ordinary language terms and concepts, and then discarding the ones that won't work, you should be starting from the fundamental terms and concepts of the theory, forgetting everything you know about your ordinary language concepts. That's why I have been encouraging you to take a geometric viewpoint; spacetime geometry is the fundamental concept of GR.

Once you have a firm understanding of the fundamental terms and concepts of the theory, you can work on building links to the ordinary language concepts. That is when you figure out which ordinary language concepts still have valid interpretations and which do not. But if you try to start from that point, you won't get anywhere.

When you say "region" here, you are referring to spacetime region not a purely spatial region, right?

Yes.

why is this region called "C" whereas the other two are called "I" and "II"?

Because I like off-the-wall naming schemes. But the scheme actually does have some basis: the labeling of Regions I and II is standard in GR for the Schwarzschild geometry; "C" stands for "collapsing matter"; and "F" in the naming scheme of the second model (with an evaporating hole) stands for "future".

if all of the collapsing matter was located within the event horizon of the *spacetime*

You're looking at it wrong. Remember, this is geometry. Geometrically, the region containing the collapsing matter, Region C, has a portion that is inside the horizon and a portion that is outside. That is all you can say. Trying to say "where" the collapsing matter is "located" is meaningless on this view; that's trying (prematurely) to link back to ordinary language concepts, where we're supposed to be working with the fundamental concepts of the theory, i.e., geometry.

then it would be something like a primordial black hole, wouldn't it?

What do you mean by "a primordial black hole"? What spacetime geometry does that term refer to?

Is there a fundamental difference between this region and the region around any other massive spherical object that is not a black hole? I mean a difference other than that this one borders on the event horizon ...

The only real difference is where the boundary of Region I is. In the black hole case, the boundary is the event horizon; in the case of a massive spherical object, the boundary is the surface of the object. (The spacetime of a massive spherical object, if we assume for the moment that the object has always existed and never disappears or evaporates, contains only two regions: the vacuum exterior region, Region I, and the region inside the object, which we could call Region O. The surface of the object is the boundary between the two regions.)

The different location of the boundary does mean that, in the case of the massive spherical object, Region I is "smaller" than it is in the black hole case. A more precise wording of that would be: the entirety of Region I in the massive object case is geometrically identical to only a portion of Region I in the black hole case (the portion with a radial coordinate greater than the radial coordinate of the surface of the massive object).

How do I distinguish a "static" geometry from one that is "not static"?

The technical definition is that a static spacetime has a timelike Killing vector field. In less technical terms, a static spacetime (or static region of spacetime) has a family of observers in it with the property that each observer, along his worldline, sees an unchanging spacetime geometry. (The geometry can change from one observer to another, but it can't change for a given observer.) In Region I of the black hole model (and of the massive spherical object model), the static observers--the ones that see an unchanging spacetime geometry all along their worldlines--are the observers who are "hovering" at a constant altitude above the horizon (or above the surface of the massive object).

 

[Smattering]

You're looking at it backwards. Instead of starting from your ordinary language terms and concepts, and then discarding the ones that won't work, you should be starting from the fundamental terms and concepts of the theory, forgetting everything you know about your ordinary language concepts.

If I forgot *everything* I know about ordinary language, then I simply would not be able to communicate with you anymore. ;-)
So I need to find out which ordinary language concepts are misleading in this context, and which are "orthogonal" enough that the can be used.

You're looking at it wrong. Remember, this is geometry. Geometrically, the region containing the collapsing matter, Region C, has a portion that is inside the horizon and a portion that is outside. That is all you can say. Trying to say "where" the collapsing matter is "located" is meaningless on this view; that's trying (prematurely) to link back to ordinary language concepts, where we're supposed to be working with the fundamental concepts of the theory, i.e., geometry.

I somehow have the impression that we got our wires crossed in this case.

When I wrote "located" I meant that in a purely geometrical sense. In the same sense that you were referring to when you wrote that one portion of the collapsing "is" inside the event horizon of the spacetime and the other "is" outside of it.

I was simply asking myself *why* there must be a portion of the collapsing matter outside of the "event horizon of the spacetime"; or respectively what would be the consequence if all the matter was inside.

What do you mean by "a primordial black hole"?

I mean a speculative black hole that did not result from the collapse of a star, but rather from some local density anomaly that originated during the big bang (hypothetically). To avoid misunderstandings: I am not asking whether such a thing exists. Just consider it a pure thought experiment.

What spacetime geometry does that term refer to?

My assumption was that this thing could correspond to a spacetime geometry where nothing is outside the "event horizon of the spacetime".

But it is entirely possible that this is just pure garbage. In this case, I simply do not understand why a portion of the collapsing mass must be outside the "event horizon of the spacetime".

The technical definition is that a static spacetime has a timelike Killing vector field. In less technical terms, a static spacetime (or static region of spacetime) has a family of observers in it with the property that each observer, along his worldline, sees an unchanging spacetime geometry. (The geometry can change from one observer to another, but it can't change for a given observer.) In Region I of the black hole model (and of the massive spherical object model), the static observers--the ones that see an unchanging spacetime geometry all along their worldlines--are the observers who are "hovering" at a constant altitude above the horizon (or above the surface of the massive object).

So the non-static geometry of region II is exactly what hinders light from getting out of the black hole?

 

[Smattering]

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.

O.k., I understand why you need a metric to distinguish a point from a line.

In my (u,v) space I gave an example where there are two possible metrics we could imagine.

But in GR, the metrics must solve the Einstein field equations, right? So apparently, the problem is not that we have multiple choices.

At a singularity, it's even worse. There is *no* possible metric that we can extend to the singularity.

O.k., but if the problem is that the metric diverges for a certain area of spacetime, then what is the boundary surface of that spacetime area?

 

[PeterDonis]

I need to find out which ordinary language concepts are misleading in this context, and which are "orthogonal" enough that the can be used.

Any ordinary language concept related to physics or gravity is misleading. That's the safest assumption you can make.

I was simply asking myself *why* there must be a portion of the collapsing matter outside of the "event horizon of the spacetime"; or respectively what would be the consequence if all the matter was inside.

There is no solution of the Einstein Field Equation that has these properties. In other words, according to GR it is not possible for Region C, containing the collapsing matter, to be entirely inside the event horizon.

One way to think of why this is true is that the event horizon is only there in the first place because the collapsing matter causes it to form--geometrically speaking, the portion of the event horizon within the collapsing matter is the boundary of the region where the collapsing matter has a density greater than a particular threshold value. But the collapsing matter can't start out with that threshold value, because that threshold value is too high for any stable configuration of matter to exist; in other words, there is no possible starting configuration for the collapse that has a high enough density for the collapse to start inside an event horizon. The only stable starting states for a collapse have densities lower than that--or, geometrically speaking, the only valid solutions of the Einstein Field Equation describing matter collapsing to a black hole must have a region containing the matter that is outside any event horizon that might be present in the spacetime.

I mean a speculative black hole that did not result from the collapse of a star, but rather from some local density anomaly that originated during the big bang (hypothetically). To avoid misunderstandings: I am not asking whether such a thing exists.

Ah, I see. Theoretically, such a thing certainly can exist, but we have no evidence that there are any primordial black holes in our actual universe. However, even if there were, what I said above would still be true; there would still have to be a portion of the region occupied by collapsing matter that was outside the event horizon of the primordial black hole.

So the non-static geometry of region II is exactly what hinders light from getting out of the black hole?

No. There are plenty of non-static spacetimes that do not have event horizons.

 

[PeterDonis]

if the problem is that the metric diverges for a certain area of spacetime

No, it doesn't diverge for an "area of spacetime". For a spherically symmetric black hole, we can assign a radial coordinate $r$ to every event in the spacetime that is greater than zero. The metric is finite at every such event. But if we take limits of quantities related to spacetime curvature as $r \rightarrow 0$, we find that the limits diverge; i.e., quantities describing spacetime curvature increase without bound as $r \rightarrow 0$. So, speaking loosely, we can describe $r = 0$ as a "singularity" where things "diverge" or "become infinite". But $r = 0$ is not in the set of events in the spacetime at all; only events with $r > 0$ are in the spacetime.

 

[Smattering]

No, it doesn't diverge for an "area of spacetime". For a spherically symmetric black hole, we can assign a radial coordinate $r$ to every event in the spacetime that is greater than zero. The metric is finite at every such event. But if we take limits of quantities related to spacetime curvature as $r \rightarrow 0$, we find that the limits diverge; i.e., quantities describing spacetime curvature increase without bound as $r \rightarrow 0$. So, speaking loosely, we can describe $r = 0$ as a "singularity" where things "diverge" or "become infinite". But $r = 0$ is not in the set of events in the spacetime at all; only events with $r > 0$ are in the spacetime.

So spacetime is defined as the domain of a metric that satisfies the Einstein field equations. And when the metric behaves not well-defined we are out of spacetime.

By the way, I just figured out that singularity is simply the English term for what we call "Definitionslücke" (literally translates as "definition gap") in German. So in English you would say that the function $f(x) = 1/x$ has a singularity at $x=0$. This is a bit embarassing, but I really wasn't aware of this.

 

[PeterDonis]

So spacetime is defined as the domain of a metric that satisfies the Einstein field equations. And when the metric behaves not well-defined we are out of spacetime.

Yes. The technical term is "manifold with metric".

 

[bcrowell]

By the way, I just figured out that singularity is simply the English term for what we call "Definitionslücke" (literally translates as "definition gap") in German. So in English you would say that the function $f(x) = 1/x$ has a singularity at $x=0$. This is a bit embarassing, but I really wasn't aware of this.

This is not really correct. Singularity is a broader term, without a single well-defined meaning that applies across all areas. See https://en.wikipedia.org/wiki/Singularity_(mathematics) . For example, a cusp is considered a singularity in a certain context: https://en.wikipedia.org/wiki/Cusp_(singularity)

 

[Smattering]

This is not really correct. Singularity is a broader term, without a single well-defined meaning that applies across all areas. See https://en.wikipedia.org/wiki/Singularity_(mathematics) . For example, a cusp is considered a singularity in a certain context: https://en.wikipedia.org/wiki/Cusp_(singularity)

Thanks for the correction.

But still, in this particular case, the statements "at $r=0$ there is a singularity" and "the domain of the metric does not include $r=0$" and "the metric is not well-defined at $r=0$" are equivalent?

 

[bcrowell]

But still, in this particular case, the statements "at $r=0$ there is a singularity" and "the domain of the metric does not include $r=0$" and "the metric is not well-defined at $r=0$" are equivalent?

No. The most fundamental reason that such a definition doesn't work is that the singularity is not a point or point-set in the spacetime manifold. Therefore it doesn't make sense to ask whether the metric is defined "there." There's "no there there" at which we could even ask whether the metric is defined. Specifying $r=0$ does not specify a set of points in the manifold; those values of the $r$ coordinate are not part of the coordinate chart.

Note that the metric could also be undefined at a certain coordinate value without there being any singularity. For example, when we express the metric of the Schwarzschild spacetime in the Schwarzschild coordinates, it's undefined at the event horizon, but this is not considered a singularity. (People describe it as a coordinate singularity, but that just means not a real singularity.)

 

[Smattering]

Yes. The technical term is "manifold with metric".

O.k., now things seem to become clearer for me.

So since this Schwarzschild case is completely symmetrical, the only parameters that the metric depends on in this particular case are that radial coordinate $r$ and some time coordinate $t$?

And inside the event horizon, there are no wordlines with constant $r$ just in the same sense as there are no wordlines with constant $t$ outside of the event horizon?

 

[PeterDonis]

the only parameters that the metric depends on in this particular case are that radial coordinate $r$ and some time coordinate ##t##?

The metric doesn't depend on $t$. The only coordinate it depends on is $r$.

inside the event horizon, there are no wordlines with constant $r$ just in the same sense as there are no wordlines with constant $t$ outside of the event horizon?

Sort of. There are no worldlines with constant $t$ inside the horizon either, so the two coordinates are not symmetric in this respect.

 

[Smattering]

No. The most fundamental reason that such a definition doesn't work is that the singularity is not a point or point-set in the spacetime manifold. Therefore it doesn't make sense to ask whether the metric is defined "there." There's "no there there" at which we could even ask whether the metric is defined. Specifying $r=0$ does not specify a set of points in the manifold; those values of the $r$ coordinate are not part of the coordinate chart.

Hm ... how does this radial coordinate $r$ translate to the spacetime manifold then?

Note that the metric could also be undefined at a certain coordinate value without there being any singularity. For example, when we express the metric of the Schwarzschild spacetime in the Schwarzschild coordinates, it's undefined at the event horizon, but this is not considered a singularity. (People describe it as a coordinate singularity, but that just means not a real singularity.)

Yes, I read that. But as far as I understood, this "coordinate singularity" can be eliminated via a coordinate transformation, whereas there exists no coordinate transformation that can eliminate the singularity at $r=0$.

 

[bcrowell]

Hm ... how does this radial coordinate $r$ translate to the spacetime manifold then?

I'm not sure what you're asking here. The coordinate r, or its specific value r=0? We can't normally cover an entire manifold with a single coordinate chart. The inability to do so is not a big crisis and occurs even in well-behaved manifolds such as a sphere.

Yes, I read that. But as far as I understood, this "coordinate singularity" can be eliminated via a coordinate transformation, whereas there exists no coordinate transformation that can eliminate the singularity at $r=0$.

Yes, that's correct, but I don't see how that affects the logic of the discussion. The point is that if you want to define a singularity in GR, it is neither necessary nor sufficient for the metric to be undefined at a particular coordinate value. This is why we define a singularity in GR in terms of geodesic incompleteness.

 

[Smattering]

I'm not sure what you're asking here. The coordinate r, or its specific value r=0? We can't normally cover an entire manifold with a single coordinate chart.

Then let me rephrase this:

1. What is the domain of the Schwarzschild metric?
2. How is this domain related to the spacetime manifold (or a specific part of it)?

Yes, that's correct, but I don't see how that affects the logic of the discussion. The point is that if you want to define a singularity in GR, it is neither necessary nor sufficient for the metric to be undefined at a particular coordinate value. This is why we define a singularity in GR in terms of geodesic incompleteness.

O.k., if that is the convention, then let's stick to it, but still I wonder whether geodesic incompleteness and the inability to eliminate a singularity by coordinate transformation is or is not equivalent.

 

[bcrowell]

Then let me rephrase this:

1. What is the domain of the Schwarzschild metric?
2. How is this domain related to the spacetime manifold (or a specific part of it)?

1. The domain is the entire spacetime manifold. The manifold is not defined in terms of a specific coordinate chart, but if you want to characterize it in terms of the Schwarzschild coordinates, $r=0$ is absent.

2. The domain is the entire spacetime manifold.

O.k., if that is the convention, then let's stick to it, but still I wonder whether geodesic incompleteness and the inability to eliminate a singularity by coordinate transformation is or is not equivalent.

You can't define a singularity as the inability to eliminate a singularity. That would be like defining a lion as a lion with four legs. Nobody would have bothered to make a definition in terms of geodesic completeness if there were some simpler definition.

The questions you're asking are all good ones, but we're spending a lot of time undoing your preconceptions (which are natural preconceptions) and reiterating material that is found in standard textbooks. There is a limit to how much we can achieve through Socratic dialog. If you could tell us a little about your background in math and physics, we could recommend a book at the right level for you that would contain a systematic treatment of these issues. A couple of good possibilities at the graduate level are Wald and Carroll. At a lower level of mathematical sophistication, my own GR book is free online: http://www.lightandmatter.com/genrel/. There is a partial version of Carroll that is free online: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html

 

[Smattering]

1. The domain is the entire spacetime manifold. The manifold is not defined in terms of a specific coordinate chart, but if you want to characterize it in terms of the Schwarzschild coordinates, $r=0$ is absent.

2. The domain is the entire spacetime manifold.

Then, I do not understand the objections you made here:
https://www.physicsforums.com/threa...hwarzschild-radius.841320/page-2#post-5282982

Because, when the domain is the entire spacetime manifold, then "not being part of the domain" should be exactly the same as "not being part of the spacetime manifold".

You can't define a singularity as the inability to eliminate a singularity. That would be like defining a lion as a lion with four legs.

But I might be able to define a coordinate singularity as a singularity that can be removed by a coordinate transformation in the same way as I can define a waterfowl as a fowl that can swim. I really do not see any principle issue here.

Nobody would have bothered to make a definition in terms of geodesic completeness if there were some simpler definition.

I am not saying that the other definition is simpler.

The questions you're asking are all good ones, but we're spending a lot of time undoing your preconceptions (which are natural preconceptions) and reiterating material that is found in standard textbooks. There is a limit to how much we can achieve through Socratic dialog. If you could tell us a little about your background in math and physics, we could recommend a book at the right level for you that would contain a systematic treatment of these issues. A couple of good possibilities at the graduate level are Wald and Carroll. At a lower level of mathematical sophistication, my own GR book is free online: http://www.lightandmatter.com/genrel/ . There is a partial version of Carroll that is free online: http://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html

O.k., thanks for the time you spent. I will have a look at this material.

I do not have any mentionable physics background apart from what I can remember from school (otherwise I would not be asking such questions, would I ;-) ), but I have the German equivalent to a Master's degree in computer science. So I have a bit of math background, but with an focus on discrete math and certainly no differential geometry.

 

[PeterDonis]

An online version of Carroll's lecture notes is here:

http://arxiv.org/abs/gr-qc/9712019

AFAIK this is the complete notes.

 

[PeterDonis]

when the domain is the entire spacetime manifold, then "not being part of the domain" should be exactly the same as "not being part of the spacetime manifold"

He wasn't objecting to that; he was objecting to your statement that "the metric is not well-defined" at the singularity. You can't even ask whether the metric is well-defined someplace that isn't part of the spacetime manifold.

 

[PeterDonis]

What is the domain of the Schwarzschild metric?

It's important to distinguish "the Schwarzschild metric" from "the Schwarzschild coordinate chart". The "Schwarzschild metric", properly speaking, is just another name for "the Schwarzschild geometry", a geometric object that exists independently of any choice of coordinates. (I prefer the term "Schwarzschild spacetime" for this object, to avoid confusion.) But many sources will use the term "the Schwarzschild metric", when what they really mean is "the Schwarzschild coordinate chart", the coordinates $t, r, \theta, \phi$, in which the line element looks like

$$
ds^2 = \left(1 - \frac{2M}{r} \right) dt^2 + \frac{1}{1 - \frac{2M}{r}} dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2
$$

These coordinates are often used, but they have a coordinate singularity at $r = 2M$, so strictly speaking there are actually two of these charts, one for $r > 2M$ and one for $0 < r < 2M$, and they are disconnected. To remove the coordinate singularity, we would need to choose a different chart which is nonsingular for all $r > 0$; examples are the Painleve chart and the Eddington-Finkelstein chart. But all of these charts only cover $r > 0$, because the spacetime itself--the geometric object that exists independently of any choice of coordinates--only contains points for which $r > 0$.

Another source of confusion is the fact that, as my last statement implied, $r$ has a double meaning. It is a coordinate which appears in all of the charts I mentioned; but it is also a physical property of a point in Schwarzschild spacetime. Every point in this spacetime lies on a 2-sphere with a well-defined physical area $A$, so we can label points by the area $A$ of the 2-sphere they lie on. For reasons which are too long to fit in the margin of this post, physicists prefer instead to use the label $r = \sqrt{A / 4 \pi}$, the "areal radius" (the radius that a 2-sphere in Euclidean 3-space with area $A$ would have). It is this second meaning of $r$ that I used in the last sentence of the previous paragraph, which could be rephrased as: Schwarzschild spacetime only contains 2-spheres with physical area $A$ that is positive.

 

[bcrowell]

Because, when the domain is the entire spacetime manifold, then "not being part of the domain" should be exactly the same as "not being part of the spacetime manifold".

Yes, that's correct.

But I might be able to define a coordinate singularity as a singularity that can be removed by a coordinate transformation in the same way as I can define a waterfowl as a fowl that can swim. I really do not see any principle issue here.

Using your analogy, you haven't defined "fowl," i.e., "singularity."

 

[bcrowell]

An online version of Carroll's lecture notes is here:

http://arxiv.org/abs/gr-qc/9712019

AFAIK this is the complete notes.

He wrote a GR textbook based on the notes. The notes are not the complete textbook.

O.k., thanks for the time you spent. I will have a look at this material.

Cool. Here are some of the topics you probably want to look at:

* manifold

* chart

* atlas

* singularity

* geodesic incompleteness

Wald introduces these very systematically and early on, but Wald is the hardest of the three books, and it's also not free online.

 

[Smattering]

He wasn't objecting to that; he was objecting to your statement that "the metric is not well-defined" at the singularity. You can't even ask whether the metric is well-defined someplace that isn't part of the spacetime manifold.

Fair enough since being "well-defined" implies that something is defined at all, but as $r=0$ is not part of the domain, the metric is simply undefined at $r=0$.

 

[Smattering]

One other question:

Is there a solution of the Einstein field equations for a completely homogeneous universe where all the mass/energy is uniformly distributed?